Coloured Noise from Stochastic Inflows in Reaction-Diffusion Systems

TL;DR

This paper develops a framework to analyze coloured noise in reaction-diffusion systems driven by extrinsic fluctuations, using algebraic and spectral methods to understand their steady states and pattern formation.

Contribution

It introduces a parameter-free algebraic approach to identify reaction systems with unique steady states under coloured noise influence.

Findings

Reaction systems with extrinsic noise can be characterized for unique steady states.

Power spectral methods can predict patterns driven by coloured noise.

The spectral relationship between coloured and white noise systems is established.

Abstract

In this paper we present a framework for investigating coloured noise in reaction-diffusion systems. We start by considering a deterministic reaction-diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases we show that the power spectrum of the coloured noise process and the power spectrum of the reaction-diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction-diffusion system.