An attribute-diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures

Based on the framework of attribute diversity (a generalization of Hill numbers of order q), we develop a class of functional diversity measures sensitive not only to species abundances but also to trait-based species-pairwise functional distances. The new method refines and improves on the conventional species-equivalent approach in three areas: (1) the conventional method often gives similar values (close to unity) to assemblages with contrasting levels of functional diversity; (2) when a distance metric is unbounded, the conventional functional diversity depends on the presence/absence of other assemblages in the study; (3) in partitioning functional gamma diversity into alpha and beta components, the conventional gamma is sometimes less than alpha. To resolve these issues, we add to the attribute-diversity framework a novel concept: tau, the threshold of functional distinctiveness between any two species; here, tau can be chosen to be any positive value. Any two species with functional distance >= tau are treated as functionally equally distinct. Our functional diversity quantifies the effective number of functionally equally distinct species (or "virtual functional groups") with all pairwise distances at least tau for different species pairs. We advocate the use of two complementary diversity profiles (tau profile and q profile), which depict functional diversity with varying levels of tau and q, respectively. Both the conventional species-equivalent method (i.e., tau is the maximum of species-pairwise distances) and classic taxonomic diversity (i.e., tau is the minimum of non-zero species-pairwise distances) are incorporated into our proposed tau profile for an assemblage. For any type of species-pairwise distance matrices, our attribute-diversity approach allows proper diversity partitioning, with the desired property gamma >= alpha and thus avoids all the restrictions that apply to the conventional diversity decomposition. Our functional alpha and gamma are interpreted as the effective numbers of functionally equally distinct species, respectively, in an assemblage and in the pooled assemblage, while beta is the effective number of equally large assemblages with no shared species and all species in the assemblages being equally distinct. The resulting beta diversity can be transformed to obtain abundance-sensitive Sorensen- and Jaccard-type functional (dis)similarity profiles. Hypothetical and real examples are used to illustrate the framework. Online software and R codes are available to facilitate computations.