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Stability Analysis and Hopf Bifurcation of a Delayed Diffusive Predator–Prey Model with a Strong Allee Effect on the Prey and the Effect of Fear on the Predator

Author

Listed:
  • Yining Xie

    (School of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China)

  • Jing Zhao

    (School of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China)

  • Ruizhi Yang

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

Abstract

In this paper, we propose a diffusive predator–prey model with a strong Allee effect and nonlocal competition in the prey and a fear effect and gestation delay in the predator. We mainly study the local stability of the coexisting equilibrium and the existence and properties of Hopf bifurcation. We provide bifurcation diagrams with the fear effect parameter ( s ) and the Allee effect parameter ( a ), showing that the stable region of the coexisting equilibrium increases (or decreases) with an increase in the fear effect parameter ( s ) (or the Allee effect parameter ( a )). We also show that gestation delay ( τ ) can affect the local stability of the coexisting equilibrium. When the delay ( τ ) is greater than the critical value, the coexistence equilibrium loses its stability, and bifurcating periodic solutions appear. Whether the bifurcated periodic solution is spatially homogeneous or inhomogeneous depends on the fear effect parameter ( s ) and the Allee effect parameter ( a ). These results show that the fear effect parameter ( s ), the Allee effect parameter ( a ), and gestation delay ( τ ) can be used to control the growth of prey and predator populations.

Suggested Citation

  • Yining Xie & Jing Zhao & Ruizhi Yang, 2023. "Stability Analysis and Hopf Bifurcation of a Delayed Diffusive Predator–Prey Model with a Strong Allee Effect on the Prey and the Effect of Fear on the Predator," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:1996-:d:1130854
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    References listed on IDEAS

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    1. Barman, Binandita & Ghosh, Bapan, 2019. "Explicit impacts of harvesting in delayed predator-prey models," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 213-228.
    2. Xinyu Song & Zhihao Ge & Jingang Wu, 2006. "Stability analysis of a ratio-dependent predator-prey system with diffusion and stage structure," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2006, pages 1-20, June.
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