So far we have considered species with an annual breeding season, and have assumed that individuals breeding in one summer die before the next, so that generations are ‘separate’, members of one generation never breeding with members of another. The appropriate mathematical formulation for such cases was in the form of a recurrence relation. We will now turn to the case where the population breeds continuously; the appropriate mathematical descriptions will then be differential equations.

Let the population density at time t be x. To start with, we will make the artificial assumption that the rate of change of the population density, dx/dt, depends only on conditions at time t, and does not depend on the past history of the population. This assumption is artificial, among other reasons, because it ignores the age structure of the population. Thus if, for example, the food supply to a population were suddenly increased, there might be a rapid increase in the rate at which adult females in the population laid eggs, but there would be an inevitable delay before these eggs hatched into feeding individuals, and a larger delay before they developed into adults themselves capable of breeding. Thus an adequate description of the population requires a knowledge, not only of the total numbers, but of the numbers in each age group. And since the number in, for example, the 30-day-old group depends on conditions 30 days ago, an adequate description cannot ignore past history.