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Stochastic periodic solution for a perturbed non-autonomous predator–prey model with generalized nonlinear harvesting and impulses

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  • Zhang, Yan
  • Chen, Shihua
  • Gao, Shujing
  • Wei, Xiang

Abstract

In this paper, stochastic non-autonomous predator–prey models with and without impulses are investigated. The effects of generalized nonlinear harvesting for prey and predator populations are considered. For the stochastic system without impulses, the existence and uniqueness of the positive solution is proven and sufficient conditions that guarantee the extinction and persistence of the population in the mean are achieved. We show the existence of a nontrivial positive periodic solution by constructing appropriate Lyapunov functions and using Khasminskii’s theory. Moreover, the global attractiveness and stochastic persistence in probability of the stochastic model are discussed. Results show that the stronger noises and nonlinear harvesting component can significantly influence the dynamics of the system and lead to the extinction of the predator population. Additionally, for the stochastic predator–prey system with impulsive effect, we prove that there exists a positive periodic solution. Numerical simulations are conducted to show the effectiveness and feasibility of the obtained results.

Suggested Citation

  • Zhang, Yan & Chen, Shihua & Gao, Shujing & Wei, Xiang, 2017. "Stochastic periodic solution for a perturbed non-autonomous predator–prey model with generalized nonlinear harvesting and impulses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 347-366.
    • Handle: RePEc:eee:phsmap:v:486:y:2017:i:c:p:347-366
    DOI: 10.1016/j.physa.2017.05.058
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    References listed on IDEAS

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    1. 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.
    2. Wang, Xiaohong & Jia, Jianwen, 2015. "Dynamic of a delayed predator–prey model with birth pulse and impulsive harvesting in a polluted environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 422(C), pages 1-15.
    3. Heggerud, Christopher M. & Lan, Kunquan, 2015. "Local stability analysis of ratio-dependent predator–prey models with predator harvesting rates," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 349-357.
    4. Liu, Meng & Bai, Chuanzhi, 2016. "Optimal harvesting of a stochastic mutualism model with Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 301-309.
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    Cited by:

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