# A unified framework for partitioning the drivers of stability of ecological communities

Jules Segrestin, Lars Götzenberger, Enrique Valencia, Francesco de Bello, Jan Lepš

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Jules Segrestin, Lars Götzenberger, Enrique Valencia, Francesco de Bello, Jan Lepš. A unified framework for partitioning the drivers of stability of ecological communities. Global Ecology and Biogeography, 2024, 33 (5), 10.1111/geb.13828. hal-04840927

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# A unified framework for partitioning the drivers of stability of ecological communities

Jules Segrestin<fnref n="1" /> ID

Lars Götzenberger<fnref n="1" />,<fnref n="2" /> ID

Enrique Valencia<fnref n="3" /> ID

Francesco de Bello<fnref n="1" />,<fnref n="4" /> ID

Jan Lepš<fnref n="1" /> ID

**Correspondence**

Jules Segrestin, Department of Botany, Faculty of Science, University of South Bohemia, České Budějovice, Czech Republic.

Email: jsegrestin@gmail.com

**Funding information**

GAČR, Grant/Award Number: 23-070875

Handling Editor: Pedro Peres-Neto

**Abstract**

**Aim:** Identifying the drivers of ecological stability is critical for ensuring the maintenance of ecosystem functioning and services, particularly in a changing world. Different ecological mechanisms by which biological communities stabilize ecosystem functions (i.e. "stabilizing effects") have been proposed, yet with various theoretical expectations and debated conclusions. Here we propose a unified framework that aims at reconciling, and combining, different approaches to reliably test the strength of three stabilizing effects on the temporal constancy of ecosystem functions: the effects of (a) dominant species, (b) species asynchrony, and (c) diversity.

**Innovation:** Compared to existing developments the approach allows, for the first time, disentangling these three stabilizing effects at the level of individual communities. So far this was not possible, and conclusions depended on indirect tests and comparative analyses across communities. We also propose a graphical representation of the relative contributions of the three stabilizing effects on a ternary plot, allowing us to easily compare communities sampled in various ecological contexts in a standardized space.

**Main conclusions:** Our study answers the current need for a unified framework to link theoretical concepts on the temporal stability of ecological communities to data analysis. The present development promises flexible tests for a deeper understanding of the ecological stabilization of biodiversity and the relative importance of its components.

**KEYWORDS**

averaging effect, community stability, compensatory dynamics, diversity-stability hypothesis, dominance effect, portfolio effect, species asynchrony, species stability, Taylor's power law, temporal variability

# 1 | INTRODUCTION

Ecological stability, and its relation to biodiversity, has been a central topic of ecology that has generated many debates in the past 70 years (Craven et al., 2018; Doak et al., 1998; MacArthur, 1955;

May, 1972; McCann, 2000; Tilman et al., 1998). Recent years have witnessed an increasing interest in a better understanding of the determinants of ecological stability and in predicting the temporal fluctuations of ecosystem functioning in a changing world (see Su et al., 2022 and references therein). Hence, a large body of literature

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

© 2024 The Authors. Global Ecology and Biogeography published by John Wiley & Sons Ltd.

<footnote n="1">¹Department of Botany, Faculty of Science, University of South Bohemia, České Budějovice, Czech Republic</footnote>

<footnote n="2">²Institute of Botany of the Czech Academy of Sciences, Třeboň, Czech Republic</footnote>

<footnote n="3">³Department of Biodiversity, Ecology and Evolution, Faculty of Biological Science, Complutense University of Madrid, Madrid, Spain</footnote>

<footnote n="4">⁴CIDE-UV-CSIC, Valencia, Spain</footnote>
investigated the effect of diversity loss (Hector et al., 2010; Loreau & de Mazancourt, 2013; Tilman et al., 2006; Wang et al., 2019), land use modification (Blüthgen et al., 2016; Hautier et al., 2020; Lepš et al., 2018; Liang et al., 2021; Zhang et al., 2016) or climate change (Ma et al., 2017; Wang et al., 2022; Zelikova et al., 2014) on the stability of ecosystem functions, in particular, net primary productivity.

A central goal of these studies is assessing the main mechanisms by which biological communities can stabilize ecosystem functions (i.e. "stabilizing effects"). In this respect, and despite great advances in our understanding of the relationships between global change drivers and stability, existing studies often use various methodolog- ical approaches, not helping to fully resolve a long-standing debate on biodiversity effects on stability (McCann, 2000). Here, we com- bined several concepts commonly used to study the stabilizing ef- fects of biological communities (see Box 1 for definitions of the key concepts), to propose a unified framework that allows a holistic de- scription of the mechanisms driving community stability. We aim at reconciling debated concepts with the definition of a comprehensive set of indices, with clear ecological interpretations, which allow, for the first time, testing the absolute and the relative strength of dif- ferent stabilizing effects directly and at the level of individual com- munities. The approach is based on a partitioning of the inter-annual constancy in an ecosystem function (also called invariability) into the constancy of an "average species" in the community and three stabilizing effects (Figure 1). We will show that such an approach provides a more complete description of the mechanisms operating simultaneously on the community stability compared to earlier at- tempts in this area and offers unprecedented flexibility in directly testing and interpreting different ecological mechanisms.

Inter-annual constancy in ecosystem functions is one of the most assessed facets of stability as its potential drivers are based on a strong theoretical foundation. Temporal constancy (stability hereafter) describes a community's ability to buffer the effects of natural fluctuations in environmental conditions and maintain con- sistent functioning around a constant baseline (Pimm, 1984). A con- sensus exists on the two main drivers of such community stability: (1) the stability of the species populations (particularly dominants, see next paragraph) composing the community and (2) the compen- satory dynamics, originating from the temporal compensations be- tween species due to their asynchronous fluctuations (Gonzalez & Loreau, 2009; Hector et al., 2010). Each of these components has been associated with specific ecological hypotheses: while the sta- bility of species populations should mainly relate to the interplay between their ecological strategies and environmental conditions (Conti et al., 2023; Májeková et al., 2014; Ren et al., 2023), compen- satory dynamics have been primarily associated with ecological mechanisms such as species interactions (positive or negative), niche differentiation or demographic stochasticity (de Bello et al., 2021; de Mazancourt et al., 2013; Loreau & de Mazancourt, 2008).

However, the two abovementioned drivers are recognized to be affected by other community characteristics. First, the effect of the species stability on community stability is modulated by their relative abundance over time: while all species can contribute

significantly to the community stability in the case of evenly distrib- uted abundances, only a few abundant species can drive the overall community stability in strongly dominated communities (Figure 1a.1 and 1a.2). Moreover, dominant species are generally found to be the most stable species of the community, as often demonstrated by Taylor's power law (Taylor, 1961; see below). Altogether, it suggests that dominant species can potentially have a strong contribution to the stability of ecological communities. However, to our knowledge, this effect has received little attention (but see Grman et al., 2010; Polley et al., 2007; Sasaki & Lauenroth, 2011; Valencia et al., 2020 for indirect estimations).

Second, high compensatory dynamics are often interpreted as an indicator of ecological processes that promote asynchronous fluctuations between species. For example, competition between species, together with their different sensitivity to environmental fluctuations, should promote anti-synchronous fluctuations (neg- ative temporal correlations) and, in turn, result in larger compen- satory dynamics. However, theoretical work has revealed that compensatory dynamics can also arise from a statistical averaging, sometimes also referred to as portfolio effect (Doak et al., 1998; Gonzalez & Loreau, 2009; Tilman et al., 1998): at a given level of asynchronous fluctuations between species, an increasing spe- cies diversity is inevitably associated with a statistical increase of compensatory dynamics (Figure 1a.4). Therefore, it is often unclear to which extent variations in compensatory dynamics are due to ecological processes affecting the degree of asynchrony between species or to variations in species diversity between communities with similar functioning.

The concept of statistical averaging has been primarily described for the simplified case of independent fluctuations between species (Doak et al., 1998; Tilman et al., 1998; see Appendix S1). However, it can be generalized to any given level of species synchrony (i.e. from perfectly synchronized fluctuations to strong anti-synchrony; Thibaut & Connolly, 2013): if all species are perfectly synchronized, there are no compensatory dynamics. Consequently, under perfect synchrony there is no statistical averaging, no matter the commu- nity diversity. At any level of asynchrony (i.e. deviation from perfect synchrony), a given increase in diversity increases the statistical av- eraging and its effect becomes larger with increasing asynchrony, particularly when species fluctuate anti-synchronously. It results in a complex interplay between the effects of species synchrony and diversity (averaging effect hereafter) on compensatory dynam- ics. Zhao et al. (2022) recently proposed a method for decompos- ing compensatory dynamics into averaging and asynchrony effects. While this seminal approach reinforces the relevance of teasing apart these effects, we will demonstrate that the method calculates a potential averaging effect rather than the realized community sta- bilization due to these components. Our framework relies on, argu- ably, a more realistic approach where the averaging effect is null in the case of perfect synchrony between species, no matter the community diversity, and its effect increases both with community diversity and species asynchrony to better reflect the complex inter- play between diversity and asynchrony on compensatory dynamics.
<figure><img src="image_2.png" /><figcaption>FIGURE 1 (a) The temporal stability of communities is simultaneously driven by four distinct components related to species characteristics and the community structure. These four components are conceptually represented in the central panels where the same community fluctuation (bold line) results from the fluctuations of different species (coloured dotted lines) representing (a.1) globally stable species, the stabilizing role of (a.2) a dominant species, or of compensatory dynamics due to (a.3) the asynchrony of species or (a.4) the averaging effect (associated with high diversity). (b) Our unified framework allows us to disentangle all four components of stability occurring simultaneously in a community and represent them on a simple graph that describes their relative importance (see Section 2 for details on the indices and Box 1 for the main definitions of the terms proposed in the framework). Briefly, CV<sub>e</sub> is the variability of a species of mean abundance (related to panel a.1) which is stabilized by dominance (Δ, related to panel a.2), asynchrony (Ψ, related to panel a.3) and averaging (ω, related to panel a.4) effects to give the community variability CV<sub>com</sub> (reciprocal of stability). CV<sub>e</sub> is the abundance-weighted mean of species CV<sub>i</sub>, and CV<sub>a</sub> is the expected CV<sub>com</sub> if the community was stabilized by species asynchrony only.</figcaption></figure>

Investigating simultaneously the role of the three stabilizing
mechanisms described above (effect of dominants, species asyn-
chrony and species diversity; Figure 1a) remains an open task for
deepening our understanding of the stabilizing mechanisms of bi-
ological communities. Recent studies have been assessing these
mechanisms mostly indirectly and through comparisons across
communities. Correlation coefficients or path coefficients in struc-
tural equation models are generally used to test the relationships

between different community characteristics and community sta-
bility (e.g. Craven et al., 2018; de Mazancourt et al., 2013; Kigel
et al., 2021; Valencia et al., 2020). These indirect approaches have
been proven successful but requires large data sets for reliable esti-
mations. Results, moreover, will depend on the type of communities
compared, and thus stabilizing effects cannot be tested effectively
and will be context dependent. A more direct and flexible approach,
as proposed here, consists of calculating different stabilizing effects
## BOX 1 Definitions of the terms proposed in our unified framework for partitioning the drivers of stability of ecological communities

**Species synchrony:** The overall covariation between species temporal fluctuations. Asynchronous communities correspond to any case of deviation from perfect covariations between all species. A higher species asynchrony means a lower overall covariation between species. For asynchronous communities, we can also distinguish communities where species are positively synchronous (positive sum of species covariances), fluctuate independently (sum of species covariances close to zero) or are anti-synchronous (negative sum of species covariances).

**Stabilization:** The extent by which community characteristics enhance the temporal stability of ecological communities. Total stabilization ($\tau$) results from the combination of dominance, asynchrony and averaging effects.

**Dominance effect ($\Delta$):** The stabilization due to the dominance of species that are relatively more stable than other species in a community. Its quantification uses, as baseline, the expected value of the community $CV_{com}$ when controlling for the dominance structure.

**Compensatory dynamics ($\sqrt{\phi}$):** The stabilization due to the temporal compensations between species. This mechanism appears as soon as species are asynchronous and increases with both species' asynchrony and community diversity. Its effect is quantified by comparing the expected community CV under the hypothetical case of perfect synchrony between species to the observed $CV_{com}$ of the community. Compensatory dynamics can be decomposed into the asynchrony and averaging effects.

**Asynchrony effect ($\psi$):** The part of compensatory dynamics associated with the observed level of asynchrony between species, controlling for the community diversity (i.e. controlling for the averaging effect, see next definition). It is quantified by the expected value of compensatory dynamics in the hypothetical case of the community having a reference diversity.

**Averaging effect ($\omega$):** The part of compensatory dynamics associated with the community diversity. It is quantified by comparing the expected compensatory dynamics for the reference diversity with the observed compensatory dynamics. It measures the realized increase in compensatory dynamics due to the community diversity.

for each community independently. However, this also requires a clarification of the indices commonly used in this type of analysis. Indices have been proposed to estimate the absolute overall compensatory dynamics, that is, including both asynchrony and

averaging components ($\phi$ index: Loreau & de Mazancourt, 2008), or species asynchrony only (Variance ratio: Schluter, 1984; Gross' index: Gross et al., 2014; improved in Blüthgen et al., 2016). Yet, no method has been proposed to estimate the averaging effect separately from asynchrony nor the comparative effect of dominants at the community level. This has resulted in a lively debate on the relative importance of these components on the stability of natural communities: while Polley et al. (2007) and Sasaki and Lauenroth (2011) highlight the prominent effect of the stability of dominant species, Valencia et al. (2020) argue that community stability is mostly explained by the asynchrony of dominant species, and Zhao et al. (2022) claim that the averaging effect is the most important factor.

Here, the proposed framework promises a deeper understanding of the ecological mechanisms driving community stability by providing straightforward representations of the relative and simultaneous effects of all stabilizing effects. The following parts introduce our theoretical framework by describing the different stabilizing effects and their estimations, demonstrate its performance using numerical simulations, and present a case study using long-term data from a temperate meadow. Finally, we discuss how our framework can help to test long-standing ecological hypotheses related to community stability.

## 2 | GENERAL FRAMEWORK

The temporal stability of communities is commonly assessed by the coefficient of variation ($CV_{com}$, reciprocal of stability; Kéfi et al., 2019) calculated as the ratio between the standard deviation ($\sigma_{com}$) and the mean ($\mu_{com}$) of repeated measurements of annual productivity. Our framework is based on the partitioning of the community $CV_{com}$ into the CV of an "average species" in the community (see next section) and three stabilizing effects: the dominance, asynchrony and averaging effects (Figure 1b).

### 2.1 | The CV of an "average species" as a baseline to assess total stabilization

The CV of an "average species" ($CV_e$ hereafter) corresponds to the CV of a theoretical species whose abundance is equal to the average abundance over the species in the community. $CV_e$ can be derived from the observed relationship between the species abundances ($\mu_i$) and their individual coefficient of variation ($CV_i$). Indeed, according to Taylor's (1961) power law, the temporal variance of species ($\sigma_i^2$) relates positively to their abundance ($\mu_i$). In other words, we can predict the $CV_i$ of a species of a given abundance from the parameters of Taylor's power law in the community as:

$$CV_i = a^{1/2} \mu_i^{b/2-1} \quad (1)$$

where a and b are the parameters of Taylor's power law expressed as follows:

$$\sigma_i^2 = a \mu_i^b \quad (2)$$
Both parameters are usually estimated by a linear regression after the log-transformation of species means and variances. Here, we define CV<sub>e</sub>, the variability of an “average species”, as the calculated CV<sub>i</sub> of a species of abundance μ<sub>e</sub> = 1/n Σ<sub>i=1</sub><sup>n</sup> μ<sub>i</sub>, predicted by Equation 1 and the parameters of Taylor's power law estimated from Equation 2 using species values of the observed community.

We propose that CV<sub>e</sub> can serve as a meaningful starting point to assess total stabilization because it corresponds to the expected value of the community CV<sub>com</sub> if all species had the same abundance and had perfectly synchronized temporal fluctuations. CV<sub>e</sub> is expected to be higher than the observed community CV<sub>com</sub>, meaning that communities are usually more stable than the stability of their average species. We define the total stabilization (τ) as the ratio between the observed community CV<sub>com</sub> and CV<sub>e</sub>.

$$ \tau = \frac{CV_{com}}{CV_e} \qquad (3) $$

τ ranges between 0 (complete stabilization) and 1 (no stabilization), and a low value means high stabilization. τ can be further decomposed into three types of stabilization related to explicit components.

## 2.2 | The dominance effect

The dominance effect (Δ) estimates the role of dominant species on the community CV<sub>com</sub>. It is calculated by comparing CV<sub>e</sub> with the mean of species CV<sub>i</sub> weighted by their relative abundances (CṼ hereafter):

$$ \Delta = \frac{\sum_{i=1}^{n} p_i CV_i}{CV_e} = \frac{CV_e}{CV_e} \qquad (4) $$

where n is the number of species in the community and p<sub>i</sub> is the relative abundance of species i.

Compared to CV<sub>e</sub>, CṼ corresponds to the expected community CV<sub>com</sub> under the assumption of perfect synchrony between species (Thibaut & Connolly, 2013) but taking into account the relative abundances of species. Hence, Δ reflects how much the community stability is affected by the dominance structure, irrespective of other community characteristics. When the scaling coefficient of Taylor's power law (b) is lower than 2, the species stability increases with their abundance (see Equations 1 and 2). Under this condition, an increase in the community dominance should be associated with a higher contribution of stable species to CṼ while CV<sub>e</sub> remains unaffected, resulting in Δ<1. The lower b and the higher community dominance, the lower will be Δ. Therefore, a low value of Δ means a strong stabilization due to the dominance effect, that is, the community stability is mainly determined by the stability of dominant species. The effect of Taylor's power law on stability has been often discussed (Lázaro et al., 2022; Lepš, 1993; Thibaut & Connolly, 2013; Tilman, 1999) but, to our best knowledge, no index has been proposed to estimate its effect on CV<sub>com</sub>. Here, we propose the first index to quantify the

stabilizing role of dominant species arising from Taylor's power low. Note that, although b has been reported to be lower than 2 for a wide range of ecological communities (Tilman, 1999 and references therein), there are limited examples of communities where b > 2 (see Section 5.2). For these cases, Δ > 1 which can be safely interpreted as that dominant species destabilize, instead of stabilize, the community.

## 2.3 | Compensatory dynamics and the decomposition into asynchrony and averaging effects

Compensatory dynamics are also expected to be a strong stabilizing component of CV<sub>com</sub>. Indeed, any deviation from perfect synchrony (asynchrony) results in partial compensations between species temporal fluctuations, decreasing the community CV<sub>com</sub>. In this context, Loreau and de Mazancourt (2008) have defined an index of asynchrony (φ) that quantifies the relative decrease of the community variance gained from species compensations. It can be shown that the square root of φ corresponds to the stabilization of CṼ due to compensatory dynamics (see Thibaut & Connolly, 2013 for more details) as follows:

$$ CV_{com} = \sqrt{\phi} = \sqrt{\phi} \frac{\sigma_{com}}{\sum_{i=1}^{n} \sigma_i} \qquad (5) $$

$\sqrt{\phi} = 1$ (no stabilization) under perfect synchrony, is decreasing with increasing compensations, and is 0 in the case of perfect compensations between species (resulting in CV<sub>com</sub>=0). An issue that should be considered is that φ is not only affected by the degree of temporal correlations between species (asynchrony) but is decreasing with species diversity. This makes it difficult to distinguish the “pure” asynchrony effect from the averaging effects and the interplay of these two stabilizing effects on the overall compensatory dynamics cannot be tested (see Appendix S1).

We present a partitioning of $\sqrt{\phi}$ that separates compensatory dynamics into a component due to species asynchrony (asynchrony effect, ψ) and a component due to diversity (averaging effect, ω), so $\sqrt{\phi} = \psi\omega$. In our framework, the asynchrony effect must satisfy the following conditions: like $\sqrt{\phi}$, ψ must be 1 under perfect synchrony (no stabilization), decrease with an increasing species asynchrony, and be 0 in the case of perfect compensations between species. However, ψ must be constant under independent species fluctuations, no matter the community diversity, unlike $\sqrt{\phi}$, which decreases with species diversity (Appendix S1). This last condition can be satisfied using a power function of $\sqrt{\phi}$ as follows:

$$ \psi = \sqrt{\phi}^{\alpha} = \left( \frac{\sigma_{com}}{\sum_{i=1}^{n} \sigma_i} \right)^{\alpha} \qquad (6) $$

The scaling coefficient α allows a rescaling of $\sqrt{\phi}$ and can be determined for each community to provide a constant ψ value under independent species fluctuations, no matter the community diversity.
This common $\psi$ value under independent species fluctuations can be fixed by selecting a null community (with a reference diversity) for which we assume that compensatory dynamics are only determined by species asynchrony, and not by the averaging effect. Applying the power transformation to respect a common $\psi$ in the case of independent fluctuation will provide, for any community, the expected compensatory dynamics associated with the observed level of species asynchrony but for the reference diversity. If we use a null community of low diversity, most natural communities will have a higher diversity than this null community. Consequently, any level of species asynchrony will be associated with higher species compensations in natural communities compared with the null community, resulting in a stabilizing effect of the averaging effect (Appendix S1). Therefore, the difference between $\sqrt{\varphi}$ and $\psi$ measures the effect of diversity on compensatory dynamics and is called averaging effect in our framework (Box 1).

Here, we propose to use a null community composed of two species with equal temporal variance (although other cases can be used as well). For this null community, compensatory dynamics are $\sqrt{\varphi} = \sqrt{1/2} (\approx 0.71)$ when the two species fluctuate independently (see Appendix S1 for details). It can be shown that the following scaling coefficient allows to scale compensatory dynamics of any community to respect $\psi = \sqrt{1/2}$ in the case of independent fluctuations between species, no matter the community diversity (see Appendix S1):

$$ \alpha = \frac{\log(1/2)}{\log\left(\frac{\sum_{i=1}^{n} \sigma_i^2}{\left(\sum_{i=1}^{n} \sigma_i\right)^2}\right)} \quad (7) $$

Using this value of $\alpha$ in Equation 6, $\psi$ has the advantage of distinguishing communities with prevalent positive synchrony ($\psi > \sqrt{1/2}$) from communities where species mostly fluctuate in an anti-synchronous fashion ($\psi < \sqrt{1/2}$). Moreover, communities with equal $\psi$ are considered to have the same level of asynchrony between species.

In most natural communities, $\sqrt{\varphi}$ will be mathematically lower (higher stabilization) than $\psi$ due to their higher diversity than in this null community. Therefore, we define the averaging effect ($\omega$) as:

$$ \omega = \frac{\sqrt{\varphi}}{\psi} \quad (8) $$

$\omega$ is always one for an observed diversity equal to the null diversity (no averaging effect) and decreases with increasing observed diversity (Appendix S1). It represents the gain of stability due to a higher community diversity compared to the null community, considering the observed level of asynchrony between species. Note that in some limited cases the averaging effect may be higher than 1. This would indicate communities with extremely low diversity and reflect the destabilizing effect of a lower diversity than the null community. This point is further developed in the discussion section and in Appendix A1.

## 2.4 | General description and relative effects

Overall, our framework decomposes community $CV_{com}$ into $CV_e$ and three stabilizing effects (Figure 1b):

$$ CV_{com} = CV_e \tau = CV_e \Delta \sqrt{\varphi} = CV_e \Delta \psi \omega \quad (9) $$

The comparison of $\Delta$, $\psi$ and $\omega$ allows, for the first time, testing the absolute strength of different stabilizing effects directly and at the level of individual communities. The three indices help to disentangle stabilizing effects occurring simultaneously and promise to deepen our understanding of the complex interplay between ecological mechanisms, such as species interactions and responses to environmental variations, and the resulting stabilizing effects on community stability.

In addition to absolute values, our framework can also offer insights into the relative contribution of each stabilizing effect to the total stabilization. While absolute values can be informative enough to study the role of each stabilizing effect on the stability of individual communities, relative effects provide additional information to address other ecological questions. These values can help us, for example, to investigate the main determinants of community stability in different ecological contexts. Using the fact that $log(\tau) = log(\Delta) + log(\psi) + log(\omega)$, we can define the relative contribution of the dominance effect as $\Delta_{rel} = \frac{log(\Delta)}{log(\tau)}$ the relative contribution of the asynchrony effect as $\psi_{rel} = \frac{log(\psi)}{log(\tau)}$ and the relative contribution of the averaging effect as $\omega_{rel} = \frac{log(\omega)}{log(\tau)}$. These three ratios sum up to 1, making it possible to represent them on a ternary plot (see Figures 2 and 4). Although relative contributions can be applied to most ecological communities, we identified rare cases when their computations cannot be interpreted. Specifically, when a stabilizing effect is found to be higher than 1, its relative effect will be a meaningless negative value and the other relative contributions cannot be interpreted as well (see Section 5.2).

The R package `comstab` allows performing the partitioning presented in the framework. It is freely available on CRAN. The package `comstab` allows calculating all indices (`partitionR` function) and producing figures (`plot` and `ternStab` functions).

# 3 | PROOF OF CONCEPT USING SIMULATIONS OF COMMUNITY TIME SERIES

## 3.1 | Simulation method

We used a simulation approach to show how differences in species richness ($n$), community evenness ($k$, reciprocal of dominance), mean-variance scaling ($b$) and asynchrony ($s$) affect community stability. We applied our framework to the simulated communities to calculate each stabilizing effect and reveal its performance to detect stabilization mechanisms. Each parameter is described in detail in Appendix S2.
<figure><img src="image_3.png" /><figcaption>FIGURE 2 Temporal fluctuation of four simulated communities representing different scenarios of species richness, dominance, mean-variance scaling and asynchrony. For each scenario, (1) the selected species richness and dominance are represented by the rank-abundance plot, (2) the log-log relationship between species means and variances shows the comparison between the selected scaling coefficient (coloured dotted line), and 2 (solid line) and (3) the annual fluctuations of all species abundances show the selected synchrony level (see main text for the exact values). Four panels show the partitioning of the corresponding community coefficient of variation (CV<sub>com</sub>) into the variability of the average species (CV<sub>e</sub>) and three stabilizing effects: the dominance (Δ), asynchrony (Ψ) and averaging (ω) effects. CV~ is the abundance-weighted mean of species CV<sub>i</sub> and CV<sub>a</sub> is the expected CV<sub>com</sub> if the community was stabilized by species asynchrony only. Y-axes of these panels are log-transformed so the strength of each stabilizing effect is represented by the slope between two successive points: a steep negative slope means a high stabilization. Finally, the ternary plot in the centre shows the relative contributions of the three stabilizing effects to the total stabilization (τ) for the four communities (symbol of the corresponding colour). Tick orientation shows how to read the values. Note that for each of the scenarios, despite their different prominent stabilizing effects, CV<sub>com</sub>, CV<sub>e</sub> and the overall stabilization are kept the same.</figcaption></figure>

The classical population models (such as Lotka-Volterra competition models, e.g., Loreau & de Mazancourt, 2008; de Mazancourt et al., 2013) simulate ecological mechanisms underlying species fluctuations and, as such, is an indirect method for manipulating communities' properties. Therefore, this method does not allow an easy manipulation of the community mean-variance scaling or the

level of species asynchrony. Instead, we used a more direct and simplistic approach that simulates communities with known *n*, *k*, *b* and *s*, but where the species fluctuations are not related to any underlying environmental factor or demographic parameters. Consequently, temporal sequences simulated for each species cannot be considered realistic. Nevertheless, this simplification has little influence
on the results since none of the indices calculated in our framework
depend on the order of the sequence, but rather describe the over-
all variability and temporal coordination of species. The simulations
were computed using the comTS function in the R package comstab
associated with this paper (see the help file in the R package and
Appendix S2 for more details). We also provide a shiny application
to run simulations with sets of parameters defined by the user and
calculate the resulting community stability and their underlying
components. The application is freely available at the following ad-
dress: https://jsegrestin.shinyapps.io/ComStab.

3.2 | Simulation results

We selected four sets of parameters that resulted in communities
with similar community stability (CV<sub>com</sub> ± sd = 0.037 ± 0.003) and simi-
lar stability of their average species (CV<sub>e</sub> = 0.13 ± 0.004). In the four
communities, CV<sub>com</sub> is c. 30% of CV<sub>e</sub>, which corresponds to a strong
stabilization (τ = 0.29 ± 0.02). However, this stabilization originated
from various combinations of the three stabilizing effects (Figure 2):

* Scenario 1 simulates a community with intermediate parameters (n = 10, k = 0.4, b = 1.4 and s = 0). It has a dominance effect of Δ = 0.7, an asynchrony effect of ψ = 0.64, and an averaging effect of ω = 0.7. Hence, the three stabilizing effects had similar contributions to the total stabilization (30%, 39% and 31% respectively).

* Scenario 2 simulates a community strongly dominated by stable species (n=10, k=0.2, b=1.2, and s=0). It has Δ=0.45, ψ=0.68 and ω=0.87. Hence, the dominance effect had a prominent contribution to the total stabilization (61%) while asynchrony and averaging effects had lower contributions (29% and 10% respectively).

* Scenario 3 simulates a community with even species abundances that randomly compensate for each other's fluctuations (n=20, k=0.75, b=1.5 and s=0). It has Δ=0.87, ψ=0.69 and ω=0.46. Hence, the averaging effect had a prominent contribution to the total stabilization (60%) while dominance and asynchrony effects had lower contributions (11%, and 29% respectively).

* Scenario 4 simulates a community with strong anti-synchronous fluctuations between species (n=5, k=0.3, b=1.5 and s=-2). It has Δ=0.82, ψ=0.46 and ω=0.77. Hence, the asynchrony effect had a prominent contribution to the total stabilization (63%) while dominance and averaging effects had lower contributions (16% and 21% respectively).

Interestingly, scenarios 3 and 4 are examples of communities
with strong compensatory dynamics but originating from differ-
ent mechanisms (prominent averaging or asynchrony effects re-
spectively). It confirms that strong compensatory dynamics do not
necessarily originate from a strong asynchrony between species.
Moreover, contrary to common expectations, a high diversity of
species fluctuating independently can even lead to higher compen-
satory dynamics (√φ = ψω = 0.31 in scenario 3) than a few species
fluctuating anti-synchronously (√φ = 0.35 in scenario 4).

Overall, the use of simulations demonstrated that high commu-
nity stabilization can be achieved by any combination of stabilizing
effects. The latter can be illustrated by the fact that a large propor-
tion of the ternary plot can be covered (using the shiny application
and see the central panel of Figure 2). Therefore, in these theoreti-
cal cases, the three stabilizing effects (dominance, asynchrony and
averaging effects) can vary relatively independently. Whether the
same conclusion applies to real communities remains to be tested.

4 | CASE STUDY: EFFECTS OF MANAGEMENT ON THE STABILITY OF A GRASSLAND COMMUNITY AND ITS COMPONENTS

4.1 | Study site and data collection

We used data from a long-term experiment conducted on a meadow
that has traditionally been extensively mown for centuries. The
study site is located 10km southeast of České Budějovice, Czech
Republic (48°570N, 14°360E, altitude 510m). The vegetation is an
oligotrophic, wet and species-rich meadow dominated by grasses,
especially *Molinia caerulea* (L.) Moench. In 1994, nine 2×2m perma-
nent plots were established and consisted of three controls, three
fertilized plots and three plots where *M. caerulea* was removed. In
the fertilization treatment, a commercial fertilizer (N-P-K: 12%-19%-19% changed to N-P-K: 8%-13%-11% since 2003) was added
annually at a dose of 65g m⁻². Since 2012, the dose was decreased
to 50g m⁻² to prevent nutrient accumulation in the soil (particularly
phosphorus). In the removal treatment, *M. caerulea* was manually
weeded in 1995 and new individuals growing afterward were re-
moved annually. During the 1999–2014 period, the plant biomass in
the central 0.25m² of each plot was measured each year in the sec-
ond half of June, at the peak of the vegetation. Biomass was clipped
simultaneously to the annual mowing of the site, sorted into species,
oven-dried and weighed. The final data set describes the temporal
variation of the above-ground biomass of all plant species in each
permanent plot over 16 years. More details on the experimental de-
sign are given in Lepš et al. (2018) and references therein.

We applied our framework to these data to investigate the ef-
fects of the treatments on community stability. Note that the low
replication level of the experiment (3 plots per treatment) resulted in
weak statistical tests. However, this section aims at demonstrating
the use of our framework rather than giving new ecological conclu-
sions. Indeed, data from this experiment have been already reported
in previous work (see Lepš et al., 2018; Májeková et al., 2014;
Segrestin & Lepš, 2022).

4.2 | Results from the framework

In controls, we found a community CV<sub>com</sub> of 0.20. We found that the
CV of an “average species” in these communities is CV<sub>e</sub> = 0.82 and
the stabilizing effects are $τ=0.24$: $Δ=0.81$, $ψ=0.73$ and $ω=0.41$ (Figure 3). Hence, the community stabilization in control plots was mainly driven by averaging effect (62%) due to their high species diversity, while dominance and asynchrony had lower contributions (15% and 23% respectively; Figure 4).

Fertilized communities tended to be less stable ($CV_{com}=0.25$) than controls (Figure 3a). This lower community stability resulted from a higher variability of their average species ($CV_e=0.95$) and a lower total stabilization ($τ=0.27$) (Figure 3a). The effect of fertilization on individual species variability has been already discussed in a previous study (Lepš et al., 2018). However, the treatment effect on stabilizing effects was only tested for the asynchrony and dominance effects, while the effect on the averaging component was speculated (Lepš et al., 2018). Here, we found that fertilization did not affect the dominance effect (similar $Δ$) but increased the stabilization due to species asynchrony (lower $ψ$) and strongly decreased the stabilization due to the averaging effect (higher $ω$) (Figure 3a). We found that species in control plots are mainly positively synchronized ($ψ>0.71$) while species in fertilized plots tend to be anti-synchronous ($ψ<0.71$, Figure 3a). Hence, one can expect higher stabilization in fertilized plots due to compensatory dynamics. However, compensatory dynamics showed the reverse pattern due to a larger opposite response of the averaging effect resulting from the strong decrease in species diversity in the fertilized treatment, which overrides the asynchrony stabilizing effect. Overall, fertilization slightly increased the contribution of dominance (17%) and asynchrony (31%) and decreased the contribution of averaging (52%) (Figure 4). It reveals the strong importance of the averaging effect in compensatory dynamics beyond asynchrony.

Removal of *M. caerulea* also tended to destabilize plant communities ($CV_{com}=0.22$) by increasing the variability of their average species ($CV_e=0.94$) while the total stabilization remained similar ($τ=0.24$) (Figure 3b). We found that the removal of the most dominant species

strongly decreased the stabilization by dominants (higher $Δ$) and increased the stabilization by averaging effect (lower $ω$) but did not influence species asynchrony (similar $ψ$) (Figure 3b). Hence, dominant removal strongly decreased the contribution of dominance (4%), slightly increased the contribution of asynchrony (23%), but strongly increased the contribution of averaging (73%) (Figure 4). Our analysis is consistent with previous findings showing that *M. caerulea* removal did not effect on the CV of other species (Májeková et al., 2014) and decreases the community dominance since none of the remaining species was able to entirely replace it (Lepš et al., 2018). Moreover, the lower dominance in communities where *M. caerulea* was removed resulted in higher diversity which, in turn, increased the stabilization due to the averaging effect. Finally, the low influence of this treatment on species synchrony suggested little change in species interactions despite the absence of the strongest species. Overall, it demonstrates the importance of a strong stable dominant for community stability.

Overall, our analyses provide a comprehensive description of the treatment effects on the temporal stability of a plant community by allowing distinguishing the direct contribution of various community properties on $CV_{com}$. In addition to their effects on the average species stability, the treatments affected simultaneously the three stabilizing effects by modifying the community species richness, evenness, mean-variance scaling and species synchrony.

# 5 | DISCUSSION AND CONCLUSIONS

## 5.1 | Linking our framework to ecological hypotheses

Our study answers the current need for a unified framework to link theoretical concepts on the temporal stability of ecological communities to data analysis. First, we provide an analytical tool

<figure><img src="image_4.png" /><figcaption>FIGURE 3 Effect of (a) fertilization and (b) dominant removal on the temporal coefficient of variation ($CV_{com}$) of a grassland community. The coefficient of variation of the average species ($CV_e$) is stabilized by the dominance ($Δ$), asynchrony ($ψ$) and averaging ($ω$) effects. $CV$ is the abundance-weighted mean of species $CV_i$ and $CV_a$ is the expected $CV_{com}$ if the community was stabilized by species asynchrony only. The total stabilizations ($τ$) are given in the panel legends. Y-axes are log-transformed so the strength of each stabilizing effect is represented by the slope between two successive points (the corresponding values are given next to each line): a steep negative slope means a high stabilization. Asynchrony effects corresponding to a prevalent positive or negative synchrony between species are represented by a (+) or (-) respectively.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

FIGURE 4 Relative contributions of dominance, asynchrony and averaging effects on the total stabilization of grassland communities in control plots (red symbol), fertilized plots (orange symbol) and plots where the most dominant species has been removed (blue symbol). The axes represent the contribution of each stabilizing effect to the total stabilization, calculated as the ratios between the corresponding log-transformed index (log(Δ), log(ψ) or log(ω)) and the log-transformed total stabilization (log(τ)). Tick orientation shows how to read the values, for example, for the orange symbol, the contribution of dominance, asynchrony and averaging effects to the total stabilization are roughly 20%, 30% and 50% respectively.

to investigate the long-standing and debated question on the rela-
tive importance of dominant species (the mass-ratio hypothesis:
Grime, 1998) versus diversity effects (the insurance hypothesis:
McNaughton, 1977; Naeem & Li, 1997) on this ecosystem function
(Grime, 1997). Indeed, the comparison between the dominance ef-
fect (Δ) and compensatory dynamics ($\sqrt{\varphi} = \psi\omega$) provides valuable
insights into the strength of the community stabilization due to
dominant species, compared to that of compensatory dynamics.

Moreover, we provide a novel and promising approach to disen-
tangle the effects of biodiversity through the insurance hypothesis
via asynchrony (McNaughton, 1977; Naeem & Li, 1997) from the
"statistical inevitability" unrelated to any biological process, that is,
the averaging effect (Doak et al., 1998). Indeed, the proposed asyn-
chrony (ψ) and averaging (ω) indices provide a partitioning of the
community stabilization by both components. Arguably, both indices
improve previous indices used in the literature:

First, we provide a new asynchrony index (ψ) with interesting
properties. Its main novelty can be particularly appreciated in its
ability to give consistent values for the cases of (i) perfect compensa-
tions between species (ψ=0), (ii) independent fluctuations between
species (ψ=√1/2), and (iii) perfect synchrony between all species
(ψ=1), no matter the community characteristics. To our knowl-
edge, none of the indices previously used have this property: the
log-transformed variance ratio (Hallett et al., 2016; Schluter, 1984)
is zero for independent fluctuations but gives increasing values

with diversity for perfect synchrony and decreasing values with
diversity for perfect compensations, the weighted Gross' index
(Blüthgen et al., 2016) is zero under independent fluctuations and
one under perfect synchrony but can be -1 even with no perfect
compensations, and the φ index (Loreau & de Mazancourt, 2008) is
zero and one under perfect compensations and perfect synchrony,
respectively, but is not constant under independent fluctuations.
Moreover, ψ is a direct estimate of the community stabilization due
to species asynchrony only and has no mathematical dependency
on the community diversity (see Appendix S3 for a comparison with
other asynchrony indices and the effect of community diversity).
Therefore, we argue that ψ is currently the most advanced index to
estimate the effect of ecological processes on the temporal synchro-
nization of species and, more importantly, on community stability
(Appendix S3).

Second, ω is currently, to our best knowledge, the most advance
index to directly measure the realized compensatory dynamics oc-
curring in a community due to its diversity. Indeed, we argue that
the averaging index recently proposed by Zhao et al. (2022) calcu-
lates a potential averaging effect rather than the realized commu-
nity stabilization due to this component. As a matter of fact, Zhao's
index gives high values for communities with high diversity, which is
logical. However, its computation does not account for the degree
of species asynchrony. Therefore, Zhao's index gives high values for
communities of high diversity, even in the case of perfect synchrony
between species. Conceptually, we expected species asynchrony
to modulate the averaging effect: if all species are perfectly syn-
chronized, there is no averaging effect, regardless of the commu-
nity diversity. More importantly, when not all species are perfectly
synchronized (i.e. any asynchronous fluctuation), an increase in di-
versity increases the statistical averaging, and its effect becomes
larger with increasing asynchrony, particularly when species fluc-
tuate anti-synchronously. This idea has already been introduced in
Doak et al. (1998), who state that correlations in the fluctuations of
species do strongly influence the strength of the stability-diversity
relationship, with positive correlations decreasing and negative cor-
relations strengthening the effect of species number on community
stability. On the other hand, ω gives consistent results compared to
theoretical expectations: it shows no stabilization in the case of per-
fect synchrony between species (ω=1), no matter the community
diversity, and its effect increases both with community diversity
and species asynchrony. Hence, ω estimates the realized averaging
effect and more accurately reflects the complex interplay between
diversity and asynchrony on compensatory dynamics.

Overall, we propose a general mathematical definition of the
total stabilization of communities (τ), that can be partitioned into
our three stabilizing effects (Δ, ψ and ω). All three are expressed on
homogenous scales (expected values between 0 and 1) allowing us
to directly compare their absolute contributions to the total stabi-
lization. Although these indices do not directly quantify the effect
of ecological processes, such as species interactions or responses
to environmental factors, on the community stability, they pro-
vide valuable tools to test clear ecological hypotheses linking these
processes to various components of the community stability. For example, it is often assumed that increasing direct competition between species should be associated with lower stability of individual species, higher dominance, anti-synchronous fluctuations between species and lower species richness. Our framework provides four indices that directly estimate the effect of each of these components on the community stability, respectively, CV<sub>e</sub>, Δ, ψ and ω, which will allow, for example, to test their responses to different levels of biotic interactions. For this reason, although it remains difficult, if not impossible, to infer processes from patterns, the proposed framework moves a step forward in the direction of a better understanding of underlying mechanisms, particularly when compared to existing approaches.

Finally, the framework allows to compute relative contributions of each stabilizing effect to the total stabilization for the majority of natural biological systems. The representation of these relative contributions on a ternary plot allows for displaying any community in a standardized space. We found that this space can be largely covered by simulated communities. Whether the same conclusions apply to natural communities remains to be tested and is a promising avenue for future research. More precisely, it is still largely unknown whether high community stabilization is achieved through any combination of dominance, asynchrony and averaging effects or if specific combinations are more common in natural communities.

## 5.2 | Recommendations for use and extreme cases

Our framework is compatible with the analysis of most natural and manipulated ecological communities. However, a few recommendations should be considered. First, special attention should be given to defining which species truly belong to the community. Indeed, we suggest that transient species, appearing rarely and not establishing locally, should be removed from the data before the analyses. Otherwise, this type of species might affect the estimation of Taylor's power law parameters and reduce the abundance of the "average species" used to calculate CV<sub>e</sub>. Moreover, such species can affect the level of asynchrony of the community, hence affecting ψ. Although transient species should have a minor impact on most communities, it can affect the estimations of CV<sub>e</sub> and stabilizing effects, especially in the case of a low diversity community. In addition to expert knowledge of data collectors, transient species can be detected by their low frequency in the time series but also using Taylor's power law. Indeed, such species should give inconsistent values compared to other species, resulting in large residuals from the general relationship. This procedure allows detecting species which are likely to be removed. Nevertheless, analyses with and without such species can be run to test for their potential effects.

We also identified two theoretical cases where stabilizing effects can give results outside of their expected ranges (values above one, meaning that the corresponding component is destabilizing rather than stabilizing the community). The first case corresponds to communities where the dominance effect is higher than one. The

latter is found in communities where the variability of individual species (CV<sub>i</sub>) is not negatively related to their abundance due, for example, to a mean-variance scaling coefficient higher than 2 in Taylor's power law. In these cases, Δ is higher than 1. Although theoretically possible, this case has been rarely reported for existing communities to our knowledge: we found only one study reporting a scaling coefficient higher than 2 when considering different coexisting species of moths (Lepš, 1993). This study used data from light traps in a ruderal site, characterized by a cultivated landscape with small wet hay meadows, clumps of trees and large temporary agroecosystems. Hence, the data contained a mixture of individuals from different communities and the analyses were based on the selection of only abundant species. Despite these limitations, the study suggests that communities dominated by species performing sporadic population outbreaks can result in a destabilizing effect of dominants (Δ> 1). The second case may arise when analysing communities with extremely low diversity. In the case of monocultures, all stabilizing effects should be fixed to one (no stabilization). However, for species-poor communities, the averaging effect can be higher than 1 when species diversity is (mathematically) lower than two species of equal (temporal) variance (see the last paragraph of Appendix S1). In addition to the rarity of this scenario, the ecological relevance of defining stabilizing effects (especially compensatory dynamics) for these types of communities appears to be questionable. It is important to stress that these cases do not weaken the generality of our framework and the proposed framework can still be applied in all cases. Absolute values of the stabilizing effects can still be interpreted correctly and allow to identify specific plots where more complex interpretations are needed. Clearly, even if relative effects can still be calculated, they will give meaningless negative values, so that the relative effects do not add up to 1 anymore.

## AUTHOR CONTRIBUTIONS

JS conceived the ideas and designed the methodology; all authors contributed to the development of the framework; LG, FdB, JL and JS designed the simulations; JS developed the R package and shiny application, JL collected the data and JS analysed the data for the case study; JS led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.

## ACKNOWLEDGEMENTS

This research was supported by GAČR 23-07087S. Open access publishing facilitated by Jihoceska Univerzita v Ceskych Budejovicich, as part of the Wiley - CzechELib agreement.

## CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

## DATA AVAILABILITY STATEMENT

The R package *comstab* includes functions for implementing our framework and is freely accessible on CRAN. Additionally, the release version *comstab* 0.0.1 is available for download from the Zenodo Digital Repository: https://doi.org/10.5281/zenodo.
10687006. All analyses conducted in this study rely on this package. The data used for the case study have already been published and is freely accessible in the Dryad Digital Repository: https://doi.org/10.5061/dryad.8gtht76r9 (Segrestin & Lepš, 2022).

ORCID

Jules Segrestin https://orcid.org/0000-0001-7661-6061

Lars Götzenberger https://orcid.org/0000-0003-3040-2900

Enrique Valencia https://orcid.org/0000-0003-3359-0759

Francesco de Bello https://orcid.org/0000-0001-9202-8198

Jan Lepš https://orcid.org/0000-0002-4822-7429

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## BIOSKETCH

All authors contributed to recent studies on the topic of ecological stability which were published in high-profile journals (PNAS, TREE, Journal of Ecology, Ecology, etc...). The research team is actively involved in the development of a global database of long-term vegetation sampling (LOTVS: https://lotvs.csic.es/) whose main purpose is to investigate the main drivers of the stability of ecological communities. Moreover, several authors are often contributing to the development of new analytical methods for the fields of community ecology and trait-based ecology.

## SUPPORTING INFORMATION

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

**How to cite this article:** Segrestin, J., Götzenberger, L., Valencia, E., de Bello, F., & Lepš, J. (2024). A unified framework for partitioning the drivers of stability of ecological communities. *Global Ecology and Biogeography*, **33**, e13828. https://doi.org/10.1111/geb.13828