Dynamic complexity in predator-prey models framed in difference equations

Abstract

THE complicated dynamics associated with simple first-order, nonlinear difference equations have received considerable attention (refs 1–4 and R. M. May and G. F. Oster, unpublished). In an ecological context, equations of this type provide a powerful and realistic means of modelling the behaviour of animal populations with ron-overlapping generations, typified by many arthropods in temperate regions. May⁴ has shown that such models, incorporating density dependence, have three regimes of dynamic solution in their parameter space, namely (1) a stable equilibrium point; (2) bifurcating cycles of period 2ⁿ, 0<n< ∞, where n is a positive integer and (3) behaviour which has been termed chaotic, that is, cycles of any integral period or complete aperiodicity, depending on the initial conditions. May⁴ has indicated that such complexity can also occur in the wider context of competition between two species, described by two first-order, nonlinear difference equations of similar form to those governing single-species growth.