Convolution-generated motion as a link between cellular automata and continuum pattern dynamics

Cellular automata have been used to model the formation and dynamics of pat terns in a variety of chemical, biological, and ecological systems. However , for patterns in which sharp interfaces form and propagate, automata simul ations can exhibit undesirable properties, including spurious anisotropy an d poor representation of interface curvature effects. These simulations are also prohibitively slow when high accuracy is required, even in two dimens ions. Also, the highly discrete nature of automata models makes theoretical analysis difficult. In this paper, we present a method for generating inte rface motions that is similar to the threshold dynamics type cellular autom ata, but based on continuous convolutions rather than discrete sums. These convolution-generated motions naturally achieve the fine-grid limit of the corresponding automata, and they are also well suited to numerical and theo retical analysis. Because of this, the desired pattern dynamics can be comp uted accurately and efficiently using adaptive resolution and fast Fourier transform techniques, and for a large class of convolutions the limiting in terface motion laws can be derived analytically. Thus convolution-generated motion provides a numerically and analytically tractable link between cell ular automata models and the smooth features of pattern dynamics. This is u seful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics. In t his latter role, it also has a number of benefits over the traditional reac tion-diffusion/Ginzburg-Landau continuum PDE models of pattern formation, w hich yield true moving interfaces only as singular limits. We illustrate th e power of this approach with convolution-generated motion models for patte rn dynamics in developmental biology and excitable media. (C) 1999 Academic Press.