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Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

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  • Feng, Xiaozhou
  • Liu, Xia
  • Sun, Cong
  • Jiang, Yaolin

Abstract

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.

Suggested Citation

  • Feng, Xiaozhou & Liu, Xia & Sun, Cong & Jiang, Yaolin, 2023. "Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:chsofr:v:174:y:2023:i:c:s0960077923006951
    DOI: 10.1016/j.chaos.2023.113794
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    References listed on IDEAS

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    1. Xiaozhou Feng & Hao Sun & Yangfan Xiao & Feng Xiao, 2020. "Stability and Coexistence of a Diffusive Predator-Prey System with Nonmonotonic Functional Response and Fear Effect," Complexity, Hindawi, vol. 2020, pages 1-10, December.
    2. Li, Yajing & He, Mengxin & Li, Zhong, 2022. "Dynamics of a ratio-dependent Leslie–Gower predator–prey model with Allee effect and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 417-439.
    3. Zongmin Yue & Wenjuan Wang, 2013. "Qualitative Analysis of a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-9, March.
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    Cited by:

    1. Qiuyue Zhao & Xinglong Niu, 2024. "Dynamics of a Stochastic Predator–Prey Model with Smith Growth Rate and Cooperative Defense," Mathematics, MDPI, vol. 12(12), pages 1-14, June.
    2. Qingyi Cui & Changjin Xu & Wei Ou & Yicheng Pang & Zixin Liu & Peiluan Li & Lingyun Yao, 2023. "Bifurcation Behavior and Hybrid Controller Design of a 2D Lotka–Volterra Commensal Symbiosis System Accompanying Delay," Mathematics, MDPI, vol. 11(23), pages 1-23, November.

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