Incorporating spatial correlations into multispecies mean-field models

In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques

Bootstrap for estimating the mean squared error of the spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay-Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the EBLUP (Empirical Best Linear Unbiased Predictor). A simulation study compares the bootstrap estimates with an asymptotic analytical approximation and studies the robustness to non-normality. Finally, two applications with real data are described

Spatial Correlations in Dynamical Mean Field Theory

We further develop an extended dynamical mean field approach introduced earlier. It goes beyond the standard $D=\infty$ dynamical mean field theory by incorporating quantum fluctuations associated with intersite (RKKY-like) interactions. This is achieved by scaling the intersite interactions to the same power in 1/D as that for the kinetic terms. In this approach, a correlated lattice problem is reduced to a single-impurity Anderson model with additional self-consistent bosonic baths. Here, we formulate the approach in terms of perturbation expansions. We show that the two-particle vertex functions are momentum-dependent, while the single-particle self-energy remains local. In spite of this, the approach is conserving. Finally, we also determine the form of a momentum-dependent dynamical susceptibility; the resulting expression relates it to the corresponding Weiss field, local correlation function and (momentum-dependent) intersite coupling.Comment: 28 pages, REVTEX, 8 figures include

Extracting Additional Information From Biotic Index Samples

Macroinvertebrates were collected from a small midwestern stream over a 3-year period as part of a non-point source pollution study. Temporal and spatial variability in standard biotic index values (BIs) were computed and compared with variability expressed by a series of additional community measurements, including the mean tolerance value of all taxa present in a sample, irrespective of the numerical abundance of individual taxa. The mean tolerance value exhibited lower spatial and temporal variability than the standard BI; therefore, mean tolerance values may be useful in estimating a stream\u27s long-term ambient water quality and its recovery potential. Computations of additional BI metrics are easily accomplished with no additional lab work required, and comparisons of mean tolerance values with standard BIs should aid investigators in interpreting changes in water quality

Role of mean free path in spatial phase correlation and nodal screening

We study the spatial correlation function of the phase and its derivative, and related, fluctuations of topological charge, in two and three dimensional random media described by Gaussian statistics. We investigate their dependence on the scattering mean free path.Comment: 7 pages, 6 figures. submitted to Europhys. Let