Long term behavior of solutions of the lotka-volterra system under small random perturbations

Abstract

A stochastic analogue of the Lotka-Volterra model for predator-prey relationship is obtained when the birth rate of the prey and the death rate of the predator are perturbed by independent white noises with intensities of order ε2, where ε > 0 is a small parameter. The evolution of this system is studied on large time intervals of O(1/ε2). It is shown that for small initial population sizes the stochastic model is adequate, whereas for large initial population sizes it is not as suitable, because it leads to ever-increasing fluctuations in population sizes, although it still precludes extinction. New results for the classical deterministic Lotka-Volterra model are obtained by a probabilistic method; we show in particular that large population sizes of predator and prey coexist only for a very short time, and most of the time one of the populations is exponentially small.