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Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Author

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  • Liyun Lai

    (College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China)

  • Zhenliang Zhu

    (College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China)

  • Fengde Chen

    (College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China)

Abstract

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

Suggested Citation

  • Liyun Lai & Zhenliang Zhu & Fengde Chen, 2020. "Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect," Mathematics, MDPI, vol. 8(8), pages 1-21, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1280-:d:393943
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    References listed on IDEAS

    as
    1. Zhang, Huisen & Cai, Yongli & Fu, Shengmao & Wang, Weiming, 2019. "Impact of the fear effect in a prey-predator model incorporating a prey refuge," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 328-337.
    2. Agus Suryanto & Isnani Darti & Syaiful Anam, 2017. "Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2017, pages 1-9, May.
    3. Xu, Junyan & Zhang, Tonghua & Han, Maoan, 2019. "A regime switching model for species subject to environmental noises and additive Allee effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 527(C).
    4. Li, Zhong & Han, Maoan & Chen, Fengde, 2014. "Almost periodic solutions of a discrete almost periodic logistic equation with delay," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 743-751.
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    1. Binhao Hong & Chunrui Zhang, 2023. "Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect," Mathematics, MDPI, vol. 11(6), pages 1-13, March.
    2. Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2023. "Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).

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