Applications

Current mathematical models of natural systems are usually based on differential equations which describe the smooth variation of one parameter as a function of a few others. Cellular automata provide alternative and in some respects complementary models, describing the discrete evolution of many (identical) components. Models based on cellular automata are typically most appropriate in highly nonlinear regimes of physical systems, and in chemical and biological systems where discrete thresholds occur. Cellular automata are particularly suitable as models when growth inhibition effects are important.

[ Figure 2 ] Evolution of small initial perturbations in cellular automata, as shown by the difference (modulo two) between patterns generated from two disordered initial states differing in the value of a single site. The examples shown illustrate the four classes of behaviour found. Information on changes in the initial state almost always propagates only a finite distance in the first two classes, but may propagate an arbitrary distance in the third and fourth classes.

As one example, cellular automata provide global models for the growth of dendritic crystals (such as snowflakes)[6]. Starting from a simple seed, sites with values representing the solid phase are aggregated according to a two-dimensional rule that accounts for the inhibition of growth near newly-aggregated sites, resulting in a fractal pattern of growth. Nonlinear chemical reaction-diffusion systems give another example [7,8]: a simple cellular automaton rule with growth inhibition captures the essential features of the usual partial differential equations, and reproduces the spatial patterns seen. Turbulent fluids may also potentially be modelled as cellular automata with local interactions between discrete vortices on lattice sites.

If probabilistic noise is added to the time evolution rule (1), then cellular automata may be identified as generalized Ising models [9,10]. Phase transitions may occur if retains some deterministic components, or in more than one dimension.

Cellular automata may serve as suitable models for a wide variety of biological systems. In particular, they may suggest mechanisms for biological pattern formation. For example, the patterns of pigmentation found on many mollusc shells bear a striking resemblance to patterns generated by class 2 and 3 cellular automata (see refs [11,12]), and cellular automaton models for the growth of some pigmentation patterns have been constructed [13].

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