A detailed analysis of the spatial dynamics of a food-chain model with Allee and fear effect

Swati Mishra; Anal Chatterjee; Ranjit Kumar Upadhyay; Mainul Haque; Swati Mishra; Anal Chatterjee; Ranjit Kumar Upadhyay; Mainul Haque

doi:10.3934/mbe.2025117

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 12: 3154-3200. doi: 10.3934/mbe.2025117

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A detailed analysis of the spatial dynamics of a food-chain model with Allee and fear effect

1.
Delhi School of Climate Change & Sustainability, Institution of Eminence, University of Delhi, Delhi 110007, India
2.
Faculty of Mathematical Sciences, South Asian University, New Delhi 110068, India
3.
Department of Mathematics, Barrackpore Rastraguru Surendranath College, Kolkata, India
4.
Department of Mathematics & computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
5.
School of Mathematical Sciences, University of Nottingham Ningbo China, Ningbo, China

Received: 04 May 2025 Revised: 22 August 2025 Accepted: 17 September 2025 Published: 03 November 2025

We investigate the spatiotemporal dynamics of a tri-trophic food chain model incorporating a strong Allee effect on the prey and a fear effect on the middle predator. The model's well-posedness is established through the positivity and boundedness of solutions. We derive all equilibria and examine their local stability, revealing saddle-node and transcritical bifurcations under varying parameter conditions. The analysis demonstrates how shifts in the Allee threshold and fear intensity induce bistability, coexistence, or extinction. Numerical simulations highlight diffusion-driven instabilities and complex Turing patterns, including labyrinthine formations and unexpected "leaser slime" structures—resembling those observed in fungi and algae in aquatic systems. These findings reveal the crucial role of behavioral and ecological feedbacks in shaping pattern formation and species persistence.
- self-diffusion and cross-diffusion,
- leaser slime type patterns,
- two-parameter bifurcation,
- chaos,
- food-chain,
- Allee and fear
Citation: Swati Mishra, Anal Chatterjee, Ranjit Kumar Upadhyay, Mainul Haque. A detailed analysis of the spatial dynamics of a food-chain model with Allee and fear effect[J]. Mathematical Biosciences and Engineering, 2025, 22(12): 3154-3200. doi: 10.3934/mbe.2025117

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Abstract

We investigate the spatiotemporal dynamics of a tri-trophic food chain model incorporating a strong Allee effect on the prey and a fear effect on the middle predator. The model's well-posedness is established through the positivity and boundedness of solutions. We derive all equilibria and examine their local stability, revealing saddle-node and transcritical bifurcations under varying parameter conditions. The analysis demonstrates how shifts in the Allee threshold and fear intensity induce bistability, coexistence, or extinction. Numerical simulations highlight diffusion-driven instabilities and complex Turing patterns, including labyrinthine formations and unexpected "leaser slime" structures—resembling those observed in fungi and algae in aquatic systems. These findings reveal the crucial role of behavioral and ecological feedbacks in shaping pattern formation and species persistence.

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