Population Growth
The first law of populations:
“A population will grow (or decline)
exponentially as long as the environment
experienced by all individuals
remains constant”
-P. Turchin. 2001. Oikos 94:17-26.
Darwin’s elephants
“…after a period of 740 to 750 years
there would be nearly 19 million
elephants alive, descended from the
first pair.”
Is this realistic?
Actually, yes.
Any population may show this
growth for short periods of time
Asking why it ISN’T happening is
extremely 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 population
growth….
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 with
respect to time is a function of the
exponential growth rate
multiplied by the population size
Can we predict population size into the
future?
dN/dt = rN
Using calculus, we can integrate this
Into 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)
or, more generally,
N(t) = λtN(0) where λ = 1 + R
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
<0
>0
1
<1
>1
stable
decreasing
increasing
How do exponential and geometric
growth differ?
The implications of continuous versus
discrete reproductive periods….
Some assumptions about exponential
and geometric growth
•Geographic closure
individuals 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 short
periods of time
•This idea is the key to natural selection
and evolution
•Clearly exists for insect pest outbreaks,
invasive exotic species, Homo sapiens
That “no structure” assumption…
(just what IS structure, anyway??)
Can we reasonably expect a population
to have neither age nor stage structure?
Sometimes, yes.
What if we can’t? How does it matter?
Consider the difference…
N = 10
N = 10
Which one has more growth potential?
Why?
Consider the difference…
N = 20
N = 20
10 yearlings
2 yearlings
5 first-year
3 first-year
5 2-year
15 2-year
Which one has more growth potential?
Why?
Structure and population growth
If individuals differ in reproductive rates
or survival rates “enough” and these
differences can be summarized by
age or stage, we can project population
growth using a structured model.
An example…
A simplified model for
Strix Occidentalis
First breeding: 2 years
Maximum 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
Structured model:
Survival and reproduction
Are stage-specific
N(t+1) = N(t) * S + N(t) * F
Add
Structure…
N(t+1) = N(t)adults * Sadults +
N(t)juveniles * Sjuveniles +
N(t)adults * fertility
Fertility: the number of young per adult
surviving to enter the population
S
F
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 us
track number of individuals in each
stage 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 on
those demographic rates
Review
Biological populations will grow at a constant
rate unless prevented from doing so
This constant growth can be modeled
as either exponential or geometric
Exponential growth: continuous time
r: the exponential growth rate
Geometric growth: discrete time steps
λ: geometric growth rate
Review
NOTE CHANGE: Exponential and Geometric
Models give same results if geometric model
Correctly accounts for reproduction (time
Step = breeding interval)
Geometric and exponential growth assume:
Constant demographic rates
No structure
Geographic closure (no I,E)
No time lags
Review
Populations differ in structure, and this
affects 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 geometric
And exponential population growth?
2). What is the first law of populations?