Spatial patterns of the Brusselator model with the asymmetric Lévy diffusion

Abstract The formation of spatial patterns is an important issue of reaction-diffusion systems. Previous studies mainly focus on spatial patterns coming from reaction-diffusion models equipped with symmetric diffusions (such as normal or fractional Laplace diffusion), namely assuming that spatial environments of the systems are homogeneous. However, the complexity and heterogeneity of spatial environments of biochemical reactions in vivo can lead to reactants’ asymmetric diffusion. Naturally, arises an open question of how the asymmetric diffusion affects dynamical behaviours of biochemical reaction systems. To answer it, we build a general asymmetric Lévy diffusion model based on the theory of the continuous time random walk (CTRW). In addition, we investigate the two-species Brusselator model with the asymmetric Lévy diffusion, and obtain a general condition of forming Turing and wave patterns. More interestingly, we find that even though the Brusselator model with the symmetric diffusion can not produce steadily spatial patterns for some parameters, the asymmetry of Lévy diffusion for this model can do wave patterns. This is different from the previous result [7] that wave instability requires at least a three-species model. In addition, the asymmetry of the Lévy diffusion can significantly affect the amplitude and frequency of spatial patters. Our results enrich the current mechanisms of pattern formation.