  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 local equilibria theory 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 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.
