Stationary distribution and ergodicity of a stochastic single-species model under regime switching in a polluted environment

Abstract

The long-term statistical rule is one of the important questions for stochastic pollution-population dynamical models, thus it would be worth looking for the stationary distribution as an indicator in analysing the effects of toxicant and noises on the variation of population in evolution process. In present paper, we investigate a stochastic single-species model under regime switching in a polluted environment. By use of the ergodic of Markov chain and constructing Lyapunov function, the sufficient conditions for the positive recurrence and ergodic property are established, which imply the existence of stationary distribution of the model. Moreover, the mean and variance of marginal stationary distribution are estimated. Our analysis indicates that the coloured noise and toxicant may play an important role in determining the shape of stationary distribution and its statistics characteristics. Finally, numerical simulations are carried out to support our theoretical results.