Population Ecology

ES 100

8/21/07

Remember from last time:

• Population ecology• Life Tables

• Cohort-based vs. Static• Identifying vulnerable growth stages• Age-specific birth rate• Computing fitness, net reproductive rate and generation

time• Population projections

Today:

Metapopulation Theory Immigration and Emigration Source and Sink Populations Maintaining Genetic Diversity

Population Models Exponential and Logistic growth

Assumptions Doubling time When should this model be used?

Is the Population Increasing, or Decreasing?

Fitness is one indication….. But… Populations vary dramatically over time

(boom/bust cycles) Individuals move in (immigration) and out

(emigration) of populationsMetapopulations (18.5 Bush)

Nt+1 = Nt + (B-D) + (I-E)

Threatened Species:Western Snowy Plover

Before 1970, 53 breeding locations in CA (including Santa Barbara)

Now, 8 breeding sites support 78% of the CA metapopulation

Populations across the landscape

Metapopulation: sum of multiple interacting sub-populations

sub-population A

sub-population B

sub-population D

sub-population C

Populations across the landscape

Genetic diversity is maintained by exchange of genes between the sub-populations

sub-population A

sub-population B

sub-population D

sub-population C

Populations across the landscape

Most mating occurs within a sub-population

sub-population A

sub-population B

sub-population D

sub-population C

Populations across the landscape

Some habitat patches are better than others

hot and dry

most ideal

manypredators

few nesting

sites

Populations across the landscape

Sub-populations can be source populations or sink populations

hot and dry

manypredators

few nesting

sitesmost ideal

source

sink

sink

sink

Populations across the landscape

In source population habitats:• living conditions are good, so births meet or exceed deaths• competition may be great, forcing some members out

hot and dry

manypredators

few nesting

sitesmost ideal

source

sink

sink

sink

Populations across the landscape

locally extinctsource of

recruits

source

sink

If a sub-population goes extinct, it can be revived by recruits from a source population….

But sinks are important too!

Controls on immigration

Distance to source population

main

lan

d

Lots of immigration

Little immigration

Obstacles• Mountains• Waterways

mountains

hills

Age Stage

sub-population A sub-population Bsub-population C Total dN/dt =

Nx-Nx-1

0-1 60 25 4 89 -----------

1-2 24 30 12 66 -23

2-3 14 26 10 50 -16

3-4 10 20 4 34 -16

4-5 7 13 1 21 -13

Number of individuals

•Is this population assessment static or cohort based?

•Which sub-population(s) are sources? Sinks?

•Can you develop a life table for each sub-population?

•Can you develop a life table for the total population?

Sample Metapopulation Data

Mathematical Models

Uses:• synthesize information• look at a system quantitatively• test your understanding• predict system dynamics• make management decisions

Population Growth

• t = time

• N = population size (number of individuals)

• = rate of change in population size (ind/time)

• r = maximum/intrinsic growth rate (1/time) = fractional increase, per unit time, when resources are unlimited

dN dt

Population Growth

• Lets build a simple model (to start)

= r * N

• Constant growth rate exponential growth• Assumptions:

• Closed population (no immigration, emigration)• Unlimited resources• No genetic structure• No age/size structure• Continuous growth with no time lags

dN dt

Projecting Population Size

Nt = N0ert

N0 = initial population size

Nt = population size at time t

e 2.7171

r = intrinsic growth rate

t = time

Doubling Time

rtdouble

)2ln(=

Let’s Try It!

The brown rat (Rattus norvegicus) is known to have an intrinsic growth rate of:

0.015 individual/individual*day

Suppose your house is infested with 20 rats. How long will it be before the population doubles? How many rats would you expect to have after 2

months?

Is the model more sensitive to N0 or r?

When Is Exponential Growth a Good Model?

•r-strategists

•Unlimited resources

•Vacant niche

Environmental Stochasticity

Our exponential growth model is deterministicOutcome is determined only by model inputs Intrinsic growth rate varies with ‘good’ and ‘bad’

environmental conditions:Often we know the mean growth rate and the

variance in the growth rate,

These can be incorporated into our model!

r2rσ

Herd Size with Environmental Stochasticity

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10

Year

Herd Size

Herd Size (Deterministic Model)

0

500

1000

1500

2000

2500

0 2 4 6 8 10

Year

Herd Size

Plover Population Model with Stochasticity

Nur, Page and Stenzel: POPULATION VIABILITY ANALYSIS FOR PACIFIC COAST SNOWY PLOVERS

What Controls Population Size and Growth Rate (dN/dt)?

• Density-dependent factors:

Population Density:# of individuals of a certain # of individuals of a certain

species in a given areaspecies in a given area

•Intra-specific competition•food•Space

•contagious disease•waste production•Interspecific competition•Other species interactions!

•Density-independent factors:•disturbance, environmental conditions

•hurricane•flood•colder than normal winter

Time (t)

Pop

ula

tion s

ize (

N)

Can the population really grow forever?

What should this curve look like to be more

realistic?

Population Growth

• Logistic growth

• Assumes that density-dependent factors affect population

• Growth rate should decline when the population size gets large

• Symmetrical S-shaped curve with an upper asymptote

Population Density:# of individuals of a certain # of individuals of a certain

species in a given areaspecies in a given area

Population Growth

How do you model logistic growth?

How do you write an equation to fit that S-shaped curve?

Start with exponential growth

= r * N= r * NdN dt

Population Growth

How do you model logistic growth?

How do you write an equation to fit that S-shaped curve?

Population growth rate (dN/dt) is limited by carrying capacity

dN dt = r * N (1 – )= r * N (1 – )N

K

What does (1-N/K) mean?

Unused Portion of K

If green area represents carrying capacity, and yellow area represents current population size…

K = 100 individualsN = 15 individuals(1-N/K) = 0.85 population is growing at 85% of the growth rate of an exponentially increasing population

Population Growth

Logistic growth Lets look at 3 cases:

N<<K (population is small compared to carrying capacity)

Result?

N=K (population size is at carrying capacity)

Result?

N>>K (population exceeds carrying capacity)

Result?

= r * N (1 – )= r * N (1 – )N K

dN dt

Population Size as a Function of Time

rtt eNNK

KN −−+

=]/)[(1 00

Last Time…Metapopulation Theory

Immigration and EmigrationSource and Sink PopulationsMaintaining Genetic Diversity

Population ModelsExponential

AssumptionsDoubling timeWhen should this model be used?

Logistic growthHow does it account for density dependent factors?What is the difference between dN/dt and r?3 cases:

N<<K (exponential growth)N=K (no growth) N>>K (exponential decline)

At What Population Size does the Population Grow Fastest?

Population growth rate (dN/dt) is slope of the S-curve

Maximum value occurs at ½ of K This value is often used to maximize sustainable

yield (# of individuals harvested)/tim

eBush pg. 225

Fisheries Management:MSY (maximum sustainable yield)

What is the maximum # of individuals that can be harvested, year after year, without lowering N?= rK/4 which is dN/dt at N= 1/2 K

What happens if a fisherman ‘cheats’?

What happens if environmental conditions fluctuate and it is a ‘bad year’ for the fishery?

Assumptions of Logistic Growth Model:

• Closed population (no immigration, emigration)• No genetic structure• No age/size structure• Continuous growth with no time lags• Constant carrying capacity• Population growth governed by intraspecific competition• “recruitment” depends on current population size

Lets Try It!

⎟⎠

⎞⎜⎝

⎛ −=K

NrN

dt

dN1 rtt eNNK

KN −−+

=]/)[(1 00

Formulas:

A fisheries biologist is maximizing her fishing yield by maintaininga population of lake trout at exactly 500 fish.

Predict the initial population growth rate if the population is stocked with an additional 600 fish. Assume that the intrinsic growth rate for trout is 0.005 individuals/individual*day .

How many fish will there be after 2 months?