Biological diversity: distinct distributions can lead to the maximization of Rao's quadratic entropy

Rao's quadratic entropy (QE) is a diversity index that includes the abundances of categories (e.g. alleles, species) and distances between them. Here we show that, once the distances between categories are fixed, QE can be maximized with a reduced number of categories and by several different distributions of relative abundances of the categories. It is shown that Rao's coefficient of distance (DISC), based on QE, can equal zero between two maximizing distributions, even if they have no categories in common. The consequences of these findings on the relevance of QE for understanding biological diversity are evaluated in three case studies. The behaviour of QE at its maximum is shown to be strongly dependent on the distance metric. We emphasize that the study of the maximization of a diversity index can bring clarity to what exactly is measured and enhance our understanding of biological diversity.