Parametric formulas for guiding through biodiversity indices

In ecology, species richness, the number of species, was a pioneer simple formula still currently most widely used as a biodiversity index. Species, functional, and phylogenetic diversity indices then emerged from the feeling that combining species richness with the distribution of species' abundances, functional and/or phylogenetic characteristics could provide indices that are more linked to ecological phenomena than species richness alone. As there are a myriad of indices developed so far, it seems impossible to explore all of them in the limited time of a research study. Selecting a single index or a limited set of indices most relevant to a given study requires methods to facilitate comparisons among indices and among their mathematical properties that reflect their different ways of summarizing biodiversity in a single numeric value. Parametric guiding indices that generalize and connect existing indices trace paths through the jungle of indices. The development of such guiding indices could thus be advantageously strengthened. As a step in this direction, I develop a parametric guiding index that generalize both well-known indices of species diversity and well-known indices of functional/phylogenetic diversity. I show under which conditions the parametric guiding index satisfies some critical mathematical properties: non-negativity; weak species monotonicity; full concentration or full similarity at minimum; evenness in species' abundance or evenness in species' effective originality at maximum; embedding of species richness and diversity. Applying the parametric guiding index on bat diversity, I show that analyzing the mathematical properties of potential biodiversity indices is critical to evaluating their relevance with regards to study objectives in ecology.