Analytical models for β-diversity and the power-law scaling of β-deviation

β-diversity is a primary biodiversity pattern for inferring community assembly. A randomized null model that generates a standardized β-deviation has been widely used for this purpose. However, the null model has been much debated and its application is limited to abundance data. Here we derive analytical models for β-diversity to address the debate, clarify the interpretation and extend the application to occurrence data. The analytical analyses show unambiguously that the standardized β-deviation is a quantification of the effect size of non-random spatial distribution of species on β-diversity for a given species abundance distribution. It robustly scales with sampling effort following a power law with exponent of 0.5. This scaling relationship offers a simple method for comparing β-diversity of communities of different sizes. Assuming log-series distribution for the metacommunity species abundance distribution, our model allows for calculation of the standardized β-deviation using occurrence data plus a datum on the total abundance. Our theoretical model justifies and generalizes the use of the β null model for inferring community assembly rules.