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.

Potential for geometric increase = exponential growth

time

nu

mber

of

indiv

iduals

2

4

8

16

32

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)

• 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 - Ddt

Births and deaths

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 = bND = dN

• So now population changes occur as a result of per capita birth and death rates:

dN = (b – d)Ndt

Births and deaths: assumptions?

dN = (b – d)Ndt

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 = 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

r = intrinsic rate of increase

dN = rNdt

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?

dN = rNdt

Adjust model so that population change reacts to K. Simplest form is called logistic growth:

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?)