Localized patterns in homogeneous networks of diffusively coupled reactors

We study the influence of network topology on instabilities of the homogeneous steady state of diffusively coupled, monostable nonlinear cells. A particular focus are diffusion-induced instabilities, i.e., Turing instabilities. We present various theorems that make it possible to determine analytically the stability properties of networks with arbitrary topologies and general monostable dynamics of the individual cells. This work aims in particular to determine those topologies that will give rise to localized stationary patterns. Specific examples focus on well-stirred chemical reactors. The reactors are coupled by diffusion-like mass transfer, and the kinetics is given by the Lengyel-Epstein model, a two-variable scheme for the chlorine dioxide-iodine-malonic acid reaction.