Coloured Noise from Stochastic Inflows in Reaction–Diffusion Systems

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. * adamer@maths.ox.ac.uk

Bespoke Turing Systems

“…In the following, we will consider two continuous populations, u and v, termed "morphogens", which is a standard and general name for the populations whose interactions may generate patterning. Critically, we are intrinsically assuming that each population is made up of a large number of individuals, so as to allow a continuum description, although there has been extensive work done on low-copy number reaction-diffusion systems (Adamer et al 2020;Woolley et al 2012).…”

“…Critically, we are intrinsically assuming that each population is made up of a large number of individuals, so as to allow a continuum description, although there has been extensive work done on low-copy number reaction–diffusion systems (Adamer et al. 2020 ; Woolley et al. 2012 ).…”

Modern perspectives on near-equilibrium analysis of Turing systems

“…Stochastic modelling in mathematical biology, and reaction-diffusion systems in particular, has seen substantial development in the past few decades [204]. Such models have also been applied to study questions regarding pattern formation, both for intrinsic noise due to small number fluctuations (analytically and via Gillespie-type algorithms) [205][206][207], and for external noise, via random forcing at the level of partial differential equations [208]. While noise has been shown to be both stabilizing and destabilizing in different contexts, it can play an important role in helping to address some aspects of the robustness problems alluded to earlier in the deterministic theory [209].…”

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

“…Stochastic modelling in mathematical biology, and reaction-diffusion systems in particular, has seen substantial development in the past few decades [50]. Such models have also been applied to study questions regarding pattern formation, both for intrinsic noise due to small number fluctuations (analytically and via Gillespie-type algorithms) [170,205,206], and for external noise, via random forcing at the level of partial differential equations [1]. While noise has been shown to be both stabilizing and destabilizing in different contexts, it can play an important role in helping to address some aspects of the robustness problems alluded to earlier in the deterministic theory [131].…”