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

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.