Patterns Turing

In this chapter, we shall describe the nature of Turing patterns and some of the systems in which they may play a role, explore why they have been so elusive, examine the experimental systems in which they have been demonstrated, and consider other systems and other methods for generating them. Much of our 

To make these conditions more precise, and to derive some important insights into what sorts of systems can exhibit Turing patterns, we consider the simplest possible class of systems capable of supporting this type of behavior, systems in one spatial dimension with two concentration variables. Let x((, t) and y(r, t) be the two concentrations whose time evolution is governed by 

We assume that the system has a spatially uniform stable steady state (x, yj such that 

We define the elements ay of the Jacobian matrix J, where all the partial 

The linear stability analysis developed in Chapter 2 implies that the steady state will be stable to spatially uniform perturbations if, and only if. 

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Diflfiisive processes nonnally operate in chemical systems so as to disperse concentration gradients. In a paper in 1952, the mathematician Alan Turing produced a remarkable prediction [37] that if selective diffiision were coupled with chemical feedback, the opposite situation may arise, with a spontaneous development of sustained spatial distributions of species concentrations from initially unifonn systems. Turmg s paper was set in the context of the development of fonn (morphogenesis) in embryos, and has been adopted in some studies of animal coat markings. With the subsequent theoretical work at Brussels [1], it became clear that oscillatory chemical systems should provide a fertile ground for the search for experimental examples of these Turing patterns. 

Figure A3.14.12. The first experimental observation of a Turing pattern in a gel strip reactor. Solutions containing separate components of the CIMA/CDIMA reaction are flowed along each edge of the strip and a spatial pattern along the horizontal axis develops for a range of experimental conditions. (Reprinted with pennission from [38], The American Physical Society.)...

Figure A3.14.13. Further examples of the various Turing patterns observable in a 2D gel reaetor. (a) and (b) spots, (e) and (d) stripes, (e) and (1) wider field of view showing long-range defeets in basie stnietine. The seale bar alongside eaeh figure represents 1 nnu. (Reprinted with pemrission from [39], The Ameriean Institute of Physios.)...

The search for Turing patterns led to the introduction of several new types of chemical reactor for studying reaction-diffusion events in feedback systems. Coupled with huge advances in imaging and data analysis capabilities, it is now possible to make detailed quantitative measurements on complex spatiotemporal behaviour. A few of the reactor configurations of interest will be mentioned here. 

Lengyel I and Epstein I R 1992 A chemical approach to designing Turing patterns in reaction-diffusion systems Proc. Natl Acad. Sc/. 89 3977-9... 

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

The Turing mechanism requires that the diffusion coefficients of the activator and inlribitor be sufficiently different but the diffusion coefficients of small molecules in solution differ very little. The chemical Turing patterns seen in the CIMA reaction used starch as an indicator for iodine. The starch indicator complexes with iodide which is the activator species in the reaction. As a result, the complexing reaction with the immobilized starch molecules must be accounted for in the mechanism and leads to the possibility of Turing pattern fonnation even if the diffusion coefficients of the activator and inlribitor species are the same 62. 

Mazouz N, Krischer K. 2000. A theoretical study on Turing patterns in electrochemical systems. J Phys Chem B 104 6081 6090. 

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... 

Although the periodate, camphor/Au system is thus far the only system in which Turing patterns were reported, the mechanism that leads to their emergence strongly suggest that they exist in all S-NDR systems in wide parameter ranges. This is quite in contrast to chemical reaction-diffusion systems in which only in exceptional cases the diffusion coefficients of the inhibitor is sufficiently larger than the one of the activator species. 

Figure 4.11 Turing patterns observed in a CIMA reaction. Light and dark colors are the two states of an indicator for the Ig concentration. From Ouyang and Swinney (1991). Reprinted by permission from Macmillan Publishers Ltd Nature 352, 610-612, copyright 1991.

Now a number of chemical and biological systems (Epstein and Pojman, 1998 Murray, 1993) are known to display Turing patterns. The mechanism has also been invoked as possible source of plankton patchiness (Levin and Segel, 1976). Although diffusion coefficients of planktonic organisms have roughly the same value for both predator... 

Q. Ouyang and H.L. Swinney. Transition from a uniform state to hexagonal and striped Turing patterns. Nature, 352 610 - 612, 1991. 

The rich consequences of adding a diffusive mechanism of transport to chemical or biological activity are described in Chapter 4. Fisher waves and other types of fronts, excitable waves, Turing patterns, and other spatiotemporal phenomena produce striking structures which are observed in chemical and biological media. Understanding them is needed before addressing the additional impact that advection has on these systems. 

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]...

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