Two-dimensional Reaction-diffusion System Research Articles

Two-dimensional model of a reaction-diffusion system as a typewriter.

Pattern formation is a common phenomenon, which appears in biological systems, especially in cell differentiation processes. The proper level for understanding the creation of patterns seems to be a physicochemical description. The most fundamental models should be based on systems, in which only chemical reactions and diffusion transport occur (reaction-diffusion systems). In order to present a richness of patterns, we show here the asymptotic patterns in the form of capital letters obtained in two-dimensional reaction-diffusion systems with zero-flux boundary conditions. All capital letters are obtained in the same model, but initial conditions and sizes of the systems are different for each letter. The chemical model consists of elementary reactions and is realistic. It can be realized experimentally in continuous-flow unstirred reactor with an enzymatic reaction allosterically inhibited by an excess of its reactant and product.

Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reaction-diffusion system.

Pattern formation in semiconductor heterostructures is studied on the basis of a spatially two-dimensional model of reaction-diffusion type. In particular, we investigate the neighborhood of a codimension-two Turing-Hopf instability by analytical methods. Amplitude equations are derived which predict the absence of mixed modes but extended ranges of bistability between homogeneous oscillatory states and hexagonal Turing patterns. Our results are confirmed by numerical simulations. The features are not confined to a neighborhood of the bifurcation point so that the conclusions of the weakly nonlinear analysis explain the observations in large portions of the parameter space at least qualitatively

Lie symmetries of nonlinear two-dimensional reaction-diffusion systems

A complete description of Lie symmetries is obtained for two-dimensional non-linear systems of two reaction-diffusion equations. Lie ansätze and exact solutions of a nonlinear reaction-diffusion system arising in mathematical biology are presented.

Explicit construction of integral manifolds with exponential tracking

A nonautonomous evolution equation in a Hilbert space is considered. A simple explicit construction of a finite-dimensional integral manifold with exponential tracking is proposed under the optimal spectral gap assumption. The dependence of the integral manifold on the initial surface is studied. Applications to one and two-dimensional nonautonomous reaction-diffusion systems are considered.

Interaction of self-organized quasiparticles in a two-dimensional reaction-diffusion system: The formation of molecules

In two-dimensional reaction-diffusion systems localized, solitary structures, that we call self-organized quasiparticles or spots, can be found as stable and stationary solutions. Combinations of two or more spots can lead to rather complex patterns, that can be understood by treating them as particles. These particles can interact with the boundaries of the system as well as with each other in different ways, that depend essentially on the parameters of the system. The interaction can be described by an approximation based on the exponential decay of the spots apart from their centers. The calculations reduce the dynamics of the system to some equations for the velocities of the spots. In particular, there is a parameter range where the interaction of two spots oscillates with their distance, which gives rise to infinitely many bounded states, resembling molecules. Investigating more than two spots molecules of numerous shapes have been obtained.

Boundary Effects on the Structural Stability of Stationary Patterns in a Bistable Reaction-Diffusion System

We study a piecewise linear version of a one-component, two-dimensional bistable reaction-diffusion system subjected to partially reflecting boundary conditions, with the aim of analyzing the structural stability of its stationary patterns. Dirichlet and Neumann boundary conditions are included as limiting cases. We find a critical line in the space of the parameters which divides different dynamical behaviors. That critical line merges as the locus of the coalescence of metastable and unstable nonuniform structures.

Scenarios of domain pattern formation in a reaction-diffusion system.

We performed an extensive numerical study of a two-dimensional reaction-diffusion system of the activator-inhibitor type in which domain patterns can form. We showed that both multidomain and labyrinthine patterns may form spontaneously as a result of Turing instability. In the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus. Depending on the parameters and the excitation level of the system stripes, spots, wriggled stripes, or labyrinthine patterns form. The labyrinthine patterns may be both connected and disconnected. In the the stable homogeneous system with the slow inhibitor one can excite self-replicating spots, breathing patterns, autowaves and turbulence. The parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101 (1996). The dynamics of the patterns observed in our simulations is very similar to that of the patterns forming in the ferrocyanide-iodate-sulfite reaction.

CLASS IV BEHAVIOR IN CELLULAR AUTOMATA MODELS OF PHYSICAL SYSTEMS

Wolfram’s cellular automata [1986] can be classified according to their asymptotic behavior: class I (homogeneous), class II (periodic), class III (chaotic) and class IV (“undecidable”, i.e. erratically changing between periodicity and chaos). While these automata are purely number-theoretical and suffer from ill-defined parametrization, we present here examples of automata describing actual physical systems and governed by well-defined control parameters: resting states between earthquakes, pigmentation on the shells of molluscs, and two-dimensional reaction–diffusion systems. We find that the dynamics of these three systems can be classified analogously to Wolfram’s automata. Moreover, we find agreement between class IV simulations and real shell patterns, indicating that these shells indeed present evidence for class IV behavior in nature. In addition, we performed an intuitively appealing quantification by averaging the fluctuations of the borders of error-propagation patterns.

Transition from stationary to traveling localized patterns in a two-dimensional reaction-diffusion system.

The bifurcation from localized stationary solutions to traveling patterns is studied in two-dimensional reaction-diffusion systems. In the case of local kinetics, which is of the Bonhoeffer--van der Pol type, the bifurcation point can be computed directly in terms of the stationary solution for arbitrary parameters. A stripelike pattern, which is the extension of a one-dimensional localized pattern into the second spatial dimension, is considered as a concrete example of such a pattern. The branching mode is computed for piecewise linear kinetics in a singular limit. The bifurcation is predominantly subcritical. Eventually the linear stability analysis of the branching traveling pattern is performed. Following the bifurcating branch, the last mode to become stable is a laterally wavy perturbation with an arbitrarily long wavelength.

Dimensional reduction of a bushfire model

Throw a match into a vegetation layer. What happens to the fire as time goes on? The spread from the ground to canopy, and along the terrain may be modelled by a simple two-dimensional reaction-diffusion system. We discuss the use of centre manifold theory to simplify a two-dimensional model of fire spread down to a one-dimensional model of fire propagation along the layer. This not only shows how to justify such simple one-dimensional models, it is also possible, for example, to determine the reaction wave speed analytically.

Impact of global interactions on patterns in a simple system

Numerical simulation by Pearson revealed a surprising variety of spatiotemporal patterns in a simple two-dimensional reaction–diffusion system. The mutual (or global) interaction between the overall reaction rate in a thin slab and the reactants concentrations in the surrounding vessel may have a strong impact on the dynamics of the system. It may lead to formation of patterns even when diffusion–reaction interactions cannot create patterns. Changes in this global interaction, which can be accomplished by changes in the reactant flow rate, may change the qualitative shape of the pattern and cause it to appear or disappear. Patterns of qualitatively different nature may be obtained for different initial conditions.

Nonlocal contour dynamics model for chemical front motion.

Pattern formation exhibited by a two-dimensional reaction-diffusion system in the fast inhibitor limit is considered from the point of view of interface motion. A dissipative nonlocal equation of motion for the boundary between high and low concentrations of the slow species is derived heuristically. Under these dynamics, a compact domain of high concentration may develop into a space-filling labyrinthine structure in which nearby fronts repel. Similar patterns have been observed recently by Lee, McCormick, Ouyang, and Swinney in a reacting chemical system.

Turbulence due to spiral breakup in a continuous excitable medium.

Excitable media are extended spatial systems, which support the propagation of waves including pulses and rotating spirals. They are well described by sets of partial differential equations involving a fast activator and a slow inhibitor variable. Here we show that spiral breakup, leading to turbulence, can occur in a two-dimensional reaction-diffusion system with delayed-inhibitor production. Upon a decrease of excitability, spirals become unstable because their wavelengths and periods are too short to be sustained in the system.

Interaction of Steady and Periodic Bifurcating Modes with Imperfection Effects in Reaction-Diffusion Systems

Asymptotic solutions for two-dimensional reaction-diffusion systems are obtained under suitable conditions. A trivial solution independent of both time and space is assumed to exist, and the linearized system about this solution is considered. The analysis is restricted to the case where this linearized system possesses one real eigenvalue and a complex pair of eigenvalues, all with real part approximately equal to zero. Steady, periodic and doubly periodic bifurcating solutions are found, with special attention given to the effects of imperfection in one of the parameters.