Two cellular automaton models for reaction-diffusion systems: theory and simulation

A cellular automaton (CA) is a discrete microscopic dynamical system widely used to investigate and understand the mechanisms of complex systems such as reaction-diffusion systems based on cell-cell interactions. We introduce two CA models for Turing-type pattern formation. These are a moving average CA and lattice-gas CA. For a moving average CA, the construction of the local CA rules from the reaction-diffusion partial differential equations relies on a moving-average procedure to implement the diffusive step and a probabilistic table lookup for the reactive step. We apply this method to the 2D Brusselator system. The corresponding 11-state CA model is able to capture the Hopf and Turing birfucation. For a lattice-gas CA, we introduce a modified reaction rule for an activator-inhibitor system and combine it with the propagation rule and shuffling rule. A variety of dynamics arise in this LGCA model. Numerical simulations of both CA models are presented and analyzed