Two-dimensional Reaction-diffusion System Research Articles

The localized meshless method of lines for the approximation of two-dimensional reaction-diffusion system

The localized meshless method of lines for the approximation of two-dimensional reaction-diffusion system

Exponential Time Differencing for Stiff Systems with Nondiagonal Linear Part

Exponential time differencing methods provide instability-free explicit schemes for systems with fast decaying or oscillating modes (stiff systems), without limitation on the time step size. Moreover, with these methods, one can drastically diminish the error accumulation rate for numerical simulation of conservative systems. The methods yield an especially large performance gain for PDEs with high order of spatial derivatives. Simultaneously, the problem of analytical calculation of coefficients of exponential time differencing schemes becomes laborious or unsolvable in the case of a nondiagonal form of the principal linear part of equations. We introduce an approach, where the scheme coefficients are obtained from the direct numerical integration of certain auxiliary problems over a short time interval—one scheme step size. The approach is universal and its implementation is illustrated with four examples: analytically solvable system of two first-order ODEs, one-dimensional reaction-diffusion system under time-dependent conditions, two-dimensional reaction-diffusion system under time-independent and time-dependent conditions, and one-dimensional Cahn–Hilliard equation with constant coefficients. The employment of an exponential time differencing method of the two-step Runge–Kutta type yields a simulation performance gain for the diffusion-type equation, with program optimization made. Without program optimization, the performance gain increases by one order with respect to the spatial step size for each order of the highest spatial derivative, and appears starting from the third order of the derivative. With the tested method, one can extend the study of an analogue of the Anderson localization to two- and three-dimensional active media and achieve an acceptable performance for the direct numerical simulations of dynamics of the probability density function for active Brownian particles.

A novel time efficient structure-preserving splitting method for the solution of two-dimensional reaction-diffusion systems

In this article, the first part is concerned with the important questions related to the existence and uniqueness of solutions for nonlinear reaction-diffusion systems. Secondly, an efficient positivity-preserving operator splitting nonstandard finite difference scheme (NSFD) is designed for such a class of systems. The presented formulation is unconditionally stable as well as implicit in nature and even time efficient. The proposed NSFD operator splitting technique also preserves all the important properties possessed by continuous systems like positivity, convergence to the fixed points of the system, and boundedness. The proposed algorithm is implicit in nature but more efficient in time than the extensively used Euler method.

Exponential time differencing for stiff systems with nondiagonal linear part

Exponential time differencing methods provide instability-free explicit schemes for systems with fast decaying or oscillating modes (stiff systems), without limitation on the time step size. Moreover, with these methods, one can drastically diminish the error accumulation rate for numerical simulation of conservative systems. The methods yield an especially large performance gain for PDEs with high order of spatial derivatives. Simultaneously, the problem of analytical calculation of coefficients of exponential time differencing schemes becomes laborious or unsolvable in the case of a nondiagonal form of the principal linear part of equations. We introduce an approach, where the scheme coefficients are obtained from the direct numerical integration of certain auxiliary problems over a short time interval—one scheme step size. The approach is universal and its implementation is illustrated with four examples: analytically solvable system of two first-order ODEs, one-dimensional reaction-diffusion system under time-dependent conditions, two-dimensional reaction-diffusion system under time-independent and time-dependent conditions, and one-dimensional Cahn–Hilliard equation with constant coefficients. The employment of an exponential time differencing method of the two-step Runge–Kutta type yields a simulation performance gain for the diffusion-type equation, with program optimization made. Without program optimization, the performance gain increases by one order with respect to the spatial step size for each order of the highest spatial derivative, and appears starting from the third order of the derivative. With the tested method, one can extend the study of an analogue of the Anderson localization to two- and three-dimensional active media and achieve an acceptable performance for the direct numerical simulations of dynamics of the probability density function for active Brownian particles.

Control of transversal instabilities in reaction-diffusion systems

In two-dimensional reaction-diffusion systems, local curvature perturbations on traveling waves are typically damped out and vanish. However, if the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated waves and to spreading spiral turbulence. Here, we propose a scheme to induce or inhibit these instabilities via a spatio-temporal feedback loop. In a piecewise-linear version of the FitzHugh–Nagumo model, transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilizing plane wave propagation. Conversely, in numerical simulations with the modified Oregonator model for the photosensitive Belousov–Zhabotinsky reaction, which does not exhibit transversal instabilities on its own, we demonstrate the feasibility of inducing transversal instabilities and study the emerging wave patterns in a well-controlled manner.

On Immovable Boundary between Coexisting Solutions and Synchronization in a Reaction–Diffusion System

In this work, the properties of several types of asymptotic solutions of a two-dimensional reaction–diffusion system with the FitzHugh–Nagumo model were studied. When several asymptotic solutions coexist in a parameter region around the bifurcation point, there is a possibility of producing the locally connecting bistable solution (LCBS). The dynamic LCBS (d-LCBS) obtained in the present study consists of two domains, the static stripe pattern and the periodically fluctuating one. It is clarified that the nearly equal stability of two global asymptotic solutions and the complete synchronization at the boundary are needed to produce the d-LCBSs. Furthermore, the dependence of the stability of solutions on the extent of the computing domain was also investigated.

A Model for Selection of Eyespots on Butterfly Wings.

Unsolved ProblemThe development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not.Key Idea and ModelWe illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature.ResultWe therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.

Superdiffusive cusp-like waves in the mercuric iodide precipitate system and their transition to regular reaction bands.

We report a two-dimensional (2D) reaction-diffusion system that exhibits a superdiffusive propagating wave with anomalous cusp-like contours. This wave results from a leading precipitation reaction (wavefront) and a trailing redissolution (waveback) between initially separated mercuric chloride and potassium iodide to produce mercuric iodide precipitate (HgI2) in a thin sheet of a solid hydrogel (agar) medium. The propagation dynamics is accompanied by continuous polymorphic transformations between the metastable yellow crystals and the stable red crystals of HgI2. We study the dynamics of wavefront and waveback propagation that reveals interesting anomalous superdiffusive behavior without the influence of external enhancement. We find that a transition from superdiffusive to subdiffusive dynamics occurs as a function of outer iodide concentration. Inner mercuric concentrations lead to the transition from the anomalous cusp-like to cusp-free regular bands. While gel concentration affects the speed of propagation of the wave, it has no effect on its shape or on its superdiffusive dynamics. Microscopically, we show that the macroscopic wave propagation and polymorphic transformations are accompanied by an Ostwald ripening mechanism in which larger red HgI2 crystals are formed at the expense of smaller yellow HgI2 crystals.

Exact Solutions for Pattern Formation in a Reaction-Diffusion System

Abstract Reaction-diffusion equations are ubiquitous as models of pattern formation in chemistry and biology. In this paper, the invariant subspace method is used to investigate exact solutions and the corresponding patterns in a reaction-diffusion system. Several forms of the exact solutions of the one-dimensional reaction-diffusion system are presented, including complexions. Furthermore, we show several patterns the two-dimensional reaction-diffusion system, involving stripes, spots and transitions between them. Additionally, we discuss the relations between patterns and exact solutions.

Turing Instability of a Chemical System with Reactants Binding to a Substrate

Relevance of Turing mechanism for biology has been questioned due to not allowing pattern formation when the diffusion constants of the two reacting chemicals are identical. The idea that binding of the activator to a substrate may effectively reduce its diffusion rate and thus destabilize a system that would otherwise be stable was formulated in an article by Lengyel and Epstein [2], where the authors reduce the original system of three linear partial differential equations to a two-dimensional reaction-diffusion system that they analyse. We question relevance of this analysis due to lack of connection between the original and the reduced model and suggest that analysing the reduced model actually does not yield any possibilities beyond the standard setting. Nevertheless, our analysis of the three dimensional system shows that one can indeed relax the standard conditions on diffusion constants that are necessary for Turing instability, in particular allow identical diffusion coefficients. Another question that we raise is whether one can relax the condition on the effective diffusion rates in the three- or four-dimensional model. This idea is supported by a previous result of Klika, Baker, Headon and Gaffney [1] that one can significantly reduce the necessary conditions for Turing instability by adding more reactants into the system. Furthermore, the approach of studying the full four-dimensional system allows relaxing the kinetic contraints, for example by permitting two activators to generate a pattern. [1] V. Klika, R. E. Baker, D. Headon, E. A. Gaffney, The Influence of Receptor-Mediated Interactions on Reaction-Diffusion Mechanisms of Cellular Self-organisation , Bull Math Biol (2012), 74 935--957. [2] I. Lengyel, I. R. Epstein, A Chemical Approach to Designing Turing Patterns in Reaction-Diffusion Systems , Proc. Natl. Acad. Sci. USA (1992) 89 3977--3979.

Information processing capacity of dynamical systems.

Many dynamical systems, both natural and artificial, are stimulated by time dependent external signals, somehow processing the information contained therein. We demonstrate how to quantify the different modes in which information can be processed by such systems and combine them to define the computational capacity of a dynamical system. This is bounded by the number of linearly independent state variables of the dynamical system, equaling it if the system obeys the fading memory condition. It can be interpreted as the total number of linearly independent functions of its stimuli the system can compute. Our theory combines concepts from machine learning (reservoir computing), system modeling, stochastic processes, and functional analysis. We illustrate our theory by numerical simulations for the logistic map, a recurrent neural network, and a two-dimensional reaction diffusion system, uncovering universal trade-offs between the non-linearity of the computation and the system's short-term memory.

Size-normalized robustness of Dpp gradient in Drosophila wing imaginal disc

Exogenous environmental changes are known to affect the intrinsic characteristics of biological organizms. For instance, the synthesis rate of the morphogen decapentaplegic (Dpp) in a Drosophila wing imaginal disc has been found to double with an increase of 5.9°C in ambient temprerature. If not compensated, such a change would alter the signaling Dpp gradient significantly and thereby the development of thewing imaginal disc. To learn how flies continue to develop "normally" under such an exogenous change, we formulate in this paper a spatially two-dimensional reaction-diffusion system of partial differential equations (PDE) that accounts for the biological processes at work in the Drosophila wing disc essential for the formation of signaling Dpp gradient. By way of this PDE model, we investigate the effect of the apical-basal thickness and antero-posterior span of the wing on the shape of signaling gradients and the robustness of wing development in an altered environment (including an enhanced morphogen synthesis rate). Our principal result is a delineation of the role of wing disc size change in maintaining the magnitude and shape of the signaling Dpp gradient. The result provides a theoretical basis for the observed robustness of wing development, preserving relative but not absolute tissue pattern, when the morphogen synthesis rate is significantly altered. A similar robustness considerqation for simultaneous changes of multiple intrinsic system characteristics is also discussed briefly.

Adaptive Multiresolution Methods for the Simulation of Waves in Excitable Media

We present fully adaptive multiresolution methods for a class of spatially two-dimensional reaction-diffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley’s approach (Barkley in Phys. D 49:61–70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform grid.

Refraction of reaction-diffusion plane wave for different diffusion coefficients

The effect of diffusion on the refraction of plane wave in two-dimensional reaction-diffusion system described by complex Ginzburg-Landau equation are numerically investigated. We derive the refractive index of plane wave from Snell's law. The numerical and theoretic results show that the refraction of plane wave in a simple purely diffusive system obeys Snell's law, i.e, diffusion only impacts the refractive index of plane wave. However, the refraction of plane wave in reaction-diffusion system obeys Snell's law only for proper diffusion coefficients and system parameters. These results show that diffusion impacts the refraction law and refractive index.

Flow-induced symmetry reduction in two-dimensional reaction–diffusion system

The influence of uniform flow on the pattern formation is investigated in a two-dimensional reaction–diffusion system. It is found that the convective flow plays a key role on pattern modulation. Both traveling and stationary periodic patterns are obtained. At moderate flow rates, the perfect hexagon, phase-shifted hexagon and stable square, which are essentially unstable in unperturbed reaction–diffusion systems, are obtained. These patterns move downstream. If the flow rate is increased further, the stationary flow-oriented stripes develop and compete with the spots. If the flow rate exceeds some critical value, the system is convectively unstable and the stationary stripes prevail against the traveling spots. The above patterns all have the same critical wavenumber associated with Turing bifurcation, which indicates that Turing instability produces the patterns while the flow induces the symmetry reduction, i.e., from six-fold symmetry to four-fold one, and to two-fold one ultimately.

Entropy production in a two-dimensional reversible Gray-Scott system

The entropy production sigma is calculated in the time evolution processes toward a Turing-like pattern and a chaotic pattern in a two-dimensional reaction-diffusion system. The contributions of reaction and diffusion to the entropy production are evaluated separately. Though its contribution to total sigma is about 5%, the entropy production in diffusion foretells the moving direction of the dots (reaction spots) and the line-shaped patterns. The entropy production of the entire system sigma depicts well the cooperative dynamics and evolution of chaotic dot patterns. It is suggested that sigma can be a scalar measure for quantitative studies of hierarchic pattern dynamics. The relation is also discussed between the bifurcation parameter and the distance from thermodynamic equilibrium.

Pattern formation in a two-dimensional reaction-diffusion channel with Poiseuille flow

We study stationary patterns arising from a combination of flow and diffusion in a two-dimensional (2D) reaction-diffusion system in a channel with Poiseuille flow. Both transverse and longitudinal modes are investigated and compared with numerical computations.

Spiral-wave meandering in reaction–diffusion models of ventricular muscle

Reentrant arrhythmias in cardiac tissue can be idealised as spiral-wave solutions of reaction–diffusion equations of excitable media. The dynamical behaviour of such solutions depends on the properties of the excitable medium. Here we characterise the effects of model-parameter changes for ionic conductances and concentrations on the meandering patterns and dynamics of spiral waves in a model of mammalian ventricular tissue. A two-dimensional reaction–diffusion system with kinetics described by the Oxsoft ® Heart equations for the single guinea-pig ventricular cell was used. We examine the development and stability of reentrant spiral-wave solutions as well as the effects of ionic conductance and concentration changes on spiral-wave meandering. Spiral-wave meandering behaviour provides a significant validation test for detailed biophysical models.

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