Transcript Ch 14: Population Growth + Regulation BRING to

Ch 14: Population Growth + Regulation
dN/dt = rN
dN/dt = rN(K-N)/K
BRING to
LECTURE:
1) PRINT of
THIS PPT
2) Pg. 79 in
Manual
Objectives
Population Structure
Population Dynamics
• Growth in unlimited environment
•
Geometric growth
Nt+1 =  Nt
•
Exponential growth Nt+1 = Ntert
•
dN/dt = rN
•
Model assumptions
• Growth in limiting environment
•
Logistic growth dN/dt = rN (K - N)/ K
•
D-D birth and death rates
•
Model assumptions
Population:
all individuals of a species in an area
Subpopulations: in different habitat patches
*** What are structures (traits) of populations?
• Size (abundance)
• Age structure
• Sex ratio
• Distribution (range)
• Density (#/unit area)
• Dispersion (spacing)
• Genetic structure
***Draw two graphs of population
growth showing:
1) Growth with unlimited resources
2) Growth with limited resources
Label axes.
Indicate carrying capacity (K).
3) What are equations representing
both types of growth:
A) exponential?
B) logistic?
Population growth predicted by the
exponential (J) vs. logistic (S) model.
Population growth can be mimicked
by simple mathematical models of
demography.
• Population growth (# ind/unit time) =
recruitment - losses
• Recruitment = ***
• Losses = ***
• Growth (g) = ***
• Growth (g) = (B - D) (in practice)
Two models of population growth with
unlimited resources :
• Geometric growth:
• Individuals added at
one time of year
(seasonal reproduction)
• Uses ***
• Exponential growth:
• individuals added to population continuously
(overlapping generations)
• Uses ***
• Both assume ***
Difference model for geometric growth
with finite amount of time
• ∆N/ ∆t = rate of ∆ = ***
• where b = finite rate of birth or
per capita birth rate/unit of time
• g = b-d, gN = ***
Projection model of geometric growth
(to predict future population size)
•
•
•
•
•
•
•
Nt+1 = Nt + gNt
=(1 + g)Nt Let  (lambda) = ***
Nt+1 =  Nt
 = ***
Proportional ∆, as opposed to finite ∆, as above
Proportional rate of ∆ / time
 = finite rate of increase, proportional/unit time
Geometric growth over many time intervals:
•
•
•
•
•
N1 =  N0
N2 =  N1 = ·  · N0
N3 = *** = ***
Nt =  t N0
Populations grow by multiplication rather
than addition (like compounding interest)
• So if know  and N0, ***
Example of geometric growth (Nt = t N0)
•
•
•
•
•
Let  =1.12 (12% per unit time)
N1 = 1.12 x 100
N2 = ***
N3 = ***
N4 = ***
N0 = 100
112
125
140
157
Geometric growth:
 > 1 and g > 0
N
 = 1 and g = 0
N0
 < 1 and g < 0
time
Values of , r, and Ro indicate whether
population is: ***
Ro < 1
Ro =1
Ro >1
Differential equation model of
exponential growth:
***
rate of
change
in
=
population
size
contribution
of each
individual X
to population
growth
number
of
individuals
in the
population
dN / dt = r N
• r = ***
• Instantaneous rate of birth and death
• r = (b - d) so r is analogous to g, but
instantaneous rates
• rates averaged over individuals (i.e. per
capita rates)
• r =***
E.g.: exponential population growth
 = 1.04
Exponential growth: Nt = ***
r>0
r=0
r<0
• Continuously accelerating curve of increase
• Slope varies directly ***
• (N) (gets steeper as size increases).
Environmental conditions and species
influence r, the intrinsic rate of increase.
Population growth rate depends on the
value of ***; ***is environmental- and
species-specific.
Value of r is unique to each set of
environmental conditions that influenced birth
and death rates…
•…but have some general expectations of pattern:
•
•
•
High rmax for organisms in ***
Low rmax for organisms in ***
habitats
habitats
Rates of population growth are
directly related to body size.
•
•
•
•
Population growth:
increases inversely with***
Mean generation time:
Increases directly with ***
Assumptions of the model
• 1. Population changes as proportion of current
population size (∆ per capita)
•
∆ x # individuals -->∆ in population;
• 2. Constant rate of ∆; constant ***
• 3. No resource limits
• 4. All individuals are the same (***
)
Sample Exam ? Problem Set 2-1 (pg. 79)
A moth species breeds in late summer and leaves
only eggs to survive the winter. The adult dies
after laying eggs. One local population of the
moth increased from 5000 to 6000 in one year.
1. Does this species have overlapping
generations? Explain.
2. What is  for this population? Show calculations.
3. Predict the population size after 3 yrs. Show
calculations.
4. What is one assumption you make in predicting
the future population size?
Review: Problem Set 1
Geometric Growth Model
Exponential Growth Model
Select correct formula…
Objectives
• Growth in unlimited environment
•
Geometric growth
Nt+1 =  Nt
•
Exponential growth Nt+1 = Ntert
•
dN/dt = rN
•
Model assumptions
• Growth in limiting environment
•
Logistic growth dN/dt = rN (K - N)/ K
•
D-D birth and death rates
•
Model assumptions
Populations have the potential to increase
rapidly…
until balanced by extrinsic factors.
Population growth rate =
Intrinsic
growth
rate at
N close
to 0
X
Population
Reduction in
size
X growth rate
due to crowding
Population growth predicted by the
***
model.
K = ***
Assumptions of the exponential model
• 1. No resource limits
• 2. Population changes as proportion of current
population size (∆ per capita)
•
∆ x # individuals -->∆ in population;
• 3. Constant rate of ∆; constant birth and death
rates
• 4. All individuals are the same (no age or size
structure)
1,2,3 are violated ***
Population growth rates become ***
population size increases.
as
• Assumption of constant birth and death rates is violated.
• Birth and/or death rates must change as pop. size changes.
Population equilibrium is reached when ***
Those rates can change with density
(=***
).
Density-dependent factors***
.
Habitat quality affects reproductive variables
affected (*** is lowered).
Reproductive variables are ***
Population size is regulated by densitydependent factors affecting birth and/or death
rates.
1) Density-dependence in plants first
decreases growth.
Size hierarchy develops.
skewed
2) Density-dependence secondly increases
some components of reproduction;
decreases others…
3) Density-dependence thirdly decreases
survival. Intraspecific competition causes
“self-thinning”.
Biomass (g)
Density of surviving plants
r (intrinsic rate of increase)
decreases as a ***
• Population growth is ***
.
.
rm
slope = rm/K
r
r0
N
K
Logistic equation
• Describes a population that experiences
***
density-dependence.
• Population size stabilizes at K = ***
• dN/dt = ****
• where rm = maximum rate of increase w/o
resource limitation
= ‘intrinsic rate of increase’
***
= carrying capacity
• ***
= environmental break (resistance)
= proportion of unused resources
Logistic (***
) growth occurs when the
population reaches a resource limit.
• ***
at K/2 separates accelerating and
decelerating phases of population growth;
point of ***
Logistic curve incorporates influences of
***
per capita growth rate and
***
population size.
Specific
Assumptions of logistic model:
• Population growth is proportional to the
remaining resources (linear response)
• All individuals can be represented by an
average (no change in age structure)
• Continuous resource renewal (constant E)
• Instantaneous responses to crowding
.
***
• K and r are specific to particular organisms
in a particular environment.
Review: Logistic Growth Model
• Problem Set 2-3 (see pg. 80)