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<p>Ecological processes are often characterised by non-local memory effects, for which time-fractional partial differential equations governed by the Caputo derivative offer a powerful mathematical description. Turing pattern formation in fractional order systems exhibits qualitatively different mechanisms compared to integer order counterparts, where spatial heterogeneity can lead to complex spatiotemporal dynamics through memory-dependent interactions. Incorporating simplified Monod-Haldane functional response, this research investigates a time-fractional order predator–prey model with cross-diffusion effects to explore pattern formation under substrate inhibition and group defense mechanisms. The stability and bifurcation analysis provides Hopf bifurcation conditions for the spatially homogeneous system and establishes critical thresholds for spatially extended equilibrium stability, identifying two distinct Turing instability mechanisms: (A1) type where complex eigenvalues cross the fractional stability boundary, and (A2) type following classical real eigenvalue behavior, leading to a comprehensive bifurcation diagram in parameter space. Numerical simulations validate theoretical predictions and demonstrate rich pattern dynamics through systematic investigation of fractional order, growth rate, cross-diffusion coefficient, and inhibition parameter, revealing morphological transitions from spots to stripes to labyrinthine structures that reflect the model’s complex dynamical behaviors. The results highlight the intricate interplay of fractional derivatives, Monod-Haldane functional responses, and cross-diffusion effects in generating diverse spatiotemporal patterns.</p>

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Bifurcation analysis and pattern formation in a time-fractional-order predator–prey model with Monod-Haldane functional response and cross-diffusion

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Yuhan Yan,
  • Feifan Zhang

摘要

Ecological processes are often characterised by non-local memory effects, for which time-fractional partial differential equations governed by the Caputo derivative offer a powerful mathematical description. Turing pattern formation in fractional order systems exhibits qualitatively different mechanisms compared to integer order counterparts, where spatial heterogeneity can lead to complex spatiotemporal dynamics through memory-dependent interactions. Incorporating simplified Monod-Haldane functional response, this research investigates a time-fractional order predator–prey model with cross-diffusion effects to explore pattern formation under substrate inhibition and group defense mechanisms. The stability and bifurcation analysis provides Hopf bifurcation conditions for the spatially homogeneous system and establishes critical thresholds for spatially extended equilibrium stability, identifying two distinct Turing instability mechanisms: (A1) type where complex eigenvalues cross the fractional stability boundary, and (A2) type following classical real eigenvalue behavior, leading to a comprehensive bifurcation diagram in parameter space. Numerical simulations validate theoretical predictions and demonstrate rich pattern dynamics through systematic investigation of fractional order, growth rate, cross-diffusion coefficient, and inhibition parameter, revealing morphological transitions from spots to stripes to labyrinthine structures that reflect the model’s complex dynamical behaviors. The results highlight the intricate interplay of fractional derivatives, Monod-Haldane functional responses, and cross-diffusion effects in generating diverse spatiotemporal patterns.