Master stability functions reveal diffusion-driven pattern formation in networks

TL;DR

This paper introduces a master stability function approach to analyze diffusion-driven pattern formation in multilayer networks, revealing deep analogies with continuous space patterns and demonstrating its utility in complex ecological models.

Contribution

It develops a novel master stability function method for multilayer networks, enabling analysis of pattern formation in complex, reaction-diffusion systems across networked habitats.

Findings

The approach uncovers how spatial structure influences ecological dynamics.

It demonstrates the method's effectiveness on a complex predator-prey meta-foodweb model.

The technique bridges network theory and pattern formation analysis.

Abstract

We study diffusion-driven pattern-formation in networks of networks, a class of multilayer systems, where different layers have the same topology, but different internal dynamics. Agents are assumed to disperse within a layer by undergoing random walks, while they can be created or destroyed by reactions between or within a layer. We show that the stability of homogeneous steady states can be analyzed with a master stability function approach that reveals a deep analogy between pattern formation in networks and pattern formation in continuous space.For illustration we consider a generalized model of ecological meta-foodwebs. This fairly complex model describes the dispersal of many different species across a region consisting of a network of individual habitats while subject to realistic, nonlinear predator-prey interactions. In this example the method reveals the intricate dependence of the dynamics on the spatial structure. The ability of the proposed approach to deal with this fairly complex system highlights it as a promising tool for ecology and other applications.