Nucleation and Relaxation from Meta-stability in Spatial Ecological Models

✍ Scribed by AMAR GANDHI; SIMON LEVIN; STEVEN ORSZAG

• Book ID: 102975857
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 625 KB
• Volume: 200
• Category: Article
• ISSN: 0022-5193
• DOI: 10.1006/jtbi.1999.0978

✦ Synopsis

We study a model for competing species that explicitly accounts for e!ects due to discreteness, stochasticity and spatial extension of populations. If a species does better locally than the other by an amount , the global outcome depends on the initial densities (uniformly distributed in space), 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-"eld approximation. The transition zone is dominated by formation of clusters and is characterized by nucleation e!ects 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}di!usion system obtained from the mean-"eld dynamics agrees well with the particle system. The statistics of clusters at an early time soon after clusterformation follow a percolation-like di!usive scaling law. We derive bounds on the time-toextinction based on cluster properties at this early time. We also deduce "nite-size scaling from in"nite system behavior.

If we observe from a frame of reference moving with speed v in the direction of the x-axis, we get

If we are moving at the same velocity as the wave, it will appear steady. That is, for that value of v, we will have

This nonlinear ordinary di!erential equation (a boundary value problem) can be solved using Newton}Raphson. The boundary conditions are 132 A. GANDHI E¹ A¸.