Chaos in perturbed Lotka-Volterra systems

Lotka-Volterra systems have been used extensively in modelling population d ynamics. gn this paper, it is shown that chaotic behaviour in the sense of Smale can exist in time periodically perturbed systems of Lotka-Volterra eq uations. First, a slowly varying three-dimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method sufficient conditi ons are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra sys tem is examined, leading finally to an application with a model for the sel f-organisation of macromolecules.