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Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

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  • Zongmin Yue
  • Xiaoqin Wang
  • Haifeng Liu

Abstract

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

Suggested Citation

  • Zongmin Yue & Xiaoqin Wang & Haifeng Liu, 2013. "Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, February.
    • Handle: RePEc:hin:jnlaaa:270191
    DOI: 10.1155/2013/270191
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    Cited by:

    1. Rana, Sourav & Bhattacharya, Sabyasachi & Samanta, Sudip, 2022. "Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 32-49.
    2. Junbo Jia & Pan Yang & Huaiping Zhu & Zhen Jin & Jinqiao Duan & Xinchu Fu, 2023. "Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems," Mathematics, MDPI, vol. 11(19), pages 1-25, September.
    3. 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).
    4. Kumar, Udai & Mandal, Partha Sarathi, 2022. "Role of Allee effect on prey–predator model with component Allee effect for predator reproduction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 623-665.

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