Population Ecology

ES 100

10/23/06

Announcements:

• Problem Set will be posted on course website today. • Start early!• Due Friday, November 3rd

• Midterm: 1 week from today• Last year’s midterm is posted on website• This year: will require a bit more thinking

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)

• = (instantaneous) rate of change in population size

• r = maximum/intrinsic growth rate (1/time) = b-d (birth rate – death rate)

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(

When Is Exponential Growth a Good Model?

•r-strategists

•Unlimited resources

•Vacant niche

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?

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

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

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?