untitled

Populations have inputs of births and immigrants and outputs of deaths and

emigrants. For simplicity we will confine discussion to a self-contained population

having only births (B) and deaths (D) per unit time.

If either the proportion of the population dying increases or the proportion enter-

ing as births decreases as population density increases then we define these changes

in proportions as being density dependent. The underlying causes for the changes

in these rates are called density-dependent factors.

Births and deaths as a proportion of the population (B/Nt, D/Nt) can be related to

the instantaneous birth (b) and death (d) rates in the following way.

The change in population per unit time is:

Nt+ 1 −Nt=B−D

the instantaneous rate of increase (r) is given by:

r=b−d

and the finite rate of increase (λ) is given by:

λ=Nt+ 1 /Nt=er

Therefore:

eb−d=(Nt+ 1 /Nt) =(B −D +Nt)/Nt

If d=0, D=0 then:

eb=(B +Nt)/Nt=[1 +(B/Nt)]

and

b=loge[1 +(B/Nt)]

Similarly if b=0, B=0, and D/Ntis much less than 1, then:

d=loge[1 +(D/Nt)]

If Band Dfall in the range of 0–20% of the population then band dare nearly lin-

ear on N, and they remain approximately linear even if Band Dare 20– 40% of N.

This range covers most of the examples we see in nature, so for our purposes we

can say that D/Ntand B/Ntchange with density in the same way as do band d, and

both go through the origin.

In Fig. 8.4a we plot bagainst density (or population size) Nas a constant so that

it is a horizontal line. If we now plot das an increasing function of density, we see

that where the two lines cross, b=d, and the population is stationary, at the equi-

librium point K. The difference between the band dlines represents r, and this declines

linearly as density increases, in the same way as it does for the logistic curve (see

Section 8.6). In Fig. 8.4a the decline in ris due solely to dbeing density dependent.

POPULATION REGULATION, FLUCTUATION, AND COMPETITION WITHIN SPECIES 111

8.3 The theory of population limitation and regulation

8.3.1Density
dependence