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.

In this paper, we investigated the nonlinear dynamics and pattern formation of a fractional-order three-variable Oregonator model. We first performed a linear stability analysis of the model without diffusion, deriving equilibrium points and Jacobian eigenvalues, and verified Matignon's stability conditions. A high-precision numerical scheme was developed, and simulations revealed that even tiny variations in fractional order produce significant changes in long-term trajectories. For the reaction-diffusion model, we analyzed Turing instability under integer-order diffusion and derived the critical wave-number conditions via Routh-Hurwitz criteria. Weakly nonlinear analysis near the Turing threshold yielded coupled amplitude equations whose coefficients predicted stripe, hexagon, and mixed patterns. Extensive two-dimensional numerical experiments confirmed the theoretical predictions: Depending on diffusion coefficients and other parameters, the model evolved into bullseye, spiral, labyrinthine, or spot-stripe mixtures.