Turing structures

The idea that spatial patterns may arise spontaneously in a system such as that under consideration here, driven by the different diffusivities of the participants (here concentration and temperature), goes back to Turing. Many of the applications of this approach have been to isothermal systems of biological importance, and the basic principles apply to all two-variable schemes. 

We start with a well-stirred system, so the diffusion terms d2a./dx2 and d20/dx2 make no contribution. The stationary states of this spatially uniform case satisfy 

The division of the parameter plane into regions of stability and instability is reproduced in Fig. 10.3. If pi and k lie within the closed region, the stationary state is unstable and spatially uniform oscillations exist. 

Now let us return to the reaction-diffusion model and imagine that the stirring is somehow stopped. The question we address is whether the region of instability can be extended beyond the closed region of Fig. 10.3, particularly when the participants diffuse at different rates so ft = 1. 

Returning to eqns (10.10) and (10.11), we look for stationary-state solutions 

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The set of the reaction-diffusion equations (78) can be solved by different methods, including bifurcation analysis [185,189-191], cellular automata simulations [192,193], or numerical integration [194—197], Recently, two-dimensional Turing structures were also successfully studied by Mecke [198,199] within the framework of integral geometry. In his works he demonstrated that using morphological measures of patterns facilitates their classification and makes possible to describe the pattern transitions quantitatively. 

Reaction-diffusion systems have been studied for about 100 years, mostly in solutions of reactants, intermediates, and products of chemical reactions [1-3]. Such systems, if initially spatially homogeneous, may develop spatial structures, called Turing structures [4-7]. Chemical waves of various types, which are traveling concentrations profiles, may also exist in such systems [2, 3, 8]. There are biological examples of chemical waves, such as in parts of glycolysis, heart... 

CHEMICAL DIFFUSION PATTERN FORMATION (TURING STRUCTURES)... 

The theoretical predictions prompted an experimental search for electrochemical Turing structures that was successful shortly after. The first experimentally observed Turing structures are reproduced in Fig. 71. They stem from the periodate reduction on Au in the presence of camphor whose homogeneous dynamics was introduced in Section 3.2.2. The patterns could be made visible with surface plasmon microscopy (cf. Fig. 54). When changing the composition of the electrolyte, the wavelength of the patterns changed, in accordance with the theoretical result that the wavelength depends on the system s parameters, but not on the dimension of the... 

The net reaction is A I B -> E I F. This reaction scheme has been developed by the Brussels School of Thermodynamics, and consists of a trimolecular collision and an autocatalytic step. This reaction may take place in a well-stirred medium leading to oscillations, or the diffusions of the components A and B may be considered. In the latter case, the system may produce Turing structures. 

Since the det is the product of the eigenvalues, if both eigenvalues are negative, then det > 0 and tr < 0, and hence the system will be stable. However, for some variation of B, if one of the eigenvalues becomes positive, then det < 0 and the system becomes unstable, and the propagating wave or the Turing structure occurs when the value of B satisfies the following condition ... 

Diffusive instability can appear in simple predator-prey models. Bartumeus et al. (2001) used the linear stability analysis and demonstrated that a simple reaction-diffusion predator-prey model with a ratio-dependent functional response for the predator can lead to Turing structures due to diffusion-driven instabilities. 

For a specified value fond and some variation of B if one of the eigenvalues becomes positive, then Det < 0, and the system becomes unstable, and the propagating wave or the Turing structure occur... 

LengyeI, I., and Epstein, I. R. (1991) Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction. Science 251, 650. 

Deposition process Residual pressure (mbar) Starting material Typical films formed Typical deposition rate Typical substrate tempera- ture Structure of the deposit Film density LTS HTS Hardness, abrasion resistance LTS HTS Sources of impurities... 

Hasslacher, B., R. Kapral A. Lawniczak. 1993. Molecular Turing structures in the biochemistry of the cell. Chaos 3 7-13. 

Tumour-suppressing gene, 455 Turing structures, 4,82,230 Two-pool model for Ca oscillations, 356, 359... 

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...

Since they are open systems that can exchange chemical species with their surrounding solvent, gels can also play the role of chemical reactors. In this framework, the design of open spatial gel reactors has allowed well controlled experimental studies of chemical patterns such as chemical waves or Turing structures (2). They are made of a thin film of gel in contact with one or two continuous stirred tank reactors that sustain controlled nonequilibrium conditions. 

Castets et al. [31] have defined Turing structure as follows These are stationary patterns originating with Sole coupling of reaction and diffusion processes. They... 

Diffusion flow-induced chemical instability Since one of the criterion for obtaining Turing structure is D activator D inhibitor, it essentially means a device to spatially disengage the counteracting species. The key species may by disengaged by an... 

If the eigenvalues of Eqn (13.10) are real and at least one of them is positive due to the controlled parameters A, B, and the diffusion coefficients, only the spatial patterns of the Turing structure of sin Xr will arise without temporal oscillations. If, on the other hand, the eigenvalues are a complex-conjugate pair, then the solutions represent spatiotemporal instabilities and lead to propagating waves. If the real part is positive, the perturbation grows (Kondepudi and Prigogine, 1999). 

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