On the relationships between rarity, uniqueness, distinctiveness, originality and functional/phylogenetic diversity

Rarity reflects the low abundance of a species while distinctiveness reflects its quality of being easy to recognize because it has unique functional characteristics and/or an isolated phylogenetic position. As such, the assemblage-level rarity of a species’ functional and phylogenetic characteristics (that we name ‘effective originality’) results from both the rarity and the distinctiveness of this species. The functional and phylogenetic diversity of an assemblage then results from a compromise between the abundances and the effective originalities of the species it contains. Although the distinctiveness of a species itself depends on the abundance of the other species in the assemblage, distinctiveness indices that are available in the ecological literature scarcely consider abundance data. We develop a unifying framework that demonstrates the direct connections between measures of diversity, rarity, distinctiveness and effective originality. While developing our framework, we discovered a family of distinctiveness indices that permit a full control of the influence one wants to give to the strict uniqueness of a species (=its smallest functional or phylogenetic distance to another species in the assemblage). Illustrating our framework with bat phylogenetic diversity along a disturbance gradient in Mexico, we show how each component of rarity, distinctiveness and originality can be controlled to obtain efficient indicators for conservation. Overall our framework is aimed to improve conservation actions directed towards highly diverse areas and/or towards species whose loss would considerably decrease biodiversity by offering flexible quantitative tools where the influence of abundant versus rare, and ordinary versus original, species is understood and controlled.


Introduction
Generally speaking, biological diversity or biodiversity is the range of many different characteristics of biological systems. In species assemblages, biodiversity thus emerges because species are not equivalent; and species diversity increases with species richness (the number of species) and the evenness (or equitability) of species abundances (e.g., Patil and Taillie, 1982). In local assemblages, often many species are rare, having small population size, and only a few species dominate in abundance (e.g., Hughes, 1986), yielding moderate levels of species diversity. However the exact shape of species abundance distribution may depend on ecological processes as for example disturbance can modify species relative abundances, and consequently species diversity (e.g., Matthews and Whittaker, 2015).
Another increasingly studied aspect of species rarity that may influence biodiversity levels is the rarity of species biological characteristics (Kondratyeva et al., 2019). Pavoine et al. (2017) defined the originality of a species in a given assemblage as 'the rarity of its biological characteristics'. As such, originality can be based on the species evolutionary history (phylogenetic originality) or on the species functional traits (functional originality) as both aspects are believed to reflect biological differences between species. Originality is related in the ecological literature to the concepts of species distinctiveness and uniqueness. Uniqueness is the quality of being unique in some way (Cambridge Dictionary, 2021), here in the functional characteristics or the phylogenetic position. Distinctiveness is the quality of being easy to recognize because of being different from other things (Cambridge Dictionary, 2021).
In our context, the distinctiveness of a species can thus be defined as its quality of being easy to recognize because it has some unique functional characteristics or as its quality of being easily found in a phylogenetic tree because it belongs to an old, species-poor clade.
Distinctiveness and uniqueness are thus two closely connected concepts as the more unique a species is the more distinct it might be. To simplify the writing, hereafter we will thus use below only the term distinctiveness. We will also use the acronym 'FP' to mean 'functional or phylogenetic'.
Originality is the quality of being special and interesting and not the same as anything or anyone else (Cambridge Dictionary, 2021). This is why the originality of a species can be 3 considered as the assemblage-level rarity of the FP-characteristics associated to this species . This originality depends on whether abundance data are considered ( Fig. 1). In absence of abundance data, the concept of originality is equivalent to that of distinctiveness (Pavoine et al., 2005). This is because the species own rarity is discarded and the rarity of a species' biological characteristics is only linked to the proportion of functional traits or phylogenetic history that are unshared with other species (see, e.g., the distinct species 1 and 2 in Fig. 1a,b,c). In global conservation studies for example abundance data are rarely considered (e.g., Gaüzère et al., 2015).
For more local ecological studies however, abundance data are often available and often reveal meaningful to analyze ecological systems (e.g. Enquist et al., 2019). Consider that a focal species j is distinct from other species in a defined species pool. If in an assemblage, its few functionally sibling or close relatives "dominate" the assemblage by their high abundance or if species j is itself very abundant, then the assemblage-level rarity of the FP-characteristics associated to species j is actually low and the originality of this species is in fact low (e.g., species 1 and 2 in Fig. 1d,e). Inversely, consider that species j has low distinctiveness in the species pool (where abundance data is discarded). Species j may still be effectively original in an assemblage where its abundance and those of its sibling, close species are all low (e.g. species 3 to 8 in Fig. 1d,e). Hereafter, we will refer to this abundance-based definition of FPoriginality as the effective FP-originality. Combining the different aspects of a species rarity (its low abundance; the distinctiveness of its traits; its isolated position in the phylogeny), for the scope of this paper, we will thus consider two intuitive conditions that a measure of effective FP-originality should respect. Consider an assemblage composed of N species with relative abundances Australia (Bouma et al., 2020), morphologically unique among extent Chelidae. The originality of this species may be due to strict trait conservatism along its evolution as fossil data indicates little morphological change for this species since the early Miocene (Zhang et al., 2017). The small population size of this endemic species (Arnall, 2018) could yield a low contribution of this species to Chelidae biodiversity. However, in contrast, due to its distinct morphological characteristics, the theoretical contribution of this species to the biodiversity of the Chelidae family is expected to be high. Moreover, its low abundance reinforces the rarity of its morphological characters and thus the key necessity to consider this species in conservation actions. Indeed, it has already been shown that rare species (of low abundance) may contribute more to ecosystem functioning and ecosystem services than their low abundance would suggest because of their unique functional role, notably via unique functional traits (Dee et al., 2019). While, if species abundance is not considered, the contribution of a species to biodiversity can directly result from its distinctiveness, considering abundance yields its contribution to result from a compromise between its distinctiveness and its abundance. However, although many species, trait and phylogenetic diversity measures do include abundance data, very few originality measures developed so far include species abundance; and the full potential, for ecological studies, of these few abundance-weighted measures of originality still needs to be emphasized.
We thus focus our study on three of these abundance-weighted measures of originality:  Rao, 1982) for the measure of Ricotta et al. We extend below these three originality indices and unify them in a common framework on the link between diversity, rarity, distinctiveness and originality.

Quadratic diversity as a mean of effective originalities
The denominator of Eq. 2, 1 ji p   , is the rarity of species j according to the well-known Simpson diversity (Simpson, 1949;Patil and Taillie, 1982 the abundance-weighted average FP-dissimilarity between any two species in the assemblage.
In contrast to FP-distinctiveness D j , effective FP-originality O j accounts for the abundance of species j itself: to evaluate the rarity of the biological characters of a focal species when abundance data are considered, then the abundance of the focal species itself has to be considered.

Parametric generalizations
We consider below, two possible parametric extensions of Rao's quadratic diversity. The parameter α of the first one controls the importance given to ordinary species in opposition to effectively original species: Eq. 4 was first developed by Ricotta and Szeidl (2006). For α = 2, 2 KQ  and for α tending to 1, it is a generalization of the Shannon index (Shannon, 1948;Ricotta and Szeidl, 2006). When parameter α in K  increases, then ordinary species (those species with low effective originality according to the quadratic diversity) are given increasingly important weights in the measurement of phylogenetic diversity. We develop in Table 1 a decomposition of K  in terms of rarity, distinctiveness and originality (see details and proofs in Appendix A).
We develop here the following alternative to index K  , named K   , where parameter α controls species' abundance instead of ordinariness, i.e. α controls the importance given to abundant species in opposition to rare species. As such, with K   , varying parameter α influences the ranking of species according to their distinctiveness and effective originality: where c|j indicates the cth closest species from species j; for c > 1 with 0|j applied to d ij = the sum of branch lengths on the shortest path from tip j to its most recent common ancestor with species i. In addition, in this particular case the associated species effective originalities (Table 2) are equivalent to those introduced by Kondratyeva et al.
(2019) who expressed index α I of phylogenetic diversity as a mean of the species originalities.
For α = 0, the phylogenetic effective originality associated with I  ( Table 2)  in the special case of a phylogenetic tree and we depict its writing in terms of rarity, distinctiveness and effective originality in Table 2. Similarly, we introduce in Table 2 a rewriting (named Y  ) of index K  and its associated indices of distinctiveness and originality, in the special case where species are tips of a phylogenetic tree and the phylogenetic distance between species j and i is calculated as the sum of branch lengths on the shortest path from tip j to its most recent common ancestor with species i. All functions of rarity, distinctiveness and effective originality discussed here are thus connected in a global framework in Tables 1 and 2, highlighting the strong links between different facets of rarity, distinctiveness, originality and diversity.
8 All these diversity indices can be easily transformed into equivalent numbers of species: the number of evenly and maximally dissimilar species needed to obtain the level of FP-diversity observed in an assemblage (Tables 1 and 2). The functions that transform the diversity indices we discussed here in terms of equivalent numbers of species do not change the way species assemblages are ranked from the least to the most diverse (Appendix A).

The special case of abundance-free distinctiveness indices
Abundance-free distinctiveness indices, particularly phylogenetic distinctiveness indices, are often used in conservation biology (Isaac et al., 2007;Redding et al., 2014). Imposing equal relative abundances for all species in our framework provides a useful family of such distinctiveness indices where abundance data is discarded as outlined below (see a complete introduction of the family in Appendix B).
In the special case of equal abundance for all species (p j =1/N for all j), the phylogenetic distinctiveness index    associated with phylogenetic diversity index  I (Table 2) becomes: where N b stands for the number of species that descend from branch b and C(j, root) for the set of branches between species j, tip of the phylogenetic tree, and the root of the tree. eq    thus provides a parametric alternative to the most widely used index of species distinctiveness named "evolutionary distinctiveness" (ED; Isaac et al., 2007) or "Fair-Proportion" (Redding et al., 2014) and whose formula is: 9 Both ED and eq j    can be seen as the sum, on the shortest path from a tip to root, of the product of a branch length times a decreasing function of the number of species descending from that branch. While by construction, the value of ED is dominated by the length of the terminal branch that connects species j to the tree (Redding et al., 2014), the parameter α in eq    allows controlling the influence of this terminal branch (see Appendix B for more details).
It can be easily shown for example that with α = 0, where ij d is the sum of branch lengths on the shortest path from tip j to its most recent common ancestor with species i.
Although the diversity indices of Tables 1 and 2  between j and any other species (α → -∞), through the average FP-dissimilarity to all other species (α = 2), to the largest FP-dissimilarity between j and any other species (α → +∞) (Appendix B).

Data
As a case study, we considered changes in bat phylogenetic diversity across a disturbance Branch lengths on the consensus tree were obtained using mean edge length, ignoring credible trees in which the branch is absent (Revell, 2012).

Analyses
We calculated, in each habitat, the phylogenetic diversity using Y  and α I, for α varying between 0 and 3. We explored then in detail the patterns of phylogenetic diversity in terms of effective originality. Additionally, we analyzed species distinctiveness in the species pool (using indices ED and eq    that both discard abundance data).

Results
In the species pool, for values of α of 1, 2 and 3, according to eq Phylogenetic diversity decreases along the disturbance gradient (from rainforest, through cacao plantation and old fields, to cornfields) whatever the value of α we considered (from 0 to 3 in Fig. 3) for I  and for high values of α (α >= 1.5 in Fig. 3) for Y  . With  Y , for low values of α (α < 1.5 in Fig. 3) the phylogenetic diversity of old fields exceeds that of cacao plantations; and as α approaches zero (α < 0.5 in Fig. 3) the phylogenetic diversity of old fields even exceeds that of the rainforest. This is because old fields contain both B.
dubiaquercus and M. keaysi, the two Vespertilionidae species with the highest average phylogenetic distance to all other species, while only one of them was observed in the rainforest and in the cacao plantation, and none in the cornfields (Fig. 4). Indeed index Y  , as K  , uses the effective originality associated with the quadratic diversity and a parameter α that controls the relative importance given to ordinary species compared to effectively original species. For low values of α, the influence of the most effectively original species in the measurement of phylogenetic diversity increases. In contrast as shown above, α in I  , as in K   , controls the importance given to abundant compared to rare species, influencing the way the effective originality of a species is perceived (with strong influence of the terminal branch for low values of α). Compared to other habitats, cornfields lacked effectively original species ( Fig. 4; see also Video S2 for a more complete visualization of variation in species originality    as a function of α). Phylogenetic diversity in cornfields was thus always the smallest (whatever α; and even strongly lower than that of other habitats when α approaches zero; Fig. 3). Species with the least effective originality were often either the Carollia species or the Artibeus species depending on the habitat and the value of α considered (Fig. 4).
However the relatively high abundance of P. parnelii observed in the rainforest, made this species perceived as one of the least effectively original in this habitat when rare species were given high weights in the measurement of phylogenetic diversity (α = 0, index α I, Fig. 4), despite its high distinctiveness ( Fig. 2 and 4; see also Video S3 for a more complete visualization of variation in species distinctiveness    as a function of α).

Discussion
The contribution of a given species to the biodiversity of an assemblage thus depends on its rarity and on the rarity of its FP-characteristics. Starting from Ricotta et al. (2016) measure of species-level originality K we have shown that quadratic diversity can be expressed as a mean of effective FP-originality values over all species in an assemblage, and that Faith's (1992) phylogenetic diversity index and Cadotte et al. (2010) and Kondratyeva et al. (2019) measures of effective phylogenetic originality can be both related to parametric extensions of quadratic diversity. This led us to develop a unified framework (summarized in Tables 1 and 2) where diversity, rarity, distinctiveness and originality measures are intrinsically linked.
The parametric indices developed in this framework allow regulating the importance given to abundant and ordinary species in FP-diversity measures. They include generalizations of the Shannon index (for α → 1) and the Simpson index (for α = 2) to functional and phylogenetic data. By increasing the value of their parameter α, the weight given to abundant and ordinary species in the measurement of FP-diversity increases in comparison with the weight given to rare and effectively original species. Low values of α may thus indicate regions with high diversity but a diversity that may be threatened by the rarity of the most effectively original species. Increasing α may reveal how much phylogenetic diversity depends on these rare species. Low values of the parameter α could thus be particularly relevant to obtain biodiversity indicators directed to the preservation of rare and distinct species while maintaining a high level of global diversity (e.g., Hidasi-Neto, Loyola and Cianciaruso, 2015).
In our case study, when effectively original species were given high weights in the measurement of diversity (according to Y  which evaluates effective originality as a function 13 of the average phylogenetic distance to all individuals observed in the habitat), the old fields had the highest measure of phylogenetic diversity. This was due to the presence of the two species from the Vespertilionidae family. Although distinct in our study area, the Vespertilionidae species represent a large family of bats at a global scale. This illustrates how the measurement of a species' distinctiveness is dependent on the reference species assemblage, on the data used to characterize species (here phylogeny), and thus on the taxonomic, phylogenetic and spatial scales of a study. A species' rarity, when measured relatively to the rarity of all other species rather than as an absolute value, is also dependent on these scales. According to the International Union for the Conservation of Nature (IUCN, 2021), all species of our case study are least-concern (i.e., neither threatened nor near threatened) at a global scale with either stable or unknown population trends except M.
megalophylla that is least-concern but with decreasing population trends, and B. dubiaquercus in plantations (e.g., Estrada et al., 2006). Other studies showed that, in addition to cave collapse, cave vandalism, threats on M. megalophylla and B. dubiaquercus also concern their sensitivity to disturbance and habitat loss (IUCN, 2021). In an urban context in the highlands of Chiapas, it was shown that abundance of B. dubiaquercus tends to diminish outside forest; and the activity of both M. megalophylla and B. dubiaquercus increases with tree density (Rodríguez-Aguilar et al., 2017).
The whole framework can thus be used in ecological studies to reveal the relative contributions of each species to biodiversity, and to depict these contributions in terms of abundance-based rarity, species-level FP-distinctiveness and effective FP-originality. It can be applied from local to global scales provided abundance data are available and the species assemblage is clearly delimited. The framework can be used to evaluate how species contributions to biodiversity vary in space and which evolutionary and/or ecological processes have shaped the biodiversity of an assemblage. Studying a process of invasion for example, 14 the framework could be used across various scales and biomes, to confront Darwin's naturalization hypothesis, specifying that the taxonomic distance (or phylogenetic and functional distinctiveness) of an invader compared with native species increases the chance of successful installation by limiting competitions, and Darwin's alternative niche-adaptation hypothesis, for which ecological (or functional) redundancy with natives is expected so that the invader is pre-adapted or tolerant to the local environmental conditions (Darwin, 1859; see, e.g., Park et al., 2020). In this context, weighting originality measures by species abundance is critical to evaluate the potential degree of competition between species individuals and how competition could influence the functional and phylogenetic compositions of an assemblage.
Our framework can be used to evaluate how changes in species contributions may impact both biodiversity levels and ecosystem functioning. Indeed, complementing abundance data with originality values is important in this context. As indicated in the introduction, rare species (of low abundance) may contribute more to ecosystem functioning and ecosystem services than their low abundance would suggest because of their unique functional characteristics (Dee et al., 2019). Depicting how rarity and functional distinctiveness influence measures of biodiversity could thus improve research on the connections between biodiversity and ecosystem functioning.
In the context of the sixth species mass extinction (Ceballos et al., 2015), our framework could be used to evaluate how changes in species contributions may impact ecosystem services, via changes in biodiversity levels. It would allow following changes in species contribution to biodiversity under novel environmental conditions, from local to global planetary-scale change. Relative abundances may change under novel environmental conditions with rare species for example benefiting from the reduction of population size of other species. If these species are functionally rare, an increase in their abundance can considerably increase the functional diversity of an assemblage and modify ecosystem functioning and the connected ecosystem services. The bioprospecting or option value associated to a species notably advocates as a precaution principle to protect species with the aim to give option to discover new uses of these species in the future, especially in medicine.
Here functionally and phylogenetically unique species may be considered as disproportionally contributing to bioprospecting value (Dee et al., 2019). Where and when they have low abundance, they might be in need of urgent conservation actions (e.g., the Western Swamp 15 Tortoise cited above, Arnall, 2018; or the Van Gelder's bat, B. dubiaquercus, of our case study). Indeed, if environmental changes inversely lead to the extinction of currently effectively original species with key role in the ecosystem, these changes could yield the biological system to collapse, with potential drastic loss of ecosystem services. Effectively original species may for example be directly threatened when they are increasingly targeted by economic activities because of their combined aspects of rarity (e.g., private collections of rare, distinct specimens; safaris spotting rare, distinct species; e.g., Holden and McDonald-Madden, 2017).
Links between the functional distinctiveness of a species and its abundance-based rarity have for example been observed in European estuarine fish communities, with identified potential consequences on the stability of these communities (Teichert et al., 2017). Links between the functional distinctiveness of a species and its risk of extinction have also sometimes been observed: e.g., among anurans in Ecuador (Menéndes-Guerrero et al., 2020); globally in mammals and birds (Cooke et al., 2020). Species which are rare both in terms of low abundance and phylogenetic distinctiveness have sometimes been found to be threatened. For example, Uchida et al. (2019) observed that, in semi-natural grasslands of south west Japan, low-abundance and phylogenetically distinct species were threatened by land-use intensification, resulting in plant phylogenetic diversity loss. In another context, at a global scale, Sol et al. (2017) found a decrease in abundance or even loss of phylogenetically distinct bird species in highly-urbanized areas compared to the surrounding natural environments.
Our framework thus allows depicting biodiversity in terms of rarity, distinctiveness, and effective originality to better identify how species together contribute to biodiversity levels. It has the potential to improve studies on the mechanisms by which global changes affect biodiversity levels, by identifying which aspect of biodiversity they impact, be it rarity, functional distinctiveness or the global effective originality of some species. Our framework unifies various measures of diversity, distinctiveness and originality, previously scattered in the literature and developed in different context. By developing it, we revealed a parametric family of phylogenetic distinctiveness indices that could complement the most currently used "evolutionary distinctiveness" index (e.g. Isaac et al., 2007;Ibáñez-álamo et al., 2017;Potter 2018;Cooke et al., 2020) whose values are strongly dominated by the independent evolutionary history of a species (length of terminal branch in a phylogenetic tree with the species as tip; Redding et al., 2014). The parametric family indeed allows controlling the 16 degree of influence of this independent evolutionary history on the distinctiveness index. We provide also an equivalent framework for functional distinctiveness. Overall, our framework helps to provide justification, explanation and order when applying a quantitative reasoning to biodiversity, improving then the research on the mechanisms than drive biodiversity changes, and contributing to the development of efficient biodiversity indicators for conservation strategies.

Data statement
All data and R scripts are currently placed in Appendixes C to F. They will soon be also placed in the adiv package of R.