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Permanence of Stochastic Lotka–Volterra Systems

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Abstract

This paper proposes a new definition of permanence for stochastic population models, which overcomes some limitations and deficiency of the existing ones. Then, we explore the permanence of two-dimensional stochastic Lotka–Volterra systems in a general setting, which models several different interactions between two species such as cooperation, competition, and predation. Sharp sufficient criteria are established with the help of the Lyapunov direct method and some new techniques. This study reveals that the stochastic noises play an essential role in the permanence and characterize the systems being permanent or not.

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Acknowledgments

We are very grateful to the anonymous referees for their careful reading and valuable comments, which greatly improved the presentation of the paper. We also thank Professor H. R. Thieme at Arizona State University for his nice comments and valuable suggestions.

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Authors and Affiliations

  1. School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, People’s Republic of China

    Meng Liu

  2. School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, 130024, Jilin, People’s Republic of China

    Meng Liu & Meng Fan

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  1. Meng Liu
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  2. Meng Fan
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Correspondence to Meng Liu.

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Communicated by Philip K. Maini.

Partially supported by the Natural Science Foundation of PR China (11671072, 11271065, 11301207), Research Fund for the Doctoral Program of Higher Education of PR China (20130043110001), Project Funded by Chinese Postdoctoral Science Foundation (2015M571349, 2016T90236), Natural Science Foundation of Jiangsu Province (BK20130411). Jiangsu Province “333 High-level Personnel Training Project” and Qing Lan Project, overseas training program for outstanding young college teachers and principals in Jiangsu Province.

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Liu, M., Fan, M. Permanence of Stochastic Lotka–Volterra Systems. J Nonlinear Sci 27, 425–452 (2017). https://doi.org/10.1007/s00332-016-9337-2

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