DAR (diversity-area relationship): Extending classic SAR (species-area relationship) for biodiversity and biogeography analyses

I extend the classic SAR, which has achieved status of ecological law and plays a critical role in global biodiversity and biogeography analyses, to general DAR (diversity-area relationship). The extension was aimed to remedy a serious application limitation of the traditional SAR that only addressed one aspect of biodiversity scaling-species richness scaling over space, but ignoring species abundance information. The extension was further inspired by a recent consensus that Hill numbers offer the most appropriate measures for alpha-diversity and multiplicative beta-diversity. In particular, Hill numbers are essentially a series of Renyi's entropy values weighted differently along the rare-common-dominant spectrum of species abundance distribution and are in the units of effective number of species (or species equivalents such as OTUs). I therefore postulate that Hill numbers should follow the same or similar law of the traditional SAR. I test the postulation with the American gut microbiome project (AGP) dataset of 1,473 healthy North American individuals. I further propose three new concepts and develop their statistical estimation formulae based on the new DAR extension, including: (i) DAR profile-z-q relationship (DAR scaling parameter z at different diversity order q), (ii) PDO (pair-wise diversity overlap) profile-g-q relationship (PDO parameter g at order q, and (iii) MAD (maximal accrual diversity: D-max) profile-D-max-q. While the classic SAR is a special case of our new DAR profile, the PDO and MAD profiles offer novel tools for analyzing biodiversity (including alpha-diversity and beta-diversity) and biogeography over space.