Quasiperiodicity and chaos in population models

Irregular fluctuations observed in natural population densities were traditionally attributed to external random influences such as climatic factors. That these may, in contrast, be the result of intrinsic nonlinearities, present even in very simple deterministic models, was first mooted by May. Since then, ‘chaos' has been shown to occur naturally in many fields of biology, ranging from physiology to the behaviour of social insects. Ecologists, however, still lack the ultimate proof of demonstrating conclusively the existence of chaotic population dynamics outside the world of equations and computer experiments. Further, it has been demonstrated that, in some population models exhibiting chaos via the period-doubling route, a constant immigration factor inhibits the onset of chaos by a process called period reversal, prompting suggestions that chaos is fragile and easily inhibited. We present three classic host-parasitoid models, whose local stability properties have been extensively studied in the past, but whose chaotic dynamics have not previously been explored. We find that chaos in this class of models is reached via quasiperiodicity, a route not normally associated with population models, and is relatively robust against reasonably small levels of immigration. This leads us to conclude that chaos could persist in some natural populations, even in the presence of such external perturbations.