Variable Reaction-Diffusion Systems

We consider the general n-variable reaction-diffusion system (2.12), 

The spatially uniform solution /o(x) = /o, where F(/o, /i) = 0, is a stationary state of (10.1) with boundary conditions (10.2a) or (10.2b). For Dirichlet boundary conditions, uniform steady states of (10.1) are only possible if 

The uniform steady state p(x) is stable against uniform perturbations, if /) is a stable steady state of the homogeneous or well-mixed system  

If all diffusion coefficients are equal, D = D, that is D = Dl is a scalar times the identity, known as a scalar matrix, then the characteristic polynomial reads (-l) det[J - (Dk + = 0. This implies that Dk + must equal the 

Corollary 10.1 No Turing bifurcation can occur in a one-variable reaction-diffusion 

---

The Turing condition for two-variable DIRWs, (10.85), has the same form as the Turing condition for two-variable reaction-diffusion systems, (10.31). Consequently, the uniform steady state (10.70) of a DIRW undergoes a Turing bifurcation with critical wavenumber... 

Spatial Hopf bifurcations or wave bifurcations can never occur in two-variable reaction-diffusion systems, see Sect. 10.1.2. This is no longer the case for reaction-transport systems with inertia. As shown in Sects. 10.2.1 and 10.2.2, spatial Hopf bifurcations are in principle possible in two-variable hyperbolic reaction-diffusions... 

To shed further light on the occurrence of spatial Hopf bifurcations in DIRWs, it is instructive to study the stability properties of the uniform steady state in one-variable systems. Consider the one-variable reaction-diffusion system... 

For one-variable reaction-diffusion systems, diffusion is always stabilizing and no spatial instability of the uniform steady state can occur, explicitly confirming Corollary 10.1. 

We impose the first inequality in (11.3), since a Turing instability can occur in a two-variable reaction-diffusion system with constant parameters only if > 1. The second inequality ensures the positivity of D (t). We assume that the system (11.1) possesses a uniform steady state, (PuU)-PvU)) = (Pu.Pv). with Ej(Pu,Pv) = E2G0U, Pv) = which fulfills the stability conditions (10.23), and U is an activator and V an inhibitor. 

This is the dispersion relation for a two-variable reaction-diffusion system with a step-function diffusivity for V. It is the analog of (10.26) for homogeneous reaction-diffusion systems and relates the growth rates X of spatial perturbations to the parameter values of the system. In contrast to the homogeneous case, the dispersion relation (11.47) is a complicated expression that cannot be solved analytically if D D. A diffusion-driven instability of the uniform steady state of the system occurs if the stability condition (10.23) is satisfied and (11.47) has solutions with a positive real part. 

The four-variable reaction-diffusion system (12.35) possesses a nontrivial uniform steady state given by... 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

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 ... 

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. 

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]. 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

In a recent numerical study (17), spontaneous chemomechanicaJ pulsations were achieved by coupling a mechanically responsive sphere of gel with a chemical reaction which cannot produce oscillations by itself. The toy model used in this study is based on a two variables model of a quadratic autocatalytic reaction described by the following reaction-diffusion system dufdt = -h V u dv/dt = (12/7)w i +... 

It is noteworthy that the model is not exceptional one. The identical patterns can be generated in many reaction-diffusion systems, provided that they have the similar qualitative properties to the presented model. Moreover, it is worth to stress that the model contains two variables only, and therefore, it is simple one. One can expect more rich patterns in systems with three, fom, and more variables. 

From the differences that arise between the dynamics of the BZ reaction in an active and a passive gel, one may conclude that the dynamics of the active gel-BZ system does not result from a simple forcing of the gel by the chemistry. Numerical results clearly show that our three-variable model (including volume fraction p) constitutes a dynamical system different from the two-variable BZ reaction-diffusion system in a passive sphere. 

The number of X molecules in each box i is an independent variable X and hence the present reaction-diffusion system is isomorphic to a multivariable homogeneous system. To evaluate in (5.15) a path of integration needs to be specified because is not a state function. 

Reaction-diffusion systems, linear or not, can be mapped into multi-variable reaction systems, as stated after (5.15). For such multi-variable reaction systems which can be linearized in the vicinity of a stable stationary state, we have at that state... 

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. 

Details of the method outlined below can be found i n [5], A stochastic system of several extensive variables X. is supposed tc be described by a master equation which can be explicitly written when the transition probabilities per unit time W( Xj Xj ) are known. In a reaction diffusion system, X. may be the number of chemical species a in a cell located by the vector r and is denoted by X. = X. Introducing the toaka tic, pot ntiai U defined by P = exp(-S - N U), where P is the probability, N is proportional to the total volume of the system and S stands for the normalization factor, we switch to the quasicontinu-ous intensive variables x = X /N, where N may be the mean number of particles in one cell of a reaction-diffusion system. If we assume that for all states for which liJ ( X j -> X1 ) are nonnegligible, x - xj is much smaller than 1, the equation for U can be expressed, at the zeroth order in 1/N, in terms of xj and 3U/9xj. liie thus obtain a Hamilton Jacobi type of equation ... 

The model which my colleagues and I have used over the past several years in our studies of spiral waves is simple, two-variable reaction-diffusion model of the Fitzhugh-Nagumo type [ 13,19]. It is a mathematical caricature of what is thought to take place in many real excitable systems. The model has the virtue of providing particularly fast time-dependent numerical simulations of... 

Fig. 5. A perspective plot of the spatio-temporal variation of the variable u x t) as computed with the reaction-diffusion system (3), with Dirichlet boundary conditions (4), the slow manifold (6) and the model parameters iii = -1.5, e = 0.01. (a) Stationary single-front pattern (u() = 1.1,D = 0.1,q = 0.01) (b) periodically oscillating single-front pattern (uo = 1.1, D = 0.045, a = 0.01) (c) periodic alternation of a single-front and a three-front pattern (u[) = 1.1, D = 0.01, a = 0.01) (d) stationary three-front pattern (uo = 0.5, D = 0.08, a = 0.2) (e) periodically oscillating three-front pattern (uq = 0.5, D = 0.06, a = 0.2) (f) periodic alternation of a single-front and a three-front pattern (uo = 0.5, D = 0.02, a = 0.2).

---