Stochastic Analysis of Predator–Prey Models under Combined Gaussian and Poisson White Noise via Stochastic Averaging Method

In the present paper, the statistical responses of two-special prey–predator type ecosystem models excited by combined Gaussian and Poisson white noise are investigated by generalizing the stochastic averaging method. First, we unify the deterministic models for the two cases where preys are abundant and the predator population is large, respectively. Then, under some natural assumptions of small perturbations and system parameters, the stochastic models are introduced. The stochastic averaging method is generalized to compute the statistical responses described by stationary probability density functions (PDFs) and moments for population densities in the ecosystems using a perturbation technique. Based on these statistical responses, the effects of ecosystem parameters and the noise parameters on the stationary PDFs and moments are discussed. Additionally, we also calculate the Gaussian approximate solution to illustrate the effectiveness of the perturbation results. The results show that the larger the mean arrival rate, the smaller the difference between the perturbation solution and Gaussian approximation solution. In addition, direct Monte Carlo simulation is performed to validate the above results.