Turing pattern in the fractional Gierer–Meinhardt model**Project supported by the National Natural Science Foundation of China (Grant Nos. 61573008 and 61703290) and Natural Science Foundation of Liaoning Province, China (Grant No. 20180550996).

Abstract

It is well-known that reaction–diffusion systems are used to describe the pattern formation models. In this paper, we will investigate the pattern formation generated by the fractional reaction–diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer–Meinhardt system. Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge–Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results.