A New Deterministic Model for the Interaction between Predator and Prey

Abstract

Listen

Translate article

SUMMARY A simple model, differing from the standard Lotka-Volterra model in the assump- tions made, is proposed to describe the interaction between predator and prey. The diffi- culties of that model are overcome without the introduction of such additional factors as immigration. in the H, P plane. The model is deterministically stable with cyclic behaviour of a common period for the sizes of the two populations. However, the stochastic variations that will occur in practice about the deterministic values of H and P produce an unstable model, as these variations will ultimately lead to certain extinction of the predator and possible extinction of the prey as well. Bartlett observes that such extinction invariably occurs after one or two cycles in laboratory experiments but that theoretical estimates of the prob- ability per cycle of an extinction have sometimes been quite small. He comments in this regard ' * * * there is no doubt from these and similar data that the Lotka-Volterra model is inappropriate because the true cycle is determin- istically unstable.' Other writers have queried the presence of oscillations in natural populations unless there are further complicating factors such as seasonal variations in food supplies or reproductive ability. Discussions are given by Kostitzin (1939), de Bach and Smith (1941), and Moran (1950). To obviate the unsatisfactory features of the Lotka-Volterra model, modifications have been tried which allow for the age structure of the popula-