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Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects

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  • Pal, Debjit
  • Kesh, Dipak
  • Mukherjee, Debasis

Abstract

The present study is focused on interacting prey–predator reaction–diffusion model with modified Leslie–Gower type functional response incorporating Allee and fear effects. We introduce self as well as cross-diffusion in the model. The equilibrium points and their stability along with saddle–node and Hopf bifurcations around steady states are investigated for non-spatial system. The conditions for Turing instability and the critical line of Hopf and Turing bifurcation in a spatial domain with zero-flux boundary conditions are determined. The parametric space for different regions is depicted and numerical simulation in Turing space is carried out. It is observed that cross diffusion has significant role in forming patterns such as spots, stripes, mixture of spots and stripes. The issue of spatiotemporal pattern controllability is also examined. The availability of cross diffusion results a paradox to the prey density with increasing level of Allee.

Suggested Citation

  • Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2023. "Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    • Handle: RePEc:eee:chsofr:v:167:y:2023:i:c:s0960077922012127
    DOI: 10.1016/j.chaos.2022.113033
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    References listed on IDEAS

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    1. Wang, Caiyun & Qi, Suying, 2018. "Spatial dynamics of a predator-prey system with cross diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 55-60.
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    5. Liyun Lai & Zhenliang Zhu & Fengde Chen, 2020. "Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect," Mathematics, MDPI, vol. 8(8), pages 1-21, August.
    6. 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.
    7. Yan, Shuixian & Jia, Dongxue & Zhang, Tonghua & Yuan, Sanling, 2020. "Pattern dynamics in a diffusive predator-prey model with hunting cooperations," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
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    Cited by:

    1. Sajan, & Anshu, & Dubey, Balram, 2024. "Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    2. Wang, Henan & Liu, Ping, 2023. "Pattern dynamics of a predator–prey system with cross-diffusion, Allee effect and generalized Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    3. Pal, Debjit & Ghorai, Santu & Kesh, Dipak & Mukherjee, Debasis, 2024. "Hopf bifurcation and patterns formation in a diffusive two prey-one predator system with fear in preys and help," Applied Mathematics and Computation, Elsevier, vol. 481(C).
    4. Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2024. "Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 128-147.
    5. Li, Shuai & Huang, Chengdai & Song, Xinyu, 2023. "Detection of Hopf bifurcations induced by pregnancy and maturation delays in a spatial predator–prey model via crossing curves method," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    6. Tian, Yuan & Li, Huanmeng & Sun, Kaibiao, 2024. "Complex dynamics of a fishery model: Impact of the triple effects of fear, cooperative hunting and intermittent harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 31-48.

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