We present a spatially-explicit, stochastic description of ecological dynamics within the framework of the Neutral Theory of Biodiversity. The overall objective is to characterize spatial aggregation in a community, which we regard as a particularly sharp test of any ecological theory, due to the wealth of existing spatially-explicit data. Our model is defined by a Langevin equation for species abundance density n(x,t), the form of which is dictated by the assumption of neutrality. Quantum Field Theory provides a powerful set of tools for the analysis of such equations, and indeed provides a framework within which one may begin to relax the assumption of neutrality, by breaking the symmetry between species or introducing ecologically-relevant interactions. The very general nature of this framework, as yet unused in the ecological applications, allows us in principle to test precisely how and where nature departs from neutrality, in a way that is not possible in existing spatially-explicity theories. In our current analysis we assume neutrality and derive the corresponding two-point correlation functions for abundances of individuals. At a species level, these correlation functions describe the nature and degree of clustering of individuals, and so are an important metric for testing intra-specific aggregation. At a community level, correlation functions for each species in the community may then be related to the spatial turnover of community composition, as characterized by F(r), the probability that two individuals separated by a distance r belong to the same species.

Results/Conclusions For relatively large mean abundances of individuals, the two-point correlation function is found to have a simple, closed form, consistent with existing derivations of F(r). However, the behaviour of the correlation function becomes more complicated when the mean abundance of a species is small, implying that very rare species may aggregate in a qualitatively different way to abundant species.
We compare these analytical results with extensive computer simulations and data from tropical forests, at both a species and community level.