Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative

Pattern is a non-uniform macro structure with some regularity in space or time, which is common in nature. In this manuscript, we introduce the Fourier transform for spatial discretization and Runge–Kutta method for time discretization to solve a class of fractional reaction–diffusion models such as Allen–Cahn model, FitzHugh–Nagumo model and Gray–Scott model with space derivatives described by the fractional Laplacian. Numerical experiments show that compared with semi-implicit Fourier spectral method, present method has higher precision and low computational complexity. The patterns of 2D FitzHugh–Nagumo model with standard diffusion obtained in this manuscript are in line with the numerical simulations and theoretical analysis that made by other academics. Then, we discuss the limit case of fractional order: the process of pattern formations of the fractional order tends to corresponding integer-order reaction–diffusion model when super diffusion tends to standard diffusion. Finally, some long time diffusion behaviors of 3D FitzHugh–Nagumo model and 3D Gray–Scott model are observed.