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

beta-diversity is a primary biodiversity pattern for inferring community assembly. A randomized null model that generates a standardized beta-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 beta-diversity to address the debate, clarify the interpretation and extend the application to occurrence data. The analytical analyses show unambiguously that the standardized beta-deviation is a quantification of the effect size of non-random spatial distribution of species on beta-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 beta-diversity of communities of different sizes. Assuming log-series distribution for the metacommunity species abundance distribution, our model allows for calculation of the standardized beta-deviation using occurrence data plus a datum on the total abundance. Our theoretical model justifies and generalizes the use of the beta null model for inferring community assembly rules.