Stochastic analysis of the Lotka-Volterra model for ecosystems.

A stochastic Lotka-Volterra-type model for the interaction between the preys and the predators in a random environment is investigated. A self-competition mechanism within the prey population itself is also included. The effect of a random environment is modeled as random variations in the birth rate of the preys and the death rate of the predators. The stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distributions of the system state variables at the state of statistical stationarity. Asymptotic behaviors of the system variables are discussed, and the mean transition time from an initial state to a critical state is obtained. Effects on the ecosystem behaviors of the self-competition term, of the random variation in the prey birth rate, and of the random variation in the predator death rate are investigated.