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<p>This study develops a fractional-order Gierer–Meinhardt reaction–diffusion system to model metamorphic pattern formation in geological media. The model integrates time-fractional derivatives to capture anomalous transport and memory effects inherent in heterogeneous rock fabrics: the Caputo derivative for sub-diffusion (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \le 1\)</EquationSource> </InlineEquation>) and the Grünwald–Letnikov formulation for super-diffusion (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha &gt; 1\)</EquationSource> </InlineEquation>). We establish rigorous conditions for the local and global asymptotic stability of the homogeneous steady state in the absence of diffusion through Lyapunov functional analysis. A subsequent linear stability analysis identifies the precise parameter regimes that induce Turing-type diffusion-driven instability, leading to spontaneous spatial pattern formation. Our numerical simulations employ an L1 time-fractional discretization for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \le 1\)</EquationSource> </InlineEquation> and the GL scheme for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha &gt; 1\)</EquationSource> </InlineEquation>, coupled with a second-order finite-difference spatial scheme, ensuring full consistency with the mathematical model. The results demonstrate that the activator diffusion coefficient (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(D_u\)</EquationSource> </InlineEquation>) and the fractional order (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> </InlineEquation>) act as key control parameters. Increasing <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(D_u\)</EquationSource> </InlineEquation> drives a morphological transition from isolated spots to labyrinthine stripes, while increasing <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\alpha &gt; 1\)</EquationSource> </InlineEquation> enhances super-diffusive memory effects, accelerating pattern evolution and modifying spot sharpness and density. This work bridges advanced fractional calculus with geological processes, providing a unified framework for understanding memory-driven spatio-temporal self-organization in complex natural systems.</p>

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Pattern formation and stability analysis in a fractional Gierer–Meinhardt model for metamorphic textures

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  • 发表日期：

    2026年2月7日

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  • Biswajit Saha,
  • Manas Kumar Roy

摘要

This study develops a fractional-order Gierer–Meinhardt reaction–diffusion system to model metamorphic pattern formation in geological media. The model integrates time-fractional derivatives to capture anomalous transport and memory effects inherent in heterogeneous rock fabrics: the Caputo derivative for sub-diffusion ( \(\alpha \le 1\) ) and the Grünwald–Letnikov formulation for super-diffusion ( \(\alpha > 1\) ). We establish rigorous conditions for the local and global asymptotic stability of the homogeneous steady state in the absence of diffusion through Lyapunov functional analysis. A subsequent linear stability analysis identifies the precise parameter regimes that induce Turing-type diffusion-driven instability, leading to spontaneous spatial pattern formation. Our numerical simulations employ an L1 time-fractional discretization for \(\alpha \le 1\) and the GL scheme for \(\alpha > 1\) , coupled with a second-order finite-difference spatial scheme, ensuring full consistency with the mathematical model. The results demonstrate that the activator diffusion coefficient ( \(D_u\) ) and the fractional order ( \(\alpha \) ) act as key control parameters. Increasing \(D_u\) drives a morphological transition from isolated spots to labyrinthine stripes, while increasing \(\alpha > 1\) enhances super-diffusive memory effects, accelerating pattern evolution and modifying spot sharpness and density. This work bridges advanced fractional calculus with geological processes, providing a unified framework for understanding memory-driven spatio-temporal self-organization in complex natural systems.