Spatial patterns of the Brusselator model with asymmetric Lévy diffusion

The formation of spatial patterns is an important issue in reaction–diffusion systems. Previous studies have mainly focused on the spatial patterns in reaction–diffusion models equipped with symmetric diffusion (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 asymmetric diffusion of reactants. Naturally, there arises an open question of how the asymmetric diffusion affects dynamical behaviors of biochemical reaction systems. To answer this, we build a general asymmetric Lévy diffusion model based on the theory of a continuous time random walk. In addition, we investigate the two-species Brusselator model with asymmetric Lévy diffusion, and obtain a general condition for the formation of Turing and wave patterns. More interestingly, we find that even though the Brusselator model with symmetric diffusion cannot produce steady spatial patterns for some parameters, the asymmetry of Lévy diffusion for this model can produce wave patterns. This is different from the previous result that wave instability requires at least a three-species model. In addition, the asymmetry of Lévy diffusion can significantly affect the amplitude and frequency of the spatial patterns. Our results enrich our knowledge of the mechanisms of pattern formation.