Delay induced Turing-like waves for one species reaction-diffusion model on a network

TL;DR

This paper investigates how time delays in a single-species reaction-diffusion model on complex networks induce Turing-like traveling waves, with theoretical predictions confirmed by numerical simulations.

Contribution

It introduces a mathematical framework using Lambert W-function to analyze delay-induced wave phenomena in reaction-diffusion systems on networks.

Findings

Delay causes symmetry-breaking instability leading to wave formation.

Theoretical conditions for instability are derived and validated.

Numerical simulations confirm the emergence of Turing-like waves in the model.

Abstract

A one species time-delay reaction-diffusion system defined on a complex networks is studied. Travelling waves are predicted to occur as follows a symmetry breaking instability of an homogenous stationary stable solution, subject to an external non homogenous perturbation. These are generalized Turing-like waves that materialize in a single species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time delayed differential equation. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz networks and with the inclusion of the delay.