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 relationshipis 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 $\varepsilon^2$, where $\varepsilon>0$ is a small parameter.The evolution of this system is studied on large time intervals of $O(1/\varepsilon^2)$. 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.