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A master equation for a spatial population model with pair interactions.

Daniel A Birch·William R Young
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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.

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AnimalsDemographyInsectaModels, StatisticalPopulation DynamicsJournal ArticleResearch Support, U.S. Gov't, Non-P.H.S.

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Daniel A Birch, William R Young. 2006-01-25. A master equation for a spatial population model with pair interactions.. https://doi.org/10.1016/j.tpb.2005.11.007

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