TABLE OF CONTENTS In this chapter we first investigate what other types of behaviour can arise in models of competitive species, and then we consider the more aggressive situation in which one species preys on the other. The simple models that we treat here are known as Lotka Volterra systems.
It is possible to treat the general model for competing species
see Exercise 33.3. However, the general treatment is much less illuminating than considering particular examples, and here we deal with two cases that have behaviour which is significantly different from that of the previous chapter.
First we consider the example,
for which there are four non-negative stationary points (where the right-hand sides are zero): if x = 0 then we could have y = 0 or y = 3; if y = 0 then we could have the additional stationary point that arises when x = 2; and finally there is an interior stationary point when x = y = 1, corresponding to a coexistent state in which there are an equal number of both species. So the four possibilities are
(0, 0), (2, 0), (0, 3) and (1, 1).
With f( x, y) = ( x(4 ? 2 x ? 2 y), y(9 ? 6 x ? 3 y)) we have
We now look at the linearisation about the four stationary points above. Near the origin we have
so the eigenvalues are 4 and 9, corresponding to...
