How local spectral gaps regulate the multistability of Turing patterns on graphs

Abstract

Due to their self-organised, collective nature, Turing patterns on graphs are an important source of information about the relationship between graph architecture and dynamics. One defining feature of these dynamics is the coexistence of multiple stable patterns (‘pattern diversity’), the dependency of which on network architecture is still not well understood. Here we create standardised situations near the Turing instability threshold and study the multistability of patterns as a function of structural perturbations of the graph. In particular, we analyse the resulting changes in pattern diversity as a binary classification problem. We find an asymmetry between lower and higher eigenvalues near the Turing instability, which can be understood in terms of the interlacing theorem, known from spectral graph theory. This allows us to derive rules governing the multistability of Turing patterns using local spectral gaps of the graph’s Laplacian as input, but also evaluate the contribution of nonlinear interactions between eigenmodes to pattern diversity, independently of the interaction model considered.