Long term behavior of solutions of the Lotka-Volterra system under small random perturbations
成果类型：
Article
署名作者：
Khasminskii, RZ; Klebaner, FC
署名单位：
Wayne State University; University of Melbourne
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
2001
页码：
952-963
关键词：
摘要：
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 epsilon (2), where epsilon > 0 is a small parameter. The evolution of this system is studied on large time intervals of O(1/epsilon (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.