TY - JOUR
T1 - Nucleation and relaxation from meta-stability in spatial ecological models
AU - Gandhi, Amar
AU - Levin, Simon Asher
AU - Orszag, Steven
N1 - Funding Information:
The authors acknowledge many useful and insightful discussions with Rick Durrett, Peter Kramer, Anil Bangia and Victor Yakhot. They also thank the reviewers for the many suggestions which improved the paper. ASG and SAO acknowledge support from ONR/DARPAURI grant N00014-92-J-1796. SAL acknowledges support from NASA, grants NAGW-4688 and NAG5-6422; Andrew W. Mellon Foundation; O$ce of Naval Research, grant ONR-URIPN00014-92-J-1527; Alfred P. Sloan Foundation, grant 97-3-5.
PY - 1999/9/21
Y1 - 1999/9/21
N2 - 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.
AB - 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.
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U2 - 10.1006/jtbi.1999.0978
DO - 10.1006/jtbi.1999.0978
M3 - Article
C2 - 10504281
AN - SCOPUS:0033592360
SN - 0022-5193
VL - 200
SP - 121
EP - 146
JO - Journal of Theoretical Biology
JF - Journal of Theoretical Biology
IS - 2
ER -