Additive partitioning of Rao's quadratic diversity: a hierarchical approach

Ecologists have long recognized three different components of species diversity: alpha or within-community diversity (α), beta or between-community diversity (β) and gamma or total species diversity in a region (γ). In this framework, β-diversity has been traditionally linked to the other diversity components through a multiplicative model so that it can be expressed as the ratio between γ-diversity and average α-diversity in a set of plots. Yet, more recently, ecologists are starting to partition diversity using the lesser known approach that α- and β-diversities sum to give the γ-diversity. This additive diversity partitioning is based on the decomposition of concave diversity measures for which the total diversity in a pooled set of communities exceeds (or equals) the average diversity within communities. In this paper, first, I shortly revise additive diversity partitioning for traditional diversity measures that are computed from species relative abundances. Next, I show that, under some specific circumstances, the same model can be extended to Rao's quadratic entropy, a measure that combines species relative abundances and pairwise interspecies differences. Finally, in the framework of taxonomic diversity, Rao's quadratic entropy has another decomposition: the sum over the Simpson indices at all the taxonomic levels. Thus, I show that, combining both partitioning models, the contribution of each level in the taxonomic hierarchy to the α- β- and γ-diversity components of Rao's quadratic entropy is made explicit. The proposed diversity decomposition is illustrated with a worked example on data from a plant community on ultramafic soils in Tuscany (central Italy).