Reaction–diffusion equations on graphs: stationary states and Lyapunov functions

Reaction–diffusion equations on graphs (networks) serve as mathematical models of various phenomena in physics and biology. We study the existence of spatially heterogeneous stationary states, provided that the diffusion coefficients are sufficiently small. We provide an easily applicable criterion for determining which of them are nonnegative. Next, we consider the problem of constructing Lyapunov functions for reaction–diffusion equations on graphs, provided that a Lyapunov function for the corresponding non-diffusive system is known. We provide an easy-to-use result applicable in situations where the non-diffusive Lyapunov function is a sum of univariate functions with nondecreasing derivatives. The results are illustrated by means of several examples from mathematical biology.