Phylogenetic beta diversity, similarity, and differentiation measures based on Hill numbers

Until now, decomposition of abundance-sensitive gamma (regional) phylogenetic diversity measures into alpha and beta (within- and between-group) components has been based on an additive partitioning of phylogenetic generalized entropies, especially Rao's quadratic entropy. This additive approach led to a phylogenetic measure of differentiation between assemblages: (gamma − alpha)/gamma. We show both empirically and theoretically that this approach inherits all of the problems recently identified in the additive partitioning of non-phylogenetic generalized entropies. When within-assemblage (alpha) quadratic entropy is high, the additive beta and the differentiation measure (gamma − alpha)/gamma always tend to zero (implying no differentiation) regardless of phylogenetic structures and differences in species abundances across assemblages. Likewise, the differentiation measure based on the phylogenetic generalization of Shannon entropy always approaches zero whenever gamma phylogenetic ent...