Pattern Formation on Networks with Reactions: A Continuous Time Random Walk Approach

TL;DR

This paper develops a framework for reaction-diffusion processes on networks using continuous time random walks, revealing how network structure influences pattern formation through derived Laplacians and Turing instability analysis.

Contribution

It introduces a novel derivation of the reaction-diffusion master equation on networks from CTRWs, incorporating reactions at the single-particle level and analyzing pattern formation mechanisms.

Findings

Pattern formation depends on the network's Laplacian structure.

Symmetric Laplacians lead to Turing instability-driven patterns.

Nonsymmetric Laplacians can produce patterns with or without Turing instability.

Abstract

We derive the generalized master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). The non-trivial incorporation of the reaction process into the CTRW is achieved by splitting the derivation into two stages. The reactions are treated as birth-death processes and the first stage of the derivation is at the single particle level, taking into account the death process, whilst the second stage considers an ensemble of these particles including the birth process. Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non \emph{ad-hoc} manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases; one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in nonsymmetric and symmetric CTRW Laplacians respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in nonsymmetric Laplacians, pattern formation may be possible with or without a Turing instability.