Abstract
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.
Highlights
The model with the predator saturation term describes a case where the preys are abundant, while the one with the predator competition term considers another case in which the predator population is very large compared with the prey population
Here, we study predator–prey models with predator saturation and predator competition terms excited by combined Gaussian and Poisson noises
We have investigated the statistical responses of a stochastic prey–predator type model for the possible situations of sufficient prey supply and a large predator population
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Since the introduction of this model, several improved LV models have been investigated with consideration of different factors Among those studies, the ones considering predator saturation and predator competition terms, respectively, have been introduced to characterize two special cases of the evolution of the ecosystems. Many investigations on the dynamics of predator–prey models with predator saturation and predator competition terms excited by stochastic perturbations have been reported [8,9,10,11,12,13,14]. The solution of the ecosystem dynamics under continuous and random jump excitation by the stochastic average method has not yet been reported.
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