Population Dynamics

Focus on births (B) & deaths (D)

B = bNt , where b = per capita rate (births per individual per time)

D = dNt

N = bNt – dNt = (b-d)Nt

Exponential Growth

• Density-independent growth models

Discrete birth intervals (Birth Pulse)

vs.

Continuous breeding (Birth Flow)

Time

0 1 2 3 4 5 6

N

0

20

40

60

80

100

120

140

Nt = N0t

= 2, N0 = 2

Geometric Growth (Birth Pulse) > 1

< 1

= 1

Nt = N0 t

Geometric Growth• When generations do not overlap, growth can be

modeled geometrically.

Nt = Noλt

– Nt = Number of individuals at time t.

– No = Initial number of individuals.

– λ = Geometric rate of increase.– t = Number of time intervals or generations.

Exponential Growth Birth Pulse Population (Geometric Growth)

e.g., woodchucks

(10 individuals to 20 indivuals)

N0 = 10, N1 = 20

N1 = N0 ,

where = growth multiplier = finite rate of increase

> 1 increase

< 1 decrease

= 1 stable population

Exponential Growth Birth Pulse Population

N2 = 40 = N1

N2 = (N0 ) = N0 2

Nt = N0 t

Nt+1 = Nt

Exponential Growth • Density-independent growth models

Discrete birth intervals (Birth Pulse)

vs.

Continuous breeding (Birth Flow)

Exponential Growth• Continuous population growth in an unlimited

environment can be modeled exponentially.

dN / dt = rN

• Appropriate for populations with overlapping generations.– As population size (N) increases, rate of population

increase (dN/dt) gets larger.

Exponential Growth• For an exponentially growing population, size at any

time can be calculated as:

Nt = Noert

• Nt = Number individuals at time t.• N0 = Initial number of individuals.• e = Base of natural logarithms = 2.718281828459 • r = Per capita rate of increase.• t = Number of time intervals.

Exponential Population Growth

Exponential Population Growth

Time

ln(N

t)

ln(N0)

slope = r

ln(Nt) = ln(N0) + rtNt = N0ert

Difference Eqn

Note: λ = er

Exponential growth and change over time

Time (t)

Nu

mb

er

(N)

N = N0ert

Number (N)S

lop

e (

dN

/dt)

dN/dt = rN

Slope = (change in N) / (change in time)

= dN / dt

ON THE MEANING OF r rm - intrinsic rate of increase – unlimited

resourses

rmax – absolute maximal rm

- also called rc = observedr

r > 0r < 0r = 0

x

x

rm > 0

rm >> 0

rm < 0

rmax

Geographic Range andValues of rm

Intrinsic Rates of Increase• On average, small organisms have higher

rates of per capita increase and more variable populations than large organisms.

Growth of a Whale Population

• Pacific gray whale (Eschrichtius robustus) divided into Western and Eastern Pacific subpopulations.– Rice and Wolman estimated average annual

mortality rate of 0.089 and calculated annual birth rate of 0.13.

0.13 - 0.089 = 0.041

– Gray Whale population growing at 4.1% per yr.

Growth of a Whale Population

• Reilly et.al. used annual migration counts from 1967-1980 to obtain 2.5% growth rate.– Thus from 1967-1980, pattern of growth in

California gray whale population fit the exponential model:

Nt = Noe0.025t

• What values of λ allow– Population Growth

– Stable Population Size

– Population Decline

• What values of r allow– Population Growth

– Stable Population Size

– Population Decline• λ < 1.0

• λ = 1.0

• λ > 1.0

• r = 0

• r > 0

• r < 0

Logistic Population Growth• As resources are depleted, population growth rate

slows and eventually stops

• Sigmoid (S-shaped) population growth curve– Carrying capacity (K): number of individuals of a

population the environment can support• Finite amount of resources can only support a finite number

of individuals

Logistic Population Growth

Logistic Population GrowthdN / dt = rN

dN/dt = rN(1-N/K)

• r = per capita rate of increase • When N nears K, the right side of the equation nears

zero– As population size increases, logistic growth rate becomes a

small fraction of growth rate• Highest when N=K/2• N/K = Environmental resistance

Exponential & Logistic Growth(J & S Curve)

Logistic Growth

Actual Growth

Populations Fluctuate

Limits to Population Growth

• Environment limits population growth by altering birth and death rates– Density-dependent factors

• Disease, Resource competition

– Density-independent factors• Natural disasters

Galapagos Finch Population Growth

Logistic Population Model

A. Discrete equationNt = 2, R = 0.15,

K = 450

Logistic Population Growth

0.0

100.0

200.0

300.0

400.0

500.0

0 20 40 60 80 100

Time

N(t)

Kr NNNN t

ttt1

1

- Built in time lag = 1

- Nt+1 depends on Nt

I. Logistic Population Model

B. Density Dependence

I. Logistic Population Model C. Assumptions

• No immigration or emigration

• No age or stage structure to influence births and deaths

• No genetic structure to influence births and deaths

• No time lags in continuous model

I. Logistic Population Model C. Assumptions

• Linear relationship of per capita growth rate and population size (linear DD)

K

I. Logistic Population Model C. Assumptions

• Linear relationship of per capita growth rate and population size (linear DD)

• Constant carrying capacity – availability of resources is constant in time and space– Reality?

I. Logistic Population Model

Discrete equation Nt = 2, r = 1.9,

K = 450Logistic Population Growth

0.0100.0200.0300.0400.0500.0600.0

0 20 40 60 80 100

Time

N(t)

Damped Oscillations

r <2.0

I. Logistic Population Model

Discrete equationNt = 2, r = 2.5,

K = 450

Stable Limit Cycles

2.0 < r < 2.57* K = midpoint

Logistic Population Growth

0.0100.0200.0300.0400.0500.0600.0

0 20 40 60 80 100

Time

N(t)

I. Logistic Population Model

Discrete equationNt = 2, r = 2.9,

K = 450

Chaos r > 2.57• Not random change• Due to DD feedback and time lag in model

Logistic Population Growth

0.0

200.0

400.0

600.0

800.0

0 20 40 60 80 100

Time

N(t)

Underpopulation or Allee Effect• Opposite type of DD

– population size down and population growth down

N

Vital rate

N* K

d

b

b=db=d

b<dr<0

I. Review of Logistic Population Model

D. Deterministic vs. Stochastic Models

Nt = 1, r = 2, K = 100

* Parameters set deterministic behavior same

I. Review of Logistic Population Model

D. Deterministic vs. Stochastic Models

Nt = 1, r = 0.15, SD = 0.1;

K = 100, SD = 20

* Stochastic model, r and K change at random each time step

I. Review of Logistic Population Model

D. Deterministic vs. Stochastic Models

Nt = 1, r = 0.15, SD = 0.1;

K = 100, SD = 20

* Stochastic model

I. Review of Logistic Population Model

D. Deterministic vs. Stochastic Models

Nt = 1, r = 0.15, SD = 0.1;

K = 100, SD = 20

* Stochastic model

II. Environmental StochasticityA. Defined

• Unpredictable change in environment occurring in time & space

• Random “good” or “bad” years in terms of changes in r and/or K

• Random variation in environmental conditions in separate populations

• Catastrophes = extreme form of environmental variation such as floods, fires, droughts

• High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

II. Environmental StochasticityA. Defined

• Unpredictable change in environment occurring in time & space

• Random “good” or “bad” years in terms of changes in r and/or K

• Random variation in environmental conditions in separate populations

• Catastrophes = extreme form of environmental variation such as floods, fires, droughts

• High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

II. Environmental StochasticityA. Defined

• Unpredictable change in environment occurring in time & space

• Random “good” or “bad” years in terms of changes in r and/or K

• Random variation in environmental conditions in separate populations

• Catastrophes = extreme form of environmental variation such as floods, fires, droughts

• High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

II. Environmental StochasiticityB. Examples – variable fecundity

Relation Dec-Apr rainfall and number

of juvenile California quail per adult (Botsford et

al. 1988 in Akcakaya et al.

1999)

II. Environmental StochasiticityB. Examples - variable

survivorship

Relation total rainfall pre-nesting and proportion of Scrub Jay nests to

fledge (Woolfenden and Fitzpatrik 1984 in Akcakaya et al.

1999)

II. Environmental StochasiticityB. Examples – variable rate of increase

Muskox population on

Nunivak Island, 1947-1964

(Akcakaya et al. 1999)

II. Environmental Stochasiticity- Example of random K

• Serengeti wildebeest data set – recovering from Rinderpest outbreak

– Fluctuations around K possibly related to rainfall

rNdt

dN

K

NrN

dt

dN1

Exponential vs. Logistic

No DDAll populations same

DDAll populations same

No Spatial component

Space Is the Final Frontier in Ecology

• History of ecology = largely nonspatiale.g., *competitors mixed perfectly with prey

*homogeneous ecosystems with uniform distributions of resources

• But ecology = fundamentally spatial– ecology = interaction of organisms with their

[spatial] environment

Incorporating Space

Metapopulation: a population of subpopulations linked by dispersal of organisms

Two processes = extinction & recolonization

• subpopulations separated by unsuitable habitat (“oceanic island-like”)

• subpopulations can differ in population size & distance between

Metapopulation Model (Look familiar?)

eppcpdt

dp 1

p = habitat patch (subpopulation)c = colonizatione = extinction

Metapopulation Model (Look familiar?)

eppcpdt

dp 1

)]/(1[)(

))]/1/((1[)(

KNNdbdt

dN

meppemdt

dp

Rescue Effect

Another Population Model

Source-sink Dynamics: grouping of multiple subpopulations, some are sinks & some are sources

Source Population = births > deaths = net exporter

Sink Population = births < deaths

<1

<1

>1

Metapopulations• Definition of Population?• Groups of populations within which there is a

significant amount of movement of individuals via dispersal

Classic Metapopulation

Metapopulation Con’t

• This kind of population structure applies when there are “groups” of populations occupying habitat that occurs in discrete patches (patchy).

• These patches are separated by areas of inhospitable habitat, but connected by routes for dispersal.

• Populations fluctuate independently of each other

The probability of dispersal from one patch to another depends on:• Distance between patches

• Nature of habitat corridors linking the patches

• Ability of the species to disperse (vagility or mobility) – dependent on body size

Who Cares?

Why bother discussing these models?

Metapopulations & Source-sink Populatons highlight the importance of:

• habitat & landscape fragmentation

• connectivity between isolated populations

• genetic diversity