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The Analysis of Stochastic Lotka–Volterra Model in Polluted Environment

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Abstract

Population systems often subject to both white noise and environment pollution, so a stochastic non-autonomous omnivory Lotka–Volterra model in a polluted environment is investigated in this paper. We establish the sufficient conditions for the existence of positive periodic solution and prove it by constructing Lyapunov function. According to Itô’s formula and the strong law of large numbers for martingales, we also discuss the extinction of population. The sufficient conditions for almost sure exponential stability of equilibrium point \(E^*(0,0,0,S^*,T^*)\) are obtained. Finally, we illustrate our results by some examples with the help of computer simulation.

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Acknowledgements

The research is supported by the National Natural Science Foundation of China (Nos. 11261043, 11461053, and 11661064) and Innovation Project of Ningxia University (GIP201624). The authors would like to thank the editors and anonymous reviewers for their valuable comments which improve the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, People’s Republic of China

    Boqiang Cao & Xining Li

  2. School of Mathematics and Information Sciences, Beifang University of Nationalities, Yinchuan, 750021, People’s Republic of China

    Qiang Li

  3. School of Information Engineering, Ningxia University, Yinchuan, 750021, People’s Republic of China

    Ying Zhang

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  1. Boqiang Cao
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  2. Xining Li
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  3. Qiang Li
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  4. Ying Zhang
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Correspondence to Xining Li.

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Cao, B., Li, X., Li, Q. et al. The Analysis of Stochastic Lotka–Volterra Model in Polluted Environment. Differ Equ Dyn Syst 26, 199–212 (2018). https://doi.org/10.1007/s12591-016-0334-6

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  • DOI: https://doi.org/10.1007/s12591-016-0334-6

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