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Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps. (English) Zbl 1338.60210
Summary: 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.

##### MSC:
 60J75 Jump processes (MSC2010) 34D05 Asymptotic properties of solutions to ordinary differential equations 34F05 Ordinary differential equations and systems with randomness 60J27 Continuous-time Markov processes on discrete state spaces 92D25 Population dynamics (general)
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