Model Schnackenberg

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 ... 

The Schnackenberg model is given by the following four reactions ... 

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... 

Another complication in the analysis of bifurcation diagrams is the ubiquitous presence of reentrant hexagonal phases. When the quadratic term F depends explicitly on the bifurcation parameter (e.g., when 7 is replaced by a function of a), the nature and stability of the hexagonal patterns can change all along the branch. In particular, the stability of the pattern can normally be lost for fi > fij and be recovered at higher values of /i since /is is now a function of /x itself A typical example has been discussed quantitatively in the Brussellator model [21]. Another example is provided by the Schnackenberg model [42] ... 

Fig. 7. Nature and stability of patterns of the Schnackenberg model in parameter space (a, b). Other parameters 7 = 10 000, d = 20. The Turing space is region T. A stable hexagons Ho A stable hexagons H r stable stripes hexagons Ho and stripes are both stable hexagons H,r and stripes are both stable. The dotted line is the stability limit of the striped patterns.

Schnackenberg (1979) considered the following hypothetical model of a chemical oscillator ... 

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