Turing pattern formation

Fig. 1.19. Coarse fluctuations support pattern formation, fine ones do not. In this parameter regime the system with Ii t) = 0 does not support Turing pattern formation. JJy = 0.5, Ar = 0.02. 2000 x 2000 points are shown. [42]...

The Turing meehanism requires that the diffusion coefficients of the activator and inhibitor be sufficiently different but the diffusion eoeffieients of small molecules in solution differ very little. The chemical Turing patterns seen in the CIMA reaetion used stareh as an indicator for iodine. The stareh indieator complexes with iodide whieh is the aetivator speeies in the reaction. As a result, the complexing reaction with the immobilized stareh molecules must be aeeounted for in the mechanism and leads to the possibility of Turing pattern formation even if the diffusion eoeffieients of the aetivator and inhibitor species are the same 62. 

Chiu, J.W., Chiam, K.H. Monte Carlo simulation and linear stability analysis of Turing pattern formation in reaction-subdiffusion systems. Phys. Rev. E 78(5), 056708 (2008). http // dx.doi.org/10.1103/PhysRevE.78.056708... 

Golovin, A.A., Matkowsky, B.J., Volpert, V.A. Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69(1), 251-272 (2008). http //link. aip.org/1ink/7SMM/69/251/1... 

If a system of the form of eq. (14.1) satisfies conditions (14.5), (14.6), (14.17), (14.19), and (14.21), then it can give rise to Turing pattern formation when the homogeneously stable steady state (x, y,) is subject to inhomogeneous perturbations whose spatial scale, is such that q satisfies eq. (14.15). In such a system, the initial, infinitesimal perturbation will grow, and the system will ultimately evolve to a stable, spatially inhomogeneous structure, the Turing pattern. 

We wish to show how, by adding diffusion terms and the complexation of the activator species (iodide in this case) by an immobile agent, such as starch, Turing patterns can be generated from an activator-inhibitor model like eqs. (14.32) and (14.33). We shall analyze a more general version of the problem, because its solution will lead us to a systematic approach to designing new svstems that exhibit Turing pattern formation. 

Intuitively, we can see the answer immediately. Formation of Turing patterns requires that concentrations of all reactants lie within ranges that allow the system to satisfy a set of conditions in the case of a two-variable activator-inhibitor system, eqs. (14.5), (14.6), (14.17), (14.19), and (14.21). Because of the way the experiment is done (recall Figure 14.2), each reactant concentration is position-dependent, ranging from its input feed value at the end of the gel where it enters to essentially zero at the other end. Clearly, the conditions for Turing pattern formation can be satisfied only in a portion of the gel, if at all. 

The two inequalities (14.45) are satisfied only within the dashed lines of Figure 14.8. Thus, Turing pattern formation can occur in at most about 20% of the thickness of the gel. If the gel were much thicker, perhaps 20-50 times the wavelength of the structures, we would almost surely find multiple layers of patterns. Mechanical problems with thick slabs of gel put a practical upper limit of a few millimeters on the usable thickness of cylindrical slabs of polyacrylamide or agar gel. 

Especially because of the wide range of applicability of ideas behind Turing pattern formation, we may expect rapid developments in this area during the coming years. Here, we suggest a few directions that seem particularly promising. 

We begin this chapter with a discussion of the automaton and present the details of the model construction in Section 2. A number of different systems has been studied using this method in order to investigate fluctuation effects on chemical wave propagation and domain growth in bistable chemical systems [6], excitable media and Turing pattern formation [3,4,7], surface catalytic oxidation processes [8], as well as oscillations and chaos [9]. Our discussions will be confined to the Willamowski-Rossler [10] reaction which displays chemical oscillations and chaos as well as a variety of spatiotemporal patterns. This reaction scheme is sufficiently rich to illustrate many of the internal noise effects we wish to present the references quoted above can be consulted for additional examples. Section 3 applies the general considerations of Section 2 to the Willamowski-Rossler reaction. Sections 4 and 5 describe a variety of aspects of the effects of fluctuations on pattern formation and reaction processes. Section 6 contains the conclusions of the study. 

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