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.
Highlights
One of the central challenges in mathematical biology is understanding mechanisms involved in development processes
Our objective is to show how extrinsic noise influences spatio-temporal reaction–diffusion patterns, in particular by developing power spectral methods to analyse the impact of coloured noise
We investigated the effect of stochastic inflows on a deterministic reaction–diffusion system
Summary
One of the central challenges in mathematical biology is understanding mechanisms involved in development processes. The stochastic driving is often represented by stochastic parameters, that is a parameter which is drawn from a certain distribution at each time step or spatial point, and referred to as extrinsic noise below. This contrasts with most previous work on stochastic pattern formation, referred to as intrinsic noise, which assumes low copy number, and it does not assume external drivers as the source of noise (Schumacher et al 2013; McKane and Newman 2005; McKane et al 2014; Biancalani et al 2011; Woolley et al 2011a, 2012). Extrinsic noise has been studied extensively in García-Ojalvo and Sancho (2012), not in the context of chemical reaction network theory and reaction–diffusion systems
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