Moment expansions in spatial ecological models and moment closure through Gaussian approximation

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. (C) 2000 Society for Mathematical Biology.