Transcript BSC 417/517 Environmental Modeling

BSC 417/517
Environmental Modeling
The Kaibab Deer Herd
Goal of Chapter 16
• Illustrate the steps of modeling discussed in
Chapter 15
• Illustrate iterative nature of modeling
process
• Learn to appreciate many decisions required
to build a model
• Do exercises which verify, apply, and
improve the model
Getting Acquainted With the System
• Kaibab Plateau is located within the Kaibab
National Forest, located north of the Colorado
River in north-central Arizona
• Approximately 60 miles long (N-S) and 45 miles
wide at its widest point
• One of the largest and best-defined “block
plateaus” in the world
• Vegetation types change with elevation and
include shrubs, sagebrush, grasslands, pinionjuniper, Ponderosa pine, and spruce-fir
Kaibab National
Forest
The Kaibab Plateau
Kaibab Plateau Deer Herd
• Kaibab plateau deer herd consists of Rocky
Mountain mule deer
• Pinion-juniper woodlands provide winter
range; summer range includes pine and
spruce-fir forests
• Deer mate in Nov/Dec; fawns arrive in
Jun/Jul; deer achieve maturity @ ca. 1.5 yr
Rocky Mountain Mule Deer
Kaibab Plateau Deer Herd
• Data on deer population size prior to 1900 is
sparse; Rasmussen (1941) estimated total size of
3000-4000 deer
• Plateau was home to several predators including
coyotes, bobcats, mountain lions, and wolves,
which kept deer populations under control
• Starting at turn of the century, predators were
systematically removed by hunting and trapping
• During 1907-1923, predator kills were estimated
at 3000 coyotes, 674 lions, 120 bobcats, and 11
wolves
Kaibab Plateau Deer Herd
• Deer population grew rapidly during decimation
of predators in the early 1900s (“irruption”)
• Rasmussen (1941) estimated deer population at ca.
100,000 in 1924
• Reconnaissance party reported that forage
conditions were deplorable
• No new growth of apsen
• White fir, typically eaten unless under stress of food
shortage, were often found “skirted”
• Condition of deer was also found to be deplorable
Kaibab Plateau Deer Herd
• Major deer die-off occurred during winters of
the years 1924-1928
• Government hunters were deployed in 1928 to
reduce the size of the deer population
• But, paradoxically, predator “control measures”
continued…
Kaibab Plateau Deer Herd
• The year 1930 was a good year for plant
growth, and deer herd began to recover and
stabilize
• By 1932, deer population was estimated at
14,000 and the range was in reasonable
conditions
• Forest service game reports declared that
the number of deer appeared “to be about
right for the range”
Be Specific About the Problem
• Develop model to gain insight into causes behind
the deer population “irrupution” and measures
that could have been used to prevent it
• Starting point: come up with a reference mode,
i.e. a target pattern for the system’s behavior
• In this case, we’re dealing with the classical
“overshoot” pattern discussed earlier in the
course
Reference Mode
Pop. peaks at ca. 100,000
Return to pseudo-stability
with government hunting or
return of predators
Initial pop.
ca. 4000
Rapid growth after
removal of predators
1900
1910
1920
1930
1940
Notes on Reference Mode
• Sketch is not a compilation of precise estimates in
terms of deer population or timing of events
• Simply a rough depiction of a likely pattern of
behavior based on accounts of various observers
• Leads to initial modeling goal of a simulating deer
population which remains stable during the initial
years, and grows rapidly when predators are
removed from the system
• Population should peak at something like 100,000
and then decline rapidly due to starvation
Specific Goals of
Modeling Exercise
1. Gain understanding of forces that led to
the overshoot and collapse
2. Explore number of predators on the
plateau as a relevant “policy variable”
which could be manipulated in order to
achieve a stable deer population
Construct An Initial Stock-andFlow Diagram
• As a first step, construct a stock-and-flow diagram
which can reproduce the reference mode
• Could attempt a predator-prey model, since we’re
dealing with deer population which is regulated (at
least originally) by predation
• However, this would lead to complexities that go
beyond the goal of the current modeling exercise
(see Chapter 18)
Design of First Model
• Allow number of predators to be specified
by the user, and set number of deer killed
per predator per unit at a constant value
• Note that for simplicity, all predators are
treated equally, i.e. coyotes, bobcats, and
mountain lions are combined into an
aggregate category, or “functional group”
Design of First Model
• Initial deer population = 4000 (spread out over
800 thousand acres = 5 deer per thousand acres)
• Net birth rate = 50%; based on favorable range
conditions
• Net birth rate comes from assumptions that
•
•
•
•
Half the deer population is female
2/3 of females are fertile at any given time
Average litter size = 1.6
Average deer life span = 15 yr => average death rate =
1/15 = 0.0667 ~ 0.07
• With the above assumptions:
• Net birth rate = [0.5 × (2/3) × 1.6] – 0.07 = 0.47 ~ 0.5
Design of First Model
• Number of deer killed per predator per yr is set at
40 based on the following assumptions
• All predators can be measured by the equivalent
number of cougars (mountain lions)
• 75% of cougar diet is mule deer
• Average cougar requires one kill per week
• With the above assumptions:
• Deer kills/predator/yr = 0.75 × 52 = 38 ~ 40
Design of First Model
• Number of predators (in cougar equivalents) in the
original ecosystem is unknown
• To get started, set equal to value that, when
multiplied by the assumed number of deer killed
per predator per year (40), produces a number of
deer deaths equal to the number of net deer births
per year at the start of the simulation (0.5 × 4000
= 2000)
• In other words, set the initial number of predators
equal to 2000/40 = 50 cougar equivalents
First Model
net births
deer population
deaths from predation
~
number of predators
net birth rate
~
deer killed per
predator per year
deer density
area
deer_killed_per_predator_per_year = GRAPH(TIME)
(0.00, 40.0), (485, 40.0), (970, 40.0), (1455, 40.0), (1940, 40.0)
number_of_predators = GRAPH(TIME)
(1900, 50.0), (1910, 50.0), (1920, 0.00), (1930, 0.00), (1940, 0.00)
Results of First Model
• Deer population is
constant for first 10 yr
• Grows exponentially after
predators are removed
during 1910-1920,
reaching 10X the initial
population by 1920
• Population goes off-scale
around 1920 and never
comes back
• Simulation clearly fails to
reproduce the reference
mode
A Second Model With Forage
• Next version of the model will keep track of the
forage requirements and the available forage on the
plateau
• Proceed with assumption that total forage
requirement is 1 MT dry biomass/deer/yr
• Estimate is based on Vallentine (1990)’s suggestion
that mule deer require ca. 23% of an animal unit
equivalent (AUE) = dry matter consumed by a 1000pound non-lactating cow (ca. 12 kg dry biomass/d)
• 0.23 × 12 kg/d × 365d/yr = 1007 kg/yr ~ 1000 kg/yr
Second Model
• Assume that plateau produces vast excess of
plant matter each year
• With all plants combined into a single
category (valid?), forage production is set at
40,000 MT/yr = 10X deer requirement
• Forage availability ratio = forage
production/forage required
Second Model
• As long as forage availability ratio > 1,
fraction of forage needs met is 100%
• If forage availability < 1, then fraction of
forage needs met = forage availability
• Fraction of forage needs met influences net
birth rate according to a graph function
Second Model
• Net birth rate = 0.5 when
0.6
0.4
Net Birth Rate
deer are meeting 100% of
their forage needs
• As fraction of forage
needs met decreases, net
birth rate declines, and
falls to zero when deer are
meeting only half of their
forage needs
• Net birth rate reaches
-40%/yr if deer meet 30%
or less of their forage
needs
0.2
0
-0.2
-0.4
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction Forage Needs Met
Note: the relationship depicted here is a “plausible
guess” only, as little info is available on deer birth
and death rates undef difficult conditions
Second Model
net births
~
deer population
deaths from pr edation
deer density
~
net birth rate
number of predators
~
deer killed per
predator per year
forage required
fr forage needs met
area
forage required per deer per yr
forage availability ratio
forage production
forage_availability_ratio = forage_production/forage_required
fr_forage_needs_met = MIN(1,forage_availability_ratio)
Second Model Results
• Deer population remains constant until
predator removal starts, then increases
rapidly to ca. 80,000
• As population grows, fraction of forage
needs met decreases rapidly to 0.5, which
causes net birth rate to go to zero, which in
turn stops growth => constant population
for the remainder of the simulation
Second Model Results
Second Model Results
• Results are closer to reference mode than the first
model, but simulation does not reproduce the
major die-off that occurred during the late 1920s
• Sensitivity analysis reveals that lack of die-off is
not caused by an erroneous value for the forage
required per deer per year: general pattern remains
the same with values of 0.75, 1.0, and 1.25 MT
dry biomass/deer/yr
• Failure to reproduce die-off is likely due to…lack
of change in forage biomass?
Second Model Results - Contd
Third Model: Forage Production
and Consumption
• Simulate growth and decay of biomass
using a simple S-shaped growth model
(check-out Ford Chapter 6 to get
reacquainted with S-shaped growth models)
• Production of new plant biomass is
dependent on ratio of current biomass to a
maximum biomass of 400,000 MT
• First-order decay of standing biomass
Third Model: Biomass Sector
forage consumpti on
standing
biomass
bio pr oductivity
addition to standing biomass
decay
new g rowth
bio decay r ate
~
intri nsic bi o productivity
prod mult from
fullness
fullness fraction
max biomass
Third Model: Biomass Sector
addition_to_standing_biomass = new_growth-forage_consumption
decay = standing_biomass*bio_decay_rate
bio_decay_rate = 0.1
bio_productivity = intrinsic_bio_productivity*prod_mult_from_fullness
forage_consumption = forage_required*fr_forage_needs_met
intrinsic_bio_productivity = 0.4
max_biomass = 400000
new_growth = standing_biomass*bio_productivity
prod mult
fullness_fraction = standing_biomass/max_biomass
1
0.8
0.6
0.4
0.2
0
0
prod_mult_from_fullness = GRAPH(fullness_fraction)
(0.00, 1.00), (0.2, 1.00), (0.4, 0.9), (0.6, 0.6), (0.8, 0.2), (1.00, 0.00)
0.2
0.4
0.6
fullness
0.8
1
Third Model: Biomass Sector
Simulation
Third Model: Full Version
Kaibab Deer Herd Third Model
Third Model Results
• Deer population increases to 80,000 by 1920,
after which net birth rate falls to slightly
below zero
• Small decrease in deer population occurs
during the 1920s and 1930, but not as
dramatic as was observed
• Alteration of annual forage rate per deer does
not change outcome
• Model still fails to reproduce reference mode
Fourth Model: Deer May
Consume Older Biomass
• Deer prefer new growth, but under stressed (i.e. starvation)
conditions will consume older biomass (=> “skirting”)
• As deer population becomes large, keep track of additional
consumption requirements which arise when the fraction of
forage needs met by new growth falls below 1
• Assume 25% of standing older biomass is available to deer, and
that the nutritional value of the old biomass is only 25% of that
of new growth
• New drainage flow must be added to depict loss of standing
biomass through consumption of older growth
Fourth Model: Key Equations
additional_con_required = forage_required-forage_consumption
stand_bio_available = standing_biomass*fr_standing_available
fr_standing_available = 0.25
old_biomass_availability_ratio = stand_bio_available/MAX(1,additional_con_required)
old_biomass_consumption = additional_con_required*fr_additional_needs_met
fr_additional_needs_met = MIN(1,old_biomass_availability_ratio)
equivalent_fraction_needs_met =
MIN(1,fr_forage_needs_met+fr_additional_needs_met*old_biomass_nutritional_factor)
old_biomass_nutritional_factor = 0.25
Fourth Model: Density-Dependent
Predator Kill Rate
kills per predator per yr
40
30
20
10
0
0
1
2
number of deer per 1000 acres
3
4
Fourth Model: Full Version
Kaibab Deer Herd Fourth Model
Fourth Model Results
• Deer population peaks at ca. 115,000 in the
early 1920s, then declines rapidly
• Net birth rate falls to zero in 1921, and
reaches –0.25 by the end of the 1920s and
remains there for the remainder of the
simulation period
• The desired overshoot pattern has been
achieved!
Fourth Model Results
• Forage variables are consistent with
expectations:
• Huge increase in forage requirements and new
forage consumption in parallel with deer
population explosion
• Old biomass consumption kicks in a few years
after start of irruption
• Standing biomass drops to low values
Fourth Model Results
• A milestone has been achieved in the
modeling process!
• Model generates the reference mode, at
least in general terms
• Improvements are possible (note that
reference mode does not depict total
decimation of the deer population), but
model is ready for sensitivity analysis
Sensitivity Analysis
• Now that the model generates the reference mode, it is
appropriate to conduct more extensive sensitivity analysis
• Purpose of the analysis is to determine if the model’s
behavior is strongly influenced by changes in the most
uncertain parameters
• If the same general pattern emerges in many different
simulations, then the model is said to “robust”
• Robust models are particularly useful in environmental
science, where models tend to contain numerous highly
uncertain parameters
Model 4 Sensitivity Analysis
• First, vary forage
requirement ± 25%
• Peak population sizes
vary considerably
• However, general
pattern of behavior is
identical
• Model is robust with
respect to changes in
forage requirement.
Model 4 Sensitivity Analysis
• Next, vary old biomass
nutritional factor from
0 to 0.75
• Peak population sizes
vary considerably, but
general pattern of
behavior is identical
• Model is robust with
respect to changes the
old biomass nutritional
factor
Model 4 Sensitivity Analysis
• Previous tests are easily implemented with
STELLA using the built-in sensitivity
analysis facility
• May also be important to test sensitivity to
changes in nonlinear functions
• To illustrate, alter the relationship between
equivalent needs met and net deer birth rate
Model 4 Sensitivity Analysis
0.6
Net birth rate
0.4
0.2
0
Run 1
-0.2
Run 2
Run 3
-0.4
-0.6
0.2
0.4
0.6
0.8
Equivalent fraction of needs met
1
Model 4 Sensitivity Analysis
Deer
160,000
140,000
Run 1
120,000
Run 2
100,000
Run 3
80,000
60,000
40,000
20,000
0
1900
1910
1920
Years
1930
1940
Model 4 Sensitivity Analysis
• Results of sensitivity analyses indicate that if we were
trying to accurately predict peak deer population, we
would not be able to proceed without more confidence in
certain parameter values
• However, our stated purpose was not to predict specific
numbers, but rather to obtain a general understanding of
the system’s tendency to overshoot
• Sensitivity analysis reveals that the same general pattern is
obtained regardless of the particular parameter values or
relationship
Model 4 Sensitivity Analysis Extended
• Conclude sensitivity analysis with a combination
of changes which stretch the value of several
parameter beyond what might be considered to be
plausible estimates
• Changes are designed to reinforce each other by
increasing the chances that the deer population
could continue to grow throughout the simulation
period
• Testing of response to extremes is designed to
learn the true extent of the model’s robustness
Model 4 Sensitivity Analysis Extended
• Changes include:
• Double foraging area from 800 to 1600 kA
• Double the initial value of standing biomass from
{
Effectively
Doubles
Plateau Size
300,000 to 600,000 MT
• Double the maximum biomass from 400,000 to
800,000 MT
• Lower food requirement from 1 to 0.5 MT/yr
• Assume that old biomass has 2X the nutritional value
compared to the base case (i.e. 0.5 vs. 0.25)
Model 4 Sensitivity Analysis Extended Results
Deer
600,000
500,000
Run 1
400,000
Extended
300,000
200,000
100,000
0
1900
1910
1920
Years
1930
1940
Model 4 Sensitivity Analysis - Summary
• Extreme testing, together with other sensitivity
analyses, indicate that the model is very robust wrt
the tendency to demonstrate overshoot once
predators are removed
• Have achieved another important milestone in the
modeling process
• May now proceed with testing the impact of
policy alternatives
First Policy Test: Predators
• Number of predators was identified as a policy
variable at the outset of the modeling exercise
• Start with assessment of how changes in the
number of predators might alter the tendency for
the deer population to overshoot
• Allow decline in predator population to occur less
rapidly (drop from 50 to 0 over 20 years rather
than 10 years)
First Policy Test Results
First Policy Test: Predators
• Deer population undergoes the same overshoot pattern
regardless of the decline in predator removal rate
• Even if number of predators is returned to 50 in 1920 after
the irruption has begun, the population still irrupts because
the deer are too numerous for the fixed number of
predators to control
• Obvious conclusion is that the predators should never have
been removed in the first place
• To more accurately test the ability of predators to control
the deer population, model would need to be expanded to
allow the number of predators to rise and fall with changes
in the deer population, predator-prey style (focus of
Chapter 18)
Second Policy Test: Fixed Deer Hunting
• Explore deer hunting as an alternative
method of controlling the deer population
• Controlled hunting is common in Europe
and North America
• Add a “deer harvest” flow to the model to
account for a policy to harvest a fixed
number of deer each year after a specified
start date
Second Policty Test: Revised
Animal Sector
deer harvest
harvest amount
net births
net birth rate
~
harvest start year
deaths from predation
deer population
number of predat ors
~
deer density
fr additional needs met
area
equivalent fraction needs met
old biomass nutritional factor
~
deer killed per
predator per year
Second Policty Test Results
Second Policy Test: Fixed Deer Hunting
• Harvest amounts of 1000-4000 are not sufficient to prevent
the irruption
• If keep increasing harvest amount, find that a value of
4700 delays the irruption by ca. 15 years, but it still occurs
• Tempting to increase harvest amount even further…but
find that a value of 4704 leads ultimately to crash of the
population after 1930
• Searching for the “ideal” harvest amount is futile: even if
the ideal harvest amount could be identified, the slightest
disturbance in any of the model variables would reveal that
the equilibrium is not a stable one
Third Policy Test:
Variable Deer Hunting
• Need a better policy for hunting, e.g. one which
incorporates information (feedback) on the size of
the deer population
• Modify model to make harvest amount dependent
on deer population by setting harvest equal to a
fixed fraction of the population
• Set harvest fraction equal to 0.5 to match the
maximum net birth rate
• Start hunting in 1915
Third Policy Test:
Variable Deer Hunting
• Deer harvest increases quickly around 1915, and
loss of deer through hunting is twice as great as
the losses to predation during the previous decade
• Deer harvest then declines and the system reaches
equilibrium
• Deer population remains at around 10,000 for the
rest of simulation
• Standing biomass is maintained at value near its
starting level
Third Policty Test Results
(Start Hunting in 1915)
Third Policy Test:
Variable Deer Hunting
• If start hunting only 3 years later (1918), equilibrium deer
•
•
•
•
population is ca. 40,000 and standing biomass declines
gradually to a new equilibrium, with 20-30% less biomass
than at the start of the simulation
If delay start of hunting to 1920: too late!!!
Deer population is already starting to decline due to food
resource depletion, and hunting only hastens the
population crash
Standing biomass does not recover
Obviously, hunting control must be implemented before
signs of severe overbrowsing are evident
Third Policty Test Results
(Start Hunting in 1918)
Third Policty Test Results
(Start Hunting in 1920)
Additional Policy Tests
• Ford identifies five additional policy tests that it would
make sense to evaluate
• Impact of lags in measuring deer population and discrepancies between
•
•
•
•
target and actual harvest
Expand hunting policy to make desired deer population size an explicit
policy variable
Impact of variable weather on the overall system, e.g. wrt biomass
productivity, biomass decay rate, and deer net birth rate
Allow hunting policy to be sensitive to the amount of standing biomass,
so as to prevent overbrowsing and associated population irruption
Alter hunting policy to include distinction between hunting of male vs.
female deer (bucks vs. does)
What About Excluded Variables?
• Easy to identify many variables and processes that are
excluded by the high level of aggregation
•
•
•
•
Influence of seasonality, snowfall
Distinctions between different types of predators
Distinctions between different types of vegetation
Impact of cattle and sheep on range forage conditions
• Should these things bother us?
• Does the simulation provide “wrong answers” because
such variables and processes were not included?
• Can the model ever be big enough to deliver the “right
answer”?
What About Excluded Variables?
• Keep in mind that computer simulation is not a magic
path to the right answer
• Modeling of environmental systems should be viewed
as a method to gain improved understanding of the
dynamics of the system
• Inclusion of additional variables and processes should
be done with caution once a working model has been
obtained: doing so may lead to confusion rather than
illumination!
Post Script
• Was removal of predators really responsible for
Kaibab Plateau deer population irruption?
• Caughley (1970) concludes that habitat alteration
by fire and grazing played a major role
• Many confounding factors were likely involved
• Botkin (1990) notes that the focus by prominent
naturalists (e.g. Aldo Leopold) on the role of
predators reveals their paradigm of a highly
ordered nature in which predators play an essential
role
Post Script
• Take home message for students of
modeling: constructing and testing a model
of the Kaibab deer herd based on the impact
of predator removal does not make the story
true
• Although model is internally consistent,
other models could be developed to account
for the population irruption