Cross-diffusion induced instability on networks

The concept of Turing instability, namely that diffusion can destabilize the homogenous steady state, is well known either in the context of partial differential equations (PDEs) or in networks of dynamical systems. Recently, reaction–diffusion equations with non-linear cross-diffusion terms have been investigated, showing an analogous effect called cross-diffusion induced instability. In this article, we consider non-linear cross-diffusion effects on networks of dynamical systems, showing that also in this framework the spectrum of the graph Laplacian determines the instability appearance, as well as the spectrum of the Laplace operator in reaction–diffusion equations. We extend to network dynamics a particular network model for competing species, coming from the PDEs context, for which the non-linear cross-diffusion terms have been justified, e.g. via a fast-reaction limit. In particular, the influence of different topology structures on the cross-diffusion induced instability is highlighted, considering regular rings and lattices, and also small-world, Erdős–Réyni, and Barabási–Albert networks.