Reaction–diffusion equations on graphs (networks) serve as mathematical models of various phenomena in physics and biology. We study the existence of spatially heterogeneous stationary states, provided that the diffusion coefficients are sufficiently small. We provide an easily applicable criterion for determining which of them are nonnegative. Next, we consider the problem of constructing Lyapunov functions for reaction–diffusion equations on graphs, provided that a Lyapunov function for the corresponding non-diffusive system is known. We provide an easy-to-use result applicable in situations where the non-diffusive Lyapunov function is a sum of univariate functions with nondecreasing derivatives. The results are illustrated by means of several examples from mathematical biology.

图（网络）上的反应扩散方程用作物理学和生物学中各种现象的数学模型。我们研究空间异质静止状态的存在，前提是扩散系数足够小。我们提供了一个易于应用的标准来确定哪些是非负的。接下来，我们考虑在图上构造反应扩散方程的李雅普诺夫函数的问题，前提是相应非扩散系统的李雅普诺夫函数是已知的。我们提供了一个易于使用的结果，适用于非扩散李雅普诺夫函数是具有非递减导数的单变量函数之和的情况。结果通过数学生物学的几个例子来说明。