ABSTRACT
Often one species uses another as food. For example, man uses fish, lions consume gazelles and some wasps eat the caterpillars of moths. The presence of one species can therefore have effects (sometimes irreversible) on another. The study of such dynamic interactions is the topic of this chapter. For simplicity, consideration will be limited to two species, though it must be understood that in nature the situation is frequently much more complicated, with several species involved and interacting in a complex manner. Nevertheless, useful conclusions can be drawn from even relatively simple models and the influence of various actions on the level of available resources ascertained. Usually, the species regarded as food is called the prey and the consum-
ing species the predator. The aim is to construct predator-prey models that are relevant, whether it is man killing deer, wasps eating caterpillars or man harvesting timber, though humans often behave differently from other organisms; some general aspects have already been treated in Section 4.5. To fix on a specific illustration for the simplest model, we shall discuss how fishing can affect the population of fish. Let N(t) be the number of fish in a designated zone at time t. It will be
assumed that N varies continuously with t; in fact, N changes only by integers but the error introduced by our approximation should not be appreciable except possibly when N is small. In the absence of fishing, new fish can be expected to be born at a rate proportional to existing numbers, say bN(t) where b is a constant. Similarly, the rate at which fish die will be taken as dN(t) with d constant. Then
dN
dt = (b − d)N. (9.1.1)
The solution of this differential equation has been found in Section 1.1 and is
N(t) = N(0)e(b−d)t (9.1.2)
where N(0) is the population at time t = 0. The solution (9.1.2) reveals that the fish population eventually disappears
if the birth rate is less than the death rate, i.e., b < d, is in equilibrium if b = d
greater the mortality rate. Exponential growth in which the population doubles in every interval 0.69/(b− d) of time is indeed exhibited by many species under ideal conditions when they have boundless space, food and the resources they need available. However, such growth cannot continue indefinitely in a finite world and there must come a time when shortages of supplies inhibit the exponential growth. The preparation of models that allow for environmental constraints has
followed a variety of routes. One of the most popular is to replace (9.1.1) by
dN
dt = a
( 1− N
N0
) N (9.1.3)
where a and N0 are positive constants. According to Section 1.3, this leads to logistic growth and the population always ends up at N0 no matter what level it started at. Thus N0 can be regarded as the maximum population that can be sustained under (9.1.3). Equation (9.1.3) often appears in ecology in the form
dN
dt = r
( 1− N
K
) N.
