Discrete Reaction-diffusion Systems Research Articles

ANALYTICAL SINGULAR VALUE DECOMPOSITION FOR A CLASS OF STOICHIOMETRY MATRICES.

We present the analytical singular value decomposition of the stoichiometry matrix for a spatially discrete reaction-diffusion system. The motivation for this work is to develop a matrix decomposition that can reveal hidden spatial flux patterns of chemical reactions. We consider a 1D domain with two subregions sharing a single common boundary. Each of the subregions is further partitioned into a finite number of compartments. Chemical reactions can occur within a compartment, whereas diffusion is represented as movement between adjacent compartments. Inspired by biology, we study both (1) the case where the reactions on each side of the boundary are different and only certain species diffuse across the boundary and (2) the case where reactions and diffusion are spatially homogeneous. We write the stoichiometry matrix for these two classes of systems using a Kronecker product formulation. For the first scenario, we apply linear perturbation theory to derive an approximate singular value decomposition in the limit as diffusion becomes much faster than reactions. For the second scenario, we derive an exact analytical singular value decomposition for all relative diffusion and reaction time scales. By writing the stoichiometry matrix using Kronecker products, we show that the singular vectors and values can also be written concisely using Kronecker products. Ultimately, we find that the singular value decomposition of the reaction-diffusion stoichiometry matrix depends on the singular value decompositions of smaller matrices. These smaller matrices represent modified versions of the reaction-only stoichiometry matrices and the analytically known diffusion-only stoichiometry matrix. Lastly, we present the singular value decomposition of the model for the Calvin cycle in cyanobacteria and demonstrate the accuracy of our formulation. The MATLAB code, available at www.github.com/MathBioCU/ReacDiffStoicSVD, provides routines for efficiently calculating the SVD for a given reaction network on a 1D spatial domain.

Propagation failure in discrete reaction-diffusion system based on the butterfly bifurcation.

Reaction-diffusion systems are used in biology, chemistry, and physics to model the interaction of spatially distributed species. Particularly of interest is the spatial replacement of one equilibrium state by another, depicted as traveling waves or fronts. Their profiles and traveling velocity depend on the nonlinearities in the reaction term and on spatial diffusion. If the reaction occurs at regularly spaced points, the velocities also depend on lattice structures and the orientation of the traveling front. Interestingly, there is a wide region of parameters where the speeds become zero and the fronts do not propagate. In this paper, we focus on systems with three stable coexisting equilibrium states that are described by the butterfly bifurcation and study to what extent the three possible 1D traveling fronts suffer from propagation failure. We demonstrate that discreteness of space affects the three fronts differently. Regions of propagation failure add a new layer of complexity to the butterfly diagram. The analysis is extended to planar fronts traveling through different orientations in regular 2D lattices. Both propagation failure and the existence of preferred orientations play a role in the transient and long-time evolution of 2D patterns.

BOUNDEDNESS OF A CLASS OF SPATIALLY DISCRETE REACTION-DIFFUSION SYSTEMS.

Although the spatially discrete reaction-diffusion equation is often used to describe biological processes, the effect of diffusion in this framework is not fully understood. In the spatially continuous case, the incorporation of diffusion can cause blow-up with respect to the norm, and criteria exist to determine whether the system is bounded for all time. However, no equivalent criteria exist for the discrete reaction-diffusion system. Due to the possible dynamical differences between these two system types and the advantage of using the spatially discrete representation to describe biological processes, it is worth examining the discrete system independently of the continuous system. Therefore, the focus of this paper is on determining sufficient conditions to guarantee that the discrete reaction-diffusion system is bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and nonnegative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discrete reaction-diffusion system is bounded. These results are considered in the context of four example systems for which Lyapunov-like functions can and cannot be found.

Kardar–Parisi–Zhang universality class for the critical dynamics of reaction–diffusion fronts

We have studied front dynamics for the discrete reaction–diffusion system, which in the continuum is described by the (stochastic) Fisher–Kolmogorov–Petrovsky–Piscunov equation. We have revisited this discrete model in two space dimensions by means of extensive numerical simulations and an improved analysis of the time evolution of the interface separating the stable and unstable phases. In particular, we have measured the full set of scaling exponents which characterize the spatio-temporal fluctuations of such front for different lattice sizes, focusing mainly in the front width and correlation length. These exponents are in very good agreement with those computed in (Moro E 2001 Phys. Rev. Lett. 87 238303) and correspond to those of the Kardar–Parisi–Zhang (KPZ) universality class for one-dimensional interfaces. Furthermore, we have studied the one-point statistics and the covariance of rescaled front fluctuations, which had remained thus far unexplored in the literature and allows for a further stringent test of KPZ universality.

Stability analysis of traveling wave solutions for lattice reaction-diffusion equations

In this work, we establish a framework to study the stability of traveling wave solutions for some lattice reaction-diffusion equations. The systems arise from epidemic, biological and many other applied models. Applying different kinds of comparison theorems, we show that all solutions of the Cauchy problem for the lattice differential equations converge exponentially to the traveling wave solutions provided that the initial perturbations around the traveling wave solutions belonging to suitable spaces. Our results can be applied to various discrete reaction-diffusion systems, e.g., the discrete multi-species Lotka-Volterra cooperative model, discrete epidemic model, three-species Lotka-Volterra competitive model, etc.

Travelling corners for spatially discrete reaction-diffusion systems

We consider reaction-diffusion equations on the planar square lattice that admit spectrally stable planar travelling wave solutions. We show that these solutions can be continued into a branch of travelling corners. As an example, we consider the monochromatic and bichromatic Nagumo lattice differential equation and show that both systems exhibit interior and exterior corners.Our result is valid in the setting where the group velocity is zero. In this case, the equations for the corner can be written as a difference equation posed on an appropriate Hilbert space. Using a non-standard global center manifold reduction, we recover a two-component difference equation that describes the behaviour of solutions that bifurcate off the planar travelling wave. The main technical complication is the lack of regularity caused by the spatial discreteness, which prevents the symmetry group from being factored out in a standard fashion.

Rapidly forming, slowly evolving, spatial patterns from quasi-cycle Mexican Hat coupling.

Alattice-indexed familyof stochasticprocesses hasquasi-cycle oscillationsif itsotherwise-damped oscillations are sustained by noise. Such a family performs the reaction part of a discrete stochastic reaction-diffusion system when we insert a local Mexican Hat-type, difference of Gaussians, coupling on a one-dimensional and on a two-dimensional lattice. Quasi-cycles are a proposed mech-anism for the production of neural oscillations, and Mexican Hat coupling is ubiquitous in the brain. Thus this combination might provide insight into the function of neural oscillations in the brain. Im-portantly, we study this system only in the transient case, on time intervals before saturation occurs. In one dimension, for weak coupling, we find that the phases of the coupled quasi-cycles synchronize (es-tablish a relatively constant relationship, or phase lock) rapidly at coupling strengths lower than those required to produce spatial patterns of their amplitudes. In two dimensions the amplitude patterns form more quickly, but there remain parameter regimes in which phase synchronization patterns form with-out being accompanied by clear amplitude patterns. At higher coupling strengths we find patterns both of phase synchronization and of amplitude (resembling Turing patterns) corresponding to the patterns of phase synchronization. Specific properties of these patterns are controlled by the parameters of the reaction and of the Mexican Hat coupling.

Effect of a modified sinusoidal forcing on spiral wave in a simulated reaction-diffusion system.

Spiral waves are often found in excitable media. In the hearts, they are abnormal forms of action potential propagation. Under an external forcing, the spiral waves drift and are subsequently terminated at the boundary. Spiral waves can be studied in simulations using a discrete reaction-diffusion system; thereby the time step must not exceed a numerical stability limit (ts). In this article, we present the dynamics of spiral waves in a simulated system under an external forcing as a modified sinusoidal function of time. The spiral waves are forced to drift along a straight line with a velocity and an angle depending on the time step. An optimization study provides the optimal time step of 0.2ts, where further reductions of the time step do not alter the drifting of the spiral waves.

Temporally Discrete Three-species Lotka-Volterra Competitive Systems with Time Delays

In this paper, we investigate the existence of traveling wave solutions for three components temporally discrete reaction-diffusion systems with delays by using the cross iteration method and Schauder's fixed point theorem. The obtained results are well applied to a temporally discrete three-species Lotka-Volterra competitive systems with delays.

Spiral patterns near Turing instability in a discrete reaction diffusion system

In this paper, linear stability analysis is applied to an exponential discrete Lotka–Volterra system, which describes the competition between two identical species. Conditions for the Turing instability are obtained and the emergence of spiral patterns is demonstrated by means of numerical simulations in the vicinity of the bifurcation point. Moreover, the impact of crucial system parameters on the stability and coherence of spiral patterns is illustrated on several examples.

Traveling wave solutions in temporally discrete reaction‐diffusion systems with delays

AbstractAn idea used by W. Li et al. (Nonlinearity 19, 1253–73, 2006) is extended to show existence of traveling wave solutions for a class of temporally discrete reaction‐diffusion systems with delays. The result is an generalization of an existing result for scalar equations to systems. As examples, the obtained abstract results will be applied to temporally discrete diffusion‐competition systems with delay. Our results suggest temporally discrete diffusion‐competition systems are dynamically consistent with the differential diffusion‐competition systems.

Switch and template pattern formation in a discrete reaction-diffusion system inspired by the Drosophila eye.

We examine a spatially discrete reaction-diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is known to be capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel "switch and template" mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly, provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation.

Explosive behavior in spatially discrete reaction-diffusion systems

Explosive instabilities in spatially discrete reaction-diffusion systems are studied. We identify classes of initial data developing singularities in finite time and obtain predictions of the blow-up times, whose accuracy is checked by comparison with numerical solutions. We present averaged and local blow-up estimates. Local blow-up results show that it is possible to have blow-up after blow-up. Conditions excluding or implying blow-up at space infinity are discussed.

UNIFORM ATTRACTORS OF NONAUTONOMOUS DISCRETE REACTION–DIFFUSION SYSTEMS IN WEIGHTED SPACES

We study the asymptotic behavior of nonautonomous discrete Reaction–Diffusion systems defined on multidimensional infinite lattices. We show that the nonautonomous systems possess uniform attractors which attract all solutions uniformly with respect to the translations of external terms when time goes to infinity. These attractors are compact subsets of weighted spaces, and contain all bounded solutions of the system. The upper semicontinuity of the uniform attractors is established when an infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems. We also examine the limiting behavior of lattice systems with almost periodic, rapidly oscillating external terms in weighted spaces. In this case, it is proved that the uniform global attractors of nonautonomous systems converge to the global attractor of an averaged autonomous system.

Exact traveling breather solutions in a discrete Klein-Gordon ring

Following an earlier work relating to traveling kinks and pulses in a discrete reaction-diffusion system, we construct exact traveling breather solutions on a closed Klein-Gordon type lattice with a double-well potential at each site. The dynamics consists of a succession of linear regimes and the problem reduces to an appropriate matching for successive regimes. The analysis shows that the so-called marginal modes are not essential for the existence of traveling breathers. Results are shown for a ring with N = 3 sites for the sake of illustration of the principles involved, while a number of basic results are also presented for the open lattice N →, see below.

Travelling fronts, pulses, and pulse trains in a 1D discrete reaction–diffusion system

Abstract Following an earlier work (briefly reviewed below) we investigate the temporal stability of an exact travelling front solution, constructed in the form of an integral expression, for a one-dimensional discrete Nagumo-like model without recovery. Since the model is a piecewise linear one with an on-site reaction function involving a Heaviside step function, a straightforward linearisation around the front solution presents problems, and we follow an alternative approach in estimating a ‘stability multiplier’ by looking at the variational problem as a succession of linear evolution of the perturbations, punctuated with ‘kicks’ of small but finite duration. Stability depends crucially on perturbations located at specific sites relative to the moving front (the ‘significant perturbations’, see below). Comparison is made with results of numerical integration of the reaction–diffusion system, whereby it appears likely that the travelling front is temporally stable for all relevant parameter values characterising the model. We modify the system by introducing a slow variation of a relevant recovery parameter and perform a leading order singular perturbation analysis to construct a pulse solution in the resulting model. In addition, we obtain a 1-parameter family of periodic pulse trains for the system, modelling re-entrant pulses in a one-dimensional ring of excitable cells.

McKean caricature of the FitzHugh-Nagumo model: traveling pulses in a discrete diffusive medium.

This paper investigates traveling wave solutions of the spatially discrete reaction-diffusion systems whose kinetics are modeled by the McKean caricature of the FitzHugh-Nagumo model. In the limit of a weak coupling strength, we construct the traveling wave solutions and obtain the critical coupling constant below which propagation failure occurs. We report the existence of two different pulse traveling waves with different propagation speeds. Analytical results on the wave speed are obtained. Earlier results on propagation in the bistable medium are found as a limiting regime of our analysis.

Waves and propagation failure in discrete space models with nonlinear coupling and feedback

Many developmental processes involve a wave of initiation of pattern formation, behind which a uniform layer of discrete cells develops a regular pattern that determines cell fates. This paper focuses on the initiation of such waves, and then on the emergence of patterns behind the wave front. I study waves in discrete space differential equation models where the coupling between sites is nonlinear. Such systems represent juxtacrine cell signalling, where cells communicate via membrane-bound molecules binding to their receptors. In this way, the signal at cell j is a nonlinear function of the average signal on neighbouring cells. Whilst considerable progress has been made in the analysis of discrete reaction diffusion systems, this paper presents a novel and detailed study of waves in juxtacrine systems. I analyse travelling wave solutions in such systems with a single variable representing activity in each cell. When there is a single stable homogeneous steady state, the wave speed is governed by the linearisation ahead of the wave front. Wave propagation (and failure) is studied when the homogeneous dynamics are bistable. Simulations show that waves may propagate in either direction, or may be pinned. A Lyapunov function is used to determine the direction of propagation of travelling waves. Pinning is studied by calculating the boundaries for propagation failure for sigmoidal and piecewise linear feedback functions, using analysis of two active sites and exact stationary solutions respectively. I then explore the calculation of travelling waves as the solution of an associated n-dimensional boundary value problem posed on [0,1], using continuation to determine the dependence of speed on model parameters. This method is shown to be very accurate, by comparison with numerical simulations. Furthermore, the method is also applicable to other discrete systems on a regular lattice, such as the discrete bistable reaction diffusion equation. Finally, I extend the study to more detailed models including the reaction kinetics of signalling, and demonstrate the same features of wave propagation. I discuss how such waves may initiate pattern formation, and the role of such mechanisms in morphogenesis.

Compactlike solutions in reaction–diffusion systems: another way to reduce the propagation failure troubles

We introduce and demonstrate theoretically the existence of discrete compact-like kink solutions in a class of discrete reaction–diffusion systems with a nonlinear coupling. Numerical simulations show that the extent of such solutions is related only to the coupling parameter, and that propagation failure mechanism occurs for nonzero critical coupling value Dc NL like for discrete reaction–diffusion systems with linear coupling. We quantify this critical coupling worth and show that the pulses stability in nonlinear coupling regime is more robust than in a linear one.

Propagation failure in discrete bistable reaction-diffusion systems: theory and experiments.

Wave front propagation failure is investigated in discrete bistable reaction-diffusion systems. We present a theoretical approach including dissipative effects and leading to an analytical expression of the critical coupling beyond which front propagation can occur as a function of the nonlinearity threshold parameter. Our theoretical predictions are confirmed by numerical simulations and experimental results on an equivalent electrical diffusive lattice.