Numerical simulation and theoretical analysis of pattern dynamics for the fractional-in-space Schnakenberg model - Wang, Ji-Lei; Han, Yu-Xing; Chen, Qing-Tong; Li, Zhi-Yuan; Du, Ming-Jing; Wang, Yu-Lan (Frontiers in Physics, 2024)
It is always a difficult and hot topic to effectively explore the pattern dynamic behavior of reaction-diffusion models. The Schnakenberg model is a famous reaction-diffusion system, which has been widely used in many fields such as physics, chemistry and biology. In this paper, we researches the stability, Turing instability and weakly nonlinear analysis for the Schnakenberg model, and pattern dynamics of the fractional-in-space Schnakenberg model was simulated numerically based on the Fourier spectral method. The pattern patterns under different parameters, different initial conditions and different disturbances are shown, including target pattern, bar pattern and dot pattern. It is found that the pattern can not only split and spread from the bar pattern to the spot pattern, but also form the bar pattern from the broken connection of the dot pattern. The effect of fractional Laplacian operator on the pattern is also shown. In most cases, the diffusion rate of fractional model is faster than that of integer model. By comparing with different methods in other literatures, it can be found that the simulated patterns are basically consistent with the results simulated by other numerical methods in literatures, which indicates that Fourier spectral method can effectively explore the dynamic behavior of fractional Schnakenberg model. Some novel pattern dynamics behavior of the fractional-in-space Schnakenberg model are shown.