# Beta redundancy for functional ecology

Carlo Ricotta, Evsey Kosman, Fabien Laroche, Sandrine Pavoine

► To cite this version:

Carlo Ricotta, Evsey Kosman, Fabien Laroche, Sandrine Pavoine. Beta redundancy for functional ecology. Methods in Ecology and Evolution, 2021, 12 (6), pp.1062-1069. 10.1111/2041-210X.13587. hal-03463567

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# Beta redundancy for functional ecology

Carlo Ricotta¹,* Evsey Kosman² Fabien Laroche³ Sandrine Pavoine⁴

¹ Department of Environmental Biology, University of Rome 'La Sapienza', Rome, Italy. ² Institute for Cereal Crops Improvement, Tel Aviv University, Tel Aviv 69978, Israel. ³ Irstea, UR EFNO, 45290 Nogent-sur-Vernisson, France. ⁴ Centre d'Ecologie et des Sciences de la Conservation (CESCO), Muséum National d'Histoire Naturelle, CNRS, Sorbonne Université, 75005 Paris, France.

*Corresponding author. E-mail: carlo.ricotta@uniroma1.it

## Abstract.

1. Functional beta redundancy has been recently defined as the fraction of species dissimilarity between two plots not expressed by functional dissimilarity. As such, it summarizes to what degree the compositional differences between two plots mirror their functional differences.

2. A fundamental condition to obtain an appropriate measure of functional beta redundancy is that the functional dissimilarity between the plots is always lower (or at least not higher) than the corresponding species dissimilarity. However, many of the extant measures of functional dissimilarity do not fulfill this requirement.

3. To overcome this problem, a class of tree-based indices of functional dissimilarity that conform to the above 'redundancy property' has been recently proposed. However, functional dissimilarity measures need not necessarily be based on a hierarchical representation of the species functional relationships.

4. In this paper we introduce an algorithmic index of functional dissimilarity that conforms to the redundancy property. Since it does not rely on a hierarchical species organization, the proposed index allows to calculate functional beta redundancy in a more suitable way to the non-hierarchical structure of the species functional relationships. The behavior of the proposed measure is illustrated with data on the species functional turnover along real and simulated ecological gradients.

**Keywords:** Algorithmic measure; Functional dissimilarity; Local vs. global scaling; Redundancy property; Standardized vs. non-standardized coefficients.

## 1. Introduction

In a recent paper, Ricotta, Laroche, Szeidl, & Pavoine (2020) introduced the notion of beta redundancy to quantify the fraction of species dissimilarity between two plots not expressed by functional or phylogenetic dissimilarity. Therefore, beta redundancy basically summarizes to what degree the species turnover between two plots is associated to a functional or phylogenetic turnover. This might help to infer the processes underpinning diversity distribution in space and time.
For a pair of plots P and Q, the chief condition to get a meaningful index of beta redundancy is that the functional or phylogenetic dissimilarity $D_F$ is always lower than the corresponding species dissimilarity $D_S$. However, many of the most popular indices of functional and phylogenetic dissimilarity violate this condition, thus leading to the paradoxical situation of a negative beta redundancy (see Ricotta et al., 2020; Appendix S1). To overcome this problem, Ricotta et al. (2020) proposed a class of tree-based indices of functional and phylogenetic dissimilarity between plots that conform to the condition $D_F \le D_S$. The indices are said ‘tree-based’ because they depend on a functional or phylogenetic tree with species as tips and branches that express functional or phylogenetic relationships between species. However, while phylogenies have a hierarchical, non-overlapping structure, functional diversity tends to have a non-hierarchical, overlapping structure (Von Euler & Svensson, 2001). Therefore, being based on a hierarchical structure of species relationships, the proposed tree-based measures are the more natural way for summarizing plot-to-plot phylogenetic dissimilarity, whereas a tree-based representation of the species functional relationships is not unanimously accepted by ecologists (see e.g. Poos, Walker & Jackson, 2009; Pavoine, 2016). For example, Maire, Grenouillet, Brosse, & Villéger (2015) and Loiseau et al. (2017) have reported that dendrograms tend to overestimate the functional distance between species leading to a biased assessment of functional relatedness.

The aim of this paper is thus to propose an abundance-based version of the algorithmic index of functional dissimilarity developed by Kosman (1996) and Gregorius, Gillet, & Ziehe (2003) that conforms to the requirement $D_F \le D_S$. Since it does not rely on a tree-based species organization, the proposed measure allows to calculate functional beta redundancy in a way that is more appropriate to the non-hierarchical structure of functional relationships among species. Two worked examples with artificial and actual data sets, both representing the species functional turnover along ecological gradients are used to illustrate our approach.

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## 2. An algorithmic measure of functional beta redundancy

Let P and Q be two plots with species relative abundance vectors $P = (p_1, ..., p_i, ..., p_S)$ and $Q = (q_1, ..., q_i, ..., q_S)$, where $0 \le p_i \le 1$, $\sum_{i=1}^{S} p_i = 1$ (similarly, $0 \le q_i \le 1$, $\sum_{i=1}^{S} q_i = 1$) and S is the total number of species in both plots. The information on the species functional organization within both plots is usually represented by a symmetric $S \times S$ matrix of pairwise functional dissimilarities $d_{ij}$ between species i and j with $0 \le d_{ij} \le 1$, $d_{ij} = d_{ji}$ and $d_{ii} = 0$ ($i, j = 1, 2, ..., S$). Also, let $D_F$ be the functional dissimilarity between P and Q and $D_S$ be the corresponding species dissimilarity. $D_F$ is typically calculated by combining the species relative abundances in both plots

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with their functional dissimilarities $d_{ij}$, whereas $D_s$ is calculated with the same species abundances as $D_F$ but considering all species equally and maximally dissimilar from each other (i.e. with $d_{ij} = 1$ for all $i \neq j$).

The basic condition to obtain a meaningful and easily interpretable index of beta redundancy in the range [0,1] is that the functional dissimilarity $D_F$ is lower (or at least not higher) than the corresponding species dissimilarity $D_s$. In this case, beta redundancy can be calculated as (Ricotta et al., 2020):

$$R_{\beta} = \frac{(D_s - D_F)}{D_s} \qquad (1)$$

while beta uniqueness, which is the complement of beta redundancy, can be calculated as:

$$U_{\beta} = 1 - R_{\beta} = \frac{D_F}{D_s} \qquad (2)$$

For $R_{\beta} = 0$, the species turnover between P and Q goes together with a corresponding functional turnover such that $D_F = D_s$. Hence, the compositional differences between P and Q mirror their functional differences. On the contrary, for $R_{\beta} = 1$, both plots are functionally identical to each other (i.e. $D_F = 0$). In this case, the species turnover between P and Q is not associated to a corresponding change in functional properties.

Kosman (1996) and Gregorius et al. (2003) independently proposed an algorithmic measure of (functional) dissimilarity which is based on the optimal overall matching between the species abundances in P and Q so as to minimize the mean functional dissimilarity between both plots. We can think of this operation as follows: for two plots of equal size P and Q with n individuals in each plot, to each individual in P an individual in Q is matched to obtain n pairs that minimize the sum of functional dissimilarities between individuals of the corresponding pairs (Kosman & Leonard, 2007). The pairs are constructed in such a way that all individuals in both plots are used only once. The dissimilarity between plots is then calculated by dividing this sum by n (the number of pairs of individuals). Finding the optimal matching between the species abundances in P and Q is known as the ‘assignment problem’ (Bellman, Cooke, & Lockett, 1970), a special case of the Transportation Problem (TP) of linear programming (Hitchcock, 1941).
Note that, since the number of individuals in both plots is usually not the same, to get a complete matching between P and Q, this operation is best performed on the species relative abundances (see Gregorius et al., 2003). Note also that, since this index is a mean dissimilarity between matched pairs of individuals, if the functional dissimilarity between each pair of individuals is in the range [0,1], the resulting mean plot-to-plot dissimilarity also ranges between 0 and 1.

In terms of relative abundances, the algorithmic index of functional dissimilarity between plots P and Q can be defined as (Gregorius et al., 2003):

$$D_{KG} = \min_{a} \sum_{i=1}^{S} \sum_{j=1}^{S} d_{ij} \times a(i, j) \quad (3)$$

where $a(i, j)$ is the proportion of abundance of species $i$ in plot P that is assigned to individuals of species $j$ in plot Q. According to Kosman & Leonard (2007), if the functional dissimilarity between species is metric, then $D_{KG}$ is also metric. Gregorius et al. (2003) and Kosman (2014) further showed that if all species are considered maximally dissimilar from each other (i.e. if $d_{ij} = 1$ for all species $i \neq j$ and $d_{ii} = 0$), $D_{KG}$ will be equal to the Rogers (1972) dissimilarity:

$$D_{R} = \frac{1}{2} \sum_{i=1}^{S} |p_{i} - q_{i}| \quad (4)$$

Otherwise, for any dissimilarity $d_{ij}$ in the range [0,1], $D_{KG}$ is always lower than $D_{R}$, thus conforming to the redundancy property that $D_{F} \le D_{S}$ (proof in Appendix 1). For those that are unfamiliar with the Rogers dissimilarity, it is worth mentioning that this coefficient is identical to the well-known Bray & Curtis (1957) dissimilarity ($D_{BC}$) computed on the species relative abundances:

$$D_{BC} = \sum_{i=1}^{S} |p_{i} - q_{i}| / \sum_{i=1}^{S} (p_{i} + q_{i}) = \sum_{i=1}^{S} |p_{i} - q_{i}| / 2.$$

Accordingly, based on the algorithmic dissimilarity $D_{KG}$, we can define a measure of beta redundancy and uniqueness as:

$$R_{\beta} = \frac{(D_{S} - D_{F})}{D_{S}} = \frac{(D_{R} - D_{KG})}{D_{R}} \quad (5)$$

and
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$$U_{\beta} = \frac{D_{F}}{D_{S}} = \frac{D_{KG}}{D_{R}} \qquad (6)$$

where beta uniqueness is the fraction of species dissimilarity that is associated to functional dissimilarity, and beta redundancy is the fraction of species dissimilarity that is not associated to functional dissimilarity.

While the algorithmic dissimilarity of Kosman (1996) and Gregorius et al. (2003) has been originally developed in the field of genetics, in community ecology, a number of non-algorithmic measures based on minimum dissimilarity between the species in both plots have been proposed (Clarke & Warwick, 1998; Izsak & Price, 2001; Clarke, Somerfield, & Chapman, 2006; Ricotta & Burrascano, 2008; Swenson, 2011; Ricotta, Podani, & Pavoine, 2016). All these measures basically calculate the functional dissimilarity between a given species in the first plot and its closest functional relative in the second plot. This procedure is repeated for all species in *P* and *Q* and then averaged over both plots. The main difference is that *D*<sub>KG</sub> is based on a (computationally intensive) optimal matching between the species abundances in both plots. This ensures that all individuals in *P* and *Q* are involved in the calculation of *D*<sub>KG</sub>. By contrast, the ‘suboptimal’ matching performed by the non-algorithmic measures retains only nearest-neighbor differences between species for the calculation of functional dissimilarity. As a result, they generally underestimate functional dissimilarity compared to *D*<sub>KG</sub>. For details, see Ricotta et al. (2020, Appendix S2).

Beta uniqueness $U_{\beta} = D_{F}/D_{S}$ and redundancy $R_{\beta} = (D_{S} - D_{F})/D_{S}$ are standardized coefficients that allow us to partition species dissimilarity into two complementary components: the degree of overlap between species dissimilarity and functional dissimilarity ($U_{\beta}$), and the fraction of species dissimilarity not expressed by functional dissimilarity ($R_{\beta}$) such that $U_{\beta} + R_{\beta} = 1$. However, these measures tell us nothing on the amount of similarity among plots. To get a more complete picture of the patterns of species and functional similarity among pairs of plots, non-standardized coefficients may be also used.

We start by noticing that if species dissimilarity $D_{S}$ is bounded between zero and one, its complement $1 - D_{S}$ represents a suitable measure of species similarity among plots. At the same time, $D_{S}$ can be also partitioned into two complementary components which represent the non-standardized versions of uniqueness and redundancy: functional dissimilarity $D_{F}$, and $D_{S} - D_{F}$ which is the (absolute) amount of species dissimilarity not expressed by functional dissimilarity. To
differentiate $D_S - D_F$ from beta redundancy, in the remainder we will call this quantity 'dissimilarity gap' (Table 1 contains a summary of terms along with their definitions).

Therefore, for any functional dissimilarity coefficient that conforms to the redundancy property, we can define three non-standardized components: functional dissimilarity ($D_F$), dissimilarity gap ($D_S - D_F$) and species similarity ($1 - D_S$) such that $D_F + (D_S - D_F) + (1 - D_S) = 1$. For the algorithmic measures of Kosman (1996) and Gregorius et al. (2003), these three non-standardized components are $D_{KG}$, $D_R - D_{KG}$ and $1 - D_R$, respectively. Note however, that the same approach can be used for any functional dissimilarity measure in the range [0, 1] that conforms to the redundancy property, including the tree-based functional and phylogenetic dissimilarity measures proposed by Ricotta et al. (2020).

## 3. Worked examples

### 3.1. Artificial data

The behavior of the proposed measures was first evaluated on a small artificial data set representing an ideal ecological gradient. The data consist of a matrix with the relative abundances of 15 species (S1-S15) in 10 plots (P1-P10). The matrix was built such that all species have unimodal abundance pattern of varying amplitude along the gradient (Table 1), while the corresponding functional dissimilarity matrix among species was built such that the interspecies dissimilarities reflect the species ecological differences along the simulated gradient (Appendix 2). To this end, the interspecies similarities were set proportional to the distance between the species optima (i.e. the locations along the gradient where the species show their maximum abundances).

To explore the response of the proposed measures to changes in species composition along the simulated gradient, we compared plot P1 with itself and with the remaining plots in terms of beta redundancy, species dissimilarity, functional dissimilarity, and dissimilarity gap. All measures were calculated with a new R function available in Appendix 3 of this paper, which uses the same TP algorithm of Gregorius et al. (2003).

### 3.2. Vegetation primary succession

We also explored the behavior of the proposed measures on Alpine vegetation sampled by Caccianiga, Luzzaro, Pierce, Ceriani, & Cerabolini (2006) along a primary succession. The data set, which has been already used in previous papers on alpha and beta redundancy (Ricotta et al., 2016; Ricotta et al., 2020) is composed of 59 plots of approximately 25 m² sampled at the foreland of the Rutor Glacier (Northern Italy). For each plot, species abundances were measured with a five-point ordinal scale transformed to ranks. Based on the age of the moraine deposits, the plots were then
classified into three successional stages: early-successional stage (17 plots), mid-successional stage (32 plots), and late-successional stage (10 plots).

For all 45 plant species in the data set, we selected six quantitative traits measured by Caccianiga et al. (2006, Table 2), which provide a good representation of the global spectrum of form and function (Díaz et al., 2016): canopy height (CH; mm), leaf dry mass content (LDMC; %), leaf dry weight (LDW; mg), specific leaf area (SLA; mm² × mg⁻¹), leaf nitrogen content (LNC; %), and leaf carbon content (LCC; %). All data are available in the data object named ‘RutorGlacier’ of the adiv package (Pavoine, 2020) of R (R Core Team, 2021).

We used the Euclidean distance to calculate an interspecies dissimilarity matrix from the six functional traits. Before calculations, all traits were standardized to zero mean and unit standard deviation. The resulting functional distances among species were then rescaled to the unit range by dividing each distance by the maximum value in the distance matrix.

Based on the scaled matrix, we calculated the standardized and non-standardized functional dissimilarity components for all pairs of plots in each successional stage. We next used the PERMDISP test (Permutational Analysis of Multivariate Dispersions) of Anderson (2006) to test for differences in the dispersion of species dissimilarity $D_R$, functional dissimilarity $D_{KG}$ and functional uniqueness $U_β$ among the three successional stages.

# 4. Results

## 4.1. Artificial data

The profile diagrams in Figure 1 (Podani & Miklós, 2002) show the response of species dissimilarity $D_R$, functional dissimilarity $D_{KG}$, beta redundancy $R_β$ and dissimilarity gap $D_R - D_{KG}$ to changes in species composition along the artificial gradient in Table 1. The comparison of the first plot with itself and all other plots shows a monotonic increase of species and functional dissimilarity along the gradient. Note however that, while for the pair of plots P1/P10, species dissimilarity reaches its maximum value (i.e. $D_R = 1$), meaning that P1 has no species in common with P10, functional dissimilarity does not reach its maximum value because the species in both plots are not completely functionally distinct. The profile diagrams in Figure 1 also show the main conceptual differences between standardized beta redundancy $R_β = (D_R - D_{KG})/D_R$ and the non-standardized dissimilarity gap $D_R - D_{KG}$. In absolute terms, $D_R - D_{KG}$ increases along the first part of the gradient where the species turnover among plot P1 and the other plots is only weakly associated to a corresponding functional turnover. Then, it decreases along the second part of the gradient, where most of the species differences among P1 and the other plots are associated to functional differences
among plots. By contrast, in relative terms, as species dissimilarity increases along the gradient, functional redundancy $R_β$ tends to be progressively replaced by functional uniqueness $U_β = 1 - R_β$.

Note that for two identical plots, species dissimilarity $D_S$ and functional dissimilarity $D_F$ are both zero. In this case, beta uniqueness $U_{PP} = D_F/D_S$ and redundancy $R_{PP} = (D_S - D_F)/D_S$ are not defined. However, considering that beta uniqueness $U_β = D_F/D_S$ is a relative measure that relates plot-to-plot functional dissimilarity to its local maximum, we set by definition $U_{PP} = 0$ if $D_S = 0$. At the other extreme, for two maximally dissimilar plots P and Q with no species in common and maximum functional dissimilarity $d_{ij} = 1$ for species i and j belonging to P and Q, respectively, we have $D_F = D_S = 1$ and hence $U_{PP} = 1$. Likewise, since beta redundancy $R_β = 1 - U_β$ it follows that $R_{PP} = 1$ if $D_S = 0$, and $R_{PP} = 0$ if $D_F = D_S = 1$.

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## 4.2. Vegetation primary succession

Caccianiga et al. (2006) showed that in the early-successional stage, the colonization of the moraine ridges of the Rutor glacier by the first pioneer species is primarily controlled by random dispersal, whereas the vegetation of the mid- and late-successional stages shows a higher level of functional homogeneity. This increase in functional homogeneity produces a similar increase in functional alpha and beta redundancy along the primary succession (Ricotta et al., 2016; 2020).

The results of the test for differences in the dispersion of $D_R$, $D_{KG}$ and $U_β$ among the three successional stages are shown in Table 3. First, based on the pairwise dissimilarities among pair of plots in each successional stage, the test calculates the dissimilarity of individual plots from the corresponding group centroid. Next, a permutational t-test with 9999 randomizations of these dissimilarities is used to test for pairwise differences in average dissimilarity of individual plots from their group centroids. For details on the PERMDISP test, see Anderson (2006).

In good agreement with the results of Ricotta et al. (2020), the successional stages do not differ significantly in species dissimilarity $D_R$, meaning that the species turnover among plots is more or less the same in all successional stages. By contrast, (non-standardized) functional dissimilarity $D_{KG}$ and (standardized) uniqueness $D_{KG}/D_R$ in the early-successional stage are both significantly higher than in the mid- and late-successional stages. That is, due to the more random dispersal mechanisms, in the early-successional stage, the species turnover is associated to a higher rate of functional turnover compared to the mid- and late-successional stages where the species in one plot tend to be replaced by functionally related species in the other plots.

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Discussion

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Ricotta et al. (2020) introduced the concept of beta redundancy to summarize the fraction of species dissimilarity $D_S$ between two plots not expressed by functional or phylogenetic dissimilarity $D_F$. From an ecological viewpoint, this index tells us to what degree the species turnover between two plots is associated to a functional or phylogenetic turnover among the species in both plots. From a technical viewpoint, a necessary condition for a suitable index of beta redundancy is that the functional or phylogenetic dissimilarity is always lower than species dissimilarity: $D_F \le D_S$. This condition prevents beta redundancy from taking negative values. Ricotta et al. (2020) also developed a new class of indices of functional and phylogenetic dissimilarity between plots that conform to the redundancy property $D_F \le D_S$. However, these indices depend on a hierarchical structure (a tree) of functional differences among species, which is not always considered adequate to portray the species functional relationships.

To develop a new index of functional beta redundancy that does not require the definition of a functional tree, our first step was to 'import' in the ecological literature an algorithmic index of functional dissimilarity $D_{KG}$ originally proposed by Kosman (1996) and Gregorius et al. (2003) for genetic studies. Given two plots P and Q, the index is based on the optimal overall matching between the species abundances in both plots, so as to minimize the mean functional dissimilarity for the matched species abundances between P and Q. Since the calculation of $D_{KG}$ does not require a tree-based species structure, the proposed measure of functional beta redundancy is now more flexible and compatible with the usual ways of representing functional relationships among species. Note that, since this index is essentially a kind of mean functional dissimilarity between the species in two plots, the practitioner can base the analysis on any functional dissimilarity measure of choice.

Being conform to the redundancy property $D_F \le D_S$, this algorithmic index enables to calculate (relative) functional redundancy and uniqueness in the usual way: $R_\beta = (D_S - D_F)/D_S$ and $U_\beta = 1 - R_\beta = D_F/D_S$. In addition to these relative coefficients, for a dissimilarity index that is bounded in the range [0, 1] the non-standardized coefficients of functional dissimilarity $D_F$ and dissimilarity gap $D_S - D_F$ may be also used.

The main difference between beta redundancy and uniqueness and their non-standardized versions, $D_S - D_F$ and $D_F$, is how the measures are scaled. For two plots with relative abundance vectors $P = (p_1, ..., p_i, ..., p_S)$ and $Q = (q_1, ..., q_j, ..., q_S)$, a measure of non-standardized functional dissimilarity $D_F$ in the range [0, 1], attains its maximum value (i.e. its global maximum) $D_F = 1$ for two maximally dissimilar plots with no species in common and maximum functional differences
$d_{ij} = 1$ for species *i* belonging to plot *P* and species *j* to plot *Q*. By contrast, to calculate $U_{\beta}$ functional dissimilarity $D_{F}$ is scaled by its local maximum $D_{S}$, i.e. by the maximum value that $D_{F}$ can reach by keeping the relative abundance vectors unchanged and by imposing $d_{ij} = 1$ for all $i \neq j$.

The ecologically interesting aspect of the non-standardized dissimilarity gap $D_S - D_F$ is related to its dual nature. In fact, $D_S - D_F$ is at the same time a component of species *dissimilarity* and of functional *similarity*. The dissimilarity gap thus summarizes to what degree the differential species in two plots are able to perform the same ecological functions. Therefore, $D_S - D_F$ and its standardized version $R_\beta$ are directly related to classical ecological processes, such as habitat filtering (Zobel, 1997), dark diversity (Pärtel, Szava-Kovats, & Zobel, 2011) and the species model (van der Maarel & Sykes, 1993).

For example, using simulated and real data, Laroche, Violle, Taudière, & Munoz (2020) illustrated how the difference between functional and compositional dissimilarities can help in identifying the processes underpinning diversity patterns at the metacommunity scale. In case of habitat filtering, species in the same habitat tend to have more similar traits that help them to cope with the local environmental conditions. Optimal traits are different from one habitat to another. Species present in the regional species pool but absent from a given habitat may thus have non-optimal traits for that habitat that enable them to colonize different habitats with different environmental conditions. If species traits are more related to their fitness than to their niche, then, competitive exclusion could lead to the absence of the less competitive species from a given site and thus to coexisting species with similar trait values. Combined together, these equalizing processes due to abiotic and biotic filtering (i.e. species share trait values that make them locally adapted to their environment and that ensure their similarity in terms of fitness; Chesson, 2000) could both lead to high $D_S$ and $D_F$, low $R_\beta$ and low $D_S - D_F$ between plots of different habitats, and to low $D_F$ and relatively high to high $R_\beta$ between plots of similar habitats depending on the level of the corresponding $D_S$.

If dispersal is high between plots regardless of habitat conditions, and if stabilizing niche differences (see Chesson, 2000) favor coexistence between species with distinct trait values allowing low overlap in resource use and stronger intraspecific competition than interspecific competition, then we expect high functional differences between locally coexisting species and high species turnover between plots because similar species cannot coexist. Accordingly, we also expect high functional overdispersion meaning that the functional differences within plots are high, and the plots are functionally similar because for each species in one plot we can expect to find a 'functionally-
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relative' species not in the same plot but in another plot. In that case, limiting similarity due to stabilizing niche differences will lead to high $D_S$, low $D_F$, and hence high $R_β$ and high $D_S - D_F$ between plots irrespective of the environmental conditions in the different plots.

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High beta redundancy between plots ($R_β$ and $D_S - D_F$), combined with high species dissimilarity (high $D_S$) could also inform priorities of conservation if some vulnerable species are absent in a site although the local ecological (biotic and abiotic) conditions would be favorable (dark diversity, Pärtel et al., 2011). This could be due to the species mobility and to stochastic colonization processes of favorable habitats (the species Carousel of van der Maarel & Sykes, 1993). However, this could also be driven by habitat fragmentation, local extinction and limited or even hampered dispersal between the remnant habitat patches. This diversity of processes emphasizes the importance of selecting an appropriate null hypothesis for analyzing beta-redundancy patterns. In this paper, we used a standard permutational approach. narrower null hypotheses, such as constrained permutations (Peres-Neto, Olden, & Jackson, 2001) or the neutral baseline approach of Laroche et al. (2020) could also be used to facilitate the distinction between alternative processes that may produce similar species distribution patterns.

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Temporal surveys, revealing increase in $D_S$ over years, accompanied by an increase in $R_β$ and $D_S - D_F$, could be even more informative alerting conservation agencies to potentially drastic environmental impact on biodiversity necessitating ecosystem restoration. Consider a hypothetical case where functional dissimilarities between plots in a habitat are proportional to species dissimilarities: the more complementary the plots are in terms of species, the more functional types they together contribute to the functional diversity of the habitat. If in addition local species extinctions are so numerous that some species become globally extinct (over all habitat patches), $R_β$ is expected to remain constant but $D_S - D_F$ to decrease, meaning that the original levels of species and functional diversity could only be reestablished by habitat restoration and species reintroductions, wherever possible.

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To conclude, functional beta redundancy allows us to explore an important aspect of the complex multidimensional space of ecological data. As highlighted by several authors (see e.g. Lavorel et al., 2008; Ricotta & Moretti, 2011), which traits are actually relevant for ecosystem functioning depends on the specific process of interest. Therefore, a critical point is the selection of an appropriate set of functional traits. The basic assumption for exploring the effects of dissimilarity on ecosystem functioning is that communities with similar functional traits have similar ecological properties. Therefore, two communities are supposed to be functionally similar with respect to some a priori-defined ecological property if they share the same traits. In principle, increasing the number of traits
leads to more accurate measures of community similarity. However, these higher-dimensional
functional spaces generally do not have any direct biological connection to the specific ecological
properties under scrutiny. Therefore, instead of building increasingly larger functional spaces, we
need to develop increasingly focused spaces that optimize their association to the property of interest
(Ricotta and Moretti 2010). The construction of such 'tailored' functional spaces may be based on
expert knowledge (Caccianiga et al., 2006), statistical methods (Ricotta & Moretti, 2010; Pierce et
al., 2017) or on modeling approaches (Petchey & Gaston, 2006). In this view, we believe that future
work combining indices of beta redundancy with the development of functional spaces strongly
connected to a priori-selected properties will greatly contribute to the understanding of the
relationships among different facets of community dissimilarity and ecosystem functioning.

**Authors contributions.** CR and SP formulated the research problem; EK and FL provided important feedback which helped shape the research; CR and SP analyzed the data; CR took the lead in writing the manuscript and SP in writing the appendices. All authors revised the manuscript critically and approved the final version.

**Data availability statement**

The R code used in this paper is deposited in the Dryad Digital Repository: https://doi.org/10.5061/dryad.b2rbnzsds (Ricotta, Kosman, Laroche, & Pavoine, 2021). The data of the Alpine vegetation are available online in the R package adiv: https://cran.r-project.org/web/packages/adiv/index.html (R Core Team, 2021). The functional dissimilarity matrix of the artificial data set is available in the supporting information of this paper (Appendix 2).
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478 **Supporting Information**

479 Additional supporting information may be found online in the Supporting Information section at the
480 end of the article.

481

482 **Appendix 1.** Proof that the algorithmic dissimilarity $D_{KG}$ is always not higher than the Rogers dissimilarity $D_R$.

484

485 **Appendix 2.** Interspecies dissimilarity matrix used for the calculation of the proposed redundancy
486 measures with the artificial data in Table 1.

487

488 **Appendix 3.** R scripts for the calculation of the proposed redundancy measures.
**Table 1.** Summary of all redundancy components used in this study. All components are formulated in terms of species and/or functional dissimilarities.

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<table><thead><tr><th>Component</th><th>Definitions and Abbreviations</th></tr></thead><tbody><tr><td>Functional Dissimilarity</td><td>D<sub>F</sub></td></tr><tr><td>Species Dissimilarity</td><td>D<sub>S</sub></td></tr><tr><td>Dissimilarity Gap</td><td>D<sub>S</sub> - D<sub>F</sub></td></tr><tr><td>Species Similarity</td><td>1 - D<sub>S</sub></td></tr><tr><td>Functional Similarity</td><td>1 - D<sub>F</sub></td></tr><tr><td>Beta Uniqueness</td><td>U<sub>β</sub> = D<sub>F</sub>/D<sub>S</sub></td></tr><tr><td>Beta Redundancy</td><td>R<sub>β</sub> = (D<sub>S</sub> - D<sub>F</sub>)/D<sub>S</sub></td></tr></tbody></table>

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498

**Table 2.** Artificial data set with the relative abundances of 15 species (S1-S15) in 10 plots (P1-P10) ordered along an ideal ecological gradient. The matrix was built such that all species have unimodal response of varying amplitude and abundance to a hypothetical one-dimensional gradient. Non-zero abundances are shown in grey. The locations of the species optima are highlighted in bold.

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<table><thead><tr><th rowspan="2">Species</th><th colspan="10">Plot</th></tr><tr><th>P1</th><th>P2</th><th>P3</th><th>P4</th><th>P5</th><th>P6</th><th>P7</th><th>P8</th><th>P9</th><th>P10</th></tr></thead><tbody><tr><td>S1</td><td><strong>0.15</strong></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S2</td><td><strong>0.4</strong></td><td>0.2</td><td>0.15</td><td>0.05</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S3</td><td>0.1</td><td><strong>0.15</strong></td><td>0.05</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S4</td><td>0.2</td><td><strong>0.35</strong></td><td>0.25</td><td>0.15</td><td>0.05</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S5</td><td>0</td><td>0</td><td><strong>0.1</strong></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S6</td><td>0</td><td>0</td><td>0</td><td><strong>0.05</strong></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S7</td><td>0.15</td><td>0.2</td><td>0.25</td><td><strong>0.35</strong></td><td>0.25</td><td>0.2</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0</td></tr><tr><td>S8</td><td>0</td><td>0.1</td><td>0.15</td><td>0.25</td><td><strong>0.35</strong></td><td>0.3</td><td>0.2</td><td>0.1</td><td>0</td><td>0</td></tr><tr><td>S9</td><td>0</td><td>0</td><td>0.05</td><td>0.15</td><td><strong>0.25</strong></td><td>0.2</td><td>0.1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S10</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.1</td><td><strong>0.2</strong></td><td>0.15</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S11</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td><strong>0.15</strong></td><td>0</td><td>0</td><td>0</td></tr><tr><td>S12</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.1</td><td>0.25</td><td><strong>0.5</strong></td><td>0.45</td><td>0.45</td></tr><tr><td>S13</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.05</td><td>0.15</td><td><strong>0.25</strong></td><td>0.05</td></tr><tr><td>S14</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.1</td><td>0.15</td><td><strong>0.3</strong></td></tr><tr><td>S15</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.05</td><td>0.05</td><td><strong>0.2</strong></td></tr></tbody></table>

506
**Table 3.** Mean (Standard deviation) dissimilarity values of the vegetation plots in each successional stage from the corresponding centroids. Pairwise differences in mean plot dissimilarity from the group centroids were tested with permutational t-tests (9999 permutations). For each dissimilarity index, numbers followed by the same letter do not differ significantly at P < 0.05. $D_R$ = species dissimilarity; $D_{KG}$ = functional dissimilarity; $U_β$ = beta uniqueness.

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<table><thead><tr><th></th><th>Early-successional stage (17 plots)</th><th>Mid-successional stage (32 plots)</th><th>Late-successional stage (10 plots)</th></tr></thead><tbody><tr><td>D<sub>R</sub></td><td>0.407 (0.075)<sup>a</sup></td><td>0.365 (0.083)<sup>a</sup></td><td>0.361 (0.106)<sup>a</sup></td></tr><tr><td>D<sub>KG</sub></td><td>0.149 (0.044)<sup>a</sup></td><td>0.095 (0.026)<sup>b</sup></td><td>0.091 (0.023)<sup>b</sup></td></tr><tr><td>U<sub>β</sub></td><td>0.253 (0.037)<sup>a</sup></td><td>0.182 (0.024)<sup>b</sup></td><td>0.170 (0.016)<sup>b</sup></td></tr></tbody></table>

515
**Figure 1.** Profile diagram showing the response of species dissimilarity $D_R$, functional dissimilarity $D_{KG}$, dissimilarity gap $D_R - D_{KG}$ and beta redundancy $R_\beta = (D_R - D_{KG})/D_R$ to changes in species composition along the artificial gradient in Table 1. The profile diagram reflects the comparison of the first plot with itself and all other plots along the gradient.

<figure><img src="image_2.png" /></figure>