Online Resource 1. Calculation of MPD, Rao and Simpson with 5 species from two functional groups (legumes=1, non-legumes=0) having either equal (Case1) or unequal (Case2) relative abundance. A trait dissimilarity (right side of the figure) is computed indicating how different species are in their traits (we consider a case that dissimilarity between functional groups is 1 and within groups is 0). In the example, pi and dij represent, as explained in the main text, respectively the relative abundance of species and the trait dissimilarity between a given pair of species.

This example shows a simple illustration of calculation of the indices mentioned in the main text of paper. Given 5 species, belonging to one of two different functional groups (e.g. legumes vs. non-legumes). First, a trait dissimilarity matrix, indicating trait differences for each pair of species is computed. This is a zero-diagonal (assuming the dissimilarity within a species is 0) symmetric matrix. It should be noted that, usually the distance between species is standardized between 0 and 1, both for single traits and for combinations of multiple traits. Under this standardization, 1 represents the maximum differentiation between species (species are functionally completely different) and 0 represents the case in which the species are functionally equivalent (Lepš et al. 2006; Pavoine et al. 2009; de Bello et al. 2013). This dissimilarity matrix can be computed using the mean trait values of species, for example with the Gower distance, which can be used for both quantitative and categorical traits (Pavoine et al. 2009). Alternatively, species dissimilarities can be computed for each trait as 1 minus the proportional overlap of the probability density functions of traits, which likewise produces a dissimilarity scale between 0 and 1(Lepš et al. 2006; de Bello et al. 2013). In this approach, as for the Gower distance, the diagonal in the dissimilarity matrix is generally set to zero, although it could be populated with dissimilarity values within species (de Bello et al. 2011).

In the example in figure above, if we assign a trait value equal to 1 to legumes and a 0 value to non-legumes, the dissimilarity between two species equals 1 when the two species are not in the same functional group (one species is a legume and the other not; the same result would be obtained with a value of 0 for legumes and a 1 value for non-legumes). Let us assume, first, that all the species have the same abundance (relative abundance for each species being in this case 1 divided by N, the number of species, i.e. 1/5=0.2). MPD, equal to 0.6, is simply the average of trait dissimilarity values extracted from one of the two symmetrical triangles composing the trait dissimilarity matrix (of course either triangle can be used to achieve the same result). Note that, as mentioned above,the diagonal of this matrix, populated by zero values, is excluded. As a second example, let us suppose that species have different abundances (the first accounting for 50% of the total abundance, the second 25%, the third and forth 10% and the fifth 5%). In this case, the difference between the first and second species (which is 1) has greater weight, because these species are more abundant. The MPD in this case is 0.72.

References

de Bello, F., Lavorel, S., Albert, C.H., Thuiller, W., Grigulis, K., Dolezal, J., Janecek, S. & Lepš, J. (2011) Quantifying the relevance of intraspecific trait variability for functional diversity. Methods in Ecology and Evolution,2, 163-174.

de Bello, F., Carmona, C.P., Mason, N.W.H., Sebastia, M.-T. & Lepš, J. (2013) Which trait dissimilarity for functional diversity: trait means or trait overlap? Journal of Vegetation Science,24, 807-819.

Lepš, J., de Bello, F., Lavorel, S. & Berman, S. (2006) Quantifying and interpreting functional diversity of natural communities: practical considerations matter. Preslia,78, 481-501.

Pavoine, S., Vallet, J., Dufour, A.B., Gachet, S. & Daniel, H. (2009) On the challenge of treating various types of variables: application for improving the measurement of functional diversity. Oikos,118, 391-402.

Online Resource 2.Results of simulations (see R code below) showing the expected relationship between the number of species on MPD and Rao, using both presence/absence data or abundance data. The script of the simulation is provided. We attributed abundance simulating possible values for the species already present in a community, i.e. ‘sample(c(1, 5, 10, 50, 100)’. As observed in natural communities, greater abundances are more rare so we gave less probability for greater values to be selected, i.e. ‘prob=c(100, 50, 10, 10, 5)’.

###script simulations - relation indices with species diversity####

library(ade4)

library(picante)

library(FD)

library(vegan)

trait<-1:100# creation of 100 species with trait value from 1 to 100

diss<-dist(trait)/99#dissimilarity matrix as in Gower#

#max(diss)

nspplots<-vector()

plots<-matrix(0, 100, 1000)

rownames(plots)<-c(1:100)

for(i in 1:1000){

nsp<-sample(3:50, 1)# richness each plot can be between 3 and 50

oneplot<-sample(c(0:1), size=100, prob=c(100-nsp, nsp), replace=T)

nspfin<-sum(oneplot)

plots[, i]<-replace(oneplot, oneplot>0, sample(c(1, 5, 10, 50, 100), nspfin, prob=c(100, 50, 10, 10, 5), replace=T))## species can have 5 abundance levels (1, 5, 10, 50, 100) with the bigger values being much less probable###

nspplots[i]<-nspfin

}

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