SECOND LAW OF POPULATIONS:

Population growth cannot go on forever

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

!?

The Basic Mathematics of Density Dependence:The Logistic Equation

We can start with the equation for exponential growth….

How does population growth changeas numbers in the population change?

dN/dt = rN

Recall the definition of r

dN/dt = rN … r is a growth rate, or thedifference between birth and death rate

So, we can write, dN/dt = (b-d)N

For the exponential equation, these birth

and death rates are constants…What if they change as a function of

population size?

We can rewrite the exponential rate equation

Let b’ and d’ represent birth and death ratesthat are NOT constant through time

So, dN/dt = (b’- d’)N

Modeling these variables…. The simplestcase is to allow them to be linear

Linear relationship:

Let b’ = b - aN and Let d’ = d - cN

N

b’

The intercept: b

The slope: a

What happens if...

The values of a or c equal zero?

This demonstrates that the exponentialequation is a special case of thelogistic equation….

Rearranging the equation...

The Carrying Capacity, K

•Definitions

•Issues

Assumptions of the logistic model

Logistic Growth, Continous Time

0

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-5 5 15 25 35

Time

Po

pu

lati

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Siz

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0.1

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te o

f C

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dN/dt

Rate of change and actual population size

What if we do have time lags?

Biologically realistic, after all (consider thestage models we’ve been working with)

What happens?

Other forms of density dependence

•Ricker model

•Beverton-Holt model

Both of these originally developed for fisheries;many more possibilities exist.

Nt+1 = Ntexp[R*(K-Nt)/K]

Nt+1 = Nt * (1 + R) 1+(R/K)*Nt

The Allee EffectMinimum density required to maintain the

population

•Defense or vigilance

•Foraging efficiency

•Mating

James F. ParnellGary Kramer

SUMMARY

•When density affects demographic rates,“density dependence”

•Many ways to model this mathematically:Logistic (linear)RickerBeverton-Holt

•In constant environment, population willstabilize at carrying capacity

SUMMARY, continued

•Unlike with density-independent models, thediscrete and continuous forms are NOTequivalent in behavior

•Discrete form can exhibit damped oscillations,stable limit cycles, or chaos

•Carrying capacity has multiple definitions,biological reality must be considered

SUMMARY, continued

Allee effect: important implications formanagement and conservation

SECOND LAW OF POPULATIONS:they can’t grow forever…