Probabilistic Limit Cycles

Abstract

Populations change. Some natural populations seem to change in a regular, cyclic manner. This report is about models of populations which so change or fluctuate. We develop a rather simple mathematical idea: Suppose a Lotka-Volterra model of a predator-prey system enjoys asymptotic stability for some population combination. However, suppose that this stability is weak in the sense that the populations, when displaced slightly from equilibrium, return slowly to levels near equilibrium and return only after oscillating around equilibrium “many” times. If such a system is given small, frequent, random perturbations it will tend not to return to equilibrium at all, but will oscillate around equilibrium in a fairly regular way, indefinitely.