Turing instability and pattern formation on directed networks

TL;DR

This paper investigates pattern formation in reaction-diffusion systems on directed networks, especially focusing on cases where the Laplacian matrix is not diagonalizable, and extends the analysis to non-autonomous and temporal networks.

Contribution

It introduces methods to detect pattern formation on directed networks with non-diagonalizable Laplacians and generalizes results to non-autonomous and temporal networks.

Findings

Patterns can form on directed networks with non-diagonalizable Laplacians.

Global reaction kinetics influence pattern formation.

Temporal network topology affects the emergence of patterns.

Abstract

Pattern formation, arising from systems of autonomous reaction-diffusion equations, on networks has become a common topic of study in the scientific literature. In this work we focus primarily on directed networks. Although some work prior has been done to understand how patterns arise on directed networks, these works have restricted their attentions to networks for whom the Laplacian matrix (corresponding to the network) is diagonalizable. Here, we address the question "how does one detect pattern formation if the Laplacian matrix is not diagonalizable?" To this end, we find it is useful to also address the related problem of pattern formation arising from systems of reaction-diffusion equations with non-local (global) reaction kinetics. These results are then generalized to include non-autonomous systems as well as temporal networks, i.e., networks whose topology is allowed to change in time.