The major step forward in the modern theory of pattern formation was given by Turing (1952), who used the linear analysis to determine the conditions necessary for the creation of spatial patterns in two-component reaction-diffusion systems. A more recent criteria for pattern formation was proposed by Koch & Meinhardt (1994) and Gierer &Meinhardt (2000) and independently by Segel and Jackson (1972). They postulate that the interplay between two antagonic feedbacks is essential for pattern formation. On one hand, the positive feedback consists in the self-enhancement or autocatalysis of one of the chemical components - generally called activator -, a reaction necessary for small perturbations to be amplified. On the other hand, the increase in activator's concentration must be complemented by a fast-diffusing response in order to obtain pattern formation.



The most studied examples of the two types of reaction-diffusion systems are the Meinhardt system (Gierer &Meinhardt 2000) and the diffusive Gray-Scott system (Pearson 1993) , respectively. The complex interplay between activator and inhibitor or substrate chemical, aided by the reaction and diffusion components create most startling spatio-temporal patterns, such as spots, stripes , travelling waves , spot replication, and spatio-temporal chaos, in a nutshell, a clear example of Turing patterns. The Turing patterns are characterized by the active role that diffusion plays in destabilizing the homogeneous steady state of the system. They emerge spontaneously as the system is driven into a state where it is unstable towards the growth of finite-wavelength stationary perturbations. Interesting enough, the replication characteristic is a particularity of the diffusive Gray-Scott model alone, which makes it the ideal model for developmental research. In such cases, cell-like localized structures grow, deform and make replica of themselves until they occupy the entire space .



The Turing patterns from the work of Pearson 1993 on the diffusive Gray-Scott model were confirmed experimentally by Lee, McCormick, Ouyang & Swinney (1993) including the spot replication - Lee, McCormick, Swinney & Pearson (1994). Theoretically, extensive work exist in the literature on the dynamics of this model concerning the ``spot replication'' in one, two and three dimensions (Muratov & Osipov 2000). Moreover, Muratov & Osipov (1999) developed a theory of rotating spiral waves for the diffusive Gray-Scott model, as an example of a vivid phenomenon observed in many models and biological systems. In addition, Nishiura &Ueyama (1999) proposed a theoretical mechanism that drives the replication dynamics itself from a global bifurcation point of view.





This model was originally introduced in Gray-Scott (1985) as an isothermal system with chemical feedback in a continuously fed, well-stirred tank reactor, where the last property implied the lack of diffusion. The analysis of the system revealed stationary states, sustained oscillations and even chaotic behavior. The model considers the chemical reactions describing the autocatalytic growth of an activator V on the continuously fed substrate, U and the decay of the former in the inert product P, subsequently removed from the system. A major development was performed by Pearson (1993) who introduced the role of space by relaxing the constraint of a well-stirred tank and studied the evolution of the concentrations of the two chemical components, u(x,y,t) and v(x,y,t) in two dimensions, in the limit of small diffusion. The system has as parameters the D u and D v , the diffusion coefficients, F, the dimensionless flow rate (the inverse of the residence time) and k, the decay constant of the activator, V. The original study involved fixed diffusion coefficients, D u = 2 x 10 -5 and D v =10 -5, with F and k being the control parameters.



For an overview of the resultant patterns, one can integrate the equations superimposing on the 2D space (X,Y) a gradient of the control parameters, k and F . In such a way, one has in each cell an approximate sample of the resultant pattern for the associated F and k parameters. The resultant ``phase diagram'' is shown on the right, where besides spot replication and stripes, the system shows for the bottom pairs of (F,k) travelling waves and spatio-temporal chaos.



As discussed also in Pearson (1993), the spots occur only for the parameter values for which the only steady state is the red one - r.h.s. of the saddle-node bifurcation curve --, and thus the gradient needed for the formation and maintenance of the blue spots is an intrinsic self-sustaining feature of the system. Once a spot of high V is formed, it is maintained by the concentration difference between the its center and the surroundings of the spot. As the concentrations are limited to the [0,1] interval, the spot can grow until its maximal gradient is not enough to achieve a maximum concentration in the center and thus its V-value starts to decrease. This induces the spot-division phase .









Mazin et al.(1996) account for and illustrate the Turing space - the region in the parameter space for which the blue state is unstable with respect to the growth of standing spatial perturbations. Their analysis clearly determines the crucial role played by the ratio of the diffusivity coefficients in the pattern formation. The increase of the diffusivity ratio beyond 2 , the value investigated in the original paper of Pearson(1993), leads to a significant extension of the Turing regime.









Lesmes et al. (2003) have carried out the first study of the noise-controlled pattern formation in the Pearson model, with emphasis on the self-replicating patterns. They found that for the chosen value of the pair (F,k) , the noise drives the system from the non-multiplicative, stripe-like pattern to the spot-multiplication one . Interesting enough, they argue in favor of an optimal noise intensity, A for which the number of spots is maximal, after a sufficiently long integration time .



In comparison with the results of Lesmes et al., ours suggest the existence of an interval of optimum noise-intensity values leading to a maximum number of spots, rather than a single optimum value. In other words, our distribution of spots shows a plateau at the level of maximum number of spots, while theirs appears to have a more bell-like shape. The plateau characteristics is more evident in the distribution of the area occupied by the two features -- spots and lines - as the noise intensity varies.













Recently, Robert Munafo drew my attention to additional interesting studies of this system.





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