The study of mathematical models for predicting the population dynamics of competing species has its origin in independent works published in the early part of the 20th century by A. J. Lotka and V. Volterra (see, for example, [Lo1], [Lo2], and [Vo]).
Consider the problem of predicting the population of two species, one of which is a predator, whose population at time t is x2(t), feeding on the other, which is the prey, whose population is x1(t). We will assume that the prey always has an adequate food supply and that its birth rate at any time is proportional to the number of prey alive at that time; that is, birth rate (prey) is k1x1(t). The death rate of the prey depends on both the number of prey and predators alive at that time. For simplicity, we assume death rate (prey) = k2x1(t)x2(t). The birth rate of the predator, on the other hand, depends on its food supply, x1(t), as well as on the number of predators available for reproduction purposes. For this reason, we assume that the birth rate (predator) is k3x1(t)x2(t). The death rate of the predator will be taken as simply proportional to the number of predators alive at the time; that is, death rate (predator) = k4x2(t).
Since x'1(t) and x' 2(t) represent the change in the prey and predator populations, respectively, with respect to time, the problem is expressed by the system of nonlinear differential equations
x'1(t) = k1x1(t) − k2x1(t)x2(t) and x'2(t) = k3x1(t)x2(t) − k4x2(t).