The linear noise approximation for reaction-diffusion systems on networks

TL;DR

This paper applies the linear noise approximation to stochastic reaction-diffusion models on networks, revealing how fluctuations can induce stochastic waves beyond deterministic predictions, supported by simulations and spectral analysis.

Contribution

It demonstrates the effectiveness of the linear noise approximation in analyzing stochastic reaction-diffusion systems on networks and introduces a spectral method for analyzing fluctuations.

Findings

Stochastic fluctuations can induce waves beyond deterministic instability regions.

The spectral method accurately predicts fluctuation peaks in simulations.

Simulations confirm theoretical predictions of stochastic wave emergence.

Abstract

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays traveling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.