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