Populations I:
a primer
Bio 415/615
5 questions
1. What is exponential growth, and why do we
care about exponential growth models?
2. How is the parameter r related to births and
deaths?
3. What is the parameter lambda (λ), and how
does it relate to r?
4. How do stochastic and deterministic models
differ?
5. What is density dependent in a logistic
growth model, and how does this relate to
carrying capacity?
How populations grow
How populations grow
Thomas Malthus (1766-1834)
English economist
An Essay on the Principle of
Population (6 eds, 1798-1826)
A population, if unchecked, increases as a geometric
rate: 2, 4, 8, 16, 32, … (Contrasted with increases in
food supply.)
Became a basis for Darwinian natural selection.
number of
individuals
Potential for geometric increase
= exponential
growth
32
16
8
4
2
time
Potential for geometric increase
= exponential growth
Nt = N0 e rt
Nt = number of individuals at time t (in the future)
No = number of individuals ‘now’
e = constant (2.71828…)
r = intrinsic rate of increase (Malthusian parameter)
t = time (beware of units)
Births and deaths
• How do populations change?
CLOSED: births (B) and deaths (D)
OPEN: add immigration (I), emigration (E)
For a closed population, a population can only
change with births and deaths:
N = B – D, or
dN = B - D
dt
Births and deaths
• Can define ‘instantaneous’ rate of births and
deaths by multiplying population size by per
capita (per individual) birth rate, death rate:
B = bN
D = dN
• So now population changes occur as a result of
per capita birth and death rates:
dN = (b – d)N
dt
Births and deaths: assumptions?
dN = (b – d)N
dt
1. Births and death occur instantly, simultaneously.
2. Birth and death rates are constant, irrespective of
N.
3. Time is continuous rather than discrete.
4. Individuals are identical (no genetic variation, no age
or size differences, no environmental differences).
These are obviously WRONG. So why use this model?
r = intrinsic rate of increase
r = difference between births and deaths (b - d)
= ‘intrinsic rate of increase’
If r > 0, population increases exponentially
r < 0, population decreases to extinction
r = 0, population doesn’t change
Note r is per capita
dN = rN
dt
r and population growth
• In the exponential
model, change is
proportional to N;
growth speeds up if r>0
• A ‘semilog’ growth
makes exponential
growth appear linear
[y axis is ln(N) ]
r and population growth
Can calculate ‘doubling time’ as:
tdouble= ln(2) / r
But if assumptions are wrong,
why care about exponential
growth?
• All populations have exponential potential
• To figure out why populations deviate
from exponential growth
– Ie, NULL MODEL
• We’ll add in complications, but we’d still
like to know about r
• ‘’Baseline model’
Relax assumption 1: discrete vs.
continuous time
• Why is time not necessarily continuous?
– Living and dying takes time! Many organisms
are on annual or multi-annual cycles of
births and deaths
• We can ask: how much did population
grow this year?
Nt+1 = Nt + rdNt
Ratio of this year’s growth
(eg, 10%)
Relax assumption 1: discrete vs.
continuous time
• Why is time not necessarily continuous?
– Living and dying takes time! Many organisms
are on annual or multi-annual cycles of
births and deaths
• We can ask: how much did population
grow this year?
Nt+1 = Nt (1 + rd)
λ = (1 + rd)
Lambda (λ) is the discrete
version of r, called the finite
rate of increase.
Relax assumption 1: discrete vs.
continuous time
Note r = ln(λ)
Populations grow if r>0 or λ>1
Populations decline if r<0 or λ<1
Populations are static if r=0 or λ=1
Relax assumption 2: population
stochasticity
• What is stochasticity?
– Deterministic processes leave nothing to
chance
– Stochastic models are, to some extent,
unpredictable
• Why do we model stochasticity?
– Because even though the expectation might
not change, outcomes can depend on amount
of uncertainty
Types of population stochasticity
• Environmental stochasticity
– Births and deaths depend on the
environment in a known way, but the
environment is itself unpredictable
• Demographic stochasticity
– Order of births and deaths may fluctuate,
even if the rate is generally constant
Stochastic parameters:
mean and variance
• Mean is the expected value; would be
the ‘typical’ outcome if you repeated the
process many times
• Variance describes how unpredictable
the expected outcome is
Stochastic parameters:
mean and variance
The outcome of stochastic population change
depends on both the expected pattern (mean)
and the amount of uncertainty involved
(variance)!
Eg, if the variance is
twice as great as the
expected (mean) value
of r, extinction is very
likely.
Stochastic parameters:
mean and variance
Is demographic stochasticity more important at
high or low population sizes? Why?
P(extinction) = (d/b)^No
Relaxing assumption 3: limited
resources and crowding
Back to Malthus:
A population, if unchecked, increases as a
geometric rate: 2, 4, 8, 16, 32, …
However, resource supplies are finite.
Relaxing assumption 3: limited
resources and crowding
• So far, birth and death rates have been
density independent; they do not vary as N
changes.
Realistic? NO!
• As populations increase and resources
become limiting, per capita death rates can go
up, and per capita birth rates can go down.
• Density dependence means vital rates
depend on N.
Relaxing assumption 3: limited
resources and crowding
Density dependence means vital
rates depend on N
Relaxing assumption 3: limited
resources and crowding
When birth rates are balanced by
death rates, the population reaches
a stable equilibrium.
Relaxing assumption 3: limited
resources and crowding
Carrying capacity (K): maximum
population size that can be
supported in a given environment.
How does K affect population
growth?
Adjust model so that population change
reacts to K. Simplest form is called logistic
growth:
dN = rN
dt
dN = rN (1 – N/K)
dt
Unused portion of K: if N=K,
growth rate becomes zero; if
N = O, growth is exponential.
Regulated population growth
K
Note decline above K is
faster than growth below K.
Growth is fastest when N=K/2.
Regulated population growth
K
What is role of r?
r-K selection…
More assumptions
• Individuals still don’t vary (more on this
next time).
• Processes occur instantaneously.
• K is constant.
• Density dependence is linear.
Time lags
• Can produce cyclic
oscillations around K,
depending on r.
• Period of cycle is 4x
lag (fits some high
latitude mammal
populations).
• Discrete logistic
growth models are
another flavor of
time lag effect. Can
become complex!
Time lags
• Whether periodic or
stochastic
fluctuations, they
tend to reduce K.
WHY? (Faster
decline above K than
growth below)
• Effect is magnified
with lower r (can
organisms track
conditions?)