19 - Non-linear coupled models

Summary

Further models of two interacting quantities are studied in this chapter. These models are distinguished from those of the previous chapter because they lead to coupled non-linear, rather than linear, differential equations. Such equations usually cannot be solved explicitly. Instead, a method of eliminating the independent variable, called the phase-plane technique, is developed for studying properties of the solution.

Non-linear equations often exhibit unusual or non-intuitive behaviour. A classic example considered in this chapter is the Lotka–Volterra equations which describe the interactions between predators and their prey. Applied to the study of fish and shark populations they predict that an increase in fishing will actually lead to an increase in the fish population.

In addition to the predator–prey system, models of combat and epidemics are presented.

Predator–prey interactions

Our concern in this chapter is with models which lead to coupled nonlinear differential equations. In this section it will be shown how a model describing the interaction of a predator (shark) and its prey (fish) leads to coupled non-linear differential equations for the shark and fish populations as functions of time.

Some unusual data

The Italian mathematician Vito Volterra (1860–1940) developed a model of predator–prey interaction in response to some unusual data. Extensive records had been kept of the yearly catches of fish and sharks at an Italian sea port (Fiume, 1914–1923). Table 19.1.1 shows the percentage ratio of sharks to total catch.