Abstract 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 (1) and the Grünwald–Letnikov formulation for super-diffusion (> 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 1 and the GL scheme for > 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 (Dᵤ) and the fractional order () act as key control parameters. Increasing Dᵤ drives a morphological transition from isolated spots to labyrinthine stripes, while increasing > 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.
Saha et al. (Sat,) studied this question.