# Non-linearity and Temporal Variability Are Overlooked Components of Global Vertebrate Population Dynamics

Maelys Boennec, Vasilis Dakos, Vincent Devictor

► To cite this version:

Maelys Boennec, Vasilis Dakos, Vincent Devictor. Non-linearity and Temporal Variability Are Overlooked Components of Global Vertebrate Population Dynamics. Diversity and Distributions, 2024, 30 (12), pp.e13932. 10.1111/ddi.13932. hal-04978114

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RESEARCH ARTICLE

OPEN ACCESS

# Non-linearity and Temporal Variability Are Overlooked Components of Global Vertebrate Population Dynamics

Maelys Boennec

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

Vasilis Dakos

Vincent Devictor

Institut des Sciences de l'Évolution de Montpellier (ISEM), Université de Montpellier, CNRS, IRD, Montpellier, France

**Correspondence:** Maelys Boennec (maelys.boennec@umontpellier.fr)

**Received:** 26 December 2023

**Revised:** 14 September 2024

**Accepted:** 24 September 2024

**Editor:** Pedro Cardoso

**Funding:** The authors received no specific funding for this work.

**Keywords:** biodiversity crisis | conservation biogeography | linear trends | non-linearity | population time-series | temporal variability

## ABSTRACT

**Aim:** Population dynamics are usually assessed through linear trend analysis, quantifying their general direction. However, linear trends may hide substantial variations in population dynamics that could reconcile apparent discrepancies when quantifying the extent of the biodiversity crisis. We seek to determine whether the use of non-linear methods and the quantification of temporal variability can offer a more complete representation of changes in global population dynamics than commonly-used linear approaches.

**Methods:** We analysed 6437 population time series from 1257 vertebrate species from the Living Planet Database over the period 1950–2020. We modelled populations through the use of second-order polynomials and classified trajectories according to their direction and acceleration. We modelled and classified these same populations using a more classical linear trend analysis. We quantified temporal variability using the mean squared error of the fitted polynomials. We then used generalised linear mixed models to test potential sources of heterogeneity in non-linear trajectories and temporal variability.

**Results:** In all, 44.8% of the analysed population time series were non-linear. Across all populations, 30% were declining, 30% were increasing, and 40% were with no linear trend. Among the population showing no linear trend, half were concave or convex. Non-linearity was expressed differently between taxonomic groups, with mammals showing higher prevalence of non-linearity. Marine and freshwater populations were more variable than terrestrial populations, and fish were more variable than other vertebrates. Differences between geographical regions were detected in both non-linearity and temporal variability, but no straightforward pattern emerged. There were no differences in both components between IUCN categories.

**Main Conclusions:** Non-linearity and temporal variability reveal usually overlooked dramatic declines or recovery signals in global population dynamics. Thus, moving beyond linearity can improve our understanding of complex population dynamics and better inform conservation decisions. In particular, populations usually classified as ‘stable’ can hide informative changes in non-linear and variability patterns that need to be considered in global biodiversity assessments.

## 1 | Introduction

Quantifying and understanding global biodiversity changes are critical research challenges. A commonly used approach

to quantifying global biodiversity changes is to rely on the linear trend in global indices derived from the combination and collation of local monitoring data distributed across realms and groups (e.g., Living Planet Index; McRae, Deinet, and

<footnote>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.</footnote>

<footnote>© 2024 The Author(s). Diversity and Distributions published by John Wiley & Sons Ltd.</footnote>
Freeman 2017; WWF 2022). More recently, the widespread availability of open-access large datasets (e.g., BiotIME; Dornelas et al. 2018), has revealed contrasting results depending on the ecological level (population versus community) (Dornelas et al. 2014; Leung et al. 2020) or metric considered (e.g., Living Planet Index vs. population trends) (Blowes et al. 2019; Antão et al. 2020; Daskalova, Myers-Smith, and Godlee 2020; Pilotto et al. 2020; Haase et al. 2023). In particular, ‘no net loss’ in the number of species or ‘no net change’ in population abundances have been reported (Vellend et al. 2013; Inger et al. 2015; Primack et al. 2018; Blowes et al. 2019; Pilotto et al. 2020; Crossley et al. 2021). While such conclusions raised many questions and caused controversies within the scientific community (Cardinale 2014; Gonzalez et al. 2016; Loreau et al. 2022; Boënnec, Dakos, and Devictor 2023; Kuczynski, Ontiveros, and Hillebrand 2023), they revealed that global biodiversity changes are complex and heterogeneous, and cannot be encapsulated in a single metric (Dornelas et al. 2023).

However, the majority of empirical studies tracking temporal changes in the abundance of individuals of different species over time rely on linear regressions (Rigal, Devictor, and Dakos 2020; Pélissié, Devictor, and Dakos 2023) or more elaborated linear models (e.g., state-space models; Leung, Greenberg, and Green 2017; Daskalova, Myers-Smith, and Godlee 2020). By focusing solely on the general direction of population trajectories (i.e., the sign of the slope) and its magnitude (i.e., the absolute value of the slope), these models could miss critical information for describing population dynamics, such as how fast and when changes occur (Rigal, Devictor, and Dakos 2020). For instance, a convex trajectory for the population size of a

threatened species (Figure 1) can testify of a successful conservation strategy and a recent improvement of its population status. On the contrary, an accelerated decrease in the population size of a species (Figure 1) can indicate an increasingly worsening situation. Such non-linear dynamics are expected to occur substantially in population dynamics given that the drivers of population trajectories themselves follow non-linear patterns. For instance, many global change pressures have accelerated synchronously in the 1980s, often labelled as a ‘great acceleration’ in the Anthropocene (Steffen et al. 2015). Besides, non-linearity has already been observed at large spatial and temporal scales in populations of arthropods (Duchenne et al. 2022), birds (Rigal, Devictor, and Dakos 2020) or fishes (Pélissié, Devictor, and Dakos 2023) or in community dynamics (Penny, Dornelas, and Magurran 2023). Yet, the extent and distribution of non-linear trajectories of populations at global scale has been, to our knowledge, largely overlooked.

Besides non-linear trajectories, temporal variability is another overlooked aspect of global biodiversity dynamics. Temporal variability is a common estimator of ecological stability, a multifaceted concept with many alternative definitions including resistance to environmental change and resilience (Pimm 1984; McCann 2000; Donohue et al. 2016). Increase in temporal variability has been long identified as a signal of a loss of stability in ecological systems (Grimm and Wissel 1997) or even a signature of abrupt ecological collapse (Scheffer et al. 2009; Hughes et al. 2013). Thus, population temporal variability was proposed as a proxy for vulnerability (Mrowicki, O'Connor, and Donohue 2016), yet it is not captured by the slope or the shape of the trajectory (even if non-linear) of a population. For instance, an apparent absence of change in regard of the overall trajectory

<figure><img src="image_3.png" /><figcaption>**FIGURE 1** | Illustrations of non-linearity and temporal variability in hypothesized population dynamics. Representation of different possible magnitude of temporal variability within population linear or non-linear trajectories. Black solid lines represent the trajectory of the time series, which might be a linear or non-linear fit. Solid grey lines represent hypothetical abundance time series. Temporal variability in populations may occur even when the apparent trajectory is null or going upwards.</figcaption></figure>
may mask strong variations in abundance (Figure 1), and there-
fore a potential risk of extinction, especially if the population's
abundance is small and exposed to demographic, environmen-
tal or genetic stochasticity (Caughley 1994). Although temporal
variability has been studied at global scale (Leung, Greenberg,
and Green 2017; Capdevila et al. 2022), how it is coupled with
non-linear population trajectories remains unclear.

Overall, a complete understanding of global biodiversity changes based on the complete description of trajectories (beyond usual linear trends) and their associated temporal variability is missing. Such comprehensive description could have implications for conservation and biodiversity assessments. Testing which habitat types, geographical regions and/or taxonomic groups are facing specific patterns of change in population abundance over time could provide key insights for improving conservation prioritisation. For instance, Global IUCN Red List Categories are often used for conservation prioritisation, yet they are based, among other factors, on simple linear population trends (IUCN 2023). Global IUCN Red List Categories could be refined by integrating recent abrupt changes within populations dynamics revealed by non-linear models or strong temporal variability exposing specific species to increasing vulnerability (Brito et al. 2010; Rueda-Cediel et al. 2018; Bachman et al. 2019; Fraixedas et al. 2022).

In this paper, we analyse vertebrate population data from the
Living Planet Database and describe their non-linear trajecto-
ries and temporal variability. We test three predictions on those
two components.

1. We assume that there would be biogeographic patterns in population non-linear trajectories and temporal variability across the planet's habitat types and regions, in line with particular regions of the world experiencing high rates of environmental change (e.g., tropical forests [Barlow et al. 2007], Arctic [IPCC 2021]). The clustered distribution of global change drivers in space has been proven to lead to heterogeneous patterns (Halpern et al. 2015; Blowes et al. 2019; He et al. 2019; Bowler et al. 2020; Jaureguiberry et al. 2022). We therefore expect regions particularly under pressure to express more non-linearity through accelerated declines and concave trajectories, given that the rate at which the changes occur is not linear. Similarly, high environmental changes are expected to result in stronger temporal variability in populations (Lawson et al. 2015).

2. We expect populations from species with life history traits related to high vulnerability to global change to express higher non-linearity and temporal variability. Species with different life history traits may respond to the environment differently or may intrinsically differ in a way that results in more non-linearity and more variable population dynamics, and this may be seen in the different taxonomic groups (Albaladejo-Robles, Böhm, and Newbold 2023). In particular, amphibians are expected to express more non-linearity and temporal variability than the other groups (Houlahan et al. 2000; Blaustein et al. 2011). We expect a higher proportion of non-linear trajectories and especially a higher diversity in the forms of these non-linear trajectories. Non-linearity may be expressed either through dramatic declines (accelerated declines or concave trajectories), along with

high temporal variability, or through recovery signals (decelerated declines, convex trajectories) if conservation actions were implemented and successful.

3. As for more vulnerable taxonomic groups, we similarly expect populations from species with a high IUCN Red List Category (being a proxy of high extinction risk) to be more likely to express non-linearity in all its forms and higher temporal variability than the populations of least concern species.

2 | Methods

## 2.1 | Ecological Time Series Data

We used population time series from the Living Planet Database (LPD). The current public version contains 32,211 population time series distributed worldwide, with populations that were monitored using different metrics of abundance or proxy of abundances (e.g., number of individuals vs. number of nests); and time series ranging from 1950 to 2020. We omitted populations that had <20 time points of monitoring data (populations were not always monitored in each intervening year, see Appendix S3, Figure S3.1), as previous studies have found that shorter time series might not capture biologically meaningful directional trends in abundance (Wauchope et al. 2019). For each population time series, habitat type (Terrestrial, Marine or Freshwater), region (e.g., Africa, North America, Europe) and taxonomic group (e.g., Birds, Mammals) is provided in the LPD. Additionally, we extracted IUCN Red List Category data for each species from the International Union for Conservation of Nature (IUCN).

## 2.2 | Modelling Populations Dynamics

2.2.1 | Fitting Linear and Non-Linear Models

We fitted both linear and non-linear models to compare the
resulting classifications of population trajectories. Before
analysis, we log transformed the abundance of each popula-
tion to make them comparable. Ecological data are often log-
transformed prior to statistical analysis as variance has the
tendency to increase with the mean abundance, and because
log-transformation allows the reduction of the outliers' weight
(Cottingham, Brown, and Lennon 2001). Population trajectories
are thus estimated in multiplicative changes and non-linearity is
referred to on a multiplicative scale.

We first fitted each population time series (log transformed abundance through time, we added 1 to every population count, to account for estimates of zero abundance in a given year) with a linear regression model to qualify the overall linear direction of the trajectory. We distinguished increasing trajectories (significantly positive slope), decreasing trajectories (significantly negative slope), and trajectories with no significant trend, classified as 'no trend'.

For the non-linear modelling, we adopted the framework pro-
posed by Rigal, Devictor, and Dakos (2020). We fitted each
population time series with a second-order polynomial model to qualify the non-linear shape of the trajectory through time:

$$Y(t) = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \varepsilon(t) \quad (1)$$

where Y is the log transformed abundance estimate (we added 1 to every population count, to account for estimates of zero abundance in a given year), t the time in years, and the process error is represented by $\varepsilon \sim N(0, \sigma^2)$. We used orthogonal polynomials so that the correlation between the first- and second-order variables is removed (Rigal, Devictor, and Dakos 2020). The significance of each coefficient ($\alpha_1$ for first order and $\alpha_2$ for second order) is therefore used to test whether the second order significantly improves the regression compared to the first order. Second-order polynomials can discriminate trajectories with no net changes—stationary processes—(if $\alpha_1$ and $\alpha_2$ are not significant); linear—or monotonous—trajectories (if only $\alpha_1$ is significant); and non-linear—or non-monotonous—trajectories (if $\alpha_2$ is significant).

The fact that we added 1 to every population count due to the log-transformation prior to the analysis could however bias the estimated population trajectories (O'Hara and Kotze 2010). To make sure this is not the case, we compared the population trajectory classifications when using the generalised linear models with polynomials on raw abundances (using a quasi-poisson distribution error) versus when using the log-transformed abundances. Appendix S2 shows that the results regarding this study were not affected overall (the difference between both methods was however more relevant for rare species, which are overall hard to monitor and evaluate).

### 2.2.2 | Classifying Population Trajectories

We used the direction and the acceleration to classify the different types of trajectories. The direction (increase, decrease or null) is null if there is no net changes, and equal to $\alpha_1$ if the trajectory is linear. If the trajectory is non-linear (if $\alpha_2$ is significant), the direction is determined by the slope of the tangent given by the linearisation around the center of the time series. The acceleration is given either by the sign of the second-order coefficient $\alpha_2$ or by the sign of $\dot{\gamma}$, the derivative of the curvature function. When the direction is null, the acceleration refers to the convexity or concavity of the trajectory and only the sign of $\alpha_2$ is needed to describe it (convex if $\alpha_2 > 0$, concave if $\alpha_2 < 0$). When the direction is an increase or a decline, the sign of $\dot{\gamma}(\alpha_2)$ is used to determine if the process is accelerated ($\dot{\gamma}(\alpha_2) < 0$) or decelerated ($\dot{\gamma}(\alpha_2) > 0$) (see Rigal, Devictor, and Dakos 2020 for details). Using the combination of direction and acceleration, one can classify any trajectory as belonging to one of the nine following trajectory types: linear decrease, accelerated decrease, decelerated decrease, no trend (linear), concave (down), convex (up), linear increase, decelerated increase and accelerated increase (Figure 2 shows examples of each trajectory type).

### 2.2.3 | Estimating Temporal Variability

Once we fitted the non-linear models, we statistically estimated the temporal variability around the calculated trajectories. We

considered the temporal variability of a population to reflect the fluctuations around the overall trajectory, that we assume to be the ‘equilibrium’.

We used the mean squared error (MSE) as a proxy of this temporal variability, which is measured as:

$$MSE = \frac{1}{n} \sum_{t=1}^{n} \left( Y_t - \hat{Y}_t \right)^2 \quad (2)$$

where $Y_t$ corresponds to the log transformed abundance value at time t and $\hat{Y}_t$ to the corresponding estimated value from the second-order polynomial fit (either the non-linear fit when the second-order term is significant or the linear one if it is not significant). The MSE is calculated from the residuals of the fitted model, it therefore allows to account for variability around the main trajectory.

The MSE is distributed on $[0, +\infty]$, with a high MSE being interpreted as a high temporal variability. However, this metric is linked to our choice of modelling via second-order polynomials, which is why we additionally used two other proxies for temporal variability to minimise potential biases associated with model and metric selection. Specifically, we used the coefficient of variation and the consecutive disparity index (Fernández-Martínez et al. 2018) and present the details regarding these analyses in Appendix S4.

## 2.3 | Identifying Determining Factors of Non-Linearity and Temporal Variability

### 2.3.1 | Non-Linearity

To determine whether non-linearity was expressed differently across habitat types, regions, taxonomic groups, and IUCN Red List Categories, we used a generalised mixed-effects model (GLMM) framework which accounts for co-variance among explanatory variables. The linear versus non-linear character of each population trajectory was encoded through a binary variable (L). We used a GLMM with a binomial error structure and a logit link function that modelled the probability ($p_{irths}$) for each time series i from region reg, habitat hab, taxonomic group tax and IUCN category of species s to not being linear. The model is structured as follows:

$$L_{irths} \sim \text{Bernoulli}(p_{irths}) \quad (3)$$

$$\log\left(\frac{p_{irths}}{1 - p_{irths}}\right) = \beta_0 + \beta_1 * \text{points}_i + \beta_{2_{reg}} + \beta_{3_{hab}} + \beta_{4_{tax}} + \beta_{5_{IUCNs}} + \theta_{\text{species}} \quad (4)$$

where $\beta_0$ is the global intercept, $\beta_1$ the effect of the number of points of the i'th time series on the probability of not being linear. $\beta_{2_{reg}}$, $\beta_{3_{hab}}$ and $\beta_{4_{tax}}$ are the effects of the regions, habitats and taxonomic groups, respectively, of the time series i. $\beta_{5_{IUCNs}}$ is the effect of the IUCN classification of species s. $\theta_{\text{species}}$ is the species random effect, accounting for the possible correlation between populations from the same species. We accounted for the time series duration in the models as duration varies among time series, and among the other factors being explored, while it also affects estimates of change on
<figure><img src="image_4.png" /><figcaption>**FIGURE 2** | Non-linear trajectories account for almost 45% of the population trajectories studied. Classification of the 6437 population trajectories (log transformed abundances) from 1950 to 2020 into nine possible shape categories. 'N' indicates the number of populations within each category. The percentages are relative to the total number of population (6437). Log transformed abundances are shown for one population of each trajectory type by a grey line and best-fitted first- or second-order polynomials are shown by a bold coloured line.</figcaption></figure>

temporal data (Wauchope et al. 2019; Duchenne et al. 2022, see Appendix S3).

### 2.3.2 | Temporal Variability

Similarly, to determine whether population temporal variability varied according to biogeographic and taxonomic patterns, we used a GLMM with a gamma error structure and a log-link function that modelled the value of the MSE for each time series i from region reg, habitat hab, taxonomic group tax and IUCN category of species s.

$$ \text{MSE}_{\text{irhts}} \sim \text{Gamma}(\mu_{\text{irhts}}, \phi) \quad (5) $$

$$ \log(\mu_{\text{irhts}}) = \beta_0 + \beta_1 * \text{points}_i + \beta_{2_{\text{reg}}} + \beta_{3_{\text{hab}}} + \beta_{4_{\text{tax}}} + \beta_{5_{\text{IUCNs}}} + \beta_{6_{\text{traj}}} + \theta_s \quad (6) $$

where $\mu_{irhts}$ in Equation (5) is the mean of the gamma distribution and $\phi$ the dispersion parameter. The $\beta_0, \beta_1, \dots, \beta_5, \theta_s$ in

Equation (6) account for the same effects than in Equation (4), and $\beta_{6_{\text{traj}}}$ accounts for the effect of the time series type of trajectory (categorical variable with nine factors corresponding to the nine types of trajectories). The aim was to investigate whether we identified significant differences in temporal variability among habitat types, regions, taxonomic groups and IUCN Red List categories even when accounting for the trajectory type, to avoid potential redundancy between the two parameters.

All analyses were performed under R version 4.3.3. We used the *glmmTMB* function from the 'glmmTMB' package to run the models (Brooks et al. 2017). We then used the Anova function from the 'car' package (Fox and Weisberg 2018) to get traditional ANOVA-style table from *glmmTMB* fits. The estimated marginal means were extracted with the 'emmeans' package (Searle, Speed, and Milliken 1980). When differences were detected, we performed post hoc tests using the *ghlt* function from the 'mult-comp' package to distinguish groups that expressed significantly different temporal variability from others (Hothorn, Bretz, and Westfall 2008). All data and scripts are available on a Zenodo repository online.
# 3 | Results

We analysed 6437 population time series from 1257 species extracted from the Living Planet Database over the period between 1950 and 2020, which reflects almost 20% of the entire database. The remaining 80% were not selected because of their small length (<20 years). The temporal (Figure S1.1), geographical and taxonomic extents (Figures S1.2, S1.3, Table S1.1) of the database we analysed are presented in Appendix S1.

## 3.1 | Non-Linear Trajectories Among Global Population Dynamics

### 3.1.1 | Classification of Population Dynamics Into Trajectory Types

We applied both the linear and non-linear classification methods on 6437 population time series. We found a broad spectrum of trends and trajectories among the studied populations (Figure 2). Across the time series we analysed, 36% (2338 time series) of populations were declining, 34% (2164 time series) were increasing and 30% (1935 time series) showed no trend in population change over time according to the linear framework. When classifying the trajectories following the non-linear framework, we found that 3550 were linear while for the other 2887 (i.e., 44.8% of the 6437 trajectories) a second-order polynomial was a better fit (Figure 2). However, the proportion of non-linearity found was conditioned by the number of years sampled and the starting year of the time series used. In particular, the detected proportion of non-linearity increased with the number of years sampled and decreased with the starting year (see Appendices S3, S5, S6, Figure S3.2).

Among the 2887 non-linear trajectories, concave (curved downwards) and convex (curved upwards) cases were the most represented, accounting for almost half of all non-linear trajectories (48%, 1378 time series) (Figure 2E,H). Non-linear declines (Figure 2D,G) and non-linear increases (Figure 2F,I) were represented in similar proportions (around 26% each). Among the 6437 population trajectories, 1939 (i.e., 30%) were classified as declining of which 554 were decelerated, 215 accelerated and 1170 were linear (Figure 2A,D,G). 1783 (i.e., 28%) were classified as increasing of which 276 were decelerated, 464 accelerated and 1043 were linear (Figure 2C,F,I). Finally, 2715 (i.e., 42%) were neither declining or increasing, of which 757 were concave, 621 were convex and 1337 showed no trend (Figure 2B,E,H).

### 3.1.2 | Cross-Comparison of Linear and Non-Linear Classifications

Interestingly, when comparing the linear versus the non-linear classifications, we found that among the 2338 populations that were classified as decreasing according to the linear framework, around 50% were classified as linearly decreasing according to the non-linear framework (Figure 3A). The other 50% were classified either as (i) decreasing non-linearly (N=768, i.e., 33% of the 2338 classified as decreasing); (ii

concave trajectories (N=201, 9%), that is, trajectories showing a recent decrease after an increase; or (iii) convex trajectories (N=203, 9%), that is, trajectories showing a recent increase after a decrease (Figure 3A). Similarly, among the 2164 populations that were classified as increasing according to the linear framework, around 50% were classified as linearly increasing according to the non-linear framework (Figure 3C). The others were showing a mix of non-linear increases, concave and convex trajectories (Figure 3C). Among the 1935 populations classified as 'stable' by the linear framework, around 70% showed no trend according to the non-linear framework, the rest 30% experiencing mostly concave or convex trajectories (Figure 3B).

### 3.1.3 | Determining Factors of Non-Linear Trajectories in Population Dynamics

Figure 4 illustrates how non-linear trajectories are distributed in space. The GLMM showed that there are significant differences on the non-linear character of the population trajectories among regions ($\chi^2=38.47$, df=7, *p* < 0.001) (Figure 5B) and taxonomic groups ($\chi^2=17.59$, df=5, *p*=0.003) (Figure 5C). Habitat types ($\chi^2=5.50$, df=2, *p*=0.06) (Figure 5A), and IUCN Red List Categories ($\chi^2=7.75$, df=5, *p*=0.17) showed no strong differences (Figure 5D).

We did not, however, identify regions expressing particularly high or low proportion of non-linearity compared to the others (Figure 5B). Among taxonomic groups, mammals were the group expressing the highest proportion of non-linearity, in particular it was significantly higher than the proportion among birds (*p*=0.005), fishes (*p*=0.04) and sharks and rays (*p*=0.01) (Figure 5C). Contrary to our expectations, amphibians showed the lowest proportion of non-linear trajectories, however the confidence level of such result remains low (Figure 5C).

Beyond the variability in the probability of non-linearity, we observed variability in the relative proportions of the different trajectory types within different habitat types (see Appendix S5, Figure S5.1A), regions (Figure S5.1B), taxonomic groups (Figure S5.2) and IUCN Red List categories (Figure S5.3). For instance, the high proportion of non-linearity in mammals' populations was associated with a high amount of concave and convex trajectories, that is, non-linear no trend trajectories (29% of the 473 mammals populations analysed) and to non-linear increases (21%).

## 3.2 | Temporal Variability Among Global Population Dynamics

We incorporated the trajectory type as a potential determining factor of temporal variability of populations along with habitat types, geographical regions, taxonomic groups and IUCN Red List categories in a GLMM. This model performed better with the trajectory type as a determining factor than without (AIC=-21516.09 against -21037.59), and there were significant differences in the temporal variability among the trajectory types ($\chi^2=563.31$, df=8, *p* < 0.001). However, there was still an effect of the habitat
<figure><img src="image_5.png" /><figcaption>**FIGURE 3** | Cross-classification of population trajectories according to the use of linear or non-linear classifications reveals potential misinterpretations of population status. Cross-classification of the populations classified as (A) linear decrease, (B) no trend, and (C) linear increase according to the use of linear models. Within each column, the same populations are classified according to the non-linear models. For example (A), among the 2338 populations classified as 'decreasing' when fitting a linear regression, 1161 are still classified as 'linear decrease' when fitting second-order polynomials. 215 are classified as 'decrease accelerated', 553 as 'decrease decelerated', 201 as 'concave' and 203 as 'convex'.</figcaption></figure>

types ($\chi^2$=563.31, df=8, $p$<0.001), the regions ($\chi^2$=563.31, df=8, $p$<0.001) and the taxonomic groups ($\chi^2$=563.31, df=8, $p$<0.001) (Figure 5). These results were consistent when using other metrics as proxy of temporal variability (see Appendix S4, Table S4.1).

Terrestrial populations were significantly less variable than marine populations ($p$ < 0.001) and freshwater populations ($p$ < 0.001). Across regions, we identified that populations from North America were significantly less variable than populations from Oceania, South America, Europe, Asia and Africa (Figure 6B). Among taxonomic groups, fishes proved to be significantly more variable than all other vertebrate groups except amphibians (Figure 6C). We observed slight differences when using other metrics as proxies of temporal variability (see Appendix S4, Figure S4.1), however the overall conclusions were consistent. Terrestrial populations were less variable than freshwater populations no matter the metric used (Figure 6A, and see Appendix S4, Figure S4.1A,B), and fishes (MSE, Figure 6C) and sharks and rays (CV, D index, see Appendix S4, Figure S4.1E,F) were more variable than populations from other groups.

When assessing the relationship between trajectory types and temporal variability, we can't simply conclude that linear trajectories are consistently more variable than non-linear ones. For example, 'no trend linear' trajectories are notably more variable

than all types of increases and decreases, but they don't exceed the variability seen in concave and convex trajectories (see Appendix S6, Figure S6).

## 4 | Discussion

In this work, we studied non-linear trajectories and temporal variability in the dynamics of 6437 populations in the last 70 years across the globe, representing 1257 unique vertebrate species, and we tested how these two components of populations dynamics were distributed among habitats, regions, taxonomic groups and conservation status. Overall, our findings reveal that adopting linear approaches and ignoring temporal variability to characterise population dynamics could mask discrepancies among biogeographical regions or habitats and taxonomic groups.

### 4.1 | The Need for Integrating Non-Linearity and Temporal Variability in Conservation and Biodiversity Assessments

Overall, our results show that almost half of the trajectories were non-linear. Of the linear trends, results were balanced with approximately 30% declines, increases or no trend.
<figure><img src="image_6.png" /><figcaption>**FIGURE 4** | Non-linearity occurs across all locations within freshwater, marine and terrestrial habitats. Figure shows the locations of vertebrate populations used in the analyses. Points are coloured according to the non-linearity of their time series trajectories. Points' shape refer to the different habitat types (freshwater, marine or terrestrial).</figcaption></figure>

Non-linearity was expressed through a broad range of patterns, nearly half were convex or concave no change trends, and the remaining classified as accelerated and decelerated declines or increases with similar proportions. That means, of all populations, ~30% are declining, 30% are increasing, while 40% have no trend. This heterogeneity could be misinterpreted as an empirical evidence that global biodiversity is not doing bad on average. However, this reasoning would imply to neglect regional and species-specific dynamics. We rather show that declining, increasing and stable trends are being differentially expressed across regions and taxonomic groups. We also reveal mixed and heterogeneous biodiversity changes evenly distributed among linear and non-linear trajectories. The comparison of population classification adopting a linear versus non-linear framework reveals that the commonly-used linear approach may miss some key information from a conservation perspective. We show that almost 18% of populations showing linear decreasing trends are in fact exhibiting recent flat or reversing curves, which may reflect successful conservation strategies. Conversely, sharp decline cases, such as accelerated declines or concave trajectories, which may require urgent conservation interventions, can't be detected by the usual linear approach. Overall, our results emphasise the risk of simplistic approaches based on averaging heterogeneous and linear trends to capture a global biodiversity trend.

We have shown that population trajectories assessment, even when taking into account the non-linear component of trajectories, returns a balanced picture of increasing and decreasing trajectories, with a majority of no trend. This further reinforces the idea that such an assessment would benefit from being

complemented by a description of temporal variability, as a 'no trend' trajectory may hide variations putting populations at risk (Capdevila et al. 2022). Instead of relying on one-dimensional approaches of population trajectories for policy making, it is more likely that a cautious consideration of both non-linearity and temporal variability might well lead to improved large-scale biodiversity monitoring. For instance, our study reveals that a high proportion of populations following a linear trajectory with no trend over time exhibits considerable temporal variability. While we cannot entirely dismiss the possibility that this temporal variability may stem from data limitations, it is imperative to highlight that, from a conservation point of view, the presence of substantial temporal fluctuations in certain populations warrants even greater attention. Indeed, temporal variability is expected to be a signal of instability or even a signal preceding ecological collapse, being either at the population (Dai et al. 2012), community (Carpenter et al. 2011), or ecosystem (Scheffer et al. 2009) level. As a result, considering only population trajectories through their averaged shape is not sufficient to monitor vulnerable populations. Furthermore, our findings illustrate that temporal variability manifests differently across habitat types, geographical regions, and taxonomic groups, even when accounting for possible correlations among similar types of trajectories. This supports further our view that temporal variability should serve as a supplementary component for global biodiversity assessments by investigating how it is distributed among subsamples of interest. Of course, the measurement of temporal variability in future research will need some adjustments to be a variability metric that could be more easily interpreted across species and time series. We could also think of metrics that group pre-established measures of
<figure><img src="image_7.png" /><figcaption>**FIGURE 5** | Non-linearity in population dynamics is heterogeneous among geographical regions and taxonomic groups, but not among habitat types and IUCN Red List Categories. Each plot represents the estimated probability of a population from each individual factor to be non-linear. The black dots are the EMMs of each factor and the blue bars are 95% confidence intervals for the EMMs calculated from the GLMM. Letters indicate the significance of pairwise comparisons, calculated with post hoc tests after running the GLMM. The letters are not shown for the Habitat types and IUCN Red List Categories as there was no effect, thus none of the differences is significant. (EMMs Estimated Marginal Means, GLMM Generalised Linear Mixed Model).</figcaption></figure>

temporal variability, from which the main dimensions could be drawn, following what has been proposed in more theoretical studies of interaction networks, for example (Domínguez-García, Dakos, and Kéfi 2019). Still, the study of variability as proposed here suggests that this often neglected dimension of population change is worth considering for conservation biogeography issues.

**4.2 | Heterogeneous Population Dynamics at Local Scales Can Be Masked by Global Biogeographic or Taxonomic Groupings**

Our results show that there is a wide range of variability in population dynamics across biogeographic regions, testifying to the diversity of ways in which populations are likely to respond to global change. In particular, we show that marine

and freshwater populations are subject to greater fluctuations in abundance over time than terrestrial populations, and fishes populations are the most variable in abundance compared to other taxonomic groups. This complements previous findings showing that marine environments, with their higher sensitivity to climate change are expected to undergo more significant and rapid biodiversity changes compared to terrestrial ones (Halpern et al. 2015; Blowes et al. 2019; Antão et al. 2020).

However, apart from those global results, we show that all types of trajectories and magnitude of temporal variability are observed across diverse regions and groups. The broad groupings into major taxonomic groups and biogeographical regions on a global scale may not reflect the drivers shaping the trajectories of the specific populations we studied (Brook, Traill, and Bradshaw 2006; Pereira, Navarro, and Martins 2012), a warning that was also flagged by studies investigating changes
<figure><img src="image_8.png" /><figcaption>**FIGURE 6** | Temporal variability in population dynamics is heterogeneous among habitat types, geographical regions and taxonomic groups, but not among IUCN Red List Categories. Each plot represents the estimated marginal mean MSE within each individual factor. The black dots are the EMMs of each factor and the blue bars are 95% confidence intervals for the EMMs calculated from the GLMM. Letters indicate the significance of pairwise comparisons, calculated with post hoc tests after running the GLMM. The letters are not shown for the IUCN Red List Categories as there was no effect, thus none of the differences is significant. (EMMs Estimated Marginal Means, GLMM Generalised Linear Mixed Model).</figcaption></figure>

through linear analyses (e.g., Daskalova, Myers-Smith, and Godlee 2020).

Furthermore, while threatened species tend to be the focus of conservation initiatives (Martín-López, González, and Montes 2011), we show that at local scales there is extensive variation in how populations are changing over time, in isolation from their overall conservation status. In fact, we were not able to detect any differences among the different IUCN Red List Categories, neither in the prevalence in non-linearity nor in temporal variability. Species' IUCN Red List Categories are determined at species level hence at a global scale, meaning that IUCN status might be unrelated to the abundance trajectories and temporal variability of individual populations at local scales (Brito et al. 2010; Rueda-Cediel et al. 2018; Bachman et al. 2019; Fraixedas et al. 2022). This decoupling further highlights the heterogeneity in local-scale population trends which can generate strong debates about the conservation status for particular groups (e.g., for Storks [Gula et al. 2023a, 2023b]).

As a result, implementing global policies, such as the post-2020 Biodiversity Framework, necessitates to focus on the diversity of responses not only at the level of species and regions but also at finer scales such as that of populations. Of course, the temporal dynamics of a population' may also be strongly influenced by its intrinsic life-history traits, such as body size or fecundity (Bird et al. 2020; Albaladejo-Robles, Böhm, and Newbold 2023). Future research should focus on how populations' dynamics are shaped by interactions that arise between species' intrinsic traits and the extrinsic factors including threats to which they may be exposed.

## 4.3 | Major Weaknesses in Global Biodiversity Changes Assessments

Analysing time series data presents caveats that are present across population data in general, including taxonomic and geographic gaps (Cordier et al. 2021; Tekwa et al. 2023;
Johnson et al. 2024). The taxonomic signals must be taken with caution, as they might undoubtedly be impacted by the lack of long-term information on the least studied taxa. Indeed, only limited numbers of amphibian and reptile populations were available in our data selection, and no invertebrate population was recorded. The patterns we identify, either in the biogeographical or taxonomic distributions of the dynamics, are likely influenced by the overrepresentation of certain groups (Birds) and regions (Europe and North America) and do not extend to the overlooked parts, notably tropical populations which comprise the species that may be at the greatest risk of decline and extinction (Butchart et al. 2010; WWF 2022). The proportion of non-linearity detected could be influenced as well by how long populations are monitored (White 2019) and by the period at which monitoring began (Mihoub et al. 2017; Duchenne et al. 2022). Our findings demonstrate that non-linear models were more likely to be a better fit for time series with longer durations (see Appendix S3). Such result is consistent with previous studies showing that trend estimates could be improved by increasing time series length (Wauchope et al. 2019). As long-term data are accumulating worldwide, this stresses even more the importance of the non-linear assessment of population changes in future research.

## 5 | Conclusion

This paper supports the view that the ongoing biodiversity crisis is far more multifaceted than a global linear and homogeneous decline, but in no way refutes the existence of global biodiversity erosion. Our results do not challenge the fundamental value of the numerous studies of global biodiversity changes using linear approaches either. We rather claim that estimating non-linear trajectories and quantifying the temporal variability of population dynamics can improve and clarify existing findings. Not only such approach clarifies how populations decline and how they increase but also it refines our approach to population 'stability'. Indeed, very informative concave or convex trajectories, as well as populations showing no trend but highly variable abundance, would be poorly characterised as 'stable' linear trends from a conservation perspective. The general perception that a population is 'stable' or 'healthy' when its dynamics cannot be characterised by a linear decline is hazardous. Similarly, a misleading conclusion would be to over-interpret the results showing equal amounts of positive and negative trends as being synonymous of a global stability for biodiversity. Another important message of our findings is that although estimating global or regional trends in biodiversity is important, monitoring species locally, regardless of their global status, should be of primary concern. Relying on large-scale average cannot be an excuse for minimising awareness and actions regarding habitat destruction, over exploitation or any causes of biodiversity loss.

## Author Contributions

Maelys Boennec performed all initial analyses under R and wrote the first draft. Vasilis Dakos and Vincent Devictor equally interpreted the results together with Maelys Boennec and contributed to the rewriting of the main text.

## Conflicts of Interest

The authors declare no conflicts of interest.

## Data Availability Statement

Raw data are available from the following websites: for population time series—https://www.livingplanetindex.org/data_portal, for IUCN Red List Categories—https://www.iucnredlist.org. Code for data processing, analyses and vizualisation are publicly available on a GitHub repository (https://anonymous.4open.science/r/Non-linearity-and-variablity-vertebrate-populations-E358/README.md),https://github.com/MaelysBoennec/Non-linearity-and-variablity-vertebrate-populations.

## Peer Review

The peer review history for this article is available at https://www.webofscience.com/api/gateway/wos/peer-review/10.1111/ddi.13932.

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## Supporting Information

Additional supporting information can be found online in the Supporting Information section.