Synchronization and coarsening (without self-organized criticality) in a forest-fire model

We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length <L> grows linearly with time as t-->infinity. The number density of patches of length L, N(L,t), scales as <L>N-2(L/<L>), and within a mean-field rate equation description we find that this scaling function decays as N(x)similar toe(-1/x) for x-->0, and as e(-x) for x-->infinity. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.