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. If a species does better locally than the other by an amount ε, the global outcome depends on the initial densities (uniformly distributed in space), ε and the size of the system. The transition point moves to lower values of the initial density of the superior species with increasing system size. Away from the transition point, the dynamics can be described by a mean-field approximation. The transition zone is dominated by formation of clusters and is characterized by nucleation effects and relaxation from meta-stability. Following cluster formation, the dynamics are dominated by motion 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 reaction–diffusion system obtained from the mean-field dynamics agrees well with the particle system. The statistics of clusters at an early time soon after cluster-formation follow a percolation-like diffusive scaling law. We derive bounds on the time-to-extinction based on cluster properties at this early time. We also deduce finite-size scaling from infinite system behavior.