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Nonlocal interaction driven pattern formation in a prey–predator model

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  • Tian, Canrong
  • Ling, Zhi
  • Zhang, Lai

Abstract

A widely observed scenario in ecological systems is that populations interact not only with those living in the same spatial location but also with those in spatially adjacent locations, a phenomenon called nonlocal interaction. In this paper, we explore the role of nonlocal interaction in the emergence of spatial patterns in a prey–predator model under the reaction–diffusion framework, which is described by two coupled integro-differential equations. We first prove the existence and uniqueness of the global solution by means of the contraction mapping theory and then conduct stability analysis of the positive equilibrium. We find that nonlocal interaction can induce Turing bifurcation and drive the formation of stationary spatial patterns. Finally we carry out numerical simulations to demonstrate our analytical findings.

Suggested Citation

  • Tian, Canrong & Ling, Zhi & Zhang, Lai, 2017. "Nonlocal interaction driven pattern formation in a prey–predator model," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 73-83.
    • Handle: RePEc:eee:apmaco:v:308:y:2017:i:c:p:73-83
    DOI: 10.1016/j.amc.2017.03.017
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    References listed on IDEAS

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    1. Banerjee, Malay & Zhang, Lai, 2014. "Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 73-81.
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    Cited by:

    1. Chen, Mengxin & Wu, Ranchao & Chen, Liping, 2020. "Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    2. Aybar, I. Kusbeyzi & Aybar, O.O. & Dukarić, M. & Ferčec, B., 2018. "Dynamical analysis of a two prey-one predator system with quadratic self interaction," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 118-132.
    3. 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.
    4. Wu, Zeyan & Li, Jianjuan & Liu, Shuying & Zhou, Liuting & Luo, Yang, 2019. "A spatial predator–prey system with non-renewable resources," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 381-391.
    5. Shivam, & Singh, Kuldeep & Kumar, Mukesh & Dubey, Ramu & Singh, Teekam, 2022. "Untangling role of cooperative hunting among predators and herd behavior in prey with a dynamical systems approach," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    6. Kalyan Manna & Vitaly Volpert & Malay Banerjee, 2020. "Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species," Mathematics, MDPI, vol. 8(1), pages 1-28, January.

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