Assessing the spatial distributions of species has strong implications for understanding the processes of community assembly and formulating effective conservation policy.    Here, we show that distance-decay relationships -- which characterize the tendency of proximate communities to share more species than distant ones -- carry detailed and useful information about the shape of spatial distributions.
Results/Conclusions

We begin by developing a model that predicts that distance-decay relationships are essentially quadratic polynomials; the model is based on the assumption that spatial distributions can be approximated by polygons.  We show that the model correctly predicts distance-decay relationships in six disparate communities.  We then show that the parameters in the model can be interpreted geometrically, being proportional to the area, perimeter, and angularity of an underlying spatial distribution.   We present numerical results confirming these conclusions.  Next, we fit the model to two microbial data sets, where the spatial distributions of the taxa are unknown.  We infer that the distributions of microbes have highly variable shapes, potentially reflecting different dispersal processes and environmental requirements.  Our approach presents a powerful tool for characterizing spatial distributions, and will yield new insight into the processes of community assembly.