Population Growth

The first law of populations:

“A population will grow (or decline)exponentially as long as the environment

experienced by all individualsremains constant”

-P. Turchin. 2001. Oikos 94:17-26.

Darwin’s elephants

“…after a period of 740 to 750 yearsthere would be nearly 19 millionelephants alive, descended from the first pair.”

Is this realistic?

Actually, yes.Any population may show thisgrowth for short periods of time

Asking why it ISN’T happening isextremely valuable

Population growth

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

Let’s assume geographic closure:

Nt+1 = Nt + B - D

Let’s also assume continuous populationgrowth….

Continuous population growth

dN/dt = B - D

“the change in population size with respect to time is due to the difference

between number of births and deaths”

b = per capita birth rate

d = per capita mortality rate

dN/dt = (b - d)N

The exponential growth rate: r

Let (b- d) = r; The equation becomes:

dN/dt = rN

The change in population size withrespect to time is a function of the

exponential growth ratemultiplied by the population size

Can we predict population size into the future?

dN/dt = rN

Using calculus, we can integrate thisInto the equation,

Nt = N0ert

•This is known as the exponential equation•It describes continuous population growth

Modeling grizzly bears in Yellowstone

Gotelli 2001 p. 7

BUT… grizzlies don’t have continuous reproduction…

We can use a discrete-time model to represent breeding seasons separated

by non-breeding seasons

N(t+1) = N(t) + bN(t) – dN(t)

N(t) = λtN(0) where λ = 1 + R

or, more generally,

The trajectory might look like this…

Gotelli 2001 p. 12

Some terminology

r = exponential growth rate

λ = geometric growth rate

Relationship between r and λ

er = λ or r = ln(λ)

If r is … then λ is….and the population is

0 1 stable<0 <1 decreasing>0 >1 increasing

How do exponential and geometric growth differ?

The implications of continuous versusdiscrete reproductive periods….

Some assumptions about exponential and geometric growth

•Geographic closureindividuals added only by birth,lost only by death

•Birth and death rates are constant This means growth rate is constant

•No age/stage structure

•No time lags

Constant growth rate??

This means (b – d) must be constant, and

it also implies that resources are unlimited

•Real populations are like this for shortperiods of time

•This idea is the key to natural selectionand evolution

•Clearly exists for insect pest outbreaks,invasive exotic species, Homo sapiens

That “no structure” assumption…

Can we reasonably expect a populationto have neither age nor stage structure?

(just what IS structure, anyway??)

Sometimes, yes.

What if we can’t? How does it matter?

Consider the difference…

Which one has more growth potential?Why?

N = 10 N = 10

Consider the difference…

10 yearlings

5 first-year

5 2-year

N = 20 N = 20

2 yearlings

3 first-year

15 2-year

Which one has more growth potential?

Why?

Structure and population growth

If individuals differ in reproductive ratesor survival rates “enough” and thesedifferences can be summarized byage or stage, we can project populationgrowth using a structured model.

An example…

A simplified model forStrix Occidentalis

First breeding: 2 yearsMaximum age: ~ 15 years

A stage-classified model:

J A

Unstructured versus stage-classified model

No structure: N(t+1) = λ * N(t)

Growth rate is the difference between births and deaths, or number surviving plus the number of their offspring that also survive…

N(t+1) = (S + F) * N(t)

N(t+1) = N(t) * S + N(t) * F

Unstructured versus stage-classified model

N(t+1) = N(t) * S + N(t) * F

Structured model:Survival and reproductionAre stage-specific

AddStructure…

N(t+1) = N(t)adults * Sadults +N(t)juveniles * Sjuveniles +N(t)adults * fertility

S

F

Fertility: the number of young per adultsurviving to enter the population

We can rewrite this:

N(t+1)juveniles = N(t)adults * F

N(t+1)adults = N(t)juveniles * Sj + N(t)adults * Sa______

N(t+1)total owls

Unstructured versus stage-classified model

•Adding structure to our model lets ustrack number of individuals in eachstage through time as well as total population numbers

•When do we need to add structure?-When demographic rates vary

by age/stage

-When we have the information onthose demographic rates

Review

Biological populations will grow at a constantrate unless prevented from doing so

This constant growth can be modeledas either exponential or geometric

Exponential growth: continuous timer: the exponential growth rate

Geometric growth: discrete time stepsλ: geometric growth rate

Review

Geometric and exponential growth assume:

Constant demographic ratesNo structureGeographic closure (no I,E)No time lags

NOTE CHANGE: Exponential and GeometricModels give same results if geometric modelCorrectly accounts for reproduction (timeStep = breeding interval)

Review

Populations differ in structure, and thisaffects their growth potential

Structured models should be used when:

-Demographic rates differ by age/stage-When the information is available

UNGRADED HOMEWORK

1). What is the difference between geometricAnd exponential population growth?

2). What is the first law of populations?