A new idea on almost sure permanence and uniform boundedness for a stochastic predator–prey model

Abstract

The existence and uniqueness of stationary distribution and ergodic properties of a stochastic system are obtained. Especially, different from the existing methods, a new method is introduced to analyze almost sure permanence and uniform boundedness of the stochastic predator–prey model. This new idea is based on geometric structure of invariant set for a stochastic system. More specifically, we obtain our main conclusions by showing the invariant set for the stochastic population system lies in the interior of the first quadrant. It is interesting and surprising that the stochastic population model can guarantee a uniform boundedness almost surely. Some numerical simulations are carried out to support our results.