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

摘要

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.