A master equation for a spatial population model with pair interactions.
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.
Birch
Daniel A
DA
Scripps Institution of Oceanography, University of California at San Diego, La Jolla, CA 92093-0213, USA. dbirch@ucsd.edu
Young
William R
WR
D000818:Animals; D003710:Demography; D007313:Insecta; D015233:Models, Statistical; D011157:Population Dynamics