Periodic travelling waves in a family of deterministic cellular automata

Reaction-diffusion equations whose kinetics contain a stable limit cycle are an established class of models for a range of biological and chemical systems. In this paper I construct a family of deterministic cellular automata, with nine states, which are qualitatively similar to oscillatory reaction-diffusion equations, in that their rules reflect both local oscillations and spatial diffusion. The automata can be crudely interpreted as models of predator-prey interactions, and I show that the behaviour following local perturbation of the prey-only state in one space dimension is very similar in the automata and in standard reaction-diffusion models for predator-prey systems. In particular, in many cases, invasion of prey by predators leaves behind periodic travelling waves in the wake of invasion. I study in detail these periodic plane waves in the automaton, by explicitly investigating periodic solutions of the difference equation governing travelling waves. I show that the automaton has many different periodic wave solutions, and I compare their properties with those of periodic wave solutions of reaction-diffusion systems. The basic conclusion is that included amongst the periodic waves in the automaton are a family of solutions which mimic quite closely the properties of reaction-diffusion periodic waves.