Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps

This paper focuses on studying the dynamics of the stochastic Gilpin-Ayala model under regime switching with jumps. The aim is to analyze what happens under the perturbations of regime switching and jumps. Some asymptotic properties are investigated and sufficient conditions for stochastic permanence, extinction, non-persistence in the mean and weak persistence are provided. The critical value among the extinction, non-persistence in the mean and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the jumps and the stationary distribution of the Markov chain.