  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.
