Fourier analysis of Turing-like pattern formation in cellular automaton models

The novelty of this paper is the study of emergence of diffusion-induced (Turing-like) patterns from a microscopic point of view, namely, in terms of cellular automata. Formally, the cellular automaton model is described in lattice-gas terminology [H. Bussemaker, A. Deutsch, E. Geigant, Phys. Rev. Lett. 78 (1997) 5018-5021]. The automaton rules capture in abstract form the essential ideas of activator-inhibitor interactions of biological systems. In spite of the automaton's simplicity, self-organised formation of stationary spatial patterns emerging from a randomly perturbed uniform state is observed. Fourier analysis of approximate mean-field kinetic difference equations [J.P. Boon, D. Dab, R. Kapral, A.T. Lawniczak, Phys. Rep. 273 (1996) 55-147] yields a critical wave length and a "Turing condition" for the onset of pattern formation. (C) 2001 Elsevier Science B.V. All rights reserved.