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Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

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  • Chunjin Wei
  • Yingjie Fu

Abstract

In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton–zooplankton system. For the autonomous system, we establish the sufficient conditions for the existence of the globally positive solution as well as the solution of population extinction and persistence in the mean. Furthermore, by constructing some suitable Lyapunov functions, we also prove that there exists a single stationary distribution which is ergodic, what is more important is that Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, some numerical simulations are introduced to illustrate our theoretical results. The results show that weaker white noise and/or toxicity will strengthen the stability of system, while stronger white noise and/or toxicity will result in the extinction of one or two populations.

Suggested Citation

  • Chunjin Wei & Yingjie Fu, 2020. "Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems," Complexity, Hindawi, vol. 2020, pages 1-15, January.
    • Handle: RePEc:hin:complx:4627571
    DOI: 10.1155/2020/4627571
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    Cited by:

    1. Xiaomei Feng & Yuan Miao & Shulin Sun & Lei Wang, 2022. "Dynamic Behaviors of a Stochastic Eco-Epidemiological Model for Viral Infection in the Toxin-Producing Phytoplankton and Zooplankton System," Mathematics, MDPI, vol. 10(8), pages 1-18, April.
    2. Chen, Zhewen & Zhang, Ruimin & Li, Jiang & Zhang, Shuwen & Wei, Chunjin, 2020. "A stochastic nutrient-phytoplankton model with viral infection and Markov switching," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).

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