Reaction diffusion system model

A Reaction>Diffusion System Model with Equal Diffusion Coefficients... 

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 ... 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

In this section we introduce several CA models of prototypical reaction-diffusion systems. Such systems, the first formal studies of which date back to Turing , often exhibit a variety of interesting spatial patterns that evolve in a self-organized fashion. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Reaction-diffusion systems can readily be modeled in thin layers using CA. Since the transition rules are simple, increases in computational power allow one to add another dimension and run simulations at a speed that should permit the simulation of meaningful behavior in three dimensions. The Zaikin-Zhabotinsky reaction is normally followed in the laboratory by studying thin films. It is difficult to determine experimentally the processes occurring in all regions of a three-dimensional segment of excitable media, but three-dimensional simulations will offer an interesting window into the behavior of such systems in the bulk. 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

Classic Turing Models. The most well studied models of spontaneous pattern formation are based on Turing s original idea that symmetry breaking in biomorphogenesis occurs via an instability in a reaction-diffusion system. In order for this to be operative the time scales for reaction and diffusion must be comparable. Since cellular systems are in general quite small... 

It is clear that all the richness of reaction-diffusion systems should persist in the more accurate nonlinear integral operator model. The important difference is, however, that very rapid variations, and posssibly discontinuities in the slope of or V, should be allowed in the integral operator description. 

Several groups have developed cellular automata models for particular reaction-diffusion systems. In particular, the Belousov-Zhabotinsky oscillating reaction has been examined in a number of studies.84-86 Attention has also been directed at the A + B —> C reaction, using both lattice-gas models 87-90 and a generalized Margolus diffusion approach.91 We developed a simple, direct cellular automaton model92 for hard-sphere bimolecular chemical reactions of the form... 

Equations (9.69) and (9.70) represent the modeling of reaction-diffusion systems with the thermodynamically coupled heat and mass flows excluding the coupling effects due to reaction. After combining Eqs. (9.64), (9.69), and (9.70) steady-state balance equations with the coupled heat and mass transfer become... 

This analysis considers the thermodynamic coupling between heat and mass flows in an industrial reaction-diffusion system with a low value of /3. Modeling with the coupling effects of Soret and Dufour opens the path to describing more complex reaction-diffusion systems by adding the two new controlling parameters s and . 

Previously, we considered the case where heat and mass flows are coupled in a reaction diffusion system with heat effects, in which the cross coefficients Zrq. Zqr. and LlS, LSl have vanished (Demirel, 2006). Here, we consider the other three cases. The first involves the stationary state balance equations. In the second case, there is no coupling between the heat flow and chemical reaction with vanishing coefficients Zrq and Zqr. Finally, in the third, there is no coupling between the mass flow and chemical reaction because of vanishing cross-coefficients of ZrS and LSl. The thermodynamically coupled modeling equations for these cases are derived and discussed briefly in the following examples. 

Example 13.4 Order in time and space with the Brusselator system The Brusselator model with unequal diffusion may produce order in time and space. When the concentrations of. 1 and B are controlled, the one-dimensional approach to complex reaction-diffusion systems with the spatial coordinate r under isothermal conditions yields the kinetic equations forXand Y (Eqs. (12.98) and (12.99))... 

In a simplified model, this representation reflects a bilinear autocatalytic reaction B + N -> 2B, and simply means that the bacteria needs nutrient (assume no shortage of nutrient) to double themselves (Muller and Saarloos, 2002). Any instability observed for k > 0 is due to the nonlinearity in the diffusion coefficient. With the relations above, the reaction-diffusion system becomes... 

The family of methods for the study of parametric information in mathematical models is called sensitivity analysis. Sensitivity analysis investigates the relationship between the parameters and the output of any model. It is usually used for two purposes first, for uncertainty analysis and, second, for gaining insight into the model. Sensitivity methods in chemical kinetics have been reviewed by Rabitz et al. [64], who concentrated mainly on the interpretation of sensitivity coefficients in reaction-diffusion systems. Turdnyi [12] considered sensitivity methods as tools for studying reaction kinetics problems and reviewed several applications. Recently, Radha-... 

Chemical systems are traditionally modeled by reaction-diffusion systems on suitable domains. As was explained above, our main modeling assumption is that the domain is actually unbounded, that is, we consider governing partial differential equations on the entire plane. This assumption may seem unrealistic neither experiments nor numerical simulations can be performed on unbounded domains. In our particular context of spiral waves in the BZ reaction, however, experiments indicate that spiral waves behave much as if there were no boundaries. Therefore until boundary annihilation sets in - typically within only one to two wavelengths from the boundary itself - we consider reaction-diffusion systems ... 

For the moment, a coupling like (3.87) remains speculation. Only reaction-diffusion systems as studied in section 3.2 provide a modeling description which is based on reasonably first principles. The mathematically proper approach to autonomous meanders and relative Hopf bifurcation, which are present in the reaction-diffusion setting, would then be... 

An even more drastic simplification of the dynamics is made in lattice-gas automaton models for fluid flow [127,128]. Here particles are placed on a suitable regular lattice so that particle positions are discrete variables. Particle velocities are also made discrete. Simple rules move particles from site to site and change discrete velocities in a manner that satisfies the basic conservation laws. Because the lattice geometry destroys isotropy, artifacts appear in the hydrodynamics equations that have limited the utility of this method. Lattice-gas automaton models have been extended to treat reaction-diffusion systems [129]. 

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