Pattern formation in a two-component reaction-diffusion system with delayed processes on a network

TL;DR

This paper analyzes how time delays and network topology influence pattern formation in reaction-diffusion systems, providing explicit conditions for Turing and wave patterns using the Lambert W-function.

Contribution

It introduces an analytical framework using Lambert W-function to determine pattern onset in delayed reaction-diffusion systems on networks, surpassing previous single-species and non-delayed models.

Findings

Explicit conditions for Turing pattern onset

Delay and topology influence pattern type

Numerical validation with Mimura-Murray model

Abstract

Reaction-diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator-inhibitor variant without delay. Numerical results gained from the Mimura-Murray model support the theoretical approach.