Stationary Turing patterns

Alan Turing s paper entitled The Chemical Basis of Morphogenesis [440] ranks without doubt among the most important papers of the last century. In that seminal work Turing laid the foundation for the theory of chemical pattern formation. Turing showed that diffusion can have nontrivial effects in nonequilibrium systems. The interplay of diffusion with nonlinear kinetics can destabilize the uniform steady state of reaction-diffusion systems and generate stable, stationary concentration patterns. To quote from the abstract,... 

Perraud, J, J. Agladze, K. Dulos, E, De Kepper, P. 1992, Stationary Turing Patterns versus Time-Dependent Structures in the Chlorite-Iodide Malonic Acid Reaction, Physica A 188, 1-16. 

Transient zig-zag patterns are commonly observed in experiments. An example obtained in the laboratory with the CIMA reaction is shown in Figure 19. The transformation of striped patterns by propagation of dislocations is also commonly seen in experiments. Nevertheless, there is not yet definite experimental evidence of induced phase instabilites for stationary Turing patterns. Since they are also observed in numerical simulations of Turing monolayers [52], this should not be out of reach in the near future. 

Waves of chemical reaction may travel through a reaction medium, but the ideas of important stationary spatial patterns are due to Turing (1952). They were at first invoked to explain the slowly developing stripes that can be exhibited by reactions like the Belousov-Zhabotinskii reaction. This (rather mathematical) chapter sets out an analysis of the physically simplest circumstances but for a system (P - A - B + heat) with thermal feedback in which the internal transport of heat and matter are wholly controlled by molecular collision processes of thermal conductivity and diffusion. After a careful study the reader should be able to ... 

The original condition for the development of a Turing pattern, namely that the inhibitor diffuses faster than the activator, can be generalized allowing also for transport processes different from diffusion, in which case this necessary condition becomes A stationary pattern with a characteristic wavelength may develop if the... 

The Turing condition C4 = 0 for hyperbolic reaction-diffusion equations leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40). In other words, the Turing condition is independent of and Ty. If inertia in the transport is modeled by HRDEs, then the inertia has no effect whatsoever on the Turing instability to stationary patterns. 

The first unambiguous observation of a Turing instability in any experimental system did not occur until 1990. That year, the Bordeaux group found convincing evidence for Turing patterns in an in vitro system, the CIMA reaction (see Sect. 1.4.9). The gap of almost 40 yr between Turing s theoretical prediction of diffusion-induced instabilities and the experimental realization of stationary chemical pattern was caused by two main factors. 

Another set of pattern formation phenomena involve stationary, or Turing patterns (77), which arise in systems where an inhibitor species diffuses much more rapidly than an activator species. These patterns, which are often invoked as a mechanism for biological pattern formation, were first found experimentally in the chlorite-iodide-malonic acid reaction (72). Examples of typical spot and stripe patterns appear in Figure 3. Recently, experiments in reverse microemulsions have given rise not only to the waves and patterns described above, but to a variety of novel behaviors, including standing waves and inwardly moving spirals, as well (75). 

Some of spatio-temporal patterns and time-evolving patterns in dissipative structures can be frozen to obtain stationary structure. Addition of Ag ions in the BZ reaction records the spiral pattern (8), and quenching the photoinduced phase separation in a polymer mixture results in a Turing-type structure (9). 

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

Transient Turing-like patterns in PA-MBO reaction (Polyacrylamide-methylene blue-sulphide-oxygen) system are obtained. Non-Turing stationary patterns in FIS reaction (Hexacyanoferrate (Il)-iodate-sulphite system) are obtained using an experimental technique similar to that used in CIMA reaction. These patterns develop through propagation of chemical fronts from the initial perturbation [54],... 

When a reaction is carried out under the condition where the thickness of the solution mixture is less, the colored spatial patterns arise due to the variations in chemical concentrations and this pattern depends on the lime. The other important pattern which is stationary in time and periodic in space, or periodic in both time and space is known as Turing structures. Turing pattern looks like very beautiful and has close resemblance with biological and chemical systems. 

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