Slide 1

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 2

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 3

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 4

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 5

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 6

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 7

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 8

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 9

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 10

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 11

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 12

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 13

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 14

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 15

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 16

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 17

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 18

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 19

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 20

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 21

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 22

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 23

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 24

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 25

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 26

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 27

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 28

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 29

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 30

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 31

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 32

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 33

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 34

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 35

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 36

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 37

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 38

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 39

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 40

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 41

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 42

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 43

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 44

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 45

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 46

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 47

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 48

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 49

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 50

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 51

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 52

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 53

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 54

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 55

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 56

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 57

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 58

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 59

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 60

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 61

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 62

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 63

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 64

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 65

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 66

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 67

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 68

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 69

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 70

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 71

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 72

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 73

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 74

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 75

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 76

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 77

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 78

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 79

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 80

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 81

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 82

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 83

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 84

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 85

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 86

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 87

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 88

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 89

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 90

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 91

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 92

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 93

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 94

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 95

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 96

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 97

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 98

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 99

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 100

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 101

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 102

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 103

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 104

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes


Slide 105

Mechanistic models for
macroecolgy:
moving beyond correlation
Nicholas J. Gotelli
Department of Biology
University of Vermont
Burlington, VT 05405

??
What causes geographic variation
in species richness
??

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Gary Entsminger
Acquired Intelligence

Nicholas Gotelli,
University of Vermont

Gary Graves
Smithsonian

Rob Colwell
University of
Connecticut

Carsten Rahbek
University of Copenhagen

Thiago Rangel

Federal University of Goiás

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Data sources
• Gridded map of domain

Avifauna of South America
“There can be no question, I
think, that South America is
the most peculiar of all the
primary regions of the globe
as to its ornithology.”
P.L. Sclater (1858)

South American Avifauna
• 2891 breeding
species
• 2248 species
endemic to South
America and
associated landbridge islands

Minimum:
18 species

Maximum:
846 species

Minimum:
18 species

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells

Anas puna

Geographic Ranges
For Individual
Species

Phalacrocorax brasilianus

Myiodoorus cardonai

Geographic Ranges

Species Richness

Geographic Ranges

Species Richness

Data sources
• Gridded map of domain
• Species occurrence records within grid
cells
• Quantitative measures of potential
predictor variables within grid cells (NPP,
temperature, habitat diversity)

Climate, Habitat Variables
Measured at Grid Cell Scale

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

How are these macroecological
data typically analyzed?

800
600

Observed Species Richness

400
200
0
0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400
200
0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

800
600
400

OLS

200

LOESS

0

Observed Species Richness

Poisson

0

2

4

6

8

10

N et Primary Produc tiv ity (Tonnes /hec tare)

12

14

How are these macroecological
data typically analyzed?

Curve-fitting!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size

Criticisms of Curve-Fitting
• “Correlation does not equal causation”
Common to all of macroecology!
• Non-linearity & non-normal, spatially correlated errors
LOESS, Poisson, Spatial Regression (SAM)
• Choosing among correlated predictor variables
Model selection strategies, stepwise regression, AIC
• Sensitivity to spatial scale, taxonomic resolution,
geographic range size
Stratify analysis

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Conceptual Weakness of
Curve-Fitting Paradigm
Potential Predictor
Variables
(tonnes/ha, C°)

??
MECHANISM
??

minimize
residuals

Predicted
Species Richness
(S / grid cell)

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

How can we build explicit
simulation models for
macroecology?

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

One-dimensional geographic domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

Species
Number

One-dimensional geographic domain
Species geographic ranges randomly
placed line segments within domain

Peak of species richness in geographic
center of domain

domain

geographic range

domain

Pancakus spp.

der Pfankuchen
Guild

Reduced species richness
at margins of the domain

Mid-domain
peak of species richness
in the center of the domain

2-dimensional MDE Model
• Random point of origination
within continent (speciation)
• Random spread of geographic
range into contiguous
unoccupied cells
• Spreading dye model (Jetz & Rahbek
2001) predicts peak richness in
center of continent (r2 = 0.17)

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

Assumptions of MDE models
• Placement of ranges within domain is
random with respect to environmental
gradients
– Controversial, but logical for a null model for
climatic effects

• Geographic ranges are cohesive within the
domain
– Rarely discussed, but important as the basis
for a mechanistic model of species richness

Range Cohesion

Range Scatter

At the 1º x 1º scale, > 95% of species of
South American birds have contiguous
geographic ranges

Causes of Range Cohesion
• Extrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes

Causes of Range Cohesion
• Extrinsic Causes
– Coarse Spatial Scale
– Spatial Autocorrelation in Environments

• Intrinsic Causes
– Limited Dispersal
– Philopatry & Site Fidelity
– Metapopulation & Source/Sink Structure
– Fine-scale Genetic Structure & Local Adaptation
– Spatially Mediated Species Interactions

Strict Range Cohesion

Stepping Stone

* The mid-domain effect does not require strict
range cohesion. A mid-domain peak in species
richness will also arise from stepping stone
models with limited dispersal and from neutral
model dynamics (Rangel & Diniz-Filho 2005)

Homogenous
Environment

Heterogeneous
Environment

Almost all MDE models have assumed a
homogeneous environment:
grid cells are equiprobable

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT
Homogeneous
Enforced
RANGE
COHESION
Relaxed

Classic MDE

Statistical Null
(slope = 0)

Heterogeneous

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

ENVIRONMENT

Enforced
RANGE
COHESION
Relaxed

Homogeneous

Heterogeneous

Classic MDE

Range Cohesion
Models

Statistical Null
(slope = 0)

Range Scatter
Models

Range Cohesion Models are a hybrid that describes a
stochastic MDE model in a more realistic heterogeneous environment.
Range Scatter Models also incorporate environmental heterogeneity, but
do not place any constraints on species geographic ranges.

Alternative Strategy:
Mechanistic Simulation Models
Potential Predictor
Variables
(tonnes/ha, C°)

Predicted
Species Richness
(S / grid cell)

mechanism

Explicit
Simulation
Model

Observed
Species Richness
(S / grid cell)

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)

Modeling Strategy
• Establish simple algorithms that describe
P(occupancy) based on environmental
variables
• Simulate origin and placement of each
species geographic range in
heterogeneous landscape (with or without
range cohesion)
• Repeat simulation to estimate predicted
species richness per grid cell

Geographic Ranges

Species Richness

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
– Temperature Kinetics (Brown et al. 2004)

What determines
P(cell occurrence)?
• Simple environmental models
P(occurrence)  Measured Environmental
Variable (NPP, Temperature, etc.)

• Formal analytical models
– Species-Energy Model (Currie et al. 2004)
P(occurrence)  (NPP)(Grid-cell Area)
– Temperature Kinetics (Brown et al. 2004)
P(occurrence)  e-E/kT

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Model-Selection
in Curve-Fitting Analyses
• Simple tests against the null hypothesis
that b=0
• No consideration of what expected slope
should be with a specific mechanism
• Least-square and AIC criteria to try and
select a subset of variables that best
account for variation in S

800
600
400
200

H0: b = 0

0

Observed Species Richness

OLS

0

2

4

6

8

10

Net Primary Productivity (Tonnes/hectare)

12

14

Model Selection with
Mechanistic Simulation Models
• Models make quantitative predictions of
expected species richness
• Test slope of observed richness versus
predicted richness
• Hypothesis of an acceptable fit H1: b = 1.0
• Rank acceptable models according to
slope, intercept, and r2
• AIC criteria not appropriate

Observed b
Theoretical b = 1.0

Observed S

Predicted S

Understanding species
richness patterns
• Data sources
• A critique of current methods
• Range cohesion and the mid-domain
effect
• Mechanistic models for species richness
• Model selection
• Summary

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)

Summary
• Curve-fitting framework does not
incorporate explicit mechanisms
• Use mechanistic simulations to define the
placement of geographic ranges in a
gridded domain
• Specify rules for P(occurrence)=
f(environmental variables)
• Test model fit against expected slope = 1.0

Criticisms & Rejoinders

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?

Criticisms & Rejoinders
• “Each species has a unique and distinctive response to different
environmental variables. Species ranges should be modeled
independently, not with a single function for all species.”
If this is true, why are there widespread repeatable patterns of
species richness (e.g., latitude, elevation, area, productivity)?
Often not enough data to model each species individually. We need
a simple framework for analysing entire floras and faunas at a
biogeographic scale.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary.”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Criticisms & Rejoinders
• “1:1 scaling of environmental variables with P(occurrence) is
unrealistic and arbitrary”
Perhaps, but this is a parsimonious mechanistic model that relates
environmental variables to geographic range placement.
Linearity in P(occurrence) is not unreasonable over the empirical
ranges of environmental variables measured in South America.
(Linearity of P(occurrence) ≠ Linearity of (Species Richness))

Mechanistic models are scarce in this literature (n = 2)! We have to
begin somewhere!

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”

Criticisms & Rejoinders
• “Many environmental variables, but especially NPP, show non-linear
relationships with peaks in richness at intermediate levels. This is
not captured by linear models.”
At least at this spatial scale, no evidence for a diversity hump of
avian species richness when plotted with NPP or other variables

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”

Criticisms & Rejoinders
• “Using slopes comparisons will not successfully distinguish between
models with intercorrelated predictor variables.”
Not a problem for these analyses. From an initial set of ~ 100
candidate models (10 variables x 2 algorithms x 5 range size
quartiles), we reduced the set down to only 4 or 5 possible
contenders.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.

Criticisms & Rejoinders
• “The model is not truly mechanistic because it does not model the
sizes of the geographic ranges, only their placement.”
True! Our model takes range sizes as a given and then uses algorithms to
place them in a heterogeneous domain. A more realistic model would
describe the processes of speciation, dispersal, and extinction of an
evolving fauna.
But how can the parameters of such a model (e.g. speciation and dispersal
rates) ever be measured in the real world? Same problems have plagued
most empirical evaluations of the neutral model.
Our models are designed to analyze the data that macroecologists typically
have: gridded maps of environmental variables and species geographic
ranges.

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”

Criticisms & Rejoinders
• “The range cohesion and range scatter models don’t’ seem like they
would give predictions that are any different from just a regression
with the underlying variables themselves. What is the added value of
these simulation models?”
The predictions are not the same. For species with large geographic
ranges, the range cohesion models always fit the data better than
the range scatter models, regardless of which environmental
variable is considered.

Key Differences
Curve-Fitting

Mechanistic
Models

Unit of Study

Species Richness

Underlying geographic ranges

Predicted values

Minimization of residuals (data
dependent)

Algorithms for origin and spread
of geographic ranges (data
independent)

Model Selection Criteria

Smallest number of variables
that reduce residual sum of
squares

Quantitative fit to model
predictions

Statistical Tests

H0: (b = 0) tests for any effect
that is larger than 0

H0: (b = 1.0) tests for
quantitative match between
observed and predicted S

To Be Continued…

Carsten Rahbek.
Perception of Species Richness Patterns:
The Role of Range Sizes