IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i18p2853-d1477899.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/18/2853/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/18/2853/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Boli Xie & Zhijun Wang & Yakui Xue & Zhenmin Zhang, 2015. "The Dynamics of a Delayed Predator-Prey Model with Double Allee Effect," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-8, October.
    2. Das, Amartya & Samanta, G.P., 2021. "Influence of environmental noises on a prey–predator species with predator-dependent carrying capacity in alpine meadow ecosystem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1294-1316.
    3. Hua Liu & Yong Ye & Yumei Wei & Weiyuan Ma & Ming Ma & Kai Zhang, 2019. "Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay," Complexity, Hindawi, vol. 2019, pages 1-14, November.
    4. Toni Bakhtiar & Ihza Rizkia Fitri & Farida Hanum & Ali Kusnanto, 2022. "Mathematical Model of Pest Control Using Different Release Rates of Sterile Insects and Natural Enemies," Mathematics, MDPI, vol. 10(6), pages 1-18, March.
    5. Gökçe, Aytül, 2022. "A dynamic interplay between Allee effect and time delay in a mathematical model with weakening memory," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    6. Li, Xiaoshuang & Pang, Danfeng & Wallhead, Philip & Bellerby, Richard Garth James, 2023. "Dynamics of an aquatic diffusive predator–prey model with double Allee effect and pH-dependent capture rate," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    7. Mandal, Dibyendu Sekhar & Chekroun, Abdennasser & Samanta, Sudip & Chattopadhyay, Joydev, 2021. "A mathematical study of a crop-pest–natural enemy model with Z-type control," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 468-488.
    8. Merdan, H. & Duman, O., 2009. "On the stability analysis of a general discrete-time population model involving predation and Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1169-1175.
    9. Duman, O. & Merdan, H., 2009. "Stability analysis of continuous population model involving predation and Allee effect," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1218-1222.
    10. Wu, Yuhang & Ni, Mingkang, 2024. "Complex dynamics in a singularly perturbed Hastings–Powell model with the additive Allee effect," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    11. Érika Diz-Pita & M. Victoria Otero-Espinar, 2021. "Predator–Prey Models: A Review of Some Recent Advances," Mathematics, MDPI, vol. 9(15), pages 1-34, July.
    12. Pal, Pallav Jyoti & Mandal, Gourav & Guin, Lakshmi Narayan & Saha, Tapan, 2024. "Allee effect and hunting-induced bifurcation inquisition and pattern formation in a modified Leslie–Gower interacting species system," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    13. Lei Shi & Jiaying Zhou & Yong Ye, 2023. "Pattern Formation in a Predator–Prey Model with Allee Effect and Hyperbolic Mortality on Multiplex Networks," Mathematics, MDPI, vol. 11(15), pages 1-15, July.
    14. Merdan, H. & Duman, O. & Akın, Ö. & Çelik, C., 2009. "Allee effects on population dynamics in continuous (overlapping) case," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1994-2001.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2853-:d:1477899. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.