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Stability and Hopf Bifurcation Analysis of a Predator–Prey Model with Weak Allee Effect Delay and Competition Delay

Author

Listed:
  • Yurong Dong

    (School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China)

  • Hua Liu

    (School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China)

  • Yumei Wei

    (Experimental Teaching Department, Northwest Minzu University, Lanzhou 730030, China)

  • Qibin Zhang

    (Gansu High-Tech Innovation Service Center, Lanzhou 730030, China)

  • Gang Ma

    (School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China)

Abstract

The purpose of this paper is to study a predator–prey model with Allee effect and double time delays. This research examines the dynamics of the model, with a focus on positivity, existence, stability and Hopf bifurcations. The stability of the periodic solution and the direction of the Hopf bifurcation are elucidated by applying the normal form theory and the center manifold theorem. To validate the correctness of the theoretical analysis, numerical simulations were conducted. The results suggest that a weak Allee effect delay can promote stability within the model, transitioning it from instability to stability. Nevertheless, the competition delay induces periodic oscillations and chaotic dynamics, ultimately resulting in the population’s collapse.

Suggested Citation

  • Yurong Dong & Hua Liu & Yumei Wei & Qibin Zhang & Gang Ma, 2024. "Stability and Hopf Bifurcation Analysis of a Predator–Prey Model with Weak Allee Effect Delay and Competition Delay," Mathematics, MDPI, vol. 12(18), pages 1-24, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2853-:d:1477899
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    References listed on IDEAS

    as
    1. Li, Danyang & Liu, Hua & Zhang, Haotian & Wei, Yumei, 2023. "Influence of multiple delays mechanisms on predator–prey model with Allee effect," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    2. Yong Ye & Hua Liu & Yu-mei Wei & Ming Ma & Kai Zhang, 2019. "Dynamic Study of a Predator-Prey Model with Weak Allee Effect and Delay," Advances in Mathematical Physics, Hindawi, vol. 2019, pages 1-15, August.
    3. Çelik, C. & Merdan, H. & Duman, O. & Akın, Ö., 2008. "Allee effects on population dynamics with delay," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 65-74.
    4. Pal, Pallav Jyoti & Saha, Tapan, 2015. "Qualitative analysis of a predator–prey system with double Allee effect in prey," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 36-63.
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    Cited by:

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    2. Ruichen Qi & Shaoyi Chen & Caiyun Huang & Qiubao Wang, 2025. "The Stochastic Hopf Bifurcation and Vibrational Response of a Double Pendulum System Under Delayed Feedback Control," Mathematics, MDPI, vol. 13(13), pages 1-25, July.

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