Abstract
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.
Highlights
This study suggests a new way of thinking about pattern formation, namely, in terms of mass redistribution that gives rise to moving local equilibria: A dissection of space into local compartments allows the spatiotemporal dynamics to be characterized on the basis of the ordinary differential equations (ODEs) phase space of local reactions
We develop a number of new theoretical concepts, exemplified by two-component mass-conserving reaction-diffusion (2C-MCRD) systems and based on simple geometric structures in the phase space of the reaction kinetics
We believe that the local equilibria theory we present here offers a new perspective on a broad class of pattern-forming systems— including intracellular pattern formation, classical chemical systems such as the BZ reaction, and even agent-based active matter systems
Summary
Nonlinear systems are as prevalent in nature as they are difficult to deal with conceptually and mathematically [1,2,3,4,5,6,7]. We go beyond this approach and gain physical insight into the global dynamics of spatially extended systems from the analysis of geometric objects in a low-dimensional phase space Such a theory should be able to explain both the dynamic process of pattern formation—initiated, for instance, by a lateral (Turing) instability—as well as the final stationary patterns in terms of the same concepts and principles. We develop a number of new theoretical concepts, exemplified by two-component mass-conserving reaction-diffusion (2C-MCRD) systems and based on simple geometric structures in the phase space of the reaction kinetics From these concepts, general geometric criteria for lateral (Turing) instability and stimulusinduced pattern formation emerge and allow us to obtain the features and bifurcations of patterns from graphical constructions. We believe that the local equilibria theory we present here offers a new perspective on a broad class of pattern-forming systems— including intracellular pattern formation, classical chemical systems such as the BZ reaction, and even agent-based active matter systems
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