Nucleation and relaxation from meta-stability in spatial ecological models

We study a model for competing species that explicitly accounts for effects due to discreteness, stochasticity and spatial extension of populations. I f a species does better locally than the other by an amount epsilon, the gl obal outcome depends on the initial densities (uniformly distributed in spa ce), epsilon and the size of the system. The transition point moves to lowe r Values of the initial density of the superior species with increasing sys tem size. Away from the transition point, the dynamics can be described by a mean-held approximation. The transition zone is dominated by formation of clusters and is characterized by nucleation effects and relaxation from me ta-stability. Following cluster formation, the dynamics are dominated by mo tion of cluster interfaces through a combination of planar wave motion and motion through mean curvature. Clusters of the superior species bigger than a certain critical threshold grow whereas smaller clusters shrink. The rea ction-diffusion system obtained from the mean-held dynamics agrees well wit h the particle system. The statistics of clusters at an early time soon aft er cluster-formation follow a percolation-like diffusive scaling law. We de rive bounds on the time-to-extinction based on cluster properties at this e arly time. We also deduce finite-size scaling from infinite system behavior . (C) 1999 Academic Press.