ABSTRACT
In Chapter 2 we met the logistic (difference) equation and explored the population dynamics of a single specie. In this chapter we extend these ideas. We start with two competing species: a predator specie P (conventionally, “foxes”) and a prey specie N (usually, “rabbits”). In the absence of foxes, the rabbits will breed and their population will
increase (just as it did for Fibonacci) at a rate proportional to their number. Foxes, however, will eat the rabbits at a rate proportional to both their populations. Conversely, in the absence of food (rabbit), the fox population will decrease. Once the foxes have something to eat, their population increases at a rate proportional to the product of the two populations. Hence, we have
dN
dt = N(a− bP ) (7.1)
dP
dt = P (cN − d), (7.2)
where a, b, c and d ≥ 0 are all positive constants. Note that if b = 0, then the fox population will be subject to Malthusian population decay; and if c = 0, the the rabbit population will grow exponentially (see Section 2.1). In fact, there is a close connection between these equations and those of Malthus and Verhulst. The predator-prey equations (7.1 and 7.2) were proposed by Vito Volterra in 1926 to study fish population dynamics. The equations of Volterra are themselves based on a model of chemical reactions proposed by Alfred Lotka. Lotka’s equations, in turn, are a variation of the logistic equation. Referring to Eqns. (7.1) and (7.2), with the four constants a, b, c and d all
non-negative, one can see that each of the four terms on the right-hand side of the equations has a specific meaning. The aN term is the natural growth of the prey population, and the −dP term is the natural decay (famine) of the predator in the absence of prey. The interaction terms −bNP and cNP model the benefit of the presence of prey to the predators, and the harm to the prey caused by the predators1. Finally, we note that by changing the signs
of the constants, it is possible to model various other ecological interactions. For example, suppose that N and P represent two species in competition
for the same resources. Then in the absence of that competition, either specie would flourish (i.e. a > 0 and d < 0), but the presence of each specie is detrimental to the other (b > 0 and b < 0). Similarly, symbiosis2 can also be modelled. Each specie will grow by itself (except in the most extreme form of interdependence) and so a > 0 and d < 0. However, each specie also benefits from the presence of the other: b < 0 and c > 0. Nonetheless, let us consider the general system of Eqns. (7.1) and (7.2)
which, if a, b, c, d > 0, models predation. The natural question is whether, and if so when, will both species exist in a happy equilibrium.
