Gray-Scott reaction

FIGURE 15 Gray/Scott reaction Bifurcation diagram at the point where a mushroom is turning into an isola. 

Example 17. Continuous Mixtures and Parallel Gray-Scott Reactions... 

The parametric approach, which is not strictly needed for a single Gray-Scott reaction, works very well for an arbitrary number of parallel reactions and for continuous mixtures. Figure 16 shows a case of two parallel reactions for which an isola and a mushroom coexist. Because the notions of continuous mixtures and reactions will be treated in Chapter 8, G H and in the group of papers listed in the Index of Subjects in Publications under the heading Continuous mixtures, we can be very brief and start with the nondimensional equations. Let x be the index of the mixture whose species are /4(x). The steady-state concentration of the material with index in (x, x + dx) is V(x)dx, the feed concentration a(x)dx and the conversion U(x) = 1 - V/(x)/a(x), the last being defined only for values of x for which a(x) is not zero. B, the autocatalytic agent, forms itself as an undifferentiated product whose concentration is W. The rate of the first reaction, and hence p,(x), depends on the... 

K. Alhumaizi and R. Aris. Surveying a Dynamical System A Study of the Gray-Scott Reaction in a Two-Phase Reactor. Longman. Harlow (1995). 

The two Proceedings of the Royal Society Papers (Reprints K and L) are a matched pair, exploring the model reaction that Schmidt and Takoudis had devised [177] A + S <-> AS, B + S BS, AS + BS + 2S -> C + 45. Here, the autocatalytic element is the vacant site, just as B is in the Gray-Scott reaction and heat is in the non-isothermal exothermic case. The two reprints, although not an absolutely comprehensive treatment of this model, have a satisfying completeness. The tale of students who worked on this class of problem includes Alhumaizi, Cordonier, Farr, Jorgenson, Kevrekidis, McKar-nin, and Takoudis their papers are listed in the Index of Co-Authors. 

Parallel Gray/Scott reactions in a stirred vessel... 

There is merit in considering at a slightly less stiff and more algebraically amenable model that has the same features as the exothermic first-order nonisothermal case. This is the Gray-Scott autocatylator, a pair of coupled reactions... 

In the case of the nonisothermal first-order exothermic reaction heat is auto catalytic, for it raises the temperature and provokes an increase of reaction rate, yet is itself a product of the reaction. In the Gray-Scott scheme, B is plainly autocatalytic and its degeneration by the second reaction plays the role of the direct cooling in the non-isothermal case. This reaction appears in the chemical engineering literature in 1983,16 and is the keynote reaction in Gray and Scott s 1990 monograph on Chemical Oscillations and Instabilities.17 A justification of the autocatalytic mechanism in terms of successive bimolecular reactions is the subject of Chapter 12. 

As the previous models, the second Schlogl model relies on the pool chemical assumption to account for the inflow of reactants. If we assume instead that this reaction scheme occurs in a CSTR, we obtain the Gray-Scott model [170, 171] ... 

The electronic states associated to the cyclic and oscillatory reactions, reactions with instabilities, etc. are obeying for the discrete state dynamics to the quantum general evolution equation (QME quantum master equation) (see Gray Scott, 1990 van Kampen, 1987 Gardiner, 1994 Risken, 1984 Haken, 1978, 1987, 1988) ... 

In this field, as in any other part of reaction kinetics, there are two approaches investigation of simple, but possibly realistic models see, for example, Caram Scriven (1976), Othmer (1976), Luss (1980, 1981) Gray Scott (1983a, b) or the example of Horn Jackson (1972, cited here as Exercise 1) or a search for general criteria that ensure or exclude multistationarity in large classes of mechanisms. This second approach was initiated by Rumschitzky Feinberg (in preparation Rumschitzky, 1984). Here we present results of this second approach, and some of the Problems contain results on specific models derived using the first approach. 

In the Gray-Scott model PI of this system, both reactions are considered to be irreversible. This reaction scheme is a simplification of the autocatalytic model of the glycolysis cycle (see Chapter 7). A is a feed term and B an inert product. PearsonI °l has shown that as a function of kinetic and diffusion parameters this system leads to the formation of local regions of concentration defined by sharp boundaries. These local regions take on cell-like characteristics, thus undergoing multiplication and division behavior. We discuss some of the results in detail, also because of the discussion in the next chapter on self replication and the origin of protocellular systems. As a function of feed (F) and rate parameter (fc), a state phase diagram can be constructed (see Fig. 8.5). 

When the law of mass action is considered for the reaction (1.4), the time evolution scheme can be draw as the system of the coupled nonlinear differential equations (Gray Scott, 1990) the present discussion follows (Putz et al., 2006a) ... 

See Gray and Scott for a detailed analysis of this (hypothetical) reaction system. 

Mauk, A. G., Scott, R. A., and Gray, H. B. (1980). Distances of electron transfer to and from metalloprotein redox sites in reactions with inorganic complexes. J. Am. Chem. Soc. 102,4360-4363. 

Gray, P. and Scott, S. K. (1983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor isolas and other forms of multistability. Chem. Eng. Sci., 38, 29-43. 

Gray, P., Showalter, K., and Scott, S. K. (1987). Propagating reaction-diffusion fronts with cubic autocatalysis the effects of reversibility. J. Chim. Phys., 84, 1329-33. 

P. Gray and S. K. Scott. Chemical Oscillations and Instabilities Nonlinear Chemical Kinetics. Oxford Clarendon Press, 1990. See also the Review Lecture by P. Gray. Instabilities and oscillations in chemical reactions in closed and open systems. Proc. Roy. Soc. Lond. A 415, 1-34 (1988). 

Reprint F is an example of analyzing a reaction in formal kinetics. Gray and Scott introduced the autocatalytic A + 2B = 3B as a simple model reaction that proved to have a rich behavior, much richer than the Brusselator for example. However, A + 2B smacks of a three-body interaction, which is a sufficiently rare occurrence as to be avoided. I had done a pseudo-steady-state analysis before I visited Leeds at Gray s invitation, and the chance of working with the fons et origo of this reaction, so to speak, was an opportunity to make sure that the limiting behavior was not lost when certain parameters were small, but not actually zero. For another analysis of autocatalytic behavior, see [107]. 

Until recently it was widely accepted that isothermal oscillatory reactions demanded rather complicated interpretations at all levels. It is now known that this is not the case. A simple representation of autocatalysis that has had great impact in recent years on a wide variety of problems, beginning with the CSTR (Gray and Scott, 1983,1984) but now extending to systems with diffusive control (Gray and Scott, 1986 Scott, 1987) and spatial non-unifor-... 

Gray, B. F., Scott, S. K. and Gray, P., 1984, Multiplicity for isothermal autocatalytic reactions in open systems the influence of reversibility and detailed balance. J. Chem. Soc. Faraday Trans. 1 80, 3409. 

---