# An a posteriori species clustering for quantifying the effects of species interactions on ecosystem functioning

Benoît Jaillard, Camille Richon, Philippe Deleporte, Michel Loreau, Cyrille Violle

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Benoît Jaillard, Camille Richon, Philippe Deleporte, Michel Loreau, Cyrille Violle. An a posteriori species clustering for quantifying the effects of species interactions on ecosystem functioning. Methods in Ecology and Evolution, 2018, 9 (3), pp.704-715. 10.1111/2041-210X.12920. hal-02350549

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
DR. BENOÎT JAILLARD (Orcid ID : 0000-0002-5156-1866)

DR. CYRILLE VIOLLE (Orcid ID : 0000-0002-2471-9226)

Handling editor: Dr. Ryan Chisholm

Article type : Research Article

**Title:** An *a posteriori* species clustering for quantifying the effects of species interactions on ecosystem functioning

**Running title:** Species interactions and ecosystem functioning

**Names of the authors:**

Benoît Jaillard

Inra, UMR1222 Ecologie fonctionnelle et Biogéochimie des Sols et Agrosystèmes
(Eco&Sols), 2 place Pierre Viala, 34060 Montpellier, France.

benoit.jaillard@inra.fr

Camille Richon

Inra, UMR1222 Ecologie fonctionnelle et Biogéochimie des Sols et Agrosystèmes
(Eco&Sols), 2 place Pierre Viala, 34060 Montpellier, France.

Current address: Laboratoire des Sciences du Climat et de l'Environnement, CEA Orme des
Merisiers, bâtiment 712, 91191 Gif-sur-Yvette, France.

camille.richon@lsce.ipsl.fr

Philippe Deleporte

Cirad, UMR1222 Ecologie fonctionnelle et Biogéochimie des Sols et Agrosystèmes

This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1111/2041-210X.12920

This article is protected by copyright. All rights reserved.
(Eco&Sols), 2 place Pierre Viala, 34060 Montpellier, France.

philippe.deleporte@cirad.fr

Michel Loreau, Centre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, 09200 Moulis, France.

michel.loreau@cnrs.fr

Cyrille Violle, UMR 5175, Centre d'Ecologie Fonctionnelle et Evolutive (CEFE), CNRS - Université de Montpellier - Université Paul Valéry - EPHE, 1919 route de Mende, 34293

Montpellier Cedex 5, France

cyrille.violle@cefe.cnrs.fr

**Abstract**

1. Quantifying the effects of species interactions is key to understanding the relationships between biodiversity and ecosystem functioning but remains elusive due to combinatorics issues. Functional groups have been commonly used to capture the diversity of forms and functions and thus simplify the reality. However, the explicit incorporation of species interactions is still lacking in functional group-based approaches. Here we propose a new approach based on an *a posteriori* clustering of species to quantify the effects of species interactions on ecosystem functioning.

2. We first decompose the observed ecosystem function using null models, in which species diversity does not affect ecosystem function, to separate the effects of species interactions and species composition. This allows the identification of *a posteriori* functional groups that have contrasting diversity effects on ecosystem functioning. We then develop a formal combinatorial model of species interactions in which an ecosystem is described as a combination of co-occurring functional groups, which we call an *assembly motif*. Each

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assembly motif corresponds to a particular biotic environment. We demonstrate the
relevance of our approach using datasets from a microbial experiment and the long-term
Cedar Creek Biodiversity II experiment.

3. We show that our *a posteriori* approach is more accurate, more efficient and more parsimonious than *a priori* approaches. The discrepancy between *a priori* and *a posteriori* approaches results from the way each clustering is set up: *a priori* approaches are based on ecosystem or species properties, such as ecosystem size (number of species or functional groups) or species' functional traits, whereas our *a posteriori* approach is based only on the observed interaction and composition effects on ecosystem functioning.

4. Our findings demonstrate that an *a posteriori* approach is highly explanatory: it identifies who interacts with whom, and quantifies the effects of species interactions on ecosystem functioning. They also highlight that a combinatorial modelling of ecosystem functioning can predict the functioning of an ecosystem without any hypothesis about the biotic or environmental determinants or any information on species functional traits. It only requires the species composition of the ecosystem and the observed functioning of others that share the same assembly motif.

**Keywords:** assembly motif, clustering, combinatorics, community, functional effect groups, modelling, theoretical ecology

**Introduction**

Species interactions strongly influence ecosystem functioning (Loreau et al. 2001; Hooper et al. 2005). However quantifying their net impact is puzzling (Huston 1997; Balvanera et al. 2006; Duffy 2009; Jiang et al. 2009). Analytical methods have been proposed to separate the net effects of species composition from those of species interactions (Wilson 1988;

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Garnier et al. 1997; Loreau & Hector 2001; Kirwan et al. 2009). These methods have successfully explained biodiversity effects in several biodiversity-ecosystem functioning experiments (Loreau & Hector 2001; Reich et al. 2012), but they have not quantified the net effect of species interactions nor identified the species that really interact. The quantification of all possible pairwise species interactions is practically infeasible in species-rich ecosystems because of the curse of dimensionality (McGill et al. 2006). The difficulty of this task lies in the fact that myriad different types of positive and negative species interactions can simultaneously occur within ecosystems. Several attempts have been proposed to frame a pragmatic and operational approach to quantify ecosystem-wide species interactions. The trait-based characterization of species' competitive abilities i.e. the search for organism traits that can explain how species' functions influence the performance of competing neighbours (Grace 1990), paved the road to a generic description of species interactions. In the last decade, this approach has been widely used to study species interactions (the "interaction milieu" sensu McGill et al. 2006) through the systematic evaluation of the community-level distribution of interaction traits. However, interaction traits are still unknown in most taxa (Violle et al. 2014).

Even if functional-group approaches simplify reality, they offer an operational and parsimonious way to analyse and model ecosystem functioning. Nevertheless, up to now this approach has mostly relied on a *priori* clustering based on expert knowledge, as illustrated by the widespread classification of plants into three groups: legumes, graminoids and forbs. An *a posteriori* clustering may be promising since it will identify functional groups based on realized effects, for instance based on the way interacting species modulate ecosystem functioning. Such clustering has hardly been applied so far. A notable exception is the study of Wright *et al*. (2006) which searched for the best species clustering to predict ecosystem biomass. More recently, Jaillard *et al*. (2014) developed a novel modelling approach to species interactions based on their effects on ecosystem functioning. This model was able to successfully reproduce all the forms of biodiversity-ecosystem functioning

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relationships reported in the literature but its flexibility remained low because it is based on a *priori* prevalent assembling rules that determine ecosystem functioning. Here we extend Jaillard *et al.* (2014)'s model by proposing an *a posteriori* clustering approach to the effects of ecosystem-wide species interactions on a given ecosystem function.

The aim of this paper is to quantify the net effects of species interactions on a given ecosystem function. As suggested by Wilson (1988), we first decompose the observed ecosystem function using null models in which species diversity does not affect ecosystem functioning. This allows us to separate the effects of species interactions from the effects of species composition on ecosystem functioning. This separation makes it possible to a *posteriori* identify functional groups (sensu Diaz & Cabido 2001 and Lavorel & Garnier 2002) that have contrasting interaction and composition effects on the ecosystem function. Then we propose a formal combinatorial framework to describe an ecosystem as a combination of co-occurring functional groups, which we call *assembly motif*, echoing the network motifs in network theory (Milo *et al.* 2002). Each assembly motif parsimoniously accounts for the observed effects of species interactions and composition on ecosystem functioning. We test this approach using two datasets: one microbial diversity experiment (Langenheder *et al.* 2010) and the Cedar Creek Biodiversity II experiment (Tilman *et al.* 2001). We demonstrate that our approach can identify (the functional structure of ecosystems, *i.e.*) who is interacting with whom, and quantify the effects of species diversity on ecosystem functioning.

## Material and Methods

### Separating the effects of species interactions and species composition on ecosystem functioning

Consider a set $S = \{1,...,s\}$ of *s* species *i* (Figure 1a). An ecosystem is defined as an

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Modelling the function of an ecosystem according to its assembly motif

An assembly motif characterizes a particular biotic environment. Next, we cluster ecosystems by assembly motifs, i.e. we cluster ecosystems that share a similar biotic environment (Figure 1b). We use the ecosystem clustering by assembly motifs to model the ecosystem function. As in equation 6, we define A<sub>i,k</sub> (with i = 1,...,s) the cluster of ecosystems A of A<sub>k</sub> that contains at least one individual of the species i of S:

$$
A_{i,k} = \{ A \in A_k \text{ such that } \exists i \in \square A \}.
$$

According to equation 7, the interaction and composition effects α(A<sub>i,k</sub>) and β(A<sub>i,k</sub>) of
ecosystems A of clusters A<sub>i,k</sub> are estimated by:

$$
\alpha(A_{i,k}) = \bar{\alpha}_{A \in A_{i,k}} (A) \quad \text{and} \quad \beta(A_{i,k}) = \bar{\beta}_{A \in A_{i,k}} (A). \tag{8}
$$

According to equations 3 and 8, the function F<sub>modelled</sub>(A) of any ecosystem A is finally
modelled as the product of the mean interaction effect, the mean composition effect and the
scale factor of ecosystems that contain the same species and share the same assembly
motif:
assembly motifs are not observed. When an assembly motif is represented by only one ecosystem, the ecosystem function cannot be independently predicted. We define the *predicting ratio* as the number of ecosystems for which the function can be predicted by the clustering model, divided by the number of all observed ecosystems.

All the computations are done using the R-software (R Development Core Team 2009).

## Biodiversity datasets

We used two datasets to test our approach. The first dataset is based on the observation of all the possible ecosystems assembled with an initial pool of microbial species. Langenheder *et al.* (2010) designed this experiment to analyse the biodiversity-productivity relationship in bacterial microcosms. They tested all the 63 possible species combinations with a pool of 6 bacteria: 6, 15, 20, 15, 6 and 1 combinations with 1, 2, 3, 4, 5 and 6 bacteria, respectively. Here we analyse the xylose oxidation (ecosystem function) after 48 hours for the 63 bacterial ecosystems, of which 6 are mono-species and 57 pluri-species cultures.

The second dataset is the Cedar Creek *Biodiversity II* experiment dataset (Tilman *et al.* 2001). This experiment was dedicated to the analysis of the biodiversity-productivity relationship in grasslands. Here the studied ecosystem function is the yearly plant aboveground biomass per unit area. The experiment contained 88 ecosystems from a pool of 16 prairie species (Tilman *et al.* 2001; Reich *et al.* 2012). Diversity treatments consisted of 1, 2, 4, 8 and 16-species plots, each replicated 34 times on average, which corresponds to 168 experimental plots. The species composition in each plot was a random draw from the experimental species pool. All species were grown in monoculture. The 16 prairie species were a priori clustered into 4 functional groups of 4 species: legumes, forbs, C3- and C4-grasses. As a case study, here we analyse the aboveground biomass harvested in the 88

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A species-centred analysis can be performed when comparing the location of ecosystems containing a given species on the α-by-β space (Figure 3). Overall, such an analysis highlights tremendous differences between species effects. For instance, in the microbial experiment, SL104 belongs to ecosystems with the highest composition effect on xylose oxidation (Figure 3a). The other species belong to ecosystems that are close to each other's (Figure 3c and S2a). In the *Biodiversity II* experiment, the functional groups defined by Tilman *et al.* (2001) are not homogeneous in their effects (Figure 3b and 3d). Among legumes, *Lupinus perennis* (Luppe) and *Lespedeza capitata* (Lesca) belong to ecosystems with the highest interaction effects (Figure 3b), whereas *Amorpha canescens* (Amocan) belongs to ecosystems with the highest composition effect (Figure 3d and S2b). Finally *Petalostemum purpureum* (Petpu) belongs to ecosystems characterized by low interaction and composition effects (Figure 3d and S2b).

The hierarchical clustering (Figure 3e and 3f) corroborates this explanatory analysis. In the microbial experiment, SL104 is first isolated in a singleton, then SL68 and SL187. Based on a 4-group clustering, the functional groups of species are namely: { SL104 }, { SL68 } and { SL187 } in singletons, then { SLWC2, SL106, SL197 } (Figure 3e). This means that, when the three species SLWC2, SL106 and SL197 co-occur with each other inside their functional group, and with species SL104, SL68 and SL187 that belong to singletons, they induce similar diversity effects on the ecosystem functioning. In the *Biodiversity II* experiment, the species are first separated in two groups, then *Lupinus perennis* is isolated from the smallest group. Based on a 3-group clustering, the functional groups of species are namely: { *Lupinus perennis* } in a singleton, { *Lespedeza capitata*, *Koeleria cristata*, *Agropyron smithii*, *Achillea millefolium*, *Poa pratensis* } and { *Amorpha canescens*, *Liatris aspera*, *Monarda fistulosa*, *Andropogon gerardi*, *Elymus canadensis*, *Asclepias tuberose*, *Panicum virgatum*, *Sorghastrum nutans*, *Petalostemum purpureum*, *Schizachyrium scoparium* } (Figure 3f). Ten species belong to the largest group, i.e. a C3-grass, two legumes, three forbs and the four C4-grasses. This clustering means that, when the ten species co-occur with each other

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inside the largest functional group, and with species of other functional groups, they induce similar diversity effects on the ecosystem functioning. Except the C4-grasses, the functional groups a priori defined by Tilman et al. (2001) are split into several clusters: C3-grasses and forbs into two, and the four legumes into three different clusters.

Modelling the ecosystem functioning based on the ecosystem clustering

Next we focus on the modelling of ecosystem function based on the ecosystem clustering by
assembly motifs, i.e. the combination of functional groups of species (see Figure 1). We
recall that each species clustering in functional groups is a model, that generates a new set
of assembly motifs. Consequently, we explore the model quality, accuracy and efficiency, by
increasing step-by-step the number of functional groups (Figure 4). The goodness-of-fit of
the model increases from a low value until one when there are as many functional groups as
species (Figure 4a and 4b). The model robustness is evaluated by its efficiency: it also
increases, but remains always lower than the goodness-of-fit of the model. In contrary, the
predicting ratio decreases with the number of functional groups, from one when all the
ecosystems are clustered together in a singleton (all ecosystem functions are predicted by
the mean function of all ecosystems), until zero when there are as many functional groups
as species (no ecosystem function can be predicted because each ecosystem cluster is a
singleton) (Figure 4c and 4d). The best number of functional groups results from a trade-off
between the accuracy (R²), the efficiency (E) and the predicting ability (predicting ratio) of
the model. In the microbial experiment, R² increases from 1 to 5 functional groups (Figure
4a). However, E presents a maximum for 4 clusters of species, which corresponds to a
predicting ratio of 53/57 ecosystems (Figure 4c). In the Biodiversity II experiment, R²
increases, with local maximums for 3, 6 and 8 species functional groups, and the predicting
ratio decreases quickly from 1 to 16 functional groups. A species clustering in 3 functional

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groups allows to predict the functions of 52 out of 53 ecosystems.

For a given species clustering, each ecosystem can be described by a unique assembly
motif, and its function can be evaluated by modelling (Figure S3-S4 for the independent
modelling of both diversity effects). In the microbial experiment, all the 63 species
combinations are observed. Species richness is the most frequently used metric of diversity
in biodiversity-ecosystem functioning research (Tilman et al. 1997, 2001; Hooper et al. 2005;
Langenheder et al. 2010; Reich et al. 2012). As a consequence, here we use species
number (ecosystem size) as a reference (Figure 5a and 5b). This modelling lead to
dispersed and overlapping diversity effects (Figure 5a), but goodness-of-fit ($R^2$ = 0.438,
p = 9.1 $10^{-7}$) and efficiency ($E$ = 0.245, p = 1.8 $10^{-3}$) are significant (Figure 5b). Based on a 4-
group species clustering, the resulting ecosystem clustering by assembly motifs is much
more structured (Figure 5c and 5d). The diversity effects of ecosystems that share an
assembly motif are close to each other, confirming the functional redundancy of clustered
species (Figure 5c). The goodness-of-fit is high ($R^2$ = 0.892, p < $10^{-16}$) and the efficiency of
this modelling remains high and highly significant ($E$ = 0.828, p < $10^{-16}$) (Figure 5d).

In the *Biodiversity II* experiment, only 53 out of 65519 (65535 - 16 monocultures) pluri-
species ecosystems are observed, thus less than 0.1 % of all possible species
assemblages. The clustering of ecosystems by their size induces dispersed and overlapping
ecosystem groups (Figure 6a), a poor goodness-of-fit ($R^2$ = 0.489, p = 1.2 $10^{-9}$) and a poor
efficiency ($E$ = 0.142, p = 1.2 $10^{-3}$) of the modelling (Figure 6b). A 4-group clustering based
on functional groups a priori defined by Tilman et al. (2001) improves significantly the
goodness-of-fit ($R^2$ = 0.771, p = 4.4 $10^{-16}$), but more weakly the efficiency of the modelling
($E$ = 0.375, p = 2.7 $10^{-8}$), and the predicting ratio equals to 46/53 (Figure 6c and 6d). An a
posteriori species clustering in only 3 groups improves strongly both the goodness-of-fit
($R^2$ = 0.863, p < $10^{-16}$) and the efficiency ($E$ = 0.717, p = 2.2 $10^{-14}$) of the modelling (Figure

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6e and 6f). Moreover, the function of 52/53 ecosystems can be predicted independently.

Discussion

Separating the interaction and composition effects: an alternative proposal

The decomposition of ecosystem functions into diversity effects has been successfully used
in the past to analyse biodiversity-ecosystem functioning experiments (Loreau & Hector
2001; Reich et al. 2012). This method follows an integrative path by computing the
covariance between the observed and expected contributions of each species to ecosystem
functioning. We here followed the same philosophy but used the ratio between observed and
expected yield (so-called "relative yield of the mixture", RYM) instead of the covariance, as
suggested by Wilson (1988). Based on this decomposition, we defined two new elementary
ecosystem-level diversity effects: interaction and composition effects. Both interaction and
composition effects are dimensionless, which allows comparisons between experiments
performed in various conditions.

The two experiments that we re-analyse here are very different: bacterial assemblages
grown for 48 hours in laboratory (Langenheder et al. 2010) and grassland mixtures cultivated
for several years in the field (Tilman et al. 2001). In the microbial experiment, the observed
function is strongly correlated to the composition effect. In both experiments, species
interactions greatly increase ecosystem function, between 1.4 and 2.1 times, indicating
strong complementary interactions (e.g., facilitation, niche differentiation) between species,
as suggested by Tilman et al. (2001) and Langenheder et al. (2010). In addition, we found
that the covariance between the interaction and composition effects is not significantly
different from zero, which corroborates Loreau & Hector (2001)'s results. These findings

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Volume 1
demonstrate that the interaction and composition effects are independent in both experiments, but the two diversity effects contribute significantly to ecosystem functioning.

## The emerging concept of assembly motifs

We propose to quantify the net effect of species interactions on ecosystem functioning through the lens of assembly motifs, i.e. a combination of co-occurring functional groups within an ecosystem. An assembly motif is a pattern of species composition that reflects a given biotic environment. Our results show that ecosystems described by the same assembly motif display diversity effects on ecosystem functioning similar to each other in terms of interaction and composition effects. An assembly motif therefore describes the biotic environment that regulates the effects of species interactions on a given ecosystem function.

The underlying condition of our modelling framework is that the presence/absence of at least one species from each functional group changes the ecosystem-wide species effect in a similar way (Jaillard et al. 2014). This condition is the simplest and the most commonly used in ecology (Diaz & Cabido 2001; Hooper et al. 2005). It implicitly assumes that the presence of particular species, thus of particular interactions, is more important than the number of interactions, the number of species or the number of functional groups. Some authors have long pointed out that the composition of an ecosystem, i.e. the identity of species or functional groups, affects its functioning more than diversity *per se* (Hooper & Vitousek 1997). Our results confirm the validity of this hypothesis in both studied experiments.

Our modelling framework is based on combinatorics through combinations and clustering of species, functional groups and ecosystems. Clustering of species or ecosystems is a simplification of the real world. It assumes that clustered species are functionally redundant,

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which means in our case that they induce roughly the same effects on the ecosystem functioning. However, the functional redundancy, and thus the species clustering, is specific to the considered experiment, to the set of species used and to the ecosystem function observed in the experiment (Loreau 2004). Species clustering is only a shortcut that should enable a better understanding of reality. The clustering of species based on their diversity effects reduces the range of possibilities to affect the ecosystem functioning. The clustering of ecosystems by assembly motifs reduces the range of effects induced by the species interactions and composition on ecosystem functioning. The variance within each ecosystem cluster is low. The clustering thus enables the *a posteriori* determination of a likely interacting schema based on parsimonious species clusters and readable ecosystem clusters. It finally sets up a comprehensive interacting organization of species within ecosystems, without any *a priori* knowledge of species properties.

## Identifying and quantifying the interacting roles of species within ecosystems

We clearly showed that an ecosystem clustering based on assembly motifs is much more suitable than the classical clustering based on the ecosystem size, such as the number of species or functional groups. Langenheder *et al.* (2010) biologically showed that the species SL104 plays a key role on xylose oxidation. Our modelling approach retrieves this key role of SL104, and specifies that the species increases the ecosystem function by composition effect rather than interaction effect. It also shows that the three species SLWC2, SL106 and SL197 are clustered in a same functional group, thus they induce similar diversity effects by interacting between them and with others.

In the *Biodiversity II* experiment, we highlight that only three *a posteriori* functional effect groups explain and successfully predict ecosystem functioning. The *a priori* species clustering based on the number of species as well as the clustering defined by Tilman *et al.*

Accepted Article
Accepted Article
Jaillard, B., Richon, C., Deleporte, P., Loreau, M., Violle, C. (2018). An a posteriori species
clustering for quantifying the effects of species interactions on ecosystem functioning.
Methods in Ecology and Evolution, 9 (3), 704-715., DOI: 10.1111/2041-210X.12920

(2001) are both based on four groups: they explain and predict less accurately the ecosystem biomass than our a posteriori approach. Our a posteriori approach is thus more accurate, more efficient and more parsimonious than both a priori approaches based on ecosystem size and functional groups a priori defined. Surprisingly the four legumes classically clustered into a single functional group (legume) were found in different functional groups based on our a posteriori clustering. *Lupinus perennis* belongs to the ecosystem cluster with the highest interaction effects. *Lupinus perennis* is a legume, the leaves and fine roots of which have a very short longevity, lesser than 4 weeks, releasing large amounts of nitrogen in soil (Craine et al. 2002). In Europe, the *Lupinus* genus is also known among legumes to produce cluster roots that release citrate and protons and mobilize poorly available nutrients such as phosphorus (Hinsinger et al. 2002; Jaillard et al. 2003; Lambers et al. 2013). The *Lupinus* species from the New World are less studied but Lambers et al. (2012) reported that *Lupinus lepidus* also produces cluster-like roots. *Lespedeza capitata* is clustered with a forb (*Achillea millefolium*) and three C3-grasses (*Koeleria cristata*, *Agropyron smithii* and *Poa pratensis*). The species cluster has high interaction effects but low composition effects. Note that *Lupinus perennis* and *Lespedeza capitata* are always isolated in small functional groups and have strong structuring effects on the ecosystem function across years (from 2001 to 2007, data not shown). *Amorpha canescens* belongs to ecosystems with the highest composition effects but low interaction effects. Our results confirm that legumes play a key role in ecosystem-level biomass production, as previously highlighted by many authors (e.g., Tilman et al. 1997, 2001), but, as suggested by Craine et al. (2002) and Wright et al. (2006), they also indicate that the 'legume' functional group is not homogeneous in the ability of legumes to affect plant biomass.
The approach developed here first separates the interaction effect from the composition
effect of diversity on an ecosystem function, then clusters species in functional groups on the
basis of these diversity effects on ecosystem functioning. However, the novelty is mainly to
combine functional groups of species and to assume that any species assemblage,
matching to all possible combinations of functional groups, i.e. all possible assembly motifs
can significantly affect, positively or negatively, the interaction and composition effects.
Jaillard et al. (2014) already showed that a combinatorial approach makes it possible to
reproduce all shapes of diversity-ecosystem functioning relationships reported in the
experimental literature, suggesting that the multi-faceted response of ecosystems to
biodiversity changes is driven by effects of assembly probabilities and variations in species
properties. We here demonstrate that a combinatorial approach also allows the modelling of
ecosystem functioning without any hypothesis about its biotic or environmental determinants,
and without any information on species functional traits. We only use the available
experimental data, i.e. the function and species composition of ecosystems, and the function
of species in monoculture. Wright et al.'s (2006) study is the only one to assess random
species clustering based on their ability to predict an ecosystem function. The main result
obtained by Wright et al. (2006) was that the explanatory powers of a priori and random
clustering were not significantly different. In contrast to the conclusion of Wright et al. (2006),
our results show that an a posteriori species clustering can be highly explanatory. It even
makes it possible to predict with accuracy the functioning of an ecosystem on the sole basis
of its species composition.

A *posteriori* approaches such as ours aim at describing the structure of raw data. In our
case, the idea is to determine the functional structure of ecosystems that best accounts for
the observed data without attempting to explain the underlying biological processes. This
approach allows testing some hypotheses, in particular it allowed us to challenge the

Accepted Article
Jaillard, B., Richon, C., Deleporte, P., Loreau, M., Violle, C. (2018). An a posteriori species clustering for quantifying the effects of species interactions on ecosystem functioning. Methods in Ecology and Evolution, 9 (3), 704-715., DOI: 10.1111/2041-210X.12920
widespread use of the legume-forb-graminoid clustering when investigating the role of
biodiversity in regulating ecosystem functioning. It also makes it possible to efficiently predict
a given ecosystem function based on a set of ecosystems in which species composition and
function have already been observed. Nevertheless, a posteriori approaches remain
statistical, phenomenological, and correlative. They should be complemented by a priori
approaches that formulate new explanatory hypotheses based on the findings of a posteriori
approaches. For instance, the identification of functional traits associated to different species
groups identified by our combinatorial approach could help to reveal the underlying biological
processes. Ultimately, a deeper understanding of biodiversity-ecosystem functioning
relationships should ideally come from a combination of a posteriori and a priori approaches:
the former to explore data and reveal real patterns, the latter to determine the functional
traits or biological processes responsible for the observed species patterns. This
combination is best suited to providing robust, general, explanatory and anticipatory
predictions of the effects of species diversity on ecosystem functioning (Mouquet et al.
2015).

Conclusion

We propose a combinatorial model to quantify the net effects of species interactions on
ecosystem functioning and an efficient method to fit species clustering to experimental
datasets. When applied to two datasets, a microbial experiment and the *Biodiversity II*
experiment, we show that assembly motifs, i.e. patterns of ecosystem assembly that reflect
patterns of species interactions, explain a large part of the biodiversity effects on ecosystem
functioning. This confirms that the existence of interactions of a certain type is more
important than the number of interactions, of species or of functional groups. Our model is
based solely on assembly motifs defined as patterns of ecosystem functional structure: it is
the first model that characterizes species by their ability to affect ecosystem functioning,

This article is protected by copyright. All rights reserved.

Accepted Article
JAillard, B., Richon, C., Deleporte, P., Loreau, M., Violle, C. (2018). An a posteriori species
clustering for quantifying the effects of species interactions on ecosystem functioning.
Methods in Ecology and Evolution, 9 (3), 704-715., DOI: 10.1111/2041-210X.12920
without explicitly accounting for their biological properties (e.g., functional traits).

## Acknowledgements

The authors gratefully thank Dr Silke Langenheder and the University of Minnesota for having put at their disposal the datasets discussed in this paper. They also thank Dr Alain Rapaport for his helpful contribution to mathematical formulations. This work was funded by the project *Effet d'échantillonnage sur la compréhension et la prédiction du fonctionnement des écosystèmes (INRA)*. ML was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41). CV was supported by the French Foundation for Research on Biodiversity (FRB; www.fondationbiodiversite.fr) in the context of the CESAB project "Assembling, analysing and sharing data on plant functional diversity to understand the effects of biodiversity on ecosystem functioning: a case study with French Permanent Grasslands" (DIVGRASS), by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Program (DiversiTraits project, no. 221060) and by the European Research Council (ERC) Starting Grant Project "Ecophysiological and biophysical constraints on domestication in crop plants" (Grant ERC-StG-2014-639706-CONSTRAINTS).

## Data accessibility

The R scripts and datasets can be found at datadryad.com, https://doi.org/10.5061/dryad.rh4v0 (Jaillard, Richon, Deleporte, Loreau & Violle, 2017)

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## Authors's contributions

B.J. conceived the research; B.J. and C.R. implemented the code; B.J. and P.D. developed the statistics; B.J., C.R., M.L. and C.V. drew the figures and wrote the manuscript.

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## Figure legends

**Figure 1: Step-by-step clustering framework to model the diversity effects on an ecosystem function.** (a) An ecosystem A is defined as a set of co-occurring species and is characterized by an aggregate function F. Species are clustered into functional groups (here $σ = 3$ for instance) on the basis of the mean diversity effects of clusters of ecosystems containing a given species. (b) Each ecosystem is then described by a combination of functional groups, i.e. an assembly motif $M_k$ (here $m = 2^σ - 1 = 7$ possible assembly motifs).

Ecosystems described by the same assembly motif $M_k$ are grouped into the clusters $A_k$ and characterized by the aggregate functions $F(A_k)$ ($k = 1...7$). The goodness-of-fit of the model clustering is evaluated by its determination coefficient and its robustness by cross validation.

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**Figure 2:** Effects of species interactions and species composition of an ecosystem function.(a) and (b) Interaction effect vs ecosystem function, (c) and (d) Composition effect vs ecosystem function. The coefficient of correlation *r*²*Pearson* of the relationship are indicated: they are significant at p < 0.001.

**Figure 3:** Species clustering based on the mean diversity effects of the clusters of ecosystems containing a given species. (a) and (b) Interaction effect vs composition effect. Example of a cluster of ecosystems containing a given species (SL104 or *Lupinus perennis*). (c) and (d) Interaction effect vs composition effect. Centroids of the clusters of ecosystems containing each species. (e) and (f) Trees of hierarchical clustering of centroids of the ecosystem clusters containing each species. The centroids are clustered by using Euclidean distance and Ward's linkage algorithm. In (d) and (f), different colours correspond to different functional groups defined by Tilman *et al.* (2001), legumes (blue), C3-grasses (red), C4-grasses (green) and forbs (gold).

**Figure 4:** Accuracy, efficiency and predicting ability of modelling versus the number of functional groups. (a) and (b) Coefficient of determination *R*² (in black) and efficiency *E* (in red) of the relationship between the ecosystem function observed vs modelled (*R*²) or predicted (*E*) by cross validation. (c) and (d) Predicting ratios, i.e. the proportion of ecosystems of which the function can be predicted by the clustering model.

**Figure 5:** Ecosystem clustering, goodness-of-fit and efficiency of the clustering model for ecosystem functioning in the microbial experiment (Langenheder *et al.* 2010). (a) and (b) Modelling based on the ecosystem size (number of species). (c) and (d) Modelling based on an *a posteriori* 4-group species clustering. (a) and (c) Interaction effect vs

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composition effect. (b) and (d) Relationship between modelled and observed ecosystem functions. Each bar corresponds to the error induced by leaving out the ecosystem to predict. Different coloured symbols correspond to different assembly motifs. Solid black line is the 1:1 line.

**Figure 6: Ecosystem clustering, goodness-of-fit and efficiency of the clustering model for ecosystem functioning in the *Biodiversity II* experiment (Tilman et al. 2001).** (a) and (b) Modelling based on the ecosystem size (number of species). (c) and (d) Modelling based on an *a priori* 4-group species clustering defined by Tilman *et al.* (2001), i.e. legumes, forbs, C3- and C4-grasses. (c) and (d) Modelling based on an *a posteriori* 3-group species clustering. (a), (c) and (e) Interaction effect vs composition effect. (b), (d) and (f) Relationship between modelled and observed ecosystem functions. Each bar corresponds to the error induced by leaving out the ecosystem to predict. Different coloured symbols correspond to different assembly motifs. Solid black line is the 1:1 line.

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## Supporting Information

**Figure S1: Decomposition of ecosystem function in interaction and composition effects.** The function $F_{observed}(A)$ of an ecosystem A is decomposed by referring to two null hypotheses $H_0$ and $G_0$: $H_0$ assumes that the species interactions have no effect on ecosystem functioning, and $G_0$ assumes that neither the species interactions nor the species composition have effect on ecosystem functioning. The two null models are computed using the functions of species in monoculture, that belong to the ecosystem A for $H_0$, that belong to the ecosystem A and the whole experimental pool S for $G_0$. The modelling allows to define an interaction effect $\alpha(A)$ and a composition effect $\beta(A)$. Here, $S = \{1, 2, 3\}$ and $A = \{2, 3\}$.

**Figure S2: Species centroids of the clusters of ecosystems containing a given species in the interaction effect x composition effect space.** The figures correspond to Figure 3b and 3c enlarged and focused on the species centroids. The centroids are clustered by using Euclidean distance and Ward's linkage algorithm. In (b), different colours correspond to different functional groups defined by Tilman et al. (2001), legumes (blue), C3-grasses (red), C4-grasses (green) and forbs (gold).

**Figure S3: Goodness-of-fit and efficiency of the clustering model for diversity effects in the microbial experiment (Langenheder et al. 2010).** (a) and (b) Modelling based on the ecosystem size (number of species). (c) and (d) Modelling based on an a posteriori 4-group species clustering. (a) and (c) Interaction effect. (b) and (d) Composition effect. Each bar corresponds to the error induced by leaving out the ecosystem to predict. Different

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coloured symbols correspond to different assembly motifs. Solid black line is the 1:1 line.

**Figure S4: Ecosystem clustering, goodness-of-fit and efficiency of the clustering model for diversity effects in the *Biodiversity II* experiment (Tilman et al. 2001).** (a) and (b) Modelling based on the ecosystem size (number of species). (c) and (d) Modelling based on an a priori 4-group species clustering defined by Tilman et al. (2001), i.e. legumes, forbs, C3- and C4-grasses. (a), (c) and (e) Interaction effect. (b), (d) and (f) Composition effect. Each bar corresponds to the error induced by leaving out the ecosystem to predict. Different coloured symbols correspond to different assembly motifs. Solid black line is the 1:1 line.

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<figure><img src="image_2.png" /><figcaption>Figure 1</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3</figcaption></figure>
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4</figcaption></figure>
Microbial experiment

Modelling outcome of an *a priori* clustering based on the ecosystem size (number of species)

<figure><img src="image_7.png" /></figure>

Modelling outcome of an *a posteriori* 4-group clustering of species

<figure><img src="image_8.png" /><figcaption>Fig. 5</figcaption></figure>
Modelling outcome of an a priori clustering based on the ecosystem size (number of species)

<figure><img src="image_9.png" /></figure>

Modelling outcome of an a priori clustering of species based on 4-functional groups defined by Tilman et al. 2001: legumes, forbs, C3- and C4-grasses

<figure><img src="image_10.png" /></figure>

Modelling outcome of an a posteriori 3-group clustering of species

<figure><img src="image_11.png" /><figcaption>Fig. 6</figcaption></figure>

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