Sufficient and Necessary Conditions of Stochastic Permanence and Extinction for Stochastic Logistic Model with Markovian Switching and Levy Noise

Abstract

In this paper, stochastic permanence and extinction of a stochastic logistic model with Markovian switching and L\'{e}vy noise are investigated by combining stochastic analytical techniques with M-matrix analysis. Sufficient and necessary conditions of stochastic permanence and extinction are obtained. In the case of stochastic permanence, both the superior limit and the inferior limit of the average in time of the sample path of the solution are estimated by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the logistic model. Finally, our conclusions are illustrated through an example.