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Stability in distribution of a stochastic predator–prey system with S-type distributed time delays

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  • Wang, Sheng
  • Hu, Guixin
  • Wei, Tengda
  • Wang, Linshan

Abstract

This paper concerns the dynamics of a stochastic predator–prey Lotka–Volterra system with S-type distributed time delays. Sufficient conditions for the stability in distribution of the solutions (SDS) to the system are obtained. The results show that the dynamic scenarios of the SDS are completely characterized by two parameters δ1>δ2, among which δ1 is just related to the environmental noise, while δ2 is closely related to both time delays and environmental noises: if δ2>0, then the distributions of prey–predator converge weakly to a unique ergodic invariant distribution (UEID); if δ1>0>δ2, then the predator goes to extinction, while the distributions of prey converge weakly to a UEID; if 0>δ1, then both the predator and prey go to extinction.

Suggested Citation

  • Wang, Sheng & Hu, Guixin & Wei, Tengda & Wang, Linshan, 2018. "Stability in distribution of a stochastic predator–prey system with S-type distributed time delays," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 919-930.
    • Handle: RePEc:eee:phsmap:v:505:y:2018:i:c:p:919-930
    DOI: 10.1016/j.physa.2018.03.078
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    References listed on IDEAS

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    1. Yu, Jingyi & Liu, Meng, 2017. "Stationary distribution and ergodicity of a stochastic food-chain model with Lévy jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 14-28.
    2. Wan, Li & Zhou, Qinghua, 2009. "Stochastic Lotka-Volterra model with infinite delay," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 698-706, March.
    3. Zhang, Qiumei & Jiang, Daqing, 2015. "The coexistence of a stochastic Lotka–Volterra model with two predators competing for one prey," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 288-300.
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    Cited by:

    1. Fu, Shuaiming & Luo, Jianfeng & Zhao, Yi, 2022. "Stability and bifurcations analysis in an ecoepidemic system with prey group defense and two infectious routes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 665-690.
    2. Khan, Hasib & Gómez-Aguilar, J.F. & Khan, Aziz & Khan, Tahir Saeed, 2019. "Stability analysis for fractional order advection–reaction diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 737-751.
    3. Zhang, Qiumei & Jiang, Daqing, 2021. "Dynamics of stochastic predator-prey systems with continuous time delay," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    4. 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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