Variable Reaction-Diffusion Equations

The diagonal elements of D are called the main-term diffusion coefficients and the off-diagonal elements are called the cross-term diffusion coefficients or crossdiffusion terms. The cross-diffusion term liiiks the gradient of species 

In this monograph, we consider only reaction-diffusion systems where the cross-diffusion terms are negligible, i.e., the diffusion matrix is a diagonal matrix, D = diag(Di. ) ), and the diffusion coefficients F),-, which must be positive, do not depend on p, 

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Calculations of three-variable reaction-diffusion equations for the concentrations of [HBr02], [Ferriin], and [Br ] are carried out where the initial excitation is in [Br j. The minimum critical radius to initiate a trigger wave by excitation of [Br ], however, is larger than the values obtained for the previously considered excitations in [HBr02] alone and in [HBr02] and [Ferriin] simultaneously. 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

We want now to see how this state of affairs is affected by the chiral perturbation of our reaction-diffusion equations [term in M in equation (29)]. To this end we follow the lines of imperfection theory (Section II.C) and expand the variables and parameters in series around X = X,. We also set the frequency fl of the solution to be identical to the external frequency w, and assume that o> is close to the linearized intrinsic frequency, ft, in the absence of the field ... 

In a spatially extended system, fluctuations that are always present cause the variables to differ somewhat in space, inducing transport processes, the most common one being diffusion. In the case of constant diffusion coefficients /), the system s dynamics is then governed by reaction-diffusion equations ... 

It is convenient to express the reaction-diffusion equation in dimensionless terms by scaling the concentration variables according to the initial reactant concentration ahead of the wave, a = [A]/[Pg.218]

We first use hyperbolic reaction-diffusion equations, see Sect. 2.2.1, to study the effect of inertia on Turing instabilities [206]. Specifically, we consider two-variable HRDEs,... 

In this section, we use Model B, see Sect. 3.4.2, to explore the effects of subdiffusion on the Turing instability. We consider the two-variable generalized reaction-diffusion equation [484],... 

Pearson has analyzed the effect of an immobile species on the Turing instability in two-variable activator-inhibitor systems for more general conditions [346]. Consider the 2+1 species system described by the following reaction-diffusion equations ... 

Vanag and Epstein have formulated a four-variable model to understand pattern formation in the BZ-AOT system [449,453]. Their model builds on the Oregonator, see Sect. 1.4.8. It assumes that the chemistry within the water core of the droplets is well described by the two-variable Oregonator rate equations (1.131). It further assumes that the species in the oil phase are inert, since they lack reaction partners, the key reactants all being confined to the aqueous core of the droplets. Consequently, only transfer reactions occur for the activator B1O2 and inhibitor Br2 in the oil phase. The rate terms for the two transfer reactions are added to the rate terms of the two-variable Oregonator model. The reaction-diffusion equations of the four-variable model of the BZ-AOT system are given in nondimensionalized form by... 

Marek and coworkers simulated the experiments using an abstract two-variable propagator-controller model (Sevcikova and Marek, 1986). To include electric field effects, they added another term to the reaction-diffusion equation ... 

In this section we compare the predictions of the thermod50iamic theory for relative stability in a two-variable example, the Selkov model, with the results obtained from numerical integration of the reaction diffusion equation. The model, constructed for early studies of glycolysis, has two variables, X and Y, and two constant concentrations. The reaction mechanism is... 

In Chap. 5 we discussed reaction diffusion systems, obtained necessary and sufficient conditions for the existence and stability of stationary states, derived criteria of relative stability of multiple stationary states, all on the basis of deterministic kinetic equations. We began this analysis in Chap. 2 for homogeneous one-variable systems, and followed it in Chap. 3 for homogeneous multi-variable systems, but now on the basis of consideration of fluctuations. In a parallel way, we now follow the discussion of the thermod3mamics of reaction diffusion equations with deterministic kinetic equations, Chap. 5, but now based on the master equation for consideration of fluctuations. 

This experimental result was compared with a calculation based on the NFT mechanism R1-R6, yet further simplified to a two-variable system [6]. Deterministic reaction-diffusion equations were solved numerically as described in Sect. 5.1.4, and the value of the flow rate at equistability was... 

Formal models, with two to four variables, the only ones that might be tractable, may have a stange chemical look. Yet, a mathematician would probably not even bother to keep any chemical structure to his models, when looking,for instance, for a particular type of bifurcation. In the most reduced form they are known to be described by normal forms" , which no longer bear any chemical appearance. Nevertheless, although chemistry, through mass action kinetics, provides all the ingredients necessary to exhibit all those exotic behaviours, it may also impose a number of constraints, that make the reaction or reaction-diffusion equations a... 

Consider the general two-variable system of reaction-diffusion equations ... 

Finally, even within the bifurcation approach, the ODE model presented is not complete. For example, including more variables in the model can affect its dynamics, as can higher-order terms in the functions /, g, and h in Equations (10). (These effects have not yet been considered because, as a first step, I wanted to consider only the simplest case.) Moreover, no direct correspondence has been established between parameters of the ODE modql and the parameters of any excitable media, though a comparison of the phase diagrams for the ODE model and the reaction-diffusion equations suggests that this might be accomplished. Nevertheless, there is every reason to believe that soon it will be possible to capture completely the dynamics of spiral waves in excitable media with a low-dimensional model similar to the ODE model considered in this chapter, in spirit if not in form. 

Solution of the reaction-diffusion equations that describe the spatial behavior of a chemically reacting system is an extremely demanding computational task. For this reason, it is a great advantage to have a simple model with the smallest possible number of variables. The availability of such a model, the Oregonator [69], is one of the reasons for the popularity of the Belousov-Zhabotinsky reaction for theoretical studies of spatial phenomena. 

Let Si (i = 1,... n) be a set of composition (intensive) variables of a chemically reacting system. Their time evolution satisfies the well-known reaction-diffusion equations... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Equation (10) is a diffusion equation which applies equally well for matter or heat. The solution of this equation has been studied by many workers. As with differential equations in general, one arbitrary constant is required for each derivative. Since the diffusion equation has partial derivatives, the arbitrary constants have to be functions of the variable which is not involved in the derivative. The diffusion equation requires two functions of time at fixed values of space coordinates (the boundary conditions) and one function of distance at a given time (the initial condition). These have already been established [eqns. (3)—(5)]. It is possible to proceed to solve the diffusion equation for p(r,t) now and then calculate the particle current of B towards each A reactant and so determine the rate of reaction. 

In (2.25a-c), the variables with the asterisk are dimensionless. Substitution into (2.24) yields a dimensionless diffusion-reaction mechanism equation. 

Spatiotemporal pattern formation at the electrode electrolyte interface is described by equations that belong in a wider sense to the class of reaction-diffusion (RD) systems. In this type of coupled partial differential equations, any sustained spatial structure comes about owing to the interplay of the homogeneous dynamics or reaction dynamics and spatial transport processes. Therefore, the evolution of each variable, such as the concentration of a reacting species, can be separated into two parts the reaction part , which depends only on the values of the other variables at one particular location, and another part accounting for transport processes that are induced by spatial variations in the variables. These latter processes constitute a spatial coupling among different locations. 

To solve highly nonlinear differential equations for systems far from global equilibrium, the method of cellular automata may be used (Ross and Vlad, 1999). For example, for nonlinear chemical reactions, the reaction space is divided into discrete cells where the time is measured, and local and state variables are attached to these cells. By introducing a set of interaction rules consistent with the macroscopic law of diffusion and with the mass action law, semimicroscopic to macroscopic rate processes or reaction-diffusion systems can be described. 

Most of the ingredients of the model described above have been formerly postulated in treatments of unimolecular reactions. In particular, the model for the barrier dynamics is inherent in the usual TST for unimolecular reactions involving polyatomic molecules, while taking the total molecular energy Ej as the important dynamic variable in the well is the underlying assumption in theories that use a master or a diffusion equation for Ej- as their starting point. 

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