Dynamics and pattern formation in a diffusive Beddington-DeAngelis predator-prey model with fear effect

Aung Zaw Myint; Aye Chan May; Mya Hnin Lwin; Toe Toe Shwe; Adisak Seesanea

doi:10.3934/dcdsb.2025113

Article Contents

2026, Volume 31: 359-376. Doi: 10.3934/dcdsb.2025113 open access

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Dynamics and pattern formation in a diffusive Beddington-DeAngelis predator-prey model with fear effect

1.
Department of Mathematics, University of Mandalay, Mandalay 05032, Myanmar
2.
Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani 12120, Thailand

^*Corresponding author: Adisak Seesanea
^*Corresponding author: Adisak Seesanea

Received: December 2024

Revised: May 2025

Early access: July 2025

Published online: July 2025

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Abstract

In this paper, dynamical properties and positive steady states of a diffusive predator-prey system with fear effect and Beddington-DeAngelis functional response subject to Neumann boundary conditions are investigated. Dynamical properties of time-dependent solutions and the stationary patterns induced by diffusion (Turing patterns) are presented.

Keywords:

Predator-prey model,
fear factor,
Beddington-DeAngelis functional response,
stationary patterns.

Mathematics Subject Classification: Primary: 35A01, 35B45; Secondary: 35B09, 92D25.

Citation:

FullText(HTML)

Figure 1. Nonconstant positive solution $ (u(x,t),v(x,t)) $ of the system (3), $ t = 200, d_{2} = 0.1, a = 0.1, \tilde{u} = 0.16608, \tilde{v} = 3.89934, $ initial data $ (u_{0},v_{0}) = (\tilde{u}+0.01\cos \frac{x}{2}, \tilde{v}+0.01\cos x,) $ $ \frac{M}{d_{1}} = 2.407\in (\mu_{1},\mu_{2}) = (1,4) $

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Figure 2. Nonconstant positive solution $ (u(x,t),v(x,t)) $ of the system (3), $ t = 200, d_{2} = 0.2, a = 0.055, \tilde{u} = 0.16675, \tilde{v} = 3.91904, $ initial data $ (u_{0},v_{0}) = (\tilde{u}+0.01\cos\frac{x}{2}, \tilde{v}+0.01\cos x,) $ $ \frac{M}{d_{1}} = 10.095\in (9, 16) $

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Figure 3. Nonconstant positive solution $ (u(x,t),v(x,t)) $ of the system (3), $ a = 0.1, \tilde{u} = 0.16608, \tilde{v} = 3.89934, $ initial data $ (u_{0},v_{0}) = (\tilde{u}+0.01\mathrm{cos}\frac{x}{2}, \tilde{v}+0.01\mathrm{cos} x), t = 50, d_{2} = 0.02, 0.05, 0.08 $ and $ 0.1 $

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Figure 4. Nonconstant positive solution $ (u(x,t),v(x,t)) $ of the system (3), $ a = 0.1, \tilde{u} = 0.16608, \tilde{v} = 3.89934, $ initial data $ (u_{0},v_{0}) = (\tilde{u}+0.01\mathrm{cos}\frac{x}{2}, \tilde{v}+0.01\mathrm{cos} x), t = 200, d_{2} = 0.02, 0.05, 0.08 $ and $ 0.1 $

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Figure 5. Nonconstant positive solution $ (u(x,t),v(x,t)) $ of the system (3), $ a = 0.055, \tilde{u} = 0.16675, \tilde{v} = 3.91904, $ initial data $ (u_{0},v_{0}) = (\tilde{u}+0.01\mathrm{cos}\frac{x}{2}, \tilde{v}+0.01\mathrm{cos} x),t = 50, d_{2} = 0.12, 0.15, 0.18 $ and $ 0.2 $

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Figure 6. Nonconstant positive solution $ (u(x,t),v(x,t)) $ of the system (3), $ a = 0.055, \tilde{u} = 0.16675, \tilde{v} = 3.91904, $ initial data $ (u_{0},v_{0}) = (\tilde{u}+0.01\mathrm{cos}\frac{x}{2}, \tilde{v}+0.01\mathrm{cos} x), t = 200, d_{2} = 0.12, 0.15, 0.18 $ and $ 0.2 $

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