Experimental studies of protein-pattern formation have stimulated new interest in the dynamics of reaction–diffusion systems. However, a comprehensive theoretical understanding of the dynamics of such highly nonlinear, spatially extended systems is still missing. Here we show how a description in phase space, which has proven invaluable in shaping our intuition about the dynamics of nonlinear ordinary differential equations, can be generalized to mass-conserving reaction–diffusion (McRD) systems. We present a comprehensive analysis of two-component McRD systems, which serve as paradigmatic minimal systems that encapsulate the core principles and concepts of the introduced in the paper. The key insight underlying this theory is that shifting local (reactive) equilibria—controlled by the local total density—give rise to concentration gradients that drive diffusive redistribution of total density. We show how this dynamic interplay can be embedded in the phase plane of the reaction kinetics in terms of simple geometric objects: the reactive nullcline (line of reactive equilibria) and the diffusive flux-balance subspace. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. The effects of nonlinearities on the global dynamics are simply encoded in the curved shape of the reactive nullcline. In particular, we show that the pattern-forming `Turing instability’ in McRD systems is a mass-redistribution instability, and that the features and bifurcations of patterns can be characterized based on regional dispersion relations, associated to distinct spatial regions (plateaus and interfaces) of the patterns. In an extensive outlook section, we detail concrete approaches to generalize local equilibria theory in several directions, including systems with more than two-components, weakly-broken mass conservation, and active matter systems.

中文翻译：

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质量守恒反应扩散动力学的相空间几何

蛋白质模式形成的实验研究激发了对反应扩散系统动力学的新兴趣。但是，仍然缺少对这种高度非线性的，空间扩展的系统的动力学的全面理论理解。在这里，我们展示了如何将相空间中的描述推广到质量守恒的反应扩散系统（McRD），该描述在塑造我们对非线性常微分方程动力学的直觉上具有不可估量的价值。我们对两部分McRD系统进行了全面的分析，它们是作为范式最小系统，包含了本文介绍的核心原理和概念。该理论的主要见解是，由局部总密度控制的局部（反应性）平衡移动会导致浓度梯度上升，从而驱动总密度的扩散性重新分布。我们展示了如何通过简单的几何对象将这种动态相互作用嵌入反应动力学的相平面中：反应零位线（反应平衡线）和扩散通量平衡子空间。在此相空间级别上，可以从几何准则和图形构造中获得物理洞察力。非线性对全局动力学的影响可以简单地编码为反应零曲线的弯曲形状。特别是，我们表明McRD系统中的模式形成“ Turing不稳定性”是质量重新分布不稳定性，并且可以基于与图案的不同空间区域（高原和界面）相关联的区域色散关系来表征图案的特征和分支。在广泛的展望部分中，我们详细介绍了在多个方向上推广局部均衡理论的具体方法，包括具有两个以上成分的系统，质量守恒性较弱的系统以及活性物质系统。