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Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey

Author

Listed:
  • Wenqi Zhang

    (Department of Mathematics, Northeast Forestry University, Harbin 150040, China)

  • Dan Jin

    (Department of Mathematics, Northeast Forestry University, Harbin 150040, China)

  • Ruizhi Yang

    (Department of Mathematics, Northeast Forestry University, Harbin 150040, China)

Abstract

In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf bifurcation is investigated by the theory of the centre manifold and normal form method. Through the numerical simulations, it is observed that the anti-predator behaviour parameter η , the memory-based diffusion coefficient parameter d , and memory delay τ can affect the stability of the coexisting equilibrium under some parameters and cause the spatially inhomogeneous oscillation of prey and predator’s densities.

Suggested Citation

  • Wenqi Zhang & Dan Jin & Ruizhi Yang, 2023. "Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey," Mathematics, MDPI, vol. 11(3), pages 1-12, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:556-:d:1042598
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    References listed on IDEAS

    as
    1. Yang, Ruizhi & Ma, Jian, 2018. "Analysis of a diffusive predator-prey system with anti-predator behaviour and maturation delay," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 128-139.
    2. Ruizhi Yang & Xiao Zhao & Yong An, 2022. "Dynamical Analysis of a Delayed Diffusive Predator–Prey Model with Additional Food Provided and Anti-Predator Behavior," Mathematics, MDPI, vol. 10(3), pages 1-18, January.
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