Numerical simulation and theoretical analysis of pattern dynamics for the fractional-in-space Schnakenberg model

Effective exploration of the pattern dynamic behaviors of reaction–diffusion models is a popular but difficult topic. The Schnakenberg model is a famous reaction–diffusion system that has been widely used in many fields, such as physics, chemistry, and biology. Herein, we explore the stability, Turing instability, and weakly non-linear analysis of the Schnakenberg model; further, the pattern dynamics of the fractional-in-space Schnakenberg model was simulated numerically based on the Fourier spectral method. The patterns under different parameters, initial conditions, and perturbations are shown, including the target, bar, and dot patterns. It was found that the pattern not only splits and spreads from the bar to spot pattern but also forms a bar pattern from the broken connections of the dot pattern. The effects of the fractional Laplacian operator on the pattern are also shown. In most cases, the diffusion rate of the fractional model was higher than that of the integer model. By comparing with different methods in literature, it was found that the simulated patterns were consistent with the results obtained with other numerical methods in literature, indicating that the Fourier spectral method can be used to effectively explore the dynamic behaviors of the fractional Schnakenberg model. Some novel pattern dynamics behaviors of the fractional-in-space Schnakenberg model are also demonstrated.