Linear difference equations have the advantage that a closed-form solution can be easily obtained. But, in many cases, the behaviour of linear difference equation models is not consistent with observation. This is true in many areas of biology, and particularly in studies of populations, where non-linear models are better.

In this chapter non-linear models are developed which describe how a population of individuals grows over time. Difference equation models are appropriate when a species has a distinct breeding season. The simplest non-linear model, the ‘logistic equation’ is studied in detail. Similar ideas are also used to model a measles epidemic. This involves iterating a pair of simultaneous difference equations for the number of those who can infect others and for the number of those susceptible to being infected.

Closed-form solutions usually cannot be found for non-linear difference equations. Thus to interpret the models one has to resort to numerical simulation or devise approximate closed-form solutions to the equations. Both approaches are developed here. The ideas rely on concepts developed in Chapters 7 and 8.

Linear models for population growth

Many people are very interested in the way populations grow and in determining what factors influence their growth. Knowledge of this kind is important in studies of bacterial growth, wildlife management, ecology and harvesting. In this section a very simple model is formulated for a population which breeds at fixed time intervals.