Moment expansions in spatial ecological models and moment closure through Gaussian approximation |
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| Authors: | Amar Gandhi Simon Levin Steven Orszag |
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| Institution: | (1) Program in Applied & Computational Mathematics, and Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA;(2) Present address: Microsoft Corporation, One Microsoft Way, Redmond, WA 98052, USA;(3) Present address: Department of Mathematics, Yale University, P. O. Box 208283, New Haven, CT 06520-8283, USA |
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| Abstract: | We describe the dynamics of competing species in terms of interactions between spatial moments. We close the moment hierarchy
by employing a Gaussian approximation which assumes that fluctuations are independent and distributed normally about the mean
values. The Gaussian approximation provides the lowest-order systematic correction to the mean-field approximation by incorporating
the effect of fluctuations. When there are no fluctuations in the system, the mean equations agree with the Gaussian approximation
as the fluctuations are weak. As the fluctuations gain strength, they influence the mean quantities and hence the Gaussian
approximation departs from the mean-field approximation. At large fluctuation levels, the Gaussian approximation breaks down,
as may be explained by the bimodality and skewness of the fluctuation distribution of the partial differential equation. |
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