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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. Ghorai, Santu & Poria, Swarup, 2016. "Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 421-429.
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    5. 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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    5. 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.
    6. Carfora, M.F. & Iovanna, F. & Torcicollo, I., 2025. "Turing patterns in an intraguild predator–prey model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 232(C), pages 192-210.
    7. Mu, Yu & Lo, Wing-Cheong & Tan, Yuanshun & Liu, Zijian, 2025. "Hybrid control for the prey in a spatial prey-predator model with cooperative hunting and fear effect time lag," Applied Mathematics and Computation, Elsevier, vol. 491(C).
    8. 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).
    9. Menon, Sidharth & Kumari, Sangeeta, 2025. "Impact of cross-diffusion and Allee effect on modified Leslie–Gower model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 236(C), pages 183-199.
    10. 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).
    11. 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.

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