Numerical simulation of the space fractional $(3+1)$-dimensional Gray-Scott models with the Riesz fractional derivative

The reaction-diffusion process always behaves extremely magically, and any a differential model cannot reveal all of its mechanism. Here we show the patterns behavior can be described well by the fractional reaction-diffusion model (FRDM), which has unique properties that the integer model does not have. Numerical simulation is carried out to elucidate the attractive properties of the fractional (3+1)-dimensional Gray-Scott model, which is to model a chemical reaction with oscillation. The Fourier transform for spatial discretization and fourth-order Runge-Kutta method for time discretization are employed to illustrate the fractal reaction-diffusion process.