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We revisit the problem of pinning a reaction-diffusion front by a defect, in particular by a reaction-free region. Using collective variables for the front and numerical simulations, we compare the behaviors of a bistable and monostable front. A bistable front can be pinned as confirmed by a pinning criterion, the analysis of the time independant problem and simulations. Conversely, a monostable front can never be pinned, it gives rise to a secondary pulse past the defect and we calculate the time this pulse takes to appear. These radically different behaviors of bistable and monostable fronts raise issues for modelers in particular areas of biology, as for example, the study of tumor growth in the presence of different tissues.
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Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of `open reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization, and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions, and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain, and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions, and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.
We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A-priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that in the original norm, initially small perturbations to the front remain bounded, while in the exponentially weighted norm, they algebraically decay in time.
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.
Wavelength selection in reaction--diffusion systems can be understood as a coarsening process that is interrupted by counteracting processes at certain wavelengths. We first show that coarsening in mass-conserving systems is driven by self-amplifying mass transport between neighboring high-density domains. We derive a general coarsening criterion and show that coarsening is generically uninterrupted in two-component systems that conserve mass. The theory is then generalized to study interrupted coarsening and anti-coarsening due to weakly-broken mass conservation, providing a general path to analyze wavelength selection in pattern formation far from equilibrium.
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.
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