TURING PATTERN FORMATION IN A TWO-SPECIES NEGATIVE FEEDBACK SYSTEM WITH CROSS-DIFFUSION

Abstract

Pattern formation is a ubiquitous phenomenon in the natural world. Previous studies showed that for an activator–inhibitor system without cross-diffusion, spatial patterns can be formed only when the diffusion of the inhibitor is significantly faster than that of the activator. However, cross-diffusion exists extensively in real systems, especially in biological systems. Here, we study a prototypic two-species negative feedback system with cross-diffusion. By performing stability analysis of equilibrium state, we find sufficient conditions for Turing instability. Both analytical and numerical results demonstrate that mutual diffusions of the two species can lead to the Turing pattern formation regardless of differences in self-diffusion coefficients. However, in the absence of the mutual diffusion or even if there is the cross-diffusion of only one species, the system cannot exhibit Turing patterns. Our results reveal the mechanism of Turing pattern formation in a class of reaction–diffusion systems, where mutual diffusion between species plays a key role.