Critical Threshold and Stability of Cluster Solutions for Large Reaction-Diffusion Systems in R

Abstract

We study a large reaction-diffusion system which arises in the modeling of catalytic networks and describes the emerging of cluster states. We construct single cluster solutions on the real line and then establish their stability or instability in terms of the number N of components and the connection matrix. We provide a rigorous analysis around the single cluster solutions, which is new for systems of this kind. Our results show that for N\leq 4 the hypercycle system is linearly stable while for N\geq 5 the hypercycle system is linearly unstable.