Eur. Phys. J. B (2013) 86: 476
https://doi.org/10.1140/epjb/e2013-40570-8

Regular Article

The linear noise approximation for reaction-diffusion systems on networks

¹ Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy
² Theoretical Physics, School of Physics and Astronomy, University of Manchester, M13 9 PL Manchester, UK
³ Dipartimento di Fisica e Astronomia, Università di Firenze and INFN, via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy

^a e-mail: duccio.fanelli@gmail.com

Received: 14 June 2013
Received in final form: 25 September 2013
Published online: 18 November 2013

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 travelling 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.