Decomposing phylogenetic entropy into α, β and γ components

Measuring the phylogenetic diversity of communities has become a key issue for biogeography and conservation. However, most diversity indices that rely on interspecies phylogenetic distances may increase with species loss and thus violate the principle of weak monotonicity. Moreover, most published phylogenetic diversity indices ignore the abundance distribution along phylogenetic trees, even though lineage abundances are crucial components of biodiversity. The recently introduced concept of phylogenetic entropy overcomes these limitations, but has not been decomposed across scales, i.e. into alpha, beta and gamma components. A full understanding of mechanisms sustaining biological diversity within and between communities needs such decomposition. Here, we propose an additive decomposition framework for estimating alpha, beta and gamma components of phylogenetic entropy. Based on simulated trees, we demonstrate its robustness to phylogenetic tree shape and species richness. Our decomposition fulfils the requirements of both independence between components and weak monotonicity. Finally, our decomposition can also be adapted to the partitioning of functional diversity across different scales with the same desirable properties.