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Long-time behaviors of a stochastic cooperative Lotka–Volterra system with distributed delay

Author

Listed:
  • Zuo, Wenjie
  • Jiang, Daqing
  • Sun, Xinguo
  • Hayat, Tasawar
  • Alsaedi, Ahmed

Abstract

This article focuses on a stochastic cooperative Lotka–Volterra system with distributed delay. We first transfer the stochastic system with weak kernel into a degenerate stochastic system made up of four equations. For the deterministic system, global stability of the positive equilibrium is investigated. For the stochastic system with distributed delay, sharp sufficient conditions for the persistence of two species are established. What is more, we obtain the existence and uniqueness of the stationary distribution by constructing suitable Lyapunov function and proving the global attraction of the positive solution. The results show that, the weaker white noises can ensure the existence of a unique stationary distribution and the stronger white noises can result in the extinction of one or two species, though the positive equilibrium is globally stable without white noises.

Suggested Citation

  • Zuo, Wenjie & Jiang, Daqing & Sun, Xinguo & Hayat, Tasawar & Alsaedi, Ahmed, 2018. "Long-time behaviors of a stochastic cooperative Lotka–Volterra system with distributed delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 542-559.
    • Handle: RePEc:eee:phsmap:v:506:y:2018:i:c:p:542-559
    DOI: 10.1016/j.physa.2018.03.071
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    References listed on IDEAS

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    1. Rudnicki, Ryszard, 2003. "Long-time behaviour of a stochastic prey-predator model," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 93-107, November.
    2. Jiang, Daqing & Zuo, Wenjie & Hayat, Tasawar & Alsaedi, Ahmed, 2016. "Stationary distribution and periodic solutions for stochastic Holling–Leslie predator–prey systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 16-28.
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    Cited by:

    1. Wenxu Ning & Zhijun Liu & Lianwen Wang & Ronghua Tan, 2021. "Analysis of a Stochastic Competitive Model with Saturation Effect and Distributed Delay," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1435-1459, December.
    2. Lu, Chun, 2021. "Dynamics of a stochastic Markovian switching predator–prey model with infinite memory and general Lévy jumps," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 316-332.
    3. Sun, Xinguo & Zuo, Wenjie & Jiang, Daqing & Hayat, Tasawar, 2018. "Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 864-881.
    4. Zhang, Qiumei & Jiang, Daqing, 2021. "Dynamics of stochastic predator-prey systems with continuous time delay," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Cao, Nan & Fu, Xianlong, 2023. "Stationary distribution and extinction of a Lotka–Volterra model with distribute delay and nonlinear stochastic perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    6. Shao, Yuanfu, 2022. "Global stability of a delayed predator–prey system with fear and Holling-type II functional response in deterministic and stochastic environments," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 65-77.

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