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4.10 Numerical Modeling: From the Generation of Button Textures up to Pattern Formation in Ecological Systems
The variety of reaction-diffusion simulations performed during recent decades has been extremely broad. It should not be forgotten that the first fundamental work in this field was the article by Alan Turing entitled "The chemical basis of morpho-genesis" (1952). It contained simulations of the differentiation processes in living entities based on nonlinear kinetic equations describing a set of coupled chemical reactions. This article was also one of the first examples showing that nonlinear dynamic mechanisms could be responsible for the high behavioral complexity of a biological system.This biological research using reaction-diffusion simulations covers a wide range of problems beginning from models of biological pattern formation such as giraffe or zebra patterns up to applications in population dynamics.In the early seventies, Gierer and Meinhardt (1972) used a reaction-diffusion approach to elaborate a theory of biological pattern formation. This approach was later further developed (Murray 1981; Meinhardt and Klinger 1987), particularly by Meinhardt and Klinger, who offered a model for pattern formation on the shells of mollusks. A similar technique was used for purely practical purposes, such as for computer graphics. It gave the opportunity to generate different reaction-diffusion textures for picturesque buttons, specific art painting, and so on (Turk 1991; Witkin and Kass 1991). It should be mentioned that the high behavioral complexity of reaction-diffusion media as revealed in textures seems to be a basis of the human sense of beauty.Also very important were repeated approaches to simulate regimes inherent in Belousov-Zhabotinsky and other reaction-diffusion media. This enabled the calculation of different reaction-diffusion patterns including evolution of breathing spots, spiral and labyrinthine patterns (see, for instance, Hagberg and Meron 1994; Haim et al. 1996).Simulations of reaction-diffusion regimes in three-dimensional media have been the standard method up to now because to obtain experimental information is extremely difficult. The simulations seem to indicate that increasing the structural complexity of the system by going from two to three dimensions leads to a large increase in behavioral complexity of the medium (Winfree and Strogatz 1983, 1984).Simulating the image processing capabilities of reaction-diffusion systems has been another remarkable field of interest for different research groups. Yakhno and Colleagues (Nuidel and Yakhno 1989; Belliustin et al. 1991) used a set of integrodifferential equations that describe media having short-range nonlocal interactions. They found out that their simulations were capable of describing a number of image processing operations (figure 4.27). They embrace:
Figure 4.27: Numerical simulation of image processing operations based on the reaction-diffusion equation. Initial halftone picture is shown in upper part of the figure. Results of simulation which can be different due to diverse choice of coupling functions (see text): enhancement of thin and thick contours, contrast enhancement, enhancement of lines having different slope and corners of the image, and skeleton of the image.
Contour enhancement
Transformation of halftone images into high-contrast ones
Image skeletonizing
Extraction of lines having a given direction
Calculation of invariant features
Some of these results were obtained a little earlier by Price, Wambacq, and Oosterlinck (1990).These simulations are in good correspondence with experimental results conforming at the theoretical level, thus confirming the close correlation between the behavioral complexity of a system and its ability to solve problems of high computational complexity.The development of cellular automata techniques has also been an important part of numerical simulations of reaction-diffusion media. Between different realizations of this technique, two directions of investigations should be mentioned:Tyson and his coworkers (Gerhardt, Schuster, and Tyson 1990; Weimar, Tyson, and Watson, 1992) succeeded in simulating complicated modes of Belousov-Zhabotinsky-type media. Markus and Hess (1990) have also offered a very interesting isotropic cellular automaton model for modeling excitable media.Adamatzky (1994, 1995, 1996b) used cellular automata calculations to analyze important characteristics of reaction-diffusion media and the potentialities of their practical use.Let us mention also that many attempts have been made during the last decade to use nonlinear dynamic ideas for explaining the behavior of complex systems inherent in different fields of human activity (neurobiology, ecology, economics, and so on). Many of these have been covered in the book Interdisciplinary Approaches to Nonlinear Complex Systems (Haken and Mikhailov 1993).
