Permanence and asymptotical behavior of stochastic prey–predator system with Markovian switching

In this paper, we investigate the stochastic permanence and extinction of a stochastic ratio-dependent prey–predator model controlled by a Markov chain. In the permanent case we estimate the superior limit and the inferior limit of the average in time of the sample path of the solution. The boundaries are related to the stationary probability distribution of the Markov chain and the parameters of the subsystems. Finally, we illustrate our main results by two examples and some numerical experiments.