Modeling spatial aggregation of finite populations.

Accurate description of spatial distribution of species is essential for correctly modeling macroecological patterns and thus to infer mechanisms of species coexistence. The Poisson and negative binomial distribution (NBD) are most widely used to respectively model random and aggregated distributions of species in infinitely large areas. As a finite version of the Poisson distribution, the binomial distribution is used to model random distribution of species populations in finite areas. Despite that spatial aggregation is the most widespread pattern and no species in nature are distributed in infinitely large areas, no model is currently available to describe spatial aggregation for species distributed in finite areas. Here we develop a finite counterpart of the NBD to model aggregated species in finite landscapes. Similar to the NBD, this new model also has a parameter k measuring spatial aggregation. When k --> infinity, this model becomes the binomial distribution; when study area approaches infinite, it becomes the NBD. This model was extensively evaluated against the distributions of over 300 tree species in a 50-ha stem-mapping plot from Barro Colorado Island, Panama. The results show that when sampling area is small (relative to the study area), the new model and the NBD are of little difference. But the former correctly models spatial distribution at the finite limit at which the NBD fails. We reveal serious theoretical pathologies by using infinite models to approximate finite distribution and show the theoretical and practical advantages for using the new finite model for modeling species-area relationships, species occupancy and spatial distribution of rare species.