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December  2016, 21(10): 3743-3766. doi: 10.3934/dcdsb.2016119

Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching

Hongfu Yang 1, , Xiaoyue Li 2, and  George Yin 3,

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
2. 

Center for Mathematics and Interdisciplinary Sciences, School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
3. 

Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Received  November 2015 Revised  June 2016 Published  November 2016

This work is concerned with permanence and ergodicity of stochastic Gilpin-Ayala models involve continuous states as well as discrete events. A distinct feature is that the Gilpin-Ayala parameter and its corresponding perturbation parameter are allowed to be varying randomly in accordance with a random switching process. Necessary and sufficient conditions of the stochastic permanence and extinction are established, which are much weaker than the previous results. The existence of the unique stationary distribution is also established. Our approach treats much wider class of systems, uses much weaker conditions, and substantially generalizes previous results. It is shown that regime switching can suppress the impermanence. Furthermore, several examples and simulations are given to illustrate our main results.
Keywords: positive recurrence, Markov chain, Stochastic permanence, Gilpin-Ayala model, stationary distribution..
Mathematics Subject Classification: Primary: 60H10; Secondary: 92B0.
Citation: Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119
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show all references

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Appl. Math. Comput., 179 (2006), 55-66. doi: 10.1016/j.amc.2005.11.079.  Google Scholar

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Comput. Math. Appl., 53 (2007), 1214-1227. doi: 10.1016/j.camwa.2006.12.015.  Google Scholar

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J. Comput. Appl. Math., 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.  Google Scholar

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SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.  Google Scholar

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