On stochastic logistic equation with Markovian switching and white noise

Abstract

This paper mainly concerns with the stochastic logistic equation with Markovian switching and white noise. We represent the unique global positive solution and show some properties such as the stochastic permanence, global attractivity. The main contribution of this paper is to demonstrate that the solution is asymptotically stable in distribution. Moreover, the similar results for the stochastic Gilpin--Ayala equation with Markovian switching and white noise are yielded by the same way. Finally, we give the definition of the stochastic equilibrium solution and prove the existence and uniqueness of such equilibrium solution of stochastic logistic equation.