The Gray-Scott Reaction Reprise

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. 

Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous stirred tank reactor Isolas and other forms of multiplicity. Chem. Eng. ScL 38, 29-43 (1983) Autocatalytic reactions in the isothermal continuous stirred tank reactor Oscillations and instabilities in the system A + 2B — 3B, B — C. Chem Eng. ScL 34, 1087-1097 (1984). 

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). 

The literature on this model reaction is already vast and a complete bibliography would be of great use to the mathematical modeler. Of particular interest are A. d Anna, P. G. Lignola, and S. K. Scott. The application of singularity theory to isothermal autocatalytic open systems The elementary scheme A + mB = (m + 1) B. Proc. Roy. Soc. Lond. A 403, 341-363 (1986) and S. R. Kay, S. K. Scott, and P. G. Lignola. The application of singularity theory to isothermal autocatalytic open systems The influence of uncatalyzed reactions. Proc. Roy. Soc. Lond A 409, 433-448 (1987). 

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