Code and data from: Measuring the overall functional diversity by aggregating its multiple facets: functional richness, biomass evenness, trait evenness, and dispersion

Human activities induce environmental changes, which can affect individuals' traits and then lead to changes in functional diversity and finally in ecosystem functioning. Measuring functional diversity is thus of utmost importance to understand the consequences of such activities on ecosystem functioning. Functional diversity is composed of several facets, but these facets are almost always measured individually and we lack a metric capturing the overall, multifaceted functional diversity. We consequently developed an index K of the overall functional diversity defined as the geometric mean of four independent facets: functional richness (the classic measure of the coverage over the trait axis), biomass evenness, and trait evenness (quantifying how evenly filled the biomass and trait distributions, separately) and dispersion (quantifying the spread around the biomass-weighted mean trait, which is maximised for uniform and bimodal distributions). K and each of its underlying facets take values between 0 and 1 and assume the uniform distribution to yield maximal diversity. We compared K to other, more classic metrics measuring a single facet of functional diversity by calculating all these indices for randomly and non-randomly generated communities. We showed that K overcomes several limitations of other indices (e.g. lack of accuracy, not computable for simple communities, unclear ecological interpretation), and was well correlated with ecosystem functions in simulated predator-prey communities. In addition, decomposing K into its underlying facets revealed that ecosystem functions can be driven by different facets of K on different trophic levels. The strength of our index K lies in being the only index that measures the overall functional diversity by combining several facets and providing the option to decompose K into them. This notably yields mechanistic insights about which facets are more important for driving changes in functional diversity and ecosystem functioning.

Methods
In the manuscript related to these scripts and data, we aimed to present a new index to measure the overall functional diversity (that we call K). Shortly, K is the geometric mean of four independent facets: functional richness (the classic measure of the coverage over the trait axis; noted FRic), biomass evenness, and trait evenness (quantifying how evenly filled the biomass and trait distributions, separately; noted BE and TE respectively) and dispersion (quantifying the spread around the biomass-weighted mean trait, which is maximised for uniform and bimodal distributions; noted Dis). K and each of its underlying facets take values between 0 and 1 and assume the uniform distribution to yield maximal diversity. We used other indices published in the literature to compare our new index K: functional evenness (FEve), functional dispersion (FDis), functional divergence (FDiv), Rao's quadratic entropy (Rao), Functional extension and evenness (FEE) and its modified version (FEEc). For a complete description and mathematical equations, see the manuscript.

We used three types of datasets, which all provided information about communities composed of a given number of species,  each characterised by biomass and trait values:

defined communities corresponding to a few established test series in the literature, which enabled us to check that our newly-developed index K behaved as expected.
randomly generated communities, i.e., the biomass and trait values of each species part of a community were drawn independently from a lognormal distribution with a mean of 0 and a standard deviation of 1, and from a uniform distribution within [0,1], respectively. This dataset was used to check the intrinsic dependencies of indices of functional diversity with the number of species present in a community but also proved that the four underlying facets of K were independent.
non-randomly generated communities, i.e., we used the equations of a modified Rosenzweig-MacArthur, predator-prey model to generate communities with two trophic levels (prey and predator), where the number of species can be varied and the adaptation within species can be enabled or disabled for each trophic level, independently. In addition to getting biomass and trait values of each species present in the system, four ecosystem functions were computed at each trophic level: prey and predator total biomass, and prey and predator production. This dataset was used to explore the dependencies of indices of functional diversity among each other and with species richness under non-random processes and to estimate the predictive power of these indices for ecosystem functioning.

The data of the non-random communities were already published in another data depository, where the details of calculations and data structure used in this present study and the original study cf. scripts and data in related work). In this present study we additionally calculated several indices of functional diversity (see some detail below and detail in the manuscript) and computed the Spearman's rank correlation coefficients (rs) among these indices per trophic level, and between these indices of the two trophic levels and four ecosystem functions.

In addition, we presented a case study in the Appendix to illustrate how our index can be applied to phytoplankton communities of Lake Constance. We notably compiled biovolume and 5 traits (cell volume, longest linear dimension, maximum growth rate, phosphate affinity, and defence) time series.