# Functional and phylogenetic β diversities and their link with clustering/overdispersion and uniqueness/redundancy

Sandrine Pavoine, Carlo Ricotta

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Sandrine Pavoine, Carlo Ricotta. Functional and phylogenetic β diversities and their link with clustering/overdispersion and uniqueness/redundancy. Ecosphere, 2024, 15 (11), 10.1002/ecs2.70045. hal-05050320

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# Functional and phylogenetic β diversities and their link with clustering/overdispersion and uniqueness/redundancy

Sandrine Pavoine<fnref n="1" /> | Carlo Ricotta<fnref n="2" />

**Correspondence**

Sandrine Pavoine
Email: sandrine.pavoine@mnhn.fr

**Handling Editor:** Michael P. Perring

**Abstract**

In recent decades, research on biodiversity in community ecology has been marked by the consideration of species' evolutionary histories and functional traits. Among the different spatial levels at which functional or phylogenetic (hereafter FP) diversity can be quantified, the definition of the general concept of between-community (β) diversity has been given less attention than that of local, within-community (α) and regional, merged-community (γ) diversities. Here, we develop a new method for partitioning FP β diversity into elementary components to determine how and why FP β diversity differs from species β diversity, with the latter reflecting only differences in species' abundances between communities. As a reference example, we consider two distinct measures of FP β diversity: Rao's dissimilarity coefficient ($Q_β$), which expresses the average FP dissimilarity between communities, and its transformation ($E_β$), which expresses the effective number of distinct communities. Through analytical partitioning and simulations, we show that $Q_β$ and $E_β$ are connected differently to typical patterns of community structure. The search for the ecological and evolutionary processes that drive community assembly and the assessment of community resilience and stability have indeed revealed typical community structures: the local clustering of species with similar traits or shared evolutionary histories and the local (α) or regional (γ) presence of functionally or phylogenetically redundant versus unique species. We show that while $Q_β$ and $E_β$ are both increasing functions of species β diversity and FP γ uniqueness, $Q_β$ increases with FP clustering, while $E_β$ increases with FP α redundancy. We also show that the component of FP clustering included in $Q_β$ partitioning formula allows the detection of an overall trend of overdispersion or clustering in a dataset without the need to use null models. To facilitate and secure the selection of an index of β diversity for a given study, we call, through our study, for the development of formal and precise definitions for FP β diversity in light of the concepts of clustering versus overdispersion and redundancy versus uniqueness. In particular, we call for further research on when and why FP β diversity should increase with FP clustering.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2024 The Author(s). *Ecosphere* published by Wiley Periodicals LLC on behalf of The Ecological Society of America.

<footnote n="1">¹Centre d'Ecologie et des Sciences de la Conservation (CESCO), Muséum National d'Histoire Naturelle, CNRS, Sorbonne Université, Paris, France</footnote>

<footnote n="2">²Department of Environmental Biology, University of Rome 'La Sapienza', Rome, Italy</footnote>
KEYWORDS

biodiversity conservation, biodiversity measure, community dissimilarities, diversity partitioning, ecological and evolutionary processes, ecosystem functions, Hill numbers, quadratic entropy, resilience

INTRODUCTION

In the context of the current biodiversity crisis, the need to understand how the current amount of biodiversity in different parts of our planet varies has never been so crucial (e.g., Albert et al., 2021; Borges et al., 2020; Davis et al., 2018). A prerequisite step for such an understanding is the development of appropriate ways to measure biodiversity (Table 1). The interspecies component of biodiversity focuses on the number of species and their relative abundances and on the variation in their biological characteristics, such as their traits (well-defined, measurable phenotypic properties, McGill et al., 2006), including functional traits (traits that determine species' response to environmental factors or species' effect on ecosystem functioning, Lavorel & Garnier, 2002), and phylogeny (relationships between species reflecting evolutionary history, Tucker et al., 2019). In community ecology, the key unit for quantifying biodiversity is the community: a set of populations of the species living within an area at a particular time (Begon et al., 1990).

The choice of a biodiversity measure depends on the objective of the study requiring it. Three main nonexclusive objectives can be cited here: (1) saving biodiversity for its intrinsic value; (2) maintaining ecosystem functioning and services; and (3) understanding the processes underlying biodiversity patterns. With respect to the first objective, the main challenge for developing measures of biodiversity is to count all the elements of as many aspects of biodiversity as possible while minimizing the risk of missing and losing a facet of biodiversity. When the second objective is of interest, the main challenge is to identify which aspects of biodiversity enable stable ecosystem functions. When the third objective is of interest, the main challenge is to disentangle the different components of biodiversity to explain how and why each component varies in space and time. Clearly, the different objectives call for different ways of quantifying biodiversity.

Biodiversity measurement also depends on the scale of a study. Since Whittaker (1972), biodiversity has often been studied within communities (α diversity), between communities (β diversity), and across all communities (γ diversity). This has allowed researchers to analyze the ecological and historical processes that drive biodiversity patterns while acting either at the local level or at the

regional level (e.g., Pavoine & Bonsall, 2011). In real case studies, both α and β diversities contribute to the global γ diversity of the whole set of communities. Despite all scales of biodiversity (α, β, γ) being critical for maintaining multiple ecosystem functions and services (Pasari et al., 2013), patterns in β diversity have been the least studied among the myriad of studies on biodiversity (e.g., Mori et al., 2018), and β diversity is the component

<table><caption>TABLE 1 Glossary.</caption><thead><tr><td>Concept</td><td>Definition</td></tr></thead><tbody><tr><td>α diversity</td><td>Degree of difference between individuals within communities</td></tr><tr><td>β diversity</td><td>Degree of difference between communities</td></tr><tr><td>γ diversity</td><td>Degree of difference between individuals both within and between communities</td></tr><tr><td>FP clustering</td><td>Clustered communities are different from each other, but individuals within communities are similar</td></tr><tr><td>FP overdispersion</td><td>Over-dispersed communities are similar to each other, and individuals within a community have different biological characteristics</td></tr><tr><td>FP randomness</td><td>In cases of randomness, individuals are distributed among the communities randomly with regard to their traits or phylogenetic positions</td></tr><tr><td>β redundancy</td><td>β redundancy occurs when communities have different species but for each species in a community, there is another one with similar biological characteristics in each other community</td></tr><tr><td>β uniqueness</td><td>β uniqueness occurs when trait-based or phylogenetic β diversity mimics species β diversity</td></tr><tr><td>α and γ redundancy</td><td>Redundancy at the α (resp. γ) scale means that individuals within a community (resp. all individuals over all communities) are similar</td></tr><tr><td>α and γ uniqueness</td><td>In contrast, uniqueness at the α (resp. γ) scale means that species are very different from each other within a community (resp. all individuals over all communities)</td></tr></tbody></table>

Abbreviation: FP, (trait-based) functional or phylogenetic.
of biodiversity that is the least precisely defined as underlined in the next paragraphs.

While α and γ diversities are usually well defined, β is often relegated to a function of γ and α: knowing that both α and β contribute to γ, it has been implied that, in γ diversity, what is not α is β, yielding $γ = α + β$ (additive partitioning, e.g., Lande, 1996); alternatively, it was proposed that β represents how many times as diverse an entire dataset is compared with its constituent communities on average such that β varies between 1 and the number of sampled communities and $γ = α × β$ (multiplicative partitioning; Tuomisto, 2010; Whittaker, 1972). Jurasinski et al. (2009) referred to such definitions of β as “proportional diversity” because their associated measure results from the comparison of diversity measurements at two embedded spatial levels (α and γ) rather than being the direct result of measures of differentiation in community composition.

β diversity being thus a complex, multidimensional concept that we try to quantify indirectly from our knowledge of α and γ diversity, its measurement has raised debates (e.g., Jost, 2006; Jurasinski et al., 2009; Schmera et al., 2020; Veech & Crist, 2010). One aspect of these debates about β diversity concerns the fact that the range of variation in β must be independent of α diversity (Chao et al., 2012). We name this property range independence from α diversity (RIαD). Indices that did not have this property were transformed into other indices commonly named Hill numbers (Jost, 2006; but see Hoffmann & Hoffmann, 2008) that do have this property (e.g., Chiu et al., 2014; Ricotta & Szeidl, 2009). Hill numbers were promoted in the context of biodiversity preservation, as their unit of measurement is a number of biodiversity elements (number of species at the α and γ levels; number of communities at the β level). In addition, when used in multiplicative partitioning ($γ = α × β$), Hill numbers also satisfy the replication property that β diversity always equals the number of sampled communities as soon as communities are evenly weighted, equally diverse, and maximally dissimilar (Ricotta & Szeidl, 2009). However, even with “Hill number” indices, the two components α and β are not statistically independent, as each individual in each community contributes to both α and β diversities (see also Veech & Crist, 2010). This partial independence and the fact that even indices with the RIαD and replication properties can be “proportional diversity” indices (Jurasinski et al., 2009) imply that deciding the meaning of β diversity also concerns “Hill number” indices.

At the species level, β diversity generally represents the differences in species' presence and/or abundance between several communities. When more information on species, such as their relative phylogenetic positions

or their functional traits, is available, β diversity can reflect differences in the phylogenetic or functional compositions of the communities. Although functional trait-based diversity patterns may be signatures of local niche-based ecological processes driving community dynamics (McGill et al., 2006), patterns in phylogenetic diversity could be driven by global dispersal, speciation, and extinction events (Davies & Buckley, 2011; Eme et al., 2020). Hereafter, we will refer to these “functional or phylogenetic” aspects as FP aspects. With the addition of these FP aspects of diversity, many research studies in the last two decades have also addressed the concepts of clustering and overdispersion (Webb et al., 2002; Weiher & Keddy, 1995). If species within communities have similar traits or phylogenetic positions while species in distinct communities have more different traits or distant phylogenetic positions, then the species are said to be clustered within communities. Here, this pattern is referred to as FP clustering. If, instead, species within communities have different traits or phylogenetic positions while for each species in a community, there exists another species in another community with similar traits or a close phylogenetic position, then species are said to be over-dispersed within communities. Here, this pattern is referred to as FP overdispersion. When species disperse and colonize communities independently of their traits or phylogenetic position, the FP characteristics of species within communities are said to be randomly assembled.

In the context of understanding the processes underlying biodiversity patterns, interest in clustering and overdispersion patterns has increased because of the search for the ecological and evolutionary processes that drive community assembly. Notably, trait-based clustering was associated with environmental filtering (if traits are related to the environmental niche) and the exclusion of poor competitors (if traits are related to fitness; Mayfield & Levine, 2010). Trait-based overdispersion has been mostly associated with limiting similarity, which prevents niche overlap and competition (e.g., Bañares-de-Dios et al., 2020; Cavender-Bares et al., 2004; Ramm et al., 2018; but see Gallien, 2017). Overdispersion can also result from other processes. For example, in plant communities, it can result from colonization by distant relatives in late successional stages (Li et al., 2015). However, how β diversity directly relates to these concepts of clustering and overdispersion has not been formally explained.

In the context of maintaining ecosystem functioning and services and understanding the processes underlying biodiversity patterns, comparing FP diversity patterns to species diversity patterns at the α and γ levels may aid in both identifying assembly rules (Pavoine & Bonsall, 2011) and assessing the vulnerability of ecosystem functioning and services (Biggs et al., 2020). One way to make this
comparison is to measure uniqueness (U) indices as the ratio of FP diversity to species diversity and redundancy (R) indices as the inverse of uniqueness indices (R = 1/U) or as a complement of uniqueness indices (R = 1 - U; provided U varies in [0, 1], Ricotta et al., 2020). High FP diversity combined with high FP redundancy could ensure the resilience and stability of ecosystem functioning and associated services for human well-being if high FP diversity ensures ecosystem multifunctionalities and higher species diversity than FP diversity (i.e., high FP redundancy) allows the effect of any species loss by disturbance to be mitigated by the presence of other species that can ensure similar functions. Recently, Ricotta et al. (2020) developed the complementary concept of β uniqueness as the ratio of FP β diversity to species β diversity. However, how FP β diversity directly relates to the concepts of α and γ uniqueness and redundancy is still an open question.

Several methodological studies have attempted to partition FP diversity indices into elementary components to disentangle the parameters responsible for changes in FP diversity in light of various factors, including spatial scale, environmental change, and disturbance (e.g., Patil & Taillie, 1982; Schmera et al., 2020; Shimatani, 2001). This is because different components may reveal different mechanisms and processes of community assembly. Regarding indices of β diversity, previous studies that managed to partition β diversity into elementary components often expressed these components in terms of nestedness, richness difference (species gains/losses), turnover, or replacement (Schmera et al., 2020). Such extraction of elementary components is now frequently performed to assess species diversity (e.g., Schmera et al., 2020) and FP diversity (e.g., Melo et al., 2014). However, the influence of FP clustering versus overdispersion and that of FP redundancy versus uniqueness on FP β diversity have been far less studied.

Here, we focus on a well-used (e.g., Chen et al., 2019) and reference measure of diversity in community ecology, Rao's quadratic entropy, which is the average abundance-weighted FP dissimilarity between species in a community. Rao's quadratic entropy has been transformed to Hill numbers to achieve the RIαD and replication properties (Ricotta & Szeidl, 2009). It has also been frequently used in association with null models to detect FP clustering and overdispersion (e.g., Ren et al., 2023). The development of β uniqueness indices that compare FP β diversity with species β diversity started with Rao's quadratic entropy (Ricotta et al., 2020). We thus explored Rao's quadratic entropy and one of its transformations into Hill numbers to evaluate the meaning of its component of FP β diversity in relation to species β diversity, redundancy and uniqueness, and clustering and overdispersion.

# METHODS

## State of knowledge on the partitioning of quadratic entropy

The initial framework for α, β, and γ diversities by Rao (1982)

Consider M areas that represent M species communities, numbered 1 to M. Let $S_m$ be the number of species present in area m, for a total of $S_γ$ different species over the whole set of areas ($S_γ \le \sum_{m=1}^{M} S_m$). Let $\mathbf{D} = (d_{ij})$ be the matrix of FP dissimilarities between species ($d_{ii} = 0, d_{ij} = d_{ji}$). Let $\mathbf{p}_m = (1...p_{im}...p_{S_m})$ be the vector of relative abundances of the $S_γ$ species in area m (species absent from area m have an abundance equal to zero). Let $w_m$ be a weight attributed to area m ($\sum_{m=1}^{M} w_m = 1$) such that the global relative abundance of any species i over all areas is $\sum_{m=1}^{M} w_m p_{im}$.

The value of the quadratic entropy applied to area m is

$$Q(\mathbf{p}_m, \mathbf{D}) = \sum_{i=1}^{S_γ} \sum_{j=1}^{S_γ} p_{im} p_{jm} d_{ij}. \qquad (1)$$

Rao (1982) defined the diversity within areas (α diversity) as $Q_α = \sum_{m=1}^{M} w_m Q(\mathbf{p}_m, \mathbf{D})$, the global diversity across all areas (γ diversity) as $Q_γ = Q(\sum_{m=1}^{M} w_m \mathbf{p}_m, \mathbf{D})$, and the dissimilarity coefficient between areas (β diversity) as $Q_β = Q_γ - Q_α$ (see detailed equations in Appendix S1). While $\mathbf{D}$ informs on the PF dissimilarities between species, $\sum_{m=1}^{M} w_m \mathbf{p}_m$ is the vector of species' relative abundance at the γ level so that $Q_γ$ is the diversity within the whole region made up of all the areas studied. $Q(\mathbf{p}_m, \mathbf{D})$ is the diversity within area m so that $Q_α$ is the average diversity within areas. The formula for $Q_β$ diversity indicates that in γ diversity, what is not α diversity is β diversity. It can also be viewed as the quadratic entropy applied to the weights ($w_m$) attributed to areas and to FP dissimilarities between areas. Let $\mathbf{w}$ be the vector of $w_m$ values and $\Delta = (\delta_{mn})$ be the matrix of FP dissimilarities between areas with

$$\delta_{mn} = \sum_{i=1}^{S_γ} \sum_{j=1}^{S_γ} p_{im} p_{jn} d_{ij} - \frac{1}{2} \sum_{i=1}^{S_γ} \sum_{j=1}^{S_γ} p_{im} p_{jm} d_{ij} \\ - \frac{1}{2} \sum_{i=1}^{S_γ} \sum_{j=1}^{S_γ} p_{in} p_{jn} d_{ij} \qquad (2)$$

Then, $Q_β = Q(\mathbf{w}, \Delta)$. The dissimilarity between any two areas m and n is thus measured in $Q_β$ by the difference between the average FP dissimilarity between one species
drawn from area *m* and the other drawn from area *n* and the average FP dissimilarity between two species drawn from the same area (either *m* or *n*). This formula satisfies the condition that the dissimilarity between two identical areas is null. It contrasts with an alternative formula that would use only the average FP dissimilarity between one species drawn from area *m* and the other drawn from area *n* as a measure of FP dissimilarity between areas *m* and *n* (as, e.g., in Webb et al., 2008), which would not satisfy this elementary condition (Ricotta et al., 2015).

The equation $Q_β = Q(\mathbf{w}, \Delta)$ exemplifies that, in Rao's apportionment of quadratic entropy, species and communities are treated in the same manner. The whole $\gamma$ diversity depends on the relative abundance of species and the FP dissimilarities between them and on the weights defined for communities and the FP dissimilarities defined between them.

If $d_{ij}$ is multiplied by a constant, for example, $\lambda$, then $Q_β$ is also multiplied by this constant. This means that $Q_β$ depends on the global level of biological differences between species. For $Q_β$ to have nonnegative values, the matrix $\mathbf{D} = (d_{ij})_{1 \le i \le S_γ, 1 \le j \le S_γ}$ needs to be a squared Euclidean matrix (i.e., $S_γ$ points $M_i$, $i = 1, 2, ..., S_γ$, can be embedded in a Euclidean space such that the Euclidean distance between $M_i$ and $M_j$ is $\sqrt{d_{ij}}$; Champely & Chessel, 2002; Gower & Legendre, 1986). Without any additional restrictions on the definition of the $d_{ij}$ values, the range of possible values for $Q_α$, $Q_β$, and $Q_γ$ is $[0, \infty)$. The range of $Q_β$ is independent of $Q_α$ (i.e., $Q_β$ satisfies the RIαD property).

## The concept of maximally dissimilar species

Applications of quadratic entropy, however, generally revealed an additional restriction on the choice of $d_{ij}$: $0 \le d_{ij} \le 1$, meaning that FP dissimilarity between species is bounded (Ricotta & Szeidl, 2009). We thus hereafter consider this restriction: the maximum value of $d_{ij}$ is unity. When $d_{ij} = 1$, species *i* and *j* are “maximally dissimilar.” Considering a set of *S* species, with this restriction, the maximum possible value of the quadratic entropy applied to this set, over all matrices of distances between species and all vectors of species' abundance, is $\max_{\mathbf{p},\mathbf{D}}\{Q(\mathbf{p}, \mathbf{D})\} = (S-1)/S$. This maximum is reached when species have equal abundances ($p_i = 1/S$ for all species *i*) and are all maximally dissimilar from each other ($d_{ij} = 1$ for all $i \neq j$). This maximum tends to 1 as *S* increases. When the FP dissimilarities are bounded so that there are maximally dissimilar species, the value of the quadratic entropy is bounded between 0 and 1, which applies to both the $\alpha$ and $\gamma$ levels of diversity. As a

consequence, for a given value of $Q_α$, $Q_β$ is bounded between 0 and $1 - Q_α$; in such a case, the range of $Q_β$ depends on $Q_α$.

Ricotta and Szeidl (2009) proposed, in this particular case of bounded FP dissimilarities, to use a modification of the quadratic entropy and a new apportionment scheme so that a component of $\beta$ diversity can be defined with a range independent of $\alpha$ diversity. Let $s_{ij} = 1 - d_{ij}$ be the FP similarity between species. This yields the component of $\alpha$ diversity

$$E_{\alpha} = 1/(1 - Q_{\alpha}) = 1/\left(\sum_{m=1}^{M} w_m \sum_{i=1}^{S_\gamma} \sum_{j=1}^{S_\gamma} p_{im} p_{jm} s_{ij}\right), \quad (3)$$

the global diversity over all areas ($\gamma$ diversity)

$$E_{\gamma} = 1/(1 - Q_{\gamma}) = 1/\left[\sum_{i=1}^{S_{\gamma}} \sum_{j=1}^{S_{\gamma}} \left(\sum_{m=1}^{M} w_m p_{im}\right) \left(\sum_{m=1}^{M} w_m p_{jm}\right) s_{ij}\right], \quad (4)$$

and the diversity coefficient between areas ($\beta$ diversity)

$$E_{\beta} = E_{\gamma}/E_{\alpha} = \frac{\left(\sum_{m=1}^{M} w_m \sum_{i=1}^{S_\gamma} \sum_{j=1}^{S_\gamma} p_{im} p_{jm} s_{ij}\right)}{\sum_{i=1}^{S_\gamma} \sum_{j=1}^{S_\gamma} \left(\sum_{m=1}^{M} w_m p_{im}\right) \left(\sum_{m=1}^{M} w_m p_{jm}\right) s_{ij}}, \quad (5)$$

which is the ratio of the average FP similarity at the $\alpha$ level to that at the $\gamma$ level. While $Q_α$ and $Q_γ$ represent the average dissimilarity between species at the $\alpha$ and $\gamma$ scales, respectively, $E_α$ and $E_γ$ represent the inverse of the average similarity between species at the $\alpha$ and $\gamma$ scales.

$Q_β$ represents the gap between the dissimilarity among individuals at the $\gamma$ level and that at the $\alpha$ level. In contrast, $E_β$ measures how many times as diverse a region (set of communities, $\gamma$ level) is than its constituent communities ($\alpha$ level) are on average. The maximal value of *M* (the number of communities) is reached by $E_β$ as soon as communities are evenly weighted ($w_m = 1/M$ for all *m*) and maximally dissimilar (communities do not share species and $s_{ij} = 0$ if species *i* and *j* belong to distinct communities, regardless of the global amount of diversity at the $\gamma$ scale). $E_β$ has been associated with an effective number of communities, that is, the number of theoretical, equally weighted, maximally distinct communities that would be needed to reach the same level of $\gamma$ diversity as that of the real communities. Indeed, a set of $\psi$ equally weighted, maximally distinct communities, each with a diversity equal to $E_α$, would have a global diversity of $\psi \times E_α$ (Ricotta & Szeidl, 2009). The theoretical effective
number of equally weighted, maximally distinct, equally ($E_\alpha$)-diverse communities that would lead to the same $E_\gamma$ as that of the real communities is thus obtained when $E_\gamma = \psi \times E_\alpha$, that is, when $\psi = E_\beta = E_\gamma/E_\alpha$.

Note that transforming $Q_\beta$ into $1/(1-Q_\beta)$ would also lead to an index of FP $\beta$ diversity that varies between 1 and M. However, this alternative transformation would not lead to an index with a range of variation independent of $\alpha$ diversity. In contrast, $E_\beta$ and $Q_\beta$ are linked by the following equation:

$$E_{\beta} = \frac{1}{1 - \frac{Q_{\beta}}{1 - Q_{\alpha}}} \qquad (6)$$

The standardization of $Q_\beta$ by $1-Q_\alpha$ in Equation 6 generates the desired RI$\alpha$D property in which the range of variation of an index of $\beta$ diversity is independent of $\alpha$ diversity.

## New developments on species $\beta$ diversity

Species diversity indices generally assume that species are all maximally dissimilar from each other ($d_{ij} = 1$ for all $i \neq j$). Let $Q_{\alpha,\text{spe div}}$, $Q_{\beta,\text{spe div}}$, $Q_{\gamma,\text{spe div}}$, $E_{\alpha,\text{spe div}}$, $E_{\beta,\text{spe div}}$, and $E_{\gamma,\text{spe div}}$ be the values of $Q_\alpha$, $Q_\beta$, $Q_\gamma$, $E_\alpha$, $E_\beta$, and $E_\gamma$, respectively, obtained by forcing all species to be maximally dissimilar, although they can vary in abundance. In this case (i.e., considering $d_{ij} = 1$ for all $i \neq j$), $Q_\beta$ and $E_\beta$ reduce to

$$Q_{\beta,\text{spe div}} = \frac{1}{2} \sum_{m=1}^{M} w_m \sum_{n=1, n \neq m}^{M} w_n \sum_{i=1}^{S_\gamma} (p_{lm} - p_{in})^2 \quad (7)$$

and

$$E_{\beta,\text{spe div}} = \frac{\left( \sum_{m=1}^{M} w_m \sum_{i=1}^{S_\gamma} p_{im}^2 \right)}{\sum_{i=1}^{S_\gamma} \left( \sum_{m=1}^{M} w_m p_{im} \right)^2}, \quad (8)$$

respectively. $Q_{\beta,\text{spe div}}$ is the average dissimilarity between any two areas, with this dissimilarity measured as the squared difference in species' relative abundance between the two compared areas. This average dissimilarity is the highest when, in each area, a species dominates in abundance but the dominant species in one area is different from that in the other. $E_{\beta,\text{spe div}}$ is the ratio of the level of dominance within areas ($\alpha$ level) to the level of dominance across all combined areas ($\gamma$ level). Here, dominance is the degree to which a single species

dominates in abundance, while others are rare. $E_{\beta,\text{spe div}}$ is the highest as soon as areas do not share species and are evenly weighted (i.e., $w_m = 1/M$). When species are maximally dissimilar, $Q_\beta$ and $E_\beta$ are thus both high in the case of clustering of identical individuals (from the same species) within an area, while individuals are maximally different (from different species) from one area to another. However, $E_\beta$ is still high if individuals within an area belong to a range of distinct species, provided that individuals are maximally different from one area to another, while $Q_\beta$ is no longer high.

If species within areas also have even abundances, $Q_{\beta,\text{spe ric}}$ and $E_{\beta,\text{spe ric}}$ have the values of $Q_\beta$ and $E_\beta$, respectively, obtained by forcing species to be maximally dissimilar and have similar abundances within an area. We determined (proof in Appendix S1) that $Q_{\beta,\text{spe ric}}$ can be viewed as the proximity to a case of complete species clustering where there are as many areas as species, each area contains a single species and areas do not share species. $Q_{\beta,\text{spe ric}}$ is thus the highest in the case of complete species clustering, while $Q_{\beta,\text{spe div}}$ is high both in the case of complete species clustering (one species per area) and when species clustering is driven by uneven abundance distributions (in each area, a species dominates in abundance, but the dominant species in one area is different from that in the other areas). In contrast, $E_{\beta,\text{spe ric}}$ is maximal as soon as areas do not share species (regardless of the number of species in each area). $E_{\beta,\text{spe ric}}$ is related to a traditional family of dissimilarity indices that includes the Jaccard and Sørensen indices (e.g., Gower & Legendre, 1986; proof in Appendix S1).

This difference in behavior still exists when species are not maximally dissimilar. Indeed, $Q_\beta$ is high in the case of clustering of similar individuals (from species with similar FP characteristics) within an area while individuals are strongly different (from different species with distinct FP characteristics) from one area to another. In contrast, $E_\beta$ becomes high as soon as individuals strongly differ from one area to another, as shown below.

## New partitioning of FP $\beta$ diversity into elementary components

Below, we introduce partitioning schemes for $Q_\beta$ and $E_\beta$, where each of these indices of FP $\beta$ diversity is compared with the value it would take if species were maximally dissimilar (i.e., if it were simply reduced to species $\beta$ diversity). As shown in the previous section, the concept of clustering is crucial in the interpretation of $Q_\beta$ values but not in the interpretation of $E_\beta$ values. To ease the interpretation of $Q_\beta$, it would thus be useful to more explicitly depict how and how much clustering of FP
characteristics determines levels of β diversity according to $Q_β$. As values of $Q_β$ are driven by FP clustering, one could simply envisage the ratio $Q_β/Q_γ$ to measure the absolute level of FP clustering in a study area. Although the ratio of β diversity to γ diversity has sometimes been used as a measure of community dissimilarity, that is, as a standardized measure to compare the compositions of communities (e.g., Hardy & Senterre, 2007; Lande, 1996), the ratio $Q_β/Q_γ$ has also been interpreted as “local similarity excess,” that is, clustering (Hardy & Jost, 2008). However, maximum absolute FP clustering can be founded with species clustering that occurs when individuals from the same species are clustered within an area and are the sole members of the area or are dominant in abundance in the area (species clustering = “local species identity excess” sensu Hardy & Jost, 2008).

To measure the relative level of FP clustering given the level of species clustering, one must compare FP clustering with species clustering, yielding the following standardized coefficient:

$$C = \frac{Q_{\beta}}{Q_{\gamma}} \left| \frac{Q_{\beta,spe.div}}{Q_{\gamma,spe.div}} \right. \quad (9)$$

C measures not absolute clustering but relative clustering, which is the amount of FP clustering that can be attributed solely to the distribution of FP characteristics across areas. C values greater than 1 indicate clustering due to the FP characteristics of the species, and C values less than 1 indicate overdispersion due to these characteristics. A typical example of a case for C = 1 is obtained when $d_{ij}$ is constant over all species pairs i, j.

$Q_β$ can be apportioned into components of species β diversity ($Q_β,spe.div$), global FP uniqueness ($U_γ = Q_γ/Q_γ,spe.div$) and FP clustering (C) using the following formula:

$$Q_{\beta} = Q_{\beta,spe.div} \times U_{\gamma} \times C \quad (10)$$

$Q_β,spe.div$ and $U_γ$ vary between 0 and 1; C is also positive but can be greater than 1 in the case of FP clustering. In a real case study, we expect that various values of C will be observed depending on the ecological processes that act in the study area. We expect that $U_γ$ will be low to moderate in most case studies, as at a regional level, species from a restricted taxonomic group usually share some of their FP characteristics, although for trait-based and functional diversity, this will depend on the number of traits considered and on the level of correlations between traits. The component that we expect to often be low is $Q_β,spe.div$. Indeed, although the abundance distribution within a local community is known to be asymmetric, with a few species being the most abundant and

many other species being rare, strong abundance-driven clustering where a single species dominates in abundance in each community but the dominant species is different from one area to another is likely to be uncommon. Overall, the values of $Q_β$ are thus expected to be low in real case studies (as previously observed, e.g., by Hardy & Jost, 2008). However, comparing each component of $Q_β$ between different regions can be informative because it will reveal how regions differ in terms of relative FP clustering and FP γ uniqueness.

A simple way to start depicting the relationship between $E_β$ and $E_β,spe.div$ is to consider $E_β = U_R \times E_β,spe.div$, with:

$$U_R = \frac{E_\gamma}{E_\alpha} \left| \frac{E_{\gamma,spe.div}}{E_{\alpha,spe.div}} \right. = \frac{E_\gamma}{E_{\gamma,spe.div}} \left| \frac{E_\alpha}{E_{\alpha,spe.div}} \right. = \frac{U_\gamma^*}{U_\alpha^*} \quad (11)$$

$U_R$ is the ratio of (regional) γ uniqueness ($U_γ^*$) to (local) α uniqueness ($U_α^*$). Let $R_α^* = 1/U_α^*$ be the (local) α redundancy; then, $E_β$ can be expressed as

$$E_{\beta} = E_{\beta,spe.div} \times U_{\gamma}^* \times R_{\alpha}^* \quad (12)$$

$Q_β$ and $E_β$ can thus both be partitioned into three components, two of which are species β diversity and γ uniqueness. In addition to differing in the equations used to measure diversity and uniqueness, $Q_β$- and $E_β$-based approaches thus differ in the third component of the partition: FP clustering for $Q_β$ versus α redundancy for $E_β$.

## Case studies

We considered, for illustration, simulated theoretical datasets based on a pool of 100 species characterized by a single quantitative, discrete trait. To obtain distances between species bounded between 0 and 1, we first standardized the trait by dividing it by the range of variation (maximum–minimum) and defined the dissimilarity between two species as the absolute difference in their standardized trait values. Note that only $E_β$ requires the use of bounded distances, but for homogeneity, we also apply standardization when using $Q_β$. Let $y_i$ be the raw trait value for species i and $x_i = y_i/(\max_j(y_j) - \min_j(y_j))$ be its standardized trait value.

We simulated functional diversity in five areas. We considered a reference species pool with 100 species with raw trait values (the $y_i$ values) equal to 1 to 100 with a step of 1. The number of individuals in the species pool was not fixed (considered infinite); however, we considered species to be evenly abundant in the species pool. That way, in each scenario a species is more/less likely to enter the community based on its trait value, but not based on its regional abundance. Then, we considered
that 250 individuals colonized each area from the species pool.

1. In the first scenario (randomness), the 250 individuals were randomly drawn from the species pool. With this reference scenario, both FP and species β diversities are expected to be low, and component C of Qβ is expected to be close to unity.

2. In the second scenario (clustering), each area was associated with an optimal trait value to simulate functional clustering. The probability, $\pi_m(i)$, that an individual of species $i$ entered area $m$ was inversely linked to the difference between its trait value and the optimal trait value of area $m$ by the following formula: $\pi_m(i) = (100 - |y_i - o_m|)^q / \sum (100 - |y_i - o_m|)^q$, where $o_m$ is the optimum trait value in area $m$ and $q$ is a positive exponent that controls the strength of filtering, with $q = 1$ the minimum considered strength. We considered four values for $q$: 1, 3, 9, and 81. The optimal trait values for the five areas were defined as $o_1 = 5$, $o_2 = 28$, $o_3 = 50$, $o_4 = 72$, and $o_5 = 95$. We also considered the extreme case where only the species that had the optimal trait value was present in the area (i.e., $q \to \infty$). Index $Q\beta$ through its component C is expected to increase with clustering, while the variation of $E\beta$ with clustering is expected to be more complex and less predictable.

3. In the third scenario (overdispersion), an individual was first randomly drawn from a species pool to enter an area. New individuals were iteratively added to the area with the following constraint: the probability, $\pi(i/\Sigma)$, that a given individual was selected to enter the area is linked to the average absolute difference between its trait value and the trait values in the set $\Sigma$ of species already present in the area by the following formula: $\pi(i/\Sigma) = (\sum_{j=1}^{100} p_{j\Sigma} |y_i - y_j|)^q / (\sum_{i=1}^{100} (\sum_{j=1}^{100} p_{j\Sigma} |y_i - y_j|))^q$, where $p_{j\Sigma}$ is the relative abundance of species $j$ in the set $\Sigma$ and $q$ is a positive exponent that controls the strength of the overdispersion, with $q = 1$ the minimum considered strength. We considered four values for $q$: 1, 3, 9, and 81. Index $Q\beta$ through its component C is expected to decrease with overdispersion, while the variation of $E\beta$ with overdispersion is expected to be more complex and less predictable.

4. In the fourth scenario (outliers), the areas were subjected to different pressures: one area was composed of the first 50 species of the species pool (with trait values between 1 and 50), and the four others were composed of two atypical species that do not belong to the reference species pool, with trait values of 499 and 500, respectively; abundances were random, with the only constraint being that each area contained 250 individuals. $\gamma$ uniqueness, as measured

by $U_\gamma$ and $U_\gamma^*$, is expected to be high in presence of distinct areas with outlying trait values, which results in increased values of FP $\beta$ diversity ($Q\beta$, $E\beta$).

5. In the fifth scenario (siblings), five species were first randomly drawn from the species pool (except species with extreme trait values of 1 and 100). These five species were assigned to the first area. The composition of species in the other areas was also based on the set of five selected species, except that each of these five species could be randomly replaced by one of its closest neighbors (a species with a trait value of $y$ has two closest species with the trait values $y-1$ and $y+1$, respectively). Species with extreme trait values were excluded from the first area because they had only one closest neighbor. As in the fourth scenario, the abundances of the species in each area were randomly defined with the only constraint being that the area included 250 individuals. In the fifth scenario, FP $\beta$ diversity is expected to be close to minimum despite significant species $\beta$ diversity. Among the elementary components of $Q\beta$ partitioning (Equation 10), the level of species $\beta$ diversity is thus expected to be compensated by high overdispersion (low $C$) and moderate $\gamma$ uniqueness ($U_\gamma$); and, in $E\beta$ framework (Equation 12), the product of $\gamma$ uniqueness ($U_\gamma^*$) to $\alpha$ redundancy ($R_\alpha^*$) is expected to be inversely proportional to species $\beta$ diversity ($E_{\beta,\text{spe.div}}$) so that $E\beta$ is close to unity.

For each scenario, 1000 sets of 5 areas were simulated, and the diversities $Q\beta$ and $E\beta$ and their components $Q_{\beta,\text{spe.div}}$, $U_\gamma$, $C$, $E_{\beta,\text{spe.div}}$, $U_\gamma^*$, and $R_\alpha^*$ were computed for each set.

## RESULTS

Scenarios 1 to 3 allowed us to illustrate the general behavior of $Q\beta$, $E\beta$ and their components when the rules of species assembly changed. Functional $\beta$ diversity (as measured by $Q\beta$ and $E\beta$) increased with the strength of the clustering (Figure 1). In scenarios 1 to 3, species $\beta$ diversity increased more slowly than functional $\beta$ diversity ($Q\beta$) with the strength of the clustering when measured by $Q_{\beta,\text{spe.div}}$. In contrast, when measured by $E_{\beta,\text{spe.div}}$, it increased faster than functional $\beta$ diversity ($E\beta$). $\gamma$ uniqueness (measured by $U_\gamma$ or $U_\gamma^*$) increased as the strength of either overdispersion or clustering increased, but it increased faster with the strength of overdispersion than with the strength of clustering (Figures 1 and 2). The $\alpha$ uniqueness $U_\alpha^*$ index exhibited a more symmetric U-shaped variation along the gradient that ranged from high overdispersion to high clustering (Figure 2); as a consequence, $\gamma$ uniqueness ($U_\gamma^*$) was
<figure><img src="image_2.png" /><figcaption>**FIGURE 1** Level of functional β diversity, as measured by Qβ, and its components obtained for each scenario of the case study. From left to right and top to bottom: Functional β diversity (Qβ), species β diversity (Qβ,spe.div), functional γ uniqueness (Uγ), functional clustering (C), and log-transformed component of functional clustering (log(C)). Values are summarized by a box plot for each simulation scenario from overdispersion with decreasing strength (Scenario 3 in *Case studies*, with parameter *q* = 81, 9, 3, and 1), through random simulation (Scenario 1 in *Case studies*), to clustering with increasing strength (Scenario 2 in *Case studies*, with parameter *q* = 1, 3, 9, 81, and ∞).</figcaption></figure>

inversely, but not linearly, associated with FP α redundancy ($R_α^*$) along the gradient that goes from high overdispersion to high clustering. The *C* index increased from high overdispersion, through the random scenario, to its peak at moderate clustering and then decreased with high clustering, as functional clustering could not be distinguished from species clustering (Figure 1). When the species were randomly distributed among the areas, the *C* index, as expected, approached unity (Figure 1). In contrast to *C*, α redundancy ($R_α^*$) peaked for the random model. *C* and $R_α^*$ thus varied differently along the gradient of overdispersion–clustering.

Scenarios 4 (outliers) and 5 (siblings) are particular cases of clustering and overdispersion, respectively, such that *C* takes high (≫1) values under scenario 4 and low values (≪1) under scenario 5, and both functional β

diversity (as measured by Qβ and Eβ) and γ uniqueness (as measured by Uγ and Uγ*) were greater under scenario 4 than those under scenario 5 (Figure 3). However, scenarios 4 and 5 represent special cases of clustering and overdispersion that depart from the general tendency. Comparing scenarios 4 and 5 showed that species β diversity can, in the case of atypical patterns of clustering and overdispersion, decrease with clustering (Figure 3). This comparison also showed that clustering (component C of Qβ) and α redundancy (component $R_α^*$ of Eβ) can display opposing patterns of variation (Figure 3).

DISCUSSION

Partitioning diversity indices into
elementary components

Here, we used a new way of partitioning FP β diversity
into elementary components to reveal how FP β
diversity depends on the concepts of clustering/overdis-
persion and redundancy/uniqueness. One of these elemen-
tary components is species β diversity. The partitioning
scheme that we propose thus allows us to identify when
and why FP β diversity differs from species β diversity
through the study of FP redundancy versus uniqueness
(thanks to component Uγ for Qβ and components Uγ* and
Rα* for Eβ) and/or FP clustering versus overdispersion
(thanks to component C for Qβ). Variations in FP β diver-
sity may indeed depart from those of species β diversity
in ecological systems. At a broad geographic scale, the
combined effects of dispersal limitation, association with
a biome, a particular history of speciation and extinction,
and niche conservatism may lead to the decoupling of
phylogenetic β diversity from species β diversity (Azevedo
et al., 2021). At more local scales, the decoupling of FP
β diversity from species β diversity may be due, for exam-
ple, to disturbance that affects the level of habitat hetero-
geneity and habitat filtering (e.g., Rodrigues et al., 2017)
or to modified patterns of interspecific competition
(e.g., Wong et al., 2019). Analyzing the ratio of FP β
diversity to species β diversity in terms of elementary
components can thus help reveal which insights into β
diversity are gained by using trait-based, FP information
rather than simply considering species' abundance. Our
case study showed that component C indeed identifies
the level of FP clustering in a dataset in reference to spe-
cies clustering. It shows that C is the highest in case of
high but not extreme FP clustering while Uγ is the
highest in case of overdispersion. As C and Uγ are not
correlated, partitioning Qβ into components Qβ,spe.div, C,
and Uγ in real case studies has the potential to reveal the
relative influence of redundancy and clustering in the
<figure><img src="image_3.png" /><figcaption>**FIGURE 2** Level of functional β diversity, as measured by $E_β$, and its components obtained for each scenario of the case study. Distributions, over simulations, are given for (from left to right and top-right to bottom-right) functional β diversity ($E_β$), species β diversity ($E_{β,spe.div}$), functional uniqueness at the γ level ($U_γ^*$), and functional redundancy at the α level ($R_α^* = 1/U_α^*$). Values are summarized by a box plot for each simulation scenario from overdispersion with decreasing strength (Scenario 3 in *Case studies*, with parameter $q = 81, 9, 3$, and 1), through random simulation (Scenario 1 in *Case studies*), to clustering with increasing strength (Scenario 2 in *Case studies*, with parameter $q = 1, 3, 9, 81$, and $\infty$).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**FIGURE 3** Observed distributions, summarized by box plots, for indices of functional β diversity $Q_β$, $E_β$, and all their intrinsic components (species β diversity $Q_{β,spe.div}$ and $E_{β,spe.div}$; functional clustering $C$; functional uniqueness at the γ level $U_γ$ and $U_γ^*$; and functional redundancy at the α level $R_α^* = 1/U_α^*$). Box plots are given for two scenarios: Siblings (Scenario 5 in *Case studies*, a particular case of overdispersion) and outliers (Scenario 4 in *Case studies*, a particular case of clustering).</figcaption></figure>

level of FP β diversity. The partitioning of $E_β$ applied to our case study showed that FP γ uniqueness ($U_γ^*$) tends to be high both in case of extreme level of overdispersion

and extreme level of clustering and that it tends to be inversely, but not linearly, associated with FP α redundancy ($R_α^*$) along the gradient that goes from high
overdispersion to high clustering. Applying the partitioning of $E_β$ to real case studies could thus reveal which among the variation in FP $γ$ uniqueness or that in FP $α$ redundancy is the main driver of change in the ratio of FP $β$ diversity to species $β$ diversity.

Global evaluations of the ratio of FP $β$ diversity to species $β$ diversity have been performed at different scales in several recent studies: between continents, islands, regions, or other geographic locations (e.g., Zhao et al., 2021); between biomes, ecosystems, successional stages, habitats, or areas within a habitat (e.g., Ricotta et al., 2020); and between hosts for a parasite group (Krasnov et al., 2021). Species invasion driven by human activities, for example, can reduce functional $β$ diversity while maintaining species $β$ diversity (e.g., Wong et al., 2019). However, the reasons for this association remain unclear. Partitioning FP $β$ diversity into elementary components could help. Wong et al. notably observed the removal of functionally unique species at the $γ$ level in invaded communities. In that case, the cumulative removal of the few unique species from each invaded community yielded a substantial decrease in functional uniqueness at the $γ$ level. $γ$ uniqueness is one of the elementary components of both the $Q_β$ and $E_β$ indices analyzed above, and a decrease in functional $β$ diversity in invaded communities, leading to functional homogenization at the landscape scale, might thus result from the loss of functionally unique species in communities. Assessing the elementary components of FP $β$ diversity could thus allow a finer description of nonrandom patterns of community change due to human activities (e.g., Uchida et al., 2018).

## β diversity and clustering/overdispersion

Clustering/overdispersion patterns are usually not identified over a whole studied region but within each local area in reference to a large delimited region (e.g., Molina-Venegas et al., 2022; Ramm et al., 2018; Rocha et al., 2019; Yang et al., 2014). Their identification has thus far required the use of null models and the randomization of species occurrences or FP characteristics (e.g., Hardy & Senterre, 2007; Webb et al., 2002). In Webb et al.'s (2002) seminal paper, clustering is, among other approaches, identified in a given area if the average phylogenetic dissimilarity between two species in the area is lower than that expected under the random assembly of species in the area from a larger regional species pool (net relatedness index or NRI). In contrast, overdispersion is identified if the average dissimilarity between two species in the area is greater than that expected under the random assembly of species in the local area from the regional

species pool. The distribution of values obtained for the NRI of clustering/overdispersion can then be summarized by analyzing their means, SDs, and/or quantiles to identify general trends (e.g., Luo et al., 2021; Rocha et al., 2019; Yang et al., 2014). Webb et al. (2002) used presence-absence data. Nevertheless, the Q index, which evaluates the average abundance-weighted dissimilarity within an area, was also found to be one of the most efficient metrics for detecting clustering and overdispersion when it is associated with a relevant null model of random community assembly (Miller et al., 2017).

In that case, the amounts of clustering and overdispersion are measured as standardized effect sizes (SESs; see also Botta-Dukát, 2018). The formula for the SES associated with the Q value observed in a given area is (observed Q value – mean of Q values simulated by the null model)/(SD of Q values simulated by the null model). Negative SES values indicate clustering, close-to-zero values indicate randomness, and positive values indicate overdispersion. A p value can also be obtained for each studied community by comparing the observed Q value with the distribution of Q values simulated by the null model.

Here, we showed that $Q_β$, in the scale-based partitioning of Q, has an intrinsic component (C) that directly summarizes the global level of clustering over the studied region: C is low in the case of an overall trend toward overdispersion and high in the case of an overall trend toward clustering. Opting for the use of this component C, the detection of an overall trend of overdispersion or clustering thus does not require the use of null models. This is reassuring given the difficulty in identifying the best-performing null models (Hardy, 2008; Miller et al., 2017), the additional methodological step they represent, and the amount of calculation these null models require, particularly for large datasets.

Although the index C could also be used with null models to test the significance of observed clustering or overdispersion, it has the great advantage of evaluating levels of clustering versus overdispersion without the need for these null models. This approach can be useful in situations where the aim is to compare the level of clustering versus overdispersion of distinct sets of communities, for example, in different ecoregions (to reveal, e.g., the influence of habitat fragmentation, habitat structure, or local climate in shaping functional clustering), on different continents (to reveal, e.g., geographic or climate-related effects), or in distinct altitudinal strata (to reveal, for example, an increase in functional clustering with elevation). Indeed, in many studies, it is not so much the significance of clustering or overdispersion in each community that is of interest but rather that of the trend of variation in clustering, such as the increase in clustering along a gradient (Miller et al., 2017).
By analyzing Rao's quadratic entropy, we showed that the switch between two definitions of FP β diversity has consequences in terms of how clustering (vs. overdispersion) is expected to influence FP β diversity relative to species β diversity. Contrary to $E_{\beta}$, $Q_{\beta}$ is a linear function of a component of clustering (component C) (see Equation 10). Nevertheless, even if $E_{\beta}$ was not revealed to be a linear function of a clustering component, our case study showed that both $Q_{\beta}$ and $E_{\beta}$ are influenced by FP clustering. Whether FP β diversity should increase with FP clustering is thus an open question raised by our analyses. The response to this question will likely depend on the objective of the study.

## β diversity as a function of α and γ uniqueness

Comparing FP diversity patterns with species diversity patterns can help reveal community assembly processes (e.g., Larson et al., 2021; Pavoine & Bonsall, 2011). This is because high species diversity accompanied by relatively low FP diversity indicates FP redundancy and low species diversity compensated by relatively high FP diversity indicates FP uniqueness. Phylogenetic redundancy/uniqueness patterns can be signatures of evolutionary processes such as diversification and dispersal between regions. For example, at the regional level, phylogenetic redundancy may be due to recent speciation events or to particular species' extinction events that affected old lineages, leaving one recent clade. Conversely, phylogenetic uniqueness could be due to the colonization of an area by phylogenetically distant species with different geographic origins, to a lack of radiation, to a low speciation rate, or to frequent extinction events that pruned all clades (e.g., Qian & Deng, 2021). In addition, functional redundancy/uniqueness patterns can be signatures of ecological processes such as avoidance of niche overlap via competitive exclusion for functional uniqueness.

In undisturbed environments with unlimited resources and thus no competition, high functional redundancy can be associated with environmental filtering that drives the co-occurrence of a large set of specialist species with similar traits adapted to local favorable environmental conditions, leading to high species diversity but low functional diversity. In rich systems, functional α and γ redundancy have been associated with resilience and stability to disturbance (Biggs et al., 2020) and with the ability of a species set to perform a diverse set of ecosystem functions and services (e.g., Mori et al., 2016). Indeed, redundant species provide biological buffering capacity that leads to stable community functions. In the case of high

functional redundancy, one could indeed expect the extinction of a redundant species to be compensated for by an extant, functionally similar species (e.g., Johnson et al., 1996). For this statement to be correct, both functional diversity and species diversity must be high (e.g., Aguirre-Gutiérrez et al., 2022), and variation in functional redundancy must result from only variation in species diversity (biodiversity-ecosystem functioning insurance hypothesis, Yachi & Loreau, 1999). For example, in highly diverse ecosystems, fluctuations in species diversity during seasonal disturbances may not be accompanied by similar fluctuations in functional diversity if the decrease in abundance for certain species is compensated for by an increase in abundance of functionally redundant species (e.g., Wang et al., 2023). In that case, the accumulation of functional redundancy during favorable seasons allows for the stabilization of functional diversity during the least favorable seasons.

Here, we showed that two widely used indices of β diversity are related to measures of FP uniqueness. By our partitioning of FP β diversity indices, we highlighted that the ratio of FP β diversity to species β diversity is dependent on FP uniqueness at the γ level for $Q_{\beta}$ and is equal to the ratio of γ uniqueness to α uniqueness for $E_{\beta}$. The level of FP redundancy might indeed differ between spatial scales such that γ redundancy is likely to differ from α redundancy (e.g., Lamothe et al., 2018). Variations in FP redundancy at the α (for $E_{\beta}$) and γ levels (for $E_{\beta}$ and $Q_{\beta}$) may thus drive changes in the ratio of FP β diversity to species β diversity.

Although redundancy has rarely been discussed in the context of β diversity, we show here that this concept is implicitly included in the definition of well-used β diversity indices. The relationships among local redundancy, regional redundancy, and β diversity must thus be further explored in a variety of ecosystems in light of ecological and evolutionary processes. Finally, FP redundancy and uniqueness are known to be dependent on the number and identity of functions considered (e.g., Mori et al., 2016), further supporting the importance of trait selection in the measurement of functional β diversity. For a direct link with ecosystem functions, functional traits must be selected as the organismal traits that influence one or more aspects of the functioning of an ecosystem (Tilman, 2001). The number of ecosystem functions considered, the number of traits considered, and the correlations between these functions and traits might affect the level of redundancy observed and thus the level of functional β diversity measured by both indices that we considered, but especially according to $Q_{\beta}$, which depends on only the absolute level of γ redundancy, while in $E_{\beta}$, it is evaluated relative to the level of α redundancy. Large levels of functional α, β, and γ diversities in
reference to species α, β, and γ diversities are more likely to be observed when considering many uncorrelated traits than if only a few correlated traits are selected.

## Conservation studies versus community ecology studies

The measurement of biodiversity cannot lead to a unique mathematical formula (Hoffmann & Hoffmann, 2008). However, biodiversity indices must be backed by sound arguments (or, simply, must be reasonable; Hoffmann & Hoffmann, 2008) regarding the objective of a study. A biodiversity index is an index not of all biodiversity but of some specific aspect of biodiversity that must be clearly specified and understood to facilitate the interpretation of its values. Partitioning an index into elementary components aids in identifying the specific aspects of biodiversity that the index expresses (e.g., Patil & Taillie, 1982; Shimatani, 2001). For β diversity, the partitioning that we introduced here, where FP β diversity is expressed in terms of species β diversity among other components, complements the approaches that partition FP β diversity into gain/loss, nestedness, or turnover/replacement components. Such partitions into elementary components are useful for better identifying which aspects of FP β diversity each index expresses and searching for the mechanisms that have shaped biodiversity patterns in a study area.

Our partitioning of the related indices $Q_β$ and $E_β$ shows that $E_β$ relies on a traditional view of biodiversity in conservation biology that is based on counts (safeguarding the highest possible number of species and the highest possible number of species communities) and that $Q_β$ can be apportioned into components usually used in community ecology to detect underlying ecological processes (FP clustering and FP redundancy). Rao's apportionment of quadratic entropy deals with community systems where FP β diversity increases with species β diversity, FP clustering at the α level (i.e., within areas), and uniqueness at the γ level (i.e., over all combined areas). In the introduction, we provided three frequent objectives for measuring biodiversity: saving biodiversity for its intrinsic value, maintaining ecosystem functioning and services, and understanding the processes underlying biodiversity patterns. Clearly, the $E_β$ definition is more intuitive for communication to the public in the context of saving biodiversity because its intrinsic value is an effective number of maximally distinct, equally diverse communities. Indeed, without a thorough explanation of the $Q_β$ definition in association with the concepts of clustering and overdispersion, using it for communication to policymakers incurs a risk of misinterpreting its value.

However, knowledge of the processes involved in biodiversity dynamics is important for long-term conservation strategies. This is where research in community ecology and in conservation biology comes together: reaching the objective of saving biodiversity for its intrinsic value also requires the implementation of actions to maintain ecosystem functioning and services and increase our understanding of the processes that underlie biodiversity patterns. The redundancy components that both $Q_β$ and $E_β$ contain are particularly informative in the context of maintaining ecosystem functioning and services. For understanding the processes underlying biodiversity patterns, a critical difference between $Q_β$ and $E_β$ is that in addition to species β diversity and FP γ uniqueness, $Q_β$ also increases with FP clustering, while $E_β$ increases with FP α redundancy. Clustering and α redundancy are related but different concepts. While clustering implies that two species co-occurring in a common site tend to have more similar FP characteristics than two species from different sites, α redundancy requires only that co-occurring species tend to have similar FP characteristics. As shown in our simulations, α redundancy does not reflect clustering.

## CONCLUSION

We have shown here that opting for additive or multiplicative decomposition of quadratic entropy modifies how FP β diversity relates to species β diversity. We also highlighted that this approach modifies the definition of species β diversity itself in two ways: one for additive decomposition, based on deviation from the unrealistic case of complete species clustering where there are as many studied areas as species and areas do not share species, and another for multiplicative decomposition, based on the excess in abundance evenness among species at the γ level compared with that at the α level. More generally, ecologists must be fully aware that there is currently no single, consensual definition of FP β diversity; therefore, opting for one index over another has important consequences for the aspect of β diversity being measured.

The relationship between species diversity and FP diversity that we analyzed here is poorly documented at all scales, particularly at the β level. This comparison is, however, important both for evaluating the processes that shape community assembly and for evaluating the resilience and stability of an ecosystem at the landscape level. An alternative way to evaluate how FP β diversity differs from species β diversity would be to compare the observed value of FP β diversity with null expectations based on species β diversity (e.g., Azevedo et al., 2021).
However, the decompositions of $Q_β$ and $E_β$ that we propose here allow us to more precisely determine why FP $β$ diversity differs from species $β$ diversity. For $Q_β$, is it in terms of $γ$ uniqueness or in terms of clustering/overdispersion? For $E_β$, is it because of particularly close, or inversely particularly dissimilar, values of $γ$ uniqueness and $α$ uniqueness? According to the findings of this study, which focuses on two FP $β$ diversity indices, we call for thorough analyses of existing indices of FP $β$ diversity in terms of how much they are driven by patterns of clustering/overdispersion and/or redundancy/ uniqueness. This will ensure that the relevance of an index in light of the specified, selected objective of a study has been checked and that the values taken by the index on the basis of field data can be interpreted with the whole set of information that they convey. We also call for further studies on whether/when/why functional and/or phylogenetic $β$ diversity should increase with FP clustering.

## AUTHOR CONTRIBUTIONS

Sandrine Pavoine conceived and designed the study. Sandrine Pavoine conducted the analyses. Sandrine Pavoine and Carlo Ricotta analyzed the results and contributed to write the paper.

## CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

## DATA AVAILABILITY STATEMENT

Scripts (Pavoine & Ricotta, 2024) are available from Figshare: https://doi.org/10.6084/m9.figshare.25342990.

## ORCID

Sandrine Pavoine https://orcid.org/0000-0003-2767-6484

Carlo Ricotta https://orcid.org/0000-0003-0818-3959

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## SUPPORTING INFORMATION

Additional supporting information can be found online in the Supporting Information section at the end of this article.

**How to cite this article:** Pavoine, Sandrine, and Carlo Ricotta. 2024. "Functional and Phylogenetic β Diversities and Their Link with Clustering/overdispersion and Uniqueness/redundancy." *Ecosphere* 15(11): e70045. https://doi.org/10.1002/ecs2.70045