A master equation for a spatial population model with pair interactions
Authored by DA Birch, WR Young
Date Published: 2006
DOI: 10.1016/j.tpb.2005.11.007
Sponsors:
United States National Science Foundation (NSF)
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Model Documentation:
Other Narrative
Mathematical description
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Abstract
We derive a closed master equation for an individual-based population
model in continuous space and time. The model and master equation
include Brownian motion, reproduction via binary fission, and an
interaction-dependent death rate moderated by a competition kernel.
Using simulations we compare this individual-based model with the
simplest approximation, the spatial logistic equation. In the limit of
strong diffusion the spatial logistic equation is a good approximation
to the model. However, in the limit of weak diffusion the spatial
logistic equation is inaccurate because of spontaneous clustering driven
by reproduction. The weak-diffusion limit can be partially analyzed
using an exact solution of the master equation applicable to a
competition kernel with infinite range. This analysis shows that in the
case of a top-hat kernel, reducing the diffusion can increase the total
population. For a Gaussian kernel, reduced diffusion invariably reduces
the total population. These theoretical results are confirmed by
simulation. (c) 2006 Elsevier Inc. All rights reserved.
Tags
diffusion
Death
Birth
Moment equations