Formalizing biodiversity measures and guiding through them

Facing the myriad of indices claimed to be biodiversity indices that have been proposed by the scientific community, it becomes critical to analyze the mathematical properties that determine whether any index respond to biological necessities about biodiversity quantification. In ecology, species richness, the number of species, was a pioneer simple formula still currently most widely used as a biodiversity index. Species, functional, 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. Although there are theoretically an infinite number of potential biodiversity indices, I show here that some widely used indices of biodiversity actually take on large values in the absence of diversity. I thus claim for stricter definition of mathematical formulas that can be classified as biodiversity indices and for the use of parametric guiding indices that generalize and connect existing indices, tracing paths through the jungle of indices. As a case study I develop a parametric guiding index, and applying it on bat diversity, I show that analyzing the mathematical properties of potential biodiversity indices is critical to evaluate their relevance as regards a study objective in ecology. I come to the conclusions that to improve applications of biodiversity indices in ecology we need to: stop claiming for a single best formula; acknowledge that biodiversity indices are above all diversity indices, with the particularity that they are applied to biological data; agree on a set of minimal conditions that any biodiversity index must fulfill; integrate a large panel of diversity indices in software packages to enable ecologists to select and apply the most relevant index for their study. I propose a set of minimal conditions as a first step in this direction.