From level to level: Coarse-graining/scaling in ecology

The dynamics of competing species are investigated by detailed particle-level stochastic simulations and employing techniques from non-equilibrium statistical physics. The questions of interest at the macroscopic level are the spatial patterns observed at long times and the time taken by the system to arrive at these states. We study two cases: (i) symmetric case where the species are equally suited to the environment, and (ii) asymmetric case where one species dominates. In either case, when the initial densities are too far apart, no spatial structures form and extinction is fairly rapid. In this regime, the mean-field approximation works well. The symmetric case displays a continuous phase transition whereas the asymmetric case displays a first-order phase transition. When the system is close to the transition point, the dynamics are governed by formation of clusters and their subsequent evolution by interface motion. We show that the statistical properties of clusters at the end of cluster formation stage can be represented well by a percolation-like diffusive scaling law. The interface motion in the symmetric case is by mean curvature alone and this allows us to develop a phenomenological theory based on surface effects that predicts a scaling law for the time to extinction as a function of the initial densities. In the asymmetric case, the interface motion is by a combination of planar wave motion and mean curvature. We compute the largest hole in the system and relate the time to fill this hole to the time to extinction in asymmetric case. We also apply the Gaussian approximation to the reaction-diffusion system in order to get a systematic lowest-order correction to the mean-field dynamics.