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.