Diversity in a fuzzy world: a review of models and measures

Biological diversity metrics have evolved from simple species counts to measures that incorporate increasingly complex functional information. A critical aspect is that different families of diversity indices are grounded in distinct mathematical models for representing uncertainty, including classical set theory, probability theory, and fuzzy set theory, which shape what these measures can reveal about community structure. In this paper, we propose a unified theoretical framework that links commonly used diversity indices to their underlying models of uncertainty, thereby clarifying the ecological and mathematical meaning of the information they summarize. As one of our main results, we show that Rao’s quadratic diversity, one of the best known indices of functional diversity, can be interpreted as a measure of strife or discord for a set of functionally overlapping species. We further demonstrate that Rao’s diversity can be expressed as the mathematical expectation of the fuzzy specificity (or functional distinctiveness) of the individual species in the community, thereby revealing a closer ecological and statistical connection between these two forms of uncertainty than previously recognized. Overall, our approach provides an integrated perspective on the relationship between uncertainty theory and biodiversity measurement, offering new insights for understanding ecological processes and for developing more coherent and informative biodiversity indices.