untitled

which is the basic equation of population dynamics. If the survivorship and fecun-

dity schedules hold constant, the population’s age distribution will converge to the

constant form of:

Sx=lxe−rx

which is called the stable age distribution. Sxis the number of females in a par-

ticular age class divided by the number of females in the first age class. The basic

equation may thus be written ∑Sxmx=1. In the special case of rate of increase

being zero, the stable age distribution, now called the stationary age distribution, is

Sx=lxby virtue of e−^0 x=1. That is the justification for using such an age distribu-

tion to construct a life table. The stationary age distribution is the special case of

the stable age distribution that obtains when r=0. It has been argued that, since

fecundity and mortality schedules seldom remain constant for long, the stable age

distribution is little more than a mathematical abstraction, although a useful one.

Although the stable distribution can be attained fairly quickly (roughly two genera-

tions) after mortality and fecundity patterns stabilize, most wildlife species that have

been adequately studied have mortality and fecundity schedules that fluctuate, some-

times substantially, from year to year.

Thomas Malthus in 1798 recognized that populations have an intrinsic tendency

towards exponential or geometric growth, just as a bank account at fixed interest

grows geometrically with the amount of money in the account. The growth of

such populations can be calculated as either a continuous or a discrete process. For

simplicity, we will concentrate on discrete time representations of population growth.

Strictly speaking, such models are most applicable to organisms whose patterns of

deaths and births follow a seasonal or annual cycle of events, which includes most

wildlife species. Consider, for example, a population whose finite growth rate (λ) is

0.61 and whose initial density (N 0 ) is 1.5. The geometric growth model predicts

subsequent changes in density over time according to Nt= N 0 λt. The outcome

depends on whether λis larger or smaller than 1. When λ<1 (Fig. 6.3) there is a

decelerating pattern, while the outcome is changed to an accelerating pattern of growth

when λ>1 (Fig. 6.4).

88 Chapter 6

1.5

1.0

0.5

0

0246810

t

N

t

Fig. 6.3Population
changes according to
the geometric model
with λ=0.61.

6.8 Geometric or exponential population growth