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Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator

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  • Zhao, Xin
  • Zeng, Zhijun

Abstract

In this paper, we deal with a stochastic predator–prey model with stage structure for predator population and ratio-dependent functional response. The proposed mathematical model consists of a system of three stochastic differential equations to stimulate the interactions between prey population, immature predator and mature predator population. We first establish sufficient conditions for the existence and uniqueness of the positive solutions by constructing an appropriate Lyapunov function. Then we extend the existence of stationary distribution under certain parametric restrictions. We also obtain the sufficient conditions for extinction of the predator populations. Finally, numerical simulations have been carried out to validate our analytical findings.

Suggested Citation

  • Zhao, Xin & Zeng, Zhijun, 2020. "Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    • Handle: RePEc:eee:phsmap:v:545:y:2020:i:c:s0378437119318540
    DOI: 10.1016/j.physa.2019.123310
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    References listed on IDEAS

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    1. Shi, Hong-Bo & Li, Yan, 2015. "Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 71-77.
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    4. Lu, Chun & Ding, Xiaohua, 2019. "Periodic solutions and stationary distribution for a stochastic predator-prey system with impulsive perturbations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 313-322.
    5. Liu, Qun & Jiang, Daqing & Hayat, Tasawar & Ahmad, Bashir, 2018. "Stationary distribution and extinction of a stochastic predator–prey model with additional food and nonlinear perturbation," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 226-239.
    6. 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.
    7. 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.
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    Cited by:

    1. Haiyin Li & Xuhua Cheng, 2021. "Dynamics of Stage-Structured Predator–Prey Model with Beddington–DeAngelis Functional Response and Harvesting," Mathematics, MDPI, vol. 9(17), pages 1-15, September.
    2. 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).

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