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<p>In this paper, we investigate a predator–prey system that includes cross-diffusion effects together with harvesting. We mainly focus on how over-harvesting affects the functional response and, subsequently, how it influences the growth of predators in the spatial domain. For the temporal dynamics, the focus is on Hopf and transcritical bifurcations, revealing the parameter regimes where the system transitions between stable states or exhibits periodic oscillations. The role of cross-diffusion in promoting Turing instability is established analytically. Two-parameter bifurcation analysis is presented, supported by numerically computed bifurcation diagrams. The impact of the harvesting effort on the dynamics of pattern formation is reported. It is observed that the cross-diffusion supports the formation of various spatial and spatiotemporal patterns. A time series plot of average predator density across the spatial domain has been presented. Under the zero-flux boundary conditions, the model generates spot patterns, stripe patterns, and mixed spot–stripe patterns and also presents a surface plot. We present the emergence of three-dimensional (3D) Turing patterns near the Turing bifurcation point. It is observed that pattern formation in the spatial predator–prey model is significantly influenced by the cross-diffusion parameter and the harvesting effort.</p>

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Pattern dynamics in a diffusive predator–prey model with the effect of over-harvesting on predator growth

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Chiranjit Das,
  • Lakpa Thendup Bhutia,
  • Tapan Kumar Kar

摘要

In this paper, we investigate a predator–prey system that includes cross-diffusion effects together with harvesting. We mainly focus on how over-harvesting affects the functional response and, subsequently, how it influences the growth of predators in the spatial domain. For the temporal dynamics, the focus is on Hopf and transcritical bifurcations, revealing the parameter regimes where the system transitions between stable states or exhibits periodic oscillations. The role of cross-diffusion in promoting Turing instability is established analytically. Two-parameter bifurcation analysis is presented, supported by numerically computed bifurcation diagrams. The impact of the harvesting effort on the dynamics of pattern formation is reported. It is observed that the cross-diffusion supports the formation of various spatial and spatiotemporal patterns. A time series plot of average predator density across the spatial domain has been presented. Under the zero-flux boundary conditions, the model generates spot patterns, stripe patterns, and mixed spot–stripe patterns and also presents a surface plot. We present the emergence of three-dimensional (3D) Turing patterns near the Turing bifurcation point. It is observed that pattern formation in the spatial predator–prey model is significantly influenced by the cross-diffusion parameter and the harvesting effort.