Computing β-diversity with Rao's quadratic entropy: a change of perspective
Ecologists have traditionally viewed β-diversity as the ratio between γ-diversity and average α-diversity. More recently, an alternative way of partitioning diversity has been proposed for which β-diversity is obtained as the difference between γ-diversity and average α-diversity. Although this additive model of diversity decomposition is generally considered superior to its multiplicative counterpart, in both models β-diversity is a formally derived quantity without any self-contained ecological meaning; it simply quantifies the diversity excess of γ-diversity with respect to average α-diversity. Taking this excess as an index of β-diversity is a questionable operation. In this paper, we show that a particular family of α-diversity measures, the most celebrated of which is Rao's quadratic entropy, can be adequately used for summarizing β-diversity. Our proposal naturally leads to a new additive model of diversity for which, given two or more sets of plots, overall plot-to-plot species variability can be additively partitioned into two non-negative components: average variability in species composition within each set of plots and the species variability between the set of plots. For conservation purposes, the suggested change of perspective in the summarization of β-diversity allows for a flexible analysis of spatial heterogeneity in ecological diversity so that different hierarchical levels of biotic relevance (i.e. from the genetic to the landscape level) can be expressed in a significant and consistent way.