Composite pattern formation in the Gierer-Meinhardt system with cross-diffusion distributions

The Gierer-Meinhardt system characterizes the fundamental dynamics of pattern formation through the interaction of two variables. Currently, research on the spatiotemporal evolution of this system at home and abroad has been limited to Turing instability and patterns, and research on composite patterns formed by non-uniform cross-diffusion distributions is not yet in-depth. This article presents a composite pattern of the system based on linear, periodic, and radial distributions of cross diffusion. We conduct a comprehensive study of the two-dimensional spatiotemporal dynamics of the Gierer-Meinhardt model, treating the cross-diffusion coefficients as bifurcation parameters. Using multiscale analysis, the amplitude equation at the Turing threshold is derived. Subsequently, the effects of the proportional-derivative controller, fractional diffusion orders, and anisotropy on system stability, pattern formation, and evolution speed are systematically investigated. Numerical simulations are provided to validate the conclusions.