Gray-Scott model

The first corresponds to the classical activator-inhibitor system, where the elements fy<0 and g > 0 represent, respectively, Y (the inhibitor) inhibiting the formation of X (the activator), and X promoting the formation of Y. The second, with the opposite sign pattern for these off-diagonal elements, corresponds to a positive-feedback system such as the Gray-Scott model, where X is the autocatalyst and Y is typically a consumable reactant. In this case, both the autocatalyst and the reactant promote the formation of the autocatalyst, and, in turn, both species participate in the consumption of the reactant. In either case, a Turing instability can exist. 

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 Gray-Scott model has a first integral expressing the conservation of mass, namely... 

For the Gray-Scott model, Sect. 1.4.7, the loss rate of the activator is /f = k2+q and the right-hand side of the Hopf condition (10.115) is given by... 

Again the spatial Hopf bifurcation does not occur, but (10.123) shows that the Gray-Scott model represents a borderline case since = 0. If this model is modified and the third-order autocatalysis replaced by a rather unrealistic fourth... 

Fig. 10.5 Plot of 9 for the Gray-Scott model with a subdiffusing inhibitor. The parameters are q = 0.15 and 2 = 0.03. Reprinted with permission from [485]. Copyright 2008 by the American Physical Society...

Figure 2. A new route to reach to a stationary structure (the Turing structure) via self-duplication of dots. Gray-Scott model (10) is usedfor calculations. The last panel is still not symmetrical because of the influence of noise initially added randomly at each pixel...

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). 

These saturation and selection mechanisms depend on the nonlinearities of the model. Most analytical and numerical nonlinear studies have been performed on model chemical schemes like the Brussellator or variants [1,9, 19-26], the Selkov model [27], the Schnackenberg model [28-30], the Gray-Scott model [31], or the Lengyel-Epstein model [32-34]. Only the last one... 

When g < 0, stripes and hexagons can bifurcate subcritically and the situation is much more complex than in the previous case. Different types of states can be simultaneously stable with the stationary state [40]. An example of a bifurcation diagram for our basic model at 7 > 7c is given in Figure 5. Moreover, the models which present these subcritical bifurcations also often present multiple homogeneous steady states like our basic model or the Gray-Scott model [31]. Actually, these fully developed structures pertain... 

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... 

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. 

The general treatment given above will now be illustrated by considering a simple two-variable chemical model. We shall examine the pattern formation that occurs in the Schnackenberg model, which is closely related to the Gray-Scott modeP and a member of the family of cubic autocatalysis models for chemical systems (a family that includes the Brusselator ). A detailed study of pattern formation in the Schnackenberg scheme has been carried out by Dufiet and Boissonade.3 ... 

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. 

Gray, P. and Scott, S. K. (1986). A new model for oscillatory behaviour in closed systems the autocatalator. Ber. Bunsenges. Phys. Chem., 90,985-96. Hanusse, P. (1973). Etude des systemes dissipatifs chimiques a deux et trois especes intermediare. C. R. Acad. Sci., Paris, C277, 263., ... 

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]. 

Modelling cubic autocatalysis by successive bi-molecular steps (with P. Gray and S.K. Scott). Chem. Eng. Sci. 43,207-211 (1988). (Reprint F)... 

J. L. Hudson and O. E. Rossler, Chaos in simple three- and four-variable systems, in Modeling of patterns in space and time (Lecture notes in biomathematics, vol. 55), W. Jager and J. D. Murray, eds., Springer, Berlin, 1984 see also Gray and Scott (ref. G2), Section 13.4. 

Gray, P. Scott, S. 1986. A New Model for Oscillatory Behaviour in Closed Systems The Autocatalator, Ber. Bunsenges. Phys. Chem. 90, 985-996. 

Isothermal Autocatalysis in Open Systems (CSTR) Simple Models and Complex Behaviour. By P. Gray, D. Knapp, and S. Scott (With 3 Figures). 55... 

Consider next a system in which explosion occurs primarly because of chemical kinetics. Real world systems of this type involve multiple steps and competition between various pathways, many of which contain autocatalytic or inhibitory effects associated with the appearance of chain reactions and free radicals [4]. Instead of developping the analysis of such a system however, we present hereafter a mathematical model which captures the essence of the phenomenon while still allowing a fairly complete mathematical treatment. The specific example we choose is the autocatalytic mechanism suggested by Schlbgl [6]. Further comments on the role of autocatalysis are to be found in the Chapters by P. Gray and S.K. Scott and by I. Epstein. 

Epstein has also reviewed the design of pH-regulated oscillators and described a general model for such systems. Noyes has also reviewed the current state of chemical oscillators. Oscillation and spatial nonuniformities in membranes have been reviewed. The proceedings of the 1989 Conference on Exotic Phenomena held at Hajduszoboszlo, Hungary, have been published and Gray and Scott have edited an issue of Philosophical Transactions devoted to this topic. A number of theoretical papers have appeared. 

---