Dynamics of a Two-Prey One-Predator System in Random Environments

Abstract

In this paper, we propose and investigate a stochastic two-prey one-predator model. Firstly, under some simple assumptions, we show that for each species x i , i=1,2,3, there is a π i which is represented by the coefficients of the model. If π i <1, then x i goes to extinction (i.e., lim t→+∞ x i (t)=0); if π i >1, then x i is stable in the mean (i.e., $\lim_{t\rightarrow+\infty}t^{-1} \int_{0}^{t}x_{i}(s)\,\mathrm {d}s=\mbox{a positive constant}$ ). Secondly, we prove that there is a stationary distribution to this model and it has the ergodic property. Thirdly, we establish the sufficient conditions for global asymptotic stability of the positive solution. Finally, we introduce some numerical simulations to illustrate the theoretical results.