POPULATION GROWTH
POPULATION GROWTH The growth of natural populations may proceed through several recognizable stages Positive growth Fluctuations Oscillations Population decline Population extinction
When resources are NOT LIMITING, natural populations will exhibit Exponential Population Growth Laboratory populations Introduced populations Populations recovering from near collapse
Northern Elephant Seal
Population Growth Rates
Assumptions of the exponential model The population is closed Constant birth and death rates No genetic variation in the population for birth and death rates No age or size structure Continuous growth with no resource limitation or time lags
Exponential Population Growth Discrete-time model number of individuals at time t
number of births during the interval t, t+1
number of deaths during the interval t, t+1
Discrete growth factor
Finite rate of increase
Initial population size
700
N0= 50 λ= 1.2
POPULATION
600 500
} ΔN
400 300 } ΔN
200 100 0 0
2
4
6
8
TIME
10
12
14
16
Continuous-time model
Intrinsic rate of increase
QUANTITATIVE ANALYSIS
Equilibrium Solution
r
Doubling Time populations experiencing exponential growth will double in a fixed amount of time
solve for t
Environment
Natality Rates
Mortality Rates
Age Distribution
Population Growth (+,-,0)
Environmental heterogeneity produces variation in the intrinsic rate of increase (r)
Environmental Stochasticity deterministic model
Let r be a random variable with µr and σr2
Demographic Stochasticity random differences among individuals in survival and reproduction within a season in natural populations the schedule of births and deaths is not deterministic suppose Births = 2Deaths BBDBBDBBDBBD (deterministic) BBBDDBDBBBBD (stochastic)
Probability of Extinction
No population can increase without limit As a population increases, the amount of essential resources per capita approaches a minimum (Liebig's Law) Resources become distributed unevenly among individuals Struggle for Existence Intraspecific Competition
Resources eventually become limiting for population growth
When resources are LIMITING, populations often exhibit Densitydependent survivorship As density increases, mortality rates increase
Uta stansburiana
Nature 406, 985-988 (31 August 2000)
When resources are limiting, populations often exhibit density-dependent growth of individuals As density increases, individuals grow more slowly and achieve a smaller final size
When resources are limiting, density can affect the size structure of a population
Density
As density increases, individuals are smaller on average
HZ I HZ II HZ III
For many types of organisms, there is often a positive relationship between body size and fecundity
When resources are limiting, populations often exhibit density-dependent reproductive rates
DENSITY SURVIVAL SIZE FECUNDITY INTRINSIC RATE OF INCREASE (r)
arctic ground squirrel (Spermophilus parryii)
Here ln(Nt+1/Nt ) is the rate of population change (Nt+1 is the population size in the spring of year t+1; Nt is the population size in the spring of year t ). Data are shown for controls and for the former experimental treatments. The equation of the dashed line is: rate of change = -1.5[log(density)] + 0.5, with r2 = 0.93, P < 0.001). Data from sites where food was added were not included in the regression, but demonstrate that rates of increase were positively influenced by added food. Nature 408, 460-463 (23 November 2000)
r0
r K
N
Continuous-time model (exponential model)
r
r0
N
If the intrinsic rate of increase (r) decreases linearly with population size (N), then the rate of population growth (dN/dt) will exhibit a quadratic relationship with N recall: exponential model
When resources are LIMITING, natural populations often exhibit Logistic Population Growth K = carrying capacity maximum number of individuals that the environment can support
K
K
QUANTITATIVE ANALYSIS Equilibrium Solutions
N* = 0 =K
r= 0.5 N0= 20 K= 200 220 200
POPULATION
180
dN
160 140
dN
120 100 80 60 40
dN
20 0 0
1
2
3
4
5
6
7
8
TIME
9
10 11 12 13 14
Assumptions of the logistic model Linear density dependence No genetic structure
r
No age structure No immigration or emigration No time lags
r0
N
A classic experiment with blowflies Nicholson (1957)
A classic experiment with blowflies Fed cultures of blowflies a fixed amount of beef liver for the larvae daily Cultures received ample sugar and water for adults Followed the number of flies in the various experimental cages
The result was clearly not logistic growth! 50g
25g
What caused time lags? When the adult population was high, more eggs/larvae were produced than the liver could support to pupation As a result, no adults were produced during this period and the adult fly population gradually decreased as adult flies died This led to fewer eggs being laid Eventually the intensity of larval competition was sufficiently reduced for a fraction of the larvae to successfully pupate These surviving larvae led to an increase in the adult fly population
An illustration Imagine that the liver can support six, and only six, larvae N=3 This population is at its carrying capacity
N=6 This population is above carrying capacity
3 adults die 2 eggs
all larvae live
2 adults die all larvae die
This population is below its carrying capacity
N=2
2 adults die all larvae die
N=4
This population is above carrying capacity
Logistic growth with time lags
Population growth is controlled by the quantity (r • τ): (r • τ) < 0.368, growth is like the usual logistic 0.368 < (r • τ) < 1.57, damped oscillations (r • τ) > 1.57, stable limit cycle
Discrete-time model
The discrete logistic equation exhibits a variety of dynamical behaviors according to the value of r
The value of the carrying capacity (K) can vary spatially and temporally and is subject to environmental stochasticity
Populations that overshoot the carrying capacity can cause environmental degradation and lower the carrying capacity
Rapidly growing populations can overshoot the carrying capacity (K) and collapse
Density-independent regulation of populations Factors that effect populations independent of population size Andrewartha and Birch (1954) The Distribution and Abundance of Animals
Herbert George Andrewartha (1907-1992)
Density-independent factors extreme weather events:
droughts floods cold snaps hurricanes natural disasters: earthquakes fire volcanoes
Density-independent factors limit the amount of time when the rate of increase (r) is positive
Age-Structured Population Growth We have considered models of the form:
We now want to track the number of individuals in different age classes
Reproduction
Survival
Leslie matrix A is an age-classified population projection matrix elements of A are:
survival probabilities fertilities
A matrix is a set of elements, organized into rows and columns rows
columns
rows columns
Matrix multiplication
In general,
Fertility of age class 2
Survival of age class 2
Leslie matrix and the life table 1 0
2 1
3 2
4 3
Age-class (i) 4
Age (x)
x
i
0
lx
mx
1.0
0
1
1
0.8
2
2
2
0.5
6
3
3
0.1
3
Pi
Fi
The Leslie Matrix can be modified to account for varied life-history patterns size structure asexual reproduction
When resources are NOT LIMITING, natural populations will exhibit Exponential Population Growth Laboratory populations Introduced populations Populations recovering from near collapse
Northern Elephant Seal
Population Growth Rates
Assumptions of the exponential model The population is closed Constant birth and death rates No genetic variation in the population for birth and death rates No age or size structure Continuous growth with no resource limitation or time lags
Exponential Population Growth Discrete-time model number of individuals at time t
number of births during the interval t, t+1
number of deaths during the interval t, t+1
Discrete growth factor
Finite rate of increase
Initial population size
700
N0= 50 λ= 1.2
POPULATION
600 500
} ΔN
400 300 } ΔN
200 100 0 0
2
4
6
8
TIME
10
12
14
16
Continuous-time model
Intrinsic rate of increase
QUANTITATIVE ANALYSIS
Equilibrium Solution
r
Doubling Time populations experiencing exponential growth will double in a fixed amount of time
solve for t
Environment
Natality Rates
Mortality Rates
Age Distribution
Population Growth (+,-,0)
Environmental heterogeneity produces variation in the intrinsic rate of increase (r)
Environmental Stochasticity deterministic model
Let r be a random variable with µr and σr2
Demographic Stochasticity random differences among individuals in survival and reproduction within a season in natural populations the schedule of births and deaths is not deterministic suppose Births = 2Deaths BBDBBDBBDBBD (deterministic) BBBDDBDBBBBD (stochastic)
Probability of Extinction
No population can increase without limit As a population increases, the amount of essential resources per capita approaches a minimum (Liebig's Law) Resources become distributed unevenly among individuals Struggle for Existence Intraspecific Competition
Resources eventually become limiting for population growth
When resources are LIMITING, populations often exhibit Densitydependent survivorship As density increases, mortality rates increase
Uta stansburiana
Nature 406, 985-988 (31 August 2000)
When resources are limiting, populations often exhibit density-dependent growth of individuals As density increases, individuals grow more slowly and achieve a smaller final size
When resources are limiting, density can affect the size structure of a population
Density
As density increases, individuals are smaller on average
HZ I HZ II HZ III
For many types of organisms, there is often a positive relationship between body size and fecundity
When resources are limiting, populations often exhibit density-dependent reproductive rates
DENSITY SURVIVAL SIZE FECUNDITY INTRINSIC RATE OF INCREASE (r)
arctic ground squirrel (Spermophilus parryii)
Here ln(Nt+1/Nt ) is the rate of population change (Nt+1 is the population size in the spring of year t+1; Nt is the population size in the spring of year t ). Data are shown for controls and for the former experimental treatments. The equation of the dashed line is: rate of change = -1.5[log(density)] + 0.5, with r2 = 0.93, P < 0.001). Data from sites where food was added were not included in the regression, but demonstrate that rates of increase were positively influenced by added food. Nature 408, 460-463 (23 November 2000)
r0
r K
N
Continuous-time model (exponential model)
r
r0
N
If the intrinsic rate of increase (r) decreases linearly with population size (N), then the rate of population growth (dN/dt) will exhibit a quadratic relationship with N recall: exponential model
When resources are LIMITING, natural populations often exhibit Logistic Population Growth K = carrying capacity maximum number of individuals that the environment can support
K
K
QUANTITATIVE ANALYSIS Equilibrium Solutions
N* = 0 =K
r= 0.5 N0= 20 K= 200 220 200
POPULATION
180
dN
160 140
dN
120 100 80 60 40
dN
20 0 0
1
2
3
4
5
6
7
8
TIME
9
10 11 12 13 14
Assumptions of the logistic model Linear density dependence No genetic structure
r
No age structure No immigration or emigration No time lags
r0
N
A classic experiment with blowflies Nicholson (1957)
A classic experiment with blowflies Fed cultures of blowflies a fixed amount of beef liver for the larvae daily Cultures received ample sugar and water for adults Followed the number of flies in the various experimental cages
The result was clearly not logistic growth! 50g
25g
What caused time lags? When the adult population was high, more eggs/larvae were produced than the liver could support to pupation As a result, no adults were produced during this period and the adult fly population gradually decreased as adult flies died This led to fewer eggs being laid Eventually the intensity of larval competition was sufficiently reduced for a fraction of the larvae to successfully pupate These surviving larvae led to an increase in the adult fly population
An illustration Imagine that the liver can support six, and only six, larvae N=3 This population is at its carrying capacity
N=6 This population is above carrying capacity
3 adults die 2 eggs
all larvae live
2 adults die all larvae die
This population is below its carrying capacity
N=2
2 adults die all larvae die
N=4
This population is above carrying capacity
Logistic growth with time lags
Population growth is controlled by the quantity (r • τ): (r • τ) < 0.368, growth is like the usual logistic 0.368 < (r • τ) < 1.57, damped oscillations (r • τ) > 1.57, stable limit cycle
Discrete-time model
The discrete logistic equation exhibits a variety of dynamical behaviors according to the value of r
The value of the carrying capacity (K) can vary spatially and temporally and is subject to environmental stochasticity
Populations that overshoot the carrying capacity can cause environmental degradation and lower the carrying capacity
Rapidly growing populations can overshoot the carrying capacity (K) and collapse
Density-independent regulation of populations Factors that effect populations independent of population size Andrewartha and Birch (1954) The Distribution and Abundance of Animals
Herbert George Andrewartha (1907-1992)
Density-independent factors extreme weather events:
droughts floods cold snaps hurricanes natural disasters: earthquakes fire volcanoes
Density-independent factors limit the amount of time when the rate of increase (r) is positive
Age-Structured Population Growth We have considered models of the form:
We now want to track the number of individuals in different age classes
Reproduction
Survival
Leslie matrix A is an age-classified population projection matrix elements of A are:
survival probabilities fertilities
A matrix is a set of elements, organized into rows and columns rows
columns
rows columns
Matrix multiplication
In general,
Fertility of age class 2
Survival of age class 2
Leslie matrix and the life table 1 0
2 1
3 2
4 3
Age-class (i) 4
Age (x)
x
i
0
lx
mx
1.0
0
1
1
0.8
2
2
2
0.5
6
3
3
0.1
3
Pi
Fi
The Leslie Matrix can be modified to account for varied life-history patterns size structure asexual reproduction
