Eigenvector estimation of phylogenetic and functional diversity

Authors


Correspondence author. E-mail: diniz@icb.ufg.br; jafdinizfilho@gmail.com

Summary

1. Although species richness is the most common metric for biodiversity, it is important to consider that among-species differences matter for many important processes such as ecosystem functioning and assembly patterns. Thus, alternative metrics have been proposed to measure phylogenetic (PD) and functional diversity (FD).

2. Here, we analysed the correlation structure and the geographic patterns of different phylogenetic and functional diversity metrics using 1000 Carnivora (mammals) assemblages distributed world-wide. We also proposed a general approach to estimate these metrics based on phylogenetic eigenvector regressions.

3. We showed that the cumulative variance of phylogenetic eigenvectors within assemblages converges to one metric of phylogenetic clustering, the phylogenetic species variability (PSV), which was recently proposed in the literature. Phylogenetic eigenvectors have also been used to model trait variation (i.e. body mass); therefore, the same reasoning allows us to decouple trait diversity (i.e. body mass) into phylogenetic [FD(P)] and specific [FD(S)] components. The cumulative variance of these components within assemblages, for a single trait or multiple traits, offers a direct estimate of the evolutionary and ecological components of functional variation. Our results indicated that the variance of the phylogenetic component of body mass [FD(P)] estimated within assemblages was highly correlated with a common measure of functional diversity (Rao’s Q).

4. Our general approach based on phylogenetic eigenvectors provides similar results when compared to other metrics of FD and PD, but also has some important advantages. First, it allows a direct interpretation of at which hierarchical level in the phylogeny (expressed by different sets of eigenvectors) the patterns in PD and FD appear. Secondly, phylogenetic components of FD can be analysed directly because of the partition of functional diversity into FD(P) and FD(S), so it eliminates the need to generate ‘a posteriori’ phylogenetic correlations between PD and FD based on independently derived metrics, as well as of these metrics with other components of environmental variation.

Introduction

Species richness has been used as a surrogate for all facets of biodiversity. Nevertheless, the appropriateness of this approach is held in check because among-species differences matter for many important processes such as ecosystem functioning and assembly patterns (Hooper et al. 2005; Petchey et al. 2007; Cianciaruso et al. 2009; Safi et al. in press). As a consequence, there is a growing interest in alternative representations of biodiversity patterns at local, regional and global scales, including patterns of functional and phylogenetic diversity (Webb et al. 2002; Petchey & Gaston 2006; Vamosi et al. 2009; Devictor et al. 2010). Although all these patterns are somehow related to each other, they can reveal different processes associated with their origin and maintenance (Wainwright 2007; Safi et al. in press).

Decoupling functional and phylogenetic diversity is a challenging task which has gained attention among ecologists and evolutionary biologists (e.g. Harmon et al. 2003; Pausas & Verdú 2010; Pillar & Duarte 2010). More than just separating the confounding effects of time and species relatedness on species’ traits and functional diversity (see Wainwright 2007), such a framework could allow us to better understand the ecological and evolutionary mechanisms determining biological diversity. It is intuitive that trait diversity among species should be affected by their evolutionary history. Indeed, there is evidence showing that niche-related ecological traits tend to be conserved among species over time (niche conservatism hypothesis; reviewed in Wiens & Graham 2005 and Wiens et al. 2010). However, this phylogenetic structure (expected from pure phylogenetic signals and/or because of niche conservatism) cannot be assumed a priori because trait convergence among distantly related species is also an important process (see Webb et al. 2002 and Pausas & Verdú 2010 for reviews). Yet, we cannot ignore the role of environment and species interactions on the levels of trait variability among species. Environmental variability can lead to phenotypic diversification and subsequent speciation by allowing different specialized types to evolve and coexist (Ackermann & Doebeli 2004). Moreover, environmental filters select those species that can persist within a community on the basis of their tolerance to the prevailing abiotic conditions, assembling species with similar niches (Weiher & Keddy 1995; Fukami et al. 2005). Knowing these patterns and the underlying processes which drive them is also important for practical applications, such as establishing conservation priorities, increasing the likelihood of conserving biodiversity patterns and evolutionary processes at multiple levels (Carvalho et al. 2010; Devictor et al. 2010).

Phylogenetic diversity (PD) reveals how evolutionary variability accumulates in a region or assemblage, and clustering patterns may be important to reveal distinct combinations of speciation and extinction processes at community and macroecological scales (Webb et al. 2002; Davies et al. 2008; Vamosi et al. 2009). More recently, PD has been associated with processes driving broad scale patterns of species richness, mainly niche conservatism (Cooper, Jetz & Freckleton 2010; Wiens et al. 2010). As a result of this process, species tend to retain their ancestral niche, which in turn determines their expansion and colonization ability under changing environments. Accordingly, diversity in new environments should appear as a product of niche differentiation followed by relatively rapid diversification of successful phenotypes (as in a process of adaptive radiation – see Schluter 2000), so that assemblages will tend to be phylogenetically clustered in these regions (low PD).

Functional diversity (FD), however, is usually estimated as the variety of ecological and life-history traits of the species in a local assemblage (Petchey & Gaston 2006). These different components of diversity are related in complex ways and across multiple geographic scales, under different environmental and evolutionary aspects. It is expected, for example, that increasing diversification time under random (neutral) diversification processes increases both richness and FD. On the other hand, under adaptive processes, the relationship between FD and PD may be more complex, as some of the phenotypes will appear more frequently; therefore, FD should present a more asymptotic relationship with PD (see Safi et al. in press).

Despite the importance of understanding these different components of biodiversity, there is still a debate on how they should be measured, and the available metrics seem to capture distinct aspects of each one. For instance, PD is frequently measured using Faith’s (1992) index, which is the sum of the branch lengths of a cladogram linking the species within a given region or assemblage (e.g. Sechrest et al. 2002; Diniz-Filho 2004). Because of this sum, a strong curvilinear (and undesirable) relationship appears between PD and species richness (see Schweiger et al. 2008 for a review). Other indices, such as the phylogenetic species variability (PSV; Helmus et al. 2007) or Delta+ (Warwick & Clarke 1998), were developed to be intrinsically independent of species richness. PSV, for example, quantifies how phylogenetic relatedness decreases the variance of a hypothetical neutral trait shared by all species in the community, so that a reduction in this total variance is expected if species in a sample are strongly phylogenetically related. Moreover, the branch lengths of a phylogenetic tree could be transformed to match more complex evolutionary processes of a trait (Diniz-Filho 2004), so that phylogenetic diversity could be weighted by a trait evolutionary model as well.

Similar discussions about the relationship between functional diversity and species richness exist when computing FD by using Petchey & Gaston’s (2002, 2006) approach (Poos, Walker & Jackson 2009). This is because the species’ traits are used to generate a hierarchical dendrogram and FD is then estimated by the sum of branch lengths linking species on this dendrogram (analogous to Faith’s PD method). Yet, there are many FD indices which were developed to estimate functional diversity independently of species richness (for example, Rao’s quadratic entropy see Botta-Dukat 2005; Fdis Laliberté & Legendre 2010). Further discussions about the advantages and disadvantages of these indices can be found elsewhere (see Schweiger et al. 2008; Petchey & Gaston 2006; Mouchet et al. 2010; for reviews).

Here, we show that both PD and FD can be calculated using simple metrics based on eigenvectors extracted from a phylogenetic distance matrix among species, which is the basis of a method called phylogenetic eigenvector regression (PVR, see Diniz-Filho, De Sant’ana & Bini 1998). We showed using empirical data that these metrics are analogous to PSV or Rao’s Q when measuring phylogenetic or trait (i.e. functional) diversity. However, rather than adding one more metric into the plethora of already existing measures of PD and FD, our purpose is to show the rationale behind using eigenvectors extracted from a phylogenetic matrix to derive more informative phylogenetic and functional diversity metrics in an integrated way. As an example, we analysed body mass patterns in terrestrial Carnivora assemblages distributed world-wide. We also show that our eigenvector approach has some important advantages. First, it allows evaluation of the hierarchical levels in the phylogeny at which the patterns of FD and PD appear (i.e. to establish if FD appears because of deep branches in the phylogeny or recent diversification events). More importantly, it is possible to calculate FD for the phylogenetic (that can also include a component attributable to niche conservatism, see Desdevises et al. 2003) and specific (unique) components of trait variation, which can then be mapped to evaluate their spatial patterns (see Diniz-Filho et al. 2009). Thus, it is not necessary to correlate and interpret the relationship between PD and FD based on independently derived metrics (since phylogenetic patterns are incorporated directly into FD).

Materials and methods

Data

To illustrate the metrics and the approach developed here, we used as an example a total of 1000 samples of terrestrial Carnivora (mammals) assemblages world-wide. These assemblages are cells of 1° latitude/longitude randomly scattered throughout the world and were randomly sampled from a total of 12 562 cells covering the globe (see Data S1, Supporting information). It is important to notice that the same results were obtained when all 12 562 cells were used (results not shown to save space; see maps in the Data S1, Supporting information). We choose to use a low number of cells (1000) to allow the use of spatial analyses (see below). Geographic ranges of 209 terrestrial Carnivora were overlaid to these cells and thus allowed us to define species composition in each one (see Diniz-Filho et al. 2009 for detail).

Measures of phylogenetic and functional diversity

We used the time-calibrated supertree of Carnivora (Bininda-Emonds, Gittleman & Purvis 1999; Bininda-Emonds et al. 2007) to assess species’ relatedness. For each assemblage, phylogenetic diversity was initially estimated by both Faith’s (1992) index (PDF) and PSV (Helmus et al. 2007). FD values were calculated with two approaches: dendrogram-based (using euclidian distance and UPGMA algorithm) following Petchey & Gaston (2002, 2006) and pairwise distance-based termed Rao’s quadratic entropy (see Botta-Dukat 2005). We refer to these metrics as FDPG and Rao’s Q, respectively, throughout the paper. Functional diversity is usually calculated for multiple traits, but here we used a single trait to illustrate our approach, the mean log-transformed body mass of the species, which tends to capture much of the ecology and life history in mammals in general, and Carnivora in particular (but see Davies et al. 2007). Due to this particular aspect, we will use the term ‘trait diversity’ from now on whenever it is applicable.

As pointed out by Helmus et al. (2007), PSV estimates phylogenetic diversity as the variance of a trait evolving under a neutral model. If the phylogenetic relationships between species are expressed as a phylogenetic correlation C (i.e. proportion of time shared between pairs of species), then the PSV is given by 1−c, where c is the mean correlation in the matrix C. Thus, if species in a sample are unrelated, their correlation is low and PSV will tend toward one, thus indicating high phylogenetic diversity because species found in this sample tend to be independent.

Measuring phylogenetic diversity by eigenfunction analysis

Following this reasoning, we initially show that PSV can be derived from the variance of eigenvectors extracted from the pairwise phylogenetic distances D (which is negatively related to C) derived from Bininda-Emonds, Gittleman & Purvis (1999) supertree (see Fig. 1). The eigenvectors of D obtained from a principal coordinate analysis (PCORD) (see Legendre & Legendre 1998) express the differences among species at distinct levels of the phylogeny and can therefore be used to evaluate phylogenetic patterns in trait variation (PVR – see Diniz-Filho, De Sant’ana & Bini 1998 and Diniz-Filho et al. 2007, 2009 for a recent application in macroecology and biogeography). As the eigenvectors describe the relationships among species at different levels of the phylogeny, the variance of eigenvectors for the species within a local assemblage is directly related to PSV. However, as each eigenvector expresses different parts of the phylogeny in a vector form, it is necessary to accumulate them to express phylogenetic diversity in the local assemblage. Here, in an attempt to validate our approach in estimating PD, we show that the relationship between the sum of variances of eigenvectors calculated for different assemblages (SUMVAR) and PSV increases with the increase in the number of eigenvectors that are used to calculate SUMVAR. Notice that if a local assemblage contains all species from the regional pool, SUMVAR will converge to the eigenvalues associated with each eigenvector, and PD will be given by the sum of eigenvalues associated with the eigenvectors (‘total inertia’) selected to represent the phylogeny. Like PSV, SUMVAR can be estimated only by using the eigenvectors extracted from a phylogenetic distance matrix (i.e. there is no need for information about any particular trait). As estimates of phylogenetic diversity, both PSV and SUMVAR assume that the phenotypes evolve under a linear (Brownian-like) model throughout the phylogeny.

Figure 1.

 Schematic representation of the methods used in this study. First, phylogenetic (PD) diversity metrics were computed for 1000 randomly selected assemblages following the methods described by Faith (1992) (PDF) and Helmus et al. (2007) (PSV). Total functional diversity [FD(T)] was also computed for the same assemblages according to Botta-Dukat (2005) and Petchey & Gaston (2002), generating the metrics Rao’s Q and FD(P&G), respectively. Our approach starts by extracting the eigenvectors of a phylogenetic distance matrix (D). The selected eigenvectors (X) are used to estimate the phylogenetic diversity (PD) considering the species present in the assemblages (i.e. for a given eigenvector a variance is estimated; then, the variances within cells are summed, generating SUMVAR). Afterward, these eigenvectors are used as predictors in a model whose response variable (Y) is a trait [the same used to estimate FD(T)]. The predicted values (Ŷ) and the residuals (ε) of this model represent the phylogenetic (P) and specific (S) components of the trait (which could be actually used to estimate, separately, any functional diversity index). These components are used to estimate the variances of P and S within cells. In this way, the total functional diversity is partitioned in two components: FD(P) and FD(S), the functional diversity that can be attributed to phylogenetic structure and the FD that can be attributed independently to each species, respectively.

Partitioning functional diversity into specific and phylogenetic components

Comparative analyses aiming to evaluate the phylogenetic structure of a trait measured across species, including the autoregressive method (ARM –Cheverud, Dow & Leutenegger 1985; Gittleman & Kot 1990) and phylogenetic eigenvector regression (PVR –Diniz-Filho, De Sant’ana & Bini 1998), partition the total variance of that trait into phylogenetic (P-component) and specific (S-component) components. The idea is that if a model was correctly fitted, the estimated values from the model capture the proportion of the variation in a trait that can be attributable to phylogenetic relationships among the species, whereas the model residuals express the part of the variation that is unique and independently distributed among species (the S-component). In both ARM and PVR, the S-component is expected to be normally distributed and independently derived among species, which can be tested using autocorrelation indexes (Gittleman & Kot 1990; Diniz-Filho, De Sant’ana & Bini 1998).

In PVR, the estimated and residuals values are obtained by a linear multiple regression model of the form

image(eqn1)

where Y is the trait under analysis, X is the matrix containing the selected eigenvectors derived from phylogenetic distances, β are the regression coefficients and ε are the residuals. The coefficient of determination (R2) of this multiple regression is an estimate of the phylogenetic signal in the data (i.e. the proportion of trait variance explained by the phylogenetic eigenvectors), and the estimated and the residuals values of this model are estimates of the P- and S-components, respectively. The critical step here is how to select eigenvectors, as using all k (k = n−1; where n = number of species in the phylogeny) eigenvectors will saturate the model (Rohlf 2001). The best approach lies in selecting the smallest number of eigenvectors that makes the residuals phylogenetically independent (Diniz-Filho & Tôrres 2002; see Griffith & Peres-Neto 2006 for a similar application in the context of spatial analyses, and Safi & Pettorelli 2010 for a recent proposal). Martins, Diniz-Filho & Housworth (2002) showed that PVR possess acceptable statistical performance using simulations, especially at small sample sizes, and that PVR can be a useful method especially because of the flexibility that eigenanalysis provides to fit models under distinct evolutionary processes.

Here, we propose to partition the functional or trait diversity from a set of species found in assemblages into P- and S-components, following the same logic described above to estimate phylogenetic diversity according to PSV. Specifically, if a set of traits is regressed against phylogenetic eigenvectors and then decoupled into P- and S-components (using the estimated and residual values, respectively), it is possible to calculate the variances of these two components for each trait in local assemblages. These variances provide direct measures of phylogenetic and specific components of FD, called FD(P) and FD(S) here, respectively (see Fig. 1). Calculating variances of P- and S-components from PVR is analogous to our PSV estimate performed previously because the P-component expresses phylogenetic variation of the traits under study, whereas the cumulative variance extracted directly from eigenvectors represents only the phylogenetic structure, in terms of dependence among species within a local assemblage. In other words, they quantify how much the species’ trait variance is explained by their ancestors (P-component, which includes ancestral adaptive components, constraints and neutral evolution) and how much is explained by recent ecological processes occurring independently in each species (S-component).

For the Carnivora body mass, we estimated P- and S-components using a total of 14 eigenvectors that were enough to minimize the autocorrelation (as measured by the Moran’s I autocorrelation coefficient) in the model residuals (see Griffith & Peres-Neto 2006 and Bini et al. 2009, for analogous reasoning in spatial analyses using eigenvectors from distance matrices). The variance of P- and S-components was then calculated for each of the 1000 assemblages, as previously described. Notice that if multiple traits were available, then FD(S) and FD(P) would also be approximated by the sum of the variances of S- and P-components for all traits within assemblages.

Spatial patterns of functional and phylogenetic diversity

For simplicity, we analysed latitudinal patterns in all metrics by relating each one with the latitudinal centroids of the assemblages distributed world-wide (see Data S1, Supporting Information). The significance levels associated with the correlation coefficients that we estimated are affected by spatial autocorrelation (see Legendre 1993), so all P-values reported in this study were obtained by using the geographically effective degrees of freedom v* based on Moran’s I correlograms (see Dutilleul 1993; Legendre & Legendre 1998).

Computational tools

PSV and PDF were calculated using the functions psv() and pd() in the package picante (Kembel et al. 2010), FDPG with the function treedive() in the package vegan (Oksanen et al. 2010), and Rao’s Q with the function dbFD() in the package FD (Laliberté & Legendre 2010) all implemented in R (R Development Core Team 2010). Phylogenetic eigenvectors, variance of FD(P), variance of FD(S) and v* were calculated with the use of spatial analysis in macroecology (sam) software, freely available at http://www.ecoevol.ufg.br/sam (see Rangel, Diniz-Filho & Bini 2006, 2010).

Results

Faith’s (1992) index (PDF) was moderately correlated with PSV (r = 0·496; P = 0·015; v* = 21). There was a high relationship between the sum of the variances of the eigenvectors in the assemblages (SUMVAR) and PSV, which tended towards 1·0 as more eigenvectors were added (Fig. 2). There was a minor local peak of correlation when 10 eigenvectors were used (i.e. at this point cumulative eigenvalues, indicating the amount of phylogenetic patterns represented by the eigenvectors increased from c. 82% to 90%), but correlation still tended to increase in general and stabilize around 0·97–0·98 when SUMVAR was obtained using more than 10 eigenvectors. Notice that because we were not analysing any particular patterns in PD here (i.e. just understanding the relationship among the different metrics), it was not necessary to select a particular set of eigenvectors and obtain a single SUMVAR. Rather, it was more interesting to evaluate how the correlations between PSV and SUMVAR changed when more eigenvectors were added. Indeed, the nonlinearity in the accumulation curve (Fig. 2) suggests a good approximation of patterns even if the first 5–10 eigenvectors were used to derive SUMVAR.

Figure 2.

 Pearson’s correlation coefficients between the variances of the eigenvectors (extracted from Carnivora supertree) and phylogenetic species variability (both estimated for each cell) as a function of the number of eigenvectors retained for the analyses (k).

The correlation between PSV and the variance calculated by using only the first eigenvector [SUMVAR(1)] across the 1000 assemblages was very high (r = 0·893; < 0·001; v* = 21). Even so, the geographic patterns in the two metrics were different (Fig. 3). There was a strong nonlinear relationship between PSV and latitude (Fig. 3a), in which northern assemblages showed a much smaller phylogenetic diversity, and PSV values were stable before a given threshold of latitude. However, SUMVAR(1) showed a less abrupt decrease of phylogenetic diversity towards northern latitudes (Fig. 3b). A nonlinear latitudinal pattern in SUMVAR (resembling a piecewise relationship), similar to that observed for PSV, was again evident when the first 10 eigenvectors were used (Fig. 3c), as expected because the two metrics became strongly correlated (see Fig. 2).

Figure 3.

 Latitudinal patterns of phylogenetic diversity as estimated by phylogenetic species variability (a), the variance of the first eigenvector (b) and the sum of the variances of the first 10 eigenvectors (c).

When extending the above reasoning to functional or trait diversity, it was first necessary to analyse the phylogenetic patterns in Carnivora body mass using PVR (i.e. a regression of body mass on eigenvectors). A total of 14 eigenvectors were necessary to reduce residual Moran’s I to a value smaller than 0·05, and the coefficient of determination of this model indicated that c. 70% of the variance in body mass was explained by the selected phylogenetic eigenvectors, revealing a very strong phylogenetic signal (see also Diniz-Filho et al. 2009). There was a nearly quadratic pattern of mean body mass across latitude, with large-bodied assemblages found in the tropics (Fig. 4a). Because of the strong phylogenetic signal, patterns in P-components were quite similar to those from mean body mass (Fig. 4b). On the other hand, patterns in the mean S-component indicated that large mean positive deviations, from body mass expected by phylogenetic relationships, appeared more often in the northern part of the globe (Fig 4c).

Figure 4.

 Latitudinal patterns of mean log-transformed body mass of Carnivora (a) and the two phylogenetic eigenvector regression components of this trait, including mean P-component (b) and S-component (c), as estimated by the phylogenetic eigenvector regression across 1000 randomly selected cells.

As expected, functional or trait diversity calculated with Petchey & Gaston’s (2002, 2006) metric was strongly correlated with richness (r = 0·73; P < 0·001; v* = 60) across assemblages because it sums the branch lengths along the functional dendrogram, whereas Rao’s Q was independent of richness (r = −0·31; P = 0·165; v* = 19). When FD was calculated using the components of body mass (S and P) obtained by PVR, the variance of P-components [FD(P)] (i.e. the variance of body mass values explained by phylogeny) was strongly and significantly correlated with Rao’s Q (r = 0·82; P < 0·001; v* = 12), whereas the correlation between Rao’s Q and the variance of S-component [FD(S)] was non-significant (r = 0·48; P = 0·068; v* = 13).

There was a latitudinal pattern of functional diversity when Rao’s Q was used (Fig. 5a), with a nonlinear increase of FD in northern latitudes. As expected because of the correlation between Rao’s Q and the FD(P) values, the phylogenetic component also increased towards northern latitudes (Fig. 5b) and a much weaker geographic signal was observed for the S-component (Fig. 5c). Indeed, RaoQ and PSV were correlated (Fig. 6) (r = −0·628; P = 0·009; v* = 14), so that regions with lower phylogenetic diversity in northern parts of the word also tend to have more functional diversity (see Safi et al. in press).

Figure 5.

 Latitudinal patterns of different metrics used to estimate functional diversity (FD) of log body mass of Carnivora: Rao’s Q (a); FD based on the phylogenetic (b) and specific (c) components of body mass.

Figure 6.

 Relationship between Rao’s Q and phylogenetic species variability. Each point represents an assemblage (n = 1000).

Discussion

There is an increasing interest in broad scale patterns in functional and phylogenetic diversity, and different metrics have been used to evaluate them (Stevens et al. 2003; Helmus et al. 2007; Petchey et al. 2007; Devictor et al. 2010). Here, we showed that the sum of the variances of phylogenetic eigenvectors (SUMVAR) also expresses phylogenetic diversity and is strongly correlated with PSV, a previously developed metric that is independent of species richness (a property usually considered important: see Schweiger et al. 2008; Poos, Walker & Jackson 2009; Mouchet et al. 2010). Although PVR does not assume an evolutionary model (but see Rohlf 2001), it is expected that phenotypic variation among species increases with time since divergence as in a Brownian motion model, and this is why the variance of eigenvectors is related to PSV (which explicitly assumes that diversity can be approximated by a Brownian motion model: see Helmus et al. 2007).

It is important to recognize that our approach can give additional insights when analysing geographic patterns in FD and PD. For instance, both PSV and SUMVAR showed a similar latitudinal trend, in which phylogenetic diversity decreased towards the northern hemisphere (Diniz-Filho 2004; Diniz-Filho et al. 2009), indicating more clustered assemblages (i.e. species in these assemblages are more phylogenetically related than expected by chance). The correlation between SUMVAR and PSV increased nonlinearly with the number of eigenvectors used to estimate SUMVAR. However, this should not be viewed as a simple methodological artefact of the way SUMVAR is calculated (i.e. summing more eigenvectors gives more information on the structure of the phylogeny, so it converges to PSV). This is because eigenvectors describe the phylogenetic relationship among species in a hierarchical way and by considering a range of time scales, in which the first eigenvectors tend to describe deeper relationships in time (i.e. variation among distantly related species or taxa). Thus, when only the first eigenvector was used to calculate SUMVAR, the phylogenetic diversity decreased more linearly with respect to latitude. However, when SUMVAR was calculated with more eigenvectors, it approached PSV and both showed a strong nonlinear geographical pattern, with a conspicuous decrease of phylogenetic diversity after c. 25°N. Thus, the different patterns depicted by SUMVARs calculated using an increasing number of eigenvectors support the idea that, at a deeper phylogenetic level, the latitudinal gradient is caused by first diversifications in the Carnivora history, at the base of the clade. Also, the nonlinearity of the spatial pattern is caused by a faster increase in the number of recent lineages in the tropics, coupled with more phylogenetic turnover in the northern temperate regions (see Valkenburgh, Wang & Damuth 2004). This is in line with recent evaluations of the evolutionary drivers of diversification across latitudinal gradients, and it is expected that these drivers are also related to changes in species’ traits, including body mass (Diniz-Filho et al. 2009). Thus, these patterns should arise in functional diversity as well if only the P-component of trait variation is taken into account, as performed here.

Our results are also clear in terms of revealing the methodological advances that can be achieved by describing directly the phylogenetic patterns in functional diversity (rather than deriving these FD and PD metrics independently and correlating them ‘a posteriori’, or using the residuals of the relationship between FD and PD (see Devictor et al. 2010 and Safi et al. in press; for examples). Previous analyses of body mass variation along local Carnivora communities using PVR (see Diniz-Filho et al. 2007, 2009) revealed interesting Bergmannian patterns, which can be directly associated with the FD patterns discussed here. Mean body mass in Carnivora assemblages does not follow the expected pattern under Bergmann’s rule, in which large-bodied species are expected to be found in northern latitudes (actually, the reverse pattern is observed, as detected for all mammals: see Rodríguez, López-Sañudo & Hawkins 2006; Rodríguez, Olalla-Tárraga & Hawkins 2008). However, as most explanations for Bergmann’s rule are based on adaptive processes driving species towards large bodies when occurring in colder parts of the world, this rule is expected to hold not for mean body mass of assemblages, but rather for mean deviations from ancestors. This is what the S-component of PVR captures, the unique component of each species’ trait after speciation. Indeed, the pattern expected under Bergmann’s rule is most clearly shown by this component (Diniz-Filho et al. 2007, 2009; Ramirez, Diniz-Filho & Hawkins 2008; Terribile et al. 2009; Olalla-Tárraga et al. 2010).

One advantage in using the eigenvector approach is that it is possible to evaluate which component (i.e. phylogenetic or specific) contributes most to overall trait diversity. By comparing the spatial patterns in trait diversity measured as the variance in body mass (which corresponds closely to Rao’s Q) and diversity calculated directly from P- and S-components [FD(P) and FD(S)], it is possible to observe that patterns in trait diversity are because of inherited phylogenetic components, and not because of the unique and recent deviations from species from ancestral conditions. Rao’s Q was much more correlated with FD(P) than with FD(S), and this is expected under the strong phylogenetic component in body mass, so that most of the variation in variance among assemblages, at least at this broad geographical scale, can be better explained by the historical patterns in body mass. The relationship between Rao’s Q and FD(P) thus suggests that the component of functional diversity is not driven by recent diversification, but rather by the presence of older lineages within assemblages that retain their ancestral functional stages. Under niche conservatism, these patterns can also be related to environmental changes through time, which in turn tend to be correlated with the current climate (thus forming a north-south cline, tracking climatic and environmental variation from tropical to temperate regions). Thus, our analyses show that historical signal in body mass is so strong that patterns at assemblage level are mainly driven by deep-time differentiation and this, coupled with north-south gradients in PSV, SUMVAR and metrics for trait diversity, corroborates the niche conservatism hypothesis (Wiens & Graham 2005).

Further analyses trying to correlate PD, and especially FD, with components of environmental variation or anthropogenic effects (see Safi et al. in press), would also be improved by the partition approach used here. For instance, PD and FD are usually correlated with environmental characteristics using comparative methods to take the phylogenetic structure into account. Also, it is possible to distinguish the effects of environmental filtering or biotic interactions (i.e. competition) when assemblages are phylogenetically structured (clustered or overdispersed) by the phylogenetic signal observed in a trait that it is assumedly important to diversity (see Webb et al. 2002). Moreover, further expansions of the approach proposed here can be foreseen. If other traits of components of environmental variation can be measured at species’ level, more complex partition approaches, such as those proposed by Desdevises et al. (2003) can be used and more components of trait or functional diversity, such as niche conservatism, can be directly estimated. Because FD(P) and FD(S) are based on the same reasoning of PSV, this approach opens the possibility of directly mapping the relative components of observed (functional) and expected (PSV) diversity in the assemblages, analogous to metrics used to evaluate phylogenetic signal and niche conservatism (i.e. Blomberg, Garland & Ives 2003K-statistics, whose denominator is the PSV – see also Hof, Rahbek & Araújo 2010; Cooper, Jetz & Freckleton 2010). Finally, it is worthwhile to notice that one can take advantage of the approach we are proposing here to calculate virtually any FD measure available in the literature. By separately using the P- and S-components as the input functional information (vectors P and S showed in Fig. 1) it is possible to produce FD measures representing how species are spread in a purely phylogenetic structured trait space and FD measures representing the purely specific (or unique) trait space, respectively. Exploring these two components of functional diversity should shed light on what the ecological and evolutionary drivers of functional diversity are.

In conclusion, our approach shows that FD and PD can be successfully estimated using eigenvectors derived from a phylogeny. The variance of eigenvectors for the species within Carnivora assemblages was highly correlated with PSV and was independent of species richness. When compared to PSV, it can be useful to show at which hierarchical level phylogenetic diversity appears and how patterns vary in geographic space, as shown for Carnivora assemblages. It is also possible to directly estimate the phylogenetic and specific components in trait variation using PVR and use the variance in these components to understand how FD evolves, without the need of correlating it a posteriori with phylogenetic diversity. Our analyses show the flexibility of the eigenvector approach and illustrate its advantages in directly estimating evolutionary scales of FD and PD and providing insights on how these patterns appear in evolutionary time.

Acknowledgements

We are indebted to Kamran Safi and three anonymous referees for their useful suggestions in an early version of the manuscript. Work by J.A.F. Diniz-Filho and L.M. Bini has been continuously supported by productivity grants from CNPq.

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