Distance-based functional diversity measures and their decomposition: a framework based on Hill numbers.

Anne Chao,Chun-Huo Chiu

doi:10.1371/journal.pone.0100014

Abstract

Hill numbers (or the “effective number of species”) are increasingly used to characterize species diversity of an assemblage. This work extends Hill numbers to incorporate species pairwise functional distances calculated from species traits. We derive a parametric class of functional Hill numbers, which quantify “the effective number of equally abundant and (functionally) equally distinct species” in an assemblage. We also propose a class of mean functional diversity (per species), which quantifies the effective sum of functional distances between a fixed species to all other species. The product of the functional Hill number and the mean functional diversity thus quantifies the (total) functional diversity, i.e., the effective total distance between species of the assemblage. The three measures (functional Hill numbers, mean functional diversity and total functional diversity) quantify different aspects of species trait space, and all are based on species abundance and species pairwise functional distances. When all species are equally distinct, our functional Hill numbers reduce to ordinary Hill numbers. When species abundances are not considered or species are equally abundant, our total functional diversity reduces to the sum of all pairwise distances between species of an assemblage. The functional Hill numbers and the mean functional diversity both satisfy a replication principle, implying the total functional diversity satisfies a quadratic replication principle. When there are multiple assemblages defined by the investigator, each of the three measures of the pooled assemblage (gamma) can be multiplicatively decomposed into alpha and beta components, and the two components are independent. The resulting beta component measures pure functional differentiation among assemblages and can be further transformed to obtain several classes of normalized functional similarity (or differentiation) measures, including N-assemblage functional generalizations of the classic Jaccard, Sørensen, Horn and Morisita-Horn similarity indices. The proposed measures are applied to artificial and real data for illustration.

Highlights

Functional diversity quantifies the diversity of species traits in biological communities, and is widely regarded as a key to understanding ecosystem processes and environmental stress or disturbance [1,2,3,4,5,6,7,8,9,10,11]
The measure based on additively partitioning quadratic entropy (Eq 2c) yields higher differentiation for Case II, we have demonstrated its counter-intuitive behavior in Appendix S5 and in Example 1
A more general property of monotonicity is proved in Appendix S2 (Proposition S2.2): any differentiation measure based on our functional beta diversity is a non-decreasing function with respect to the distance of any non-shared species pair regardless of species abundance distributions

Summary

Introduction

Functional diversity quantifies the diversity of species traits in biological communities, and is widely regarded as a key to understanding ecosystem processes and environmental stress or disturbance [1,2,3,4,5,6,7,8,9,10,11]. The measure qD(Q) can be interpreted as ‘‘the effective number of abundant and (functionally) distinct species’’ with a constant distance Q for all species pairs. If qD(Q) = v, the functional Hill number of order q of the actual assemblage is the same as that of an idealized assemblage having v abundant and distinct species with a constant distance Q for all species pairs; see Table 1 for illustration. We define the column (or row) sum as our proposed measure of mean functional diversity (per species), qMD(Q), of order q: qMDðQÞ~1⁄2qD(Q)|Q, ð4aÞ which quantifies the effective sum of pairwise distances between a fixed species and all other species (plus intraspecific distance if exists).

À 1 Á2

S1 X S2

Conclusion and Discussion