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Anti-predator behavior: A mechanism to lose localized patterns for diffusive ratio-dependent predator–prey systems

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  • Shen, Zihao
  • Xu, Yancong
  • Gai, Chunyi
  • Wei, Junjie

Abstract

In this paper, the role of anti-predator behavior in a diffusive ratio-dependent predator–prey model is investigated. The Hopf bifurcation analysis in ODEs and PDEs is carried out using the Lyapunov coefficient and center manifold theory. The one-dimensional and two-dimensional patterns are given to illustrate the impact of anti-predator behavior. Also, using weakly nonlinear analysis, we identify the codimension-two point where the Turing bifurcation shifts from supercritical to subcritical. We find that the spatial diffusion is the crucial factor to determine the occurrence of snake branch of localized patterns, and the anti-predator behavior can shrink the snaking region and shift the entire bifurcation diagram forward or backward, while preserving the overall structure of the snaking solution branch. In particular, numerical investigations illustrate that anti-predator behavior can force the predator and prey to separate from each other, which is an antagonistic mechanism to lose localized patterns for ecological systems.

Suggested Citation

  • Shen, Zihao & Xu, Yancong & Gai, Chunyi & Wei, Junjie, 2026. "Anti-predator behavior: A mechanism to lose localized patterns for diffusive ratio-dependent predator–prey systems," Chaos, Solitons & Fractals, Elsevier, vol. 202(P1).
    • Handle: RePEc:eee:chsofr:v:202:y:2026:i:p1:s0960077925014729
    DOI: 10.1016/j.chaos.2025.117459
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    References listed on IDEAS

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    1. Ahmed, Nauman & Yasin, Muhammad Waqas & Baleanu, Dumitru & Tintareanu-Mircea, Ovidiu & Iqbal, Muhammad Sajid & Akgül, Ali, 2024. "Pattern Formation and analysis of reaction–diffusion ratio-dependent prey–predator model with harvesting in predator," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    2. Tang, Biao & Xiao, Yanni, 2015. "Bifurcation analysis of a predator–prey model with anti-predator behaviour," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 58-68.
    3. Freddy Dumortier & Jaume Llibre & Joan C. Artés, 2006. "Qualitative Theory of Planar Differential Systems," Springer Books, Springer, number 978-3-540-32902-2, January.
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