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Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect

Author

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  • Yangyang Shao

    (School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China)

  • Yan Meng

    (School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China)

  • Xinyue Xu

    (School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China)

Abstract

The Allee effect is widespread among endangered plants and animals in ecosystems, suggesting that a minimum population density or size is necessary for population survival. This paper investigates the stability and pattern formation of a predator–prey model with nonlinear reactive cross-diffusion under Neumann boundary conditions, which introduces the Allee effect. Firstly, the ODE system is asymptotically stable for its positive equilibrium solution. In a reaction system with self-diffusion, the Allee effect can destabilize the system. Then, in a reaction system with cross-diffusion, through a linear stability analysis, the cross-diffusion coefficient is used as a bifurcation parameter, and instability conditions driven by the cross-diffusion are obtained. Furthermore, we show that the system (5) has at least one inhomogeneous stationary solution. Finally, our theoretical results are illustrated with numerical simulations.

Suggested Citation

  • Yangyang Shao & Yan Meng & Xinyue Xu, 2022. "Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect," Mathematics, MDPI, vol. 10(9), pages 1-15, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1500-:d:807045
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    References listed on IDEAS

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    1. Naveed Iqbal & Yeliz Karaca, 2021. "Complex Fractional-Order Hiv Diffusion Model Based On Amplitude Equations With Turing Patterns And Turing Instability," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(05), pages 1-16, August.
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    3. Yao, Shao-Wen & Ma, Zhan-Ping & Cheng, Zhi-Bo, 2019. "Pattern formation of a diffusive predator–prey model with strong Allee effect and nonconstant death rate," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 527(C).
    4. Mansouri, Djamel & Abdelmalek, Salem & Bendoukha, Samir, 2020. "Bifurcations and pattern formation in a generalized Lengyel–Epstein reaction–diffusion model," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    5. Capone, F. & Carfora, M.F. & De Luca, R. & Torcicollo, I., 2019. "Turing patterns in a reaction–diffusion system modeling hunting cooperation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 172-180.
    6. Peng, Yahong & Ling, Heyang, 2018. "Pattern formation in a ratio-dependent predator-prey model with cross-diffusion," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 307-318.
    7. Yao, Shao-Wen & Ma, Zhan-Ping & Yue, Jia-Long, 2018. "Bistability and Turing pattern induced by cross fraction diffusion in a predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 982-988.
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    Cited by:

    1. Ishtiaq Ali & Maliha Tehseen Saleem, 2023. "Spatiotemporal Dynamics of Reaction–Diffusion System and Its Application to Turing Pattern Formation in a Gray–Scott Model," Mathematics, MDPI, vol. 11(6), pages 1-17, March.

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