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.

中文翻译：

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反应扩散系统中随机流入的有色噪声

在本文中，我们提出了一个研究反应扩散系统中的有色噪声的框架。我们首先考虑确定性的反应-扩散方程，并展示外力如何导致时间相关或有色噪声。在这里，外部噪声的主要来源被认为是代表粒子流入系统的参数值的波动。首先，我们确定哪些由外在噪声驱动的反应系统只能允许一个稳态，因此我们的分析排除了随机切换等影响。为了分析反应系统的稳态行为，即使参数值发生变化，也需要一种无参数方法，这在化学反应网络理论中一直是代数分析的核心。为了确定合适的模型，我们使用来自真实代数几何的工具，将网络结构与其动态特性联系起来。然后，我们与内部噪声模型建立联系，并展示如何使用功率谱方法来预测具有有色噪声的系统中的随机驱动模式。在简单的情况下，我们展示了有色噪声过程的功率谱和用白噪声建模的反应-扩散系统的功率谱相乘以给出有色噪声反应-扩散系统的功率谱。