Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor

Yuhong Huo; Gourav Mandal; Lakshmi Narayan Guin; Santabrata Chakravarty; Renji Han; Yuhong Huo; Gourav Mandal; Lakshmi Narayan Guin; Santabrata Chakravarty; Renji Han

doi:10.3934/mbe.2023834

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 18820-18860. doi: 10.3934/mbe.2023834

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Research article

Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor

1.
School of Finance and Mathematics, Huainan Normal University, Dongshanxi Street, Huainan, 232038, China
2.
Department of Mathematics, Visva-Bharati, Santiniketan 731235, India
3.
Former Professor, Department of Mathematics, Visva-Bharati, Santiniketan 731235, India
4.
School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Academic Editor: Peter Kim

Received: 25 July 2023 Revised: 19 September 2023 Accepted: 20 September 2023 Published: 09 October 2023

In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.
- reaction-diffusion system,
- fear and Allee effect,
- normal form,
- center manifold theory,
- bifurcation theory,
- spatiotemporal pattern transition
Citation: Yuhong Huo, Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty, Renji Han. Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18820-18860. doi: 10.3934/mbe.2023834

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Abstract

In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.

References

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