Dynamics and simulations of a stochastic predator-prey model with infinite delay and impulsive perturbations

This paper considers a stochastic predator-prey model with infinite delay and impulsive perturbations. Sufficient conditions for permanence in time average are established as well as extinction, stability in time average and global attractivity of the stochasic model. Some simulation figures, which are obtained by the split-step θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-method to discretize the stochasic model, are introduced to support the analytical findings. Our results demonstrate that, firstly, impulsive perturbations which may represent human factor play a key role in maintaining ecological balance; secondly, environmental noise, which can be modelled by Brownian motion, is disadvantageous to population survival; finally, infinite delay has not affect permanence in time average, extinction, stability in time average and global attractivity of the stochasic model.