Uncovering broad macroecological patterns by comparing the shape of species’ distributions along environmental gradients

Abstract

Species’ distributions can take many different forms. For example, fat-tailed or skewed distributions are very common in nature, as these can naturally emerge as a result of individual variability and asymmetric environmental tolerances, respectively. Studying the basic shape of distributions can teach us a lot about the ways climatic processes and historical contingencies shape ecological communities. Yet we still lack a general understanding of how their shapes and properties compare to each other along gradients. Here, we use Bayesian non-linear models to quantify range shape properties in empirical plant distributions. With this approach, we are able to distil the shape of plant distributions and compare them along gradients and across species. Studying the relationship between distribution properties, we revealed the existence of broad macroecological patterns along environmental gradients—such as those expected from Rapoport’s rule and the abiotic stress limitation hypothesis. We also find that some aspects of the shape of observed ranges—such as kurtosis and skewness of the distributions—could be intrinsic properties of species or the result of their historical contexts. Overall, our modelling approach and results untangle the overall shape of plant distributions and provide a mapping of how this changes along environmental gradients.

Introduction

One of the central goals of ecology is to understand the processes structuring species distributions across space and time. Key to this goal is knowledge about the basic shape of distributions across environmental gradients; that is, in which environments species are most abundant or likely to occur, and how their abundance changes towards range edges. Ecologists have developed a plethora of statistical models to untangle the factors that play a role in defining species’ realized niches (Guisan and Zimmermann, 2000; Zurell et al., 2019). These models excel at predicting species’ geographic distributions (Ovaskainen et al., 2016), forecasting the effects of environmental disturbances such as climate change (Ehrlén and Morris, 2015), and identifying the key drivers of variation of such distributions (Engler et al., 2009). However, these models are less suitable for the study of some basic aspects of species’ distributions. One such aspect is how range shape changes along environmental gradients and across species. That is, we currently do a poor job at describing the underlying physical shape of species distributions and how this is determined by species’ biology and the environment.

The general shape of distributions along climatic or geographic axes is surprisingly unresolved. Ranges are typically caricatured as Gaussian (bell-shaped, or normal) curves, characterized by a mean position along an environmental gradient (range centre) and variance (determining range breadth). However, empirical data have shown that distributions can take many different forms (Austin, 1987, 2002; Huisman et al., 1993), and there is currently no general agreement on the basic shape of species’ realized niches (Michaelis and Diekmann, 2017; Sagarin and Gaines, 2002; Sagarin et al., 2006). Fat-tailed and skewed distributions are very common across scientific fields. The former naturally emerges as a result of processes involving intermittent (e.g. in email communications patterns; Malmgren et al. 2008) or stochastic events (e.g. in the spread of infectious diseases; Wong and Collins 2020). Indeed, species’ dispersal patterns have been shown to have fat tails due to natural variability among individuals (Petrovskii et al., 2009). Similarly, several processes can lead to skewed distributions. For example, species might have asymmetric environmental tolerances along one or multiple environmental gradients, allowing them to potentially withstand different temperature extremes (Chen and Lewis, 2022; Sunday et al., 2011). Species might also experience abiotic and biotic pressures that increase or decrease along a temperature gradient, which could result in species’ distributions presenting steeper declines towards warmer or colder environments (Normand et al., 2009).

Every aspect of the shape of distributions is important because it entails different and potentially competing hypotheses regarding the distribution of species (D’Amen et al., 2017; Maggini et al., 2011); and, comparing the shape of distributions along environmental axes and across species is interesting because it allows us to shed light on some such hypotheses. First, correlations between the shape of distributions and climatic or geographic gradients could reveal general macroecological patterns structuring ecological communities. For example, Rapoport’s rule is a classic biogeographical hypothesis that predicts species’ range size (i.e., variance) to increase with latitude or elevation (i.e. range centre) as a result of stronger climatic variability at the gradient extremes (Stevens, 1992). Similarly, the abiotic stress limitation hypothesis (Louthan et al., 2015) predicts a relationship between the center and skewness of distributions, reflecting the impact of environmental stressors such as drought or cold at one range edge and biotic pressures at the other. Both hypotheses hint at the existence of basic biogeographical constraints shaping the way species are distributed along gradients, providing insights into the way different species assemble and establish in different environments (Linder et al., 2000). Extending this rationale to examine correlations between other parameters describing range shape, such as kurtosis and skewness, could yield new insight into the forces shaping species’ distributions. Second, understanding the extent to which certain species show variation in range shape has important implications from a conservation and management perspective (Channell and Lomolino, 2000a; Stevens, 1992). Range shape has been linked to a species’ ability to respond to climatic changes (Bonachela et al., 2021) and shown to differ substantially across species (Ehnes et al., 2011; Lillie and Knowlton, 1897); therefore, understanding such differences could shed light on how climatic processes and historical contingencies singularly affect species’ realized niches (Helmuth et al., 2004; Rohde, 1992; Siefert et al., 2015). For example, the skewness or kurtosis of species distributions might reflect the effects of species’ climatic tolerances, but it could also capture the legacy of range shifts and spread, including species’ responses to climate change (Hurford et al., 2019; Patsiou et al., 2021) or their introduction status (Uden et al., 2015)—with those species expanding or retracting their niches towards opposite directions of an environmental gradient showing contrasting levels of skewness.

Uncovering and parsimoniously comparing the shape of the realized niche of many species along environmental gradients is nevertheless a challenging statistical problem. One way to address this is by using parametric models, which allow for a direct comparison of species distributions through the study of the corresponding parameter estimates. That said, despite the diversity of covariate transformations and link functions available across distribution models (Norberg et al., 2019), parametric structures often consider that species’ response curves are a function of a linear combination of different abiotic or biotic predictors (and transformations of and interactions between such predictors; Phillips et al. 2006). This structure is useful for maximizing our predictive ability and simplifying the optimization process, but it is not ideal when comparing the shape of species’ distributions along gradients and across species. First and foremost, a direct biological interpretation of parameter estimates in such models becomes increasingly difficult as one moves from unimodal and symmetric distributions (Jamil and ter Braak, 2013; ter Braak and Looman, 1986) to fat-tailed or skewed responses (Huisman et al., 1993), as any given parameter can only be interpreted in the context of the entire mathematical model. This limits our ability to understand the basic shape of distributions and its relationship with the species’ biology and environment, having to rely on post-hoc statistical tests (e.g. evaluation strip Elith et al. 2005) or indirect measures of such relationships (e.g. McCain and Knight 2013). Second, the lack of direct interpretation of the parameter estimates is also a limiting factor when accounting for any existing prior information of distributions. Empirical observations of the shape of species’ realized niches have long been documented by botanists, field ecologists (e.g. Landolt et al. 2010) and indigenous knowledge (Skroblin et al., 2021); information that could be critically accounted for in habitats lacking abundant occurrence data. However, this is rarely the case, likely because there is not a straightforward way to feed such empirical observations into the parameters of a linear model (e.g. requiring the use of community weighted means Scherrer and Guisan 2019; but see Ovaskainen et al. 2017; ter Braak and Looman 1986). Finally, there are a few non-linear model structures that have been proposed to characterize specific features of individual species’ response curves (Huisman et al., 1993); however, these are generally not designed to jointly study different species, and therefore taking full advantage of modern statistical approaches (e.g. sharing information among species or accounting for parameter uncertainty; Evans et al. 2016).

In this work, we develop a set of Bayesian hierarchical distribution models to study the shape of empirical plant distributions across whole communities. The main aim of these models is to understand the physical shape of distributions and provide a way to parsimoniously compare them along gradients and across species. We start by considering species’ response curves as Gaussian distributed, and then we adapt our model to allow non-linear responses characterizing skewed and fat-tailed distributions. This enables us to measure different properties of species’ realized niches while accounting for prior information regarding these distributions, including expert and indigenous knowledge of species’ environmental indicator values, range sizes, and plant ecological strategies. To highlight these ideas, we use a detailed dataset of the distribution of plant species along an elevation gradient in the Swiss Alps. Specifically, we ask (1) what are the typical range shapes within this regional flora, and how do range shapes change along climatic and geographic gradients? We examine correlations among the parameters describing range shape (e.g. mean position, variance, kurtosis and skewness) to ask (2) whether we can recover known macroecological patterns predicted by Rapoport’s rule (correlation of mean and variance) or the abiotic stress limitation hypothesis (correlation of mean and skewness), as well as other associations that might reflect novel macroecological patterns. Comparing such associations across species, we then ask (3) whether range shape differs between native species and those with a legacy of range expansion. Finally, we ask (4) whether or not the observed shape of distributions conforms to the idea that species are most abundant at the center of their ranges, putting to test a long-standing assumption that is central to classic modelling approaches.

Methods

Empirical data

We studied the distribution of plant communities in a mountain region of the Swiss Alps characterized by a strong elevation gradient. To do so, we combined two different datasets: i) one describing the co-occurrence of species across multiple open grasslands of this region (Dubuis et al., 2011; Randin et al., 2009), and ii) an extensive plant-attribute database containing environmental and life-history traits for all plant species across Switzerland (Landolt et al., 2010).

Distribution data

We used data describing the distribution of 798 species across 912 sites covering most of the mountain region of the Western Alps in the Canton de Vaud (Switzerland; Dubuis et al. 2011; Randin et al. 2009). Each of these sites is a (8 × 8) m² plot placed somewhere along an elevation range from 375 m to 3210 m. In all sites, presence/absence data as well as Braun-Blanquet abundance-dominance classes—i.e. an ordinal measure of plant abundance based on the percent cover in a given site (Vittoz and Guisan, 2007)—were recorded for all species.

Meteorological data provided by Scherrer and Guisan (2019) was used to define a climatic gradient to study species’ distributions. These data contain multiple variables characterizing the climate in each site at high spatial resolution (25 m). Based on 30 years (1961–1990) of records from national weather stations, this dataset was compiled using Zimmermann et al. (2007). Since most of the data are highly correlated, we used a Principal Component Analysis to calculate the main axes of variation of the following 9 scaled variables: daily minimum, maximum and average temperature; sum of growing degree-days above 5°C; mean temperature of wettest quarter; annual precipitation, precipitation seasonality, and precipitation of driest quarter (Supplementary Fig. S1). The first axis of variation is positively correlated with elevation and negatively correlated with temperature, while the second axis is positively correlated with precipitation seasonality (Supplementary Fig. S1). While many climatic or geographical gradients could be considered when studying species’ distributions, we focus on these two climatic gradients from this point on. Notice that any references to elevation in the results and figures are used only for illustrative purposes, and this variable was not used as a covariate in the model unless explicitly stated.

Plant attributes

To complement the aforementioned distribution data, we used an attribute database of around 5500 vascular plants across Switzerland. Some of the information in this database has been previously shown to account for unexplained variation when used as explanatory variables in species’ distribution models (Scherrer and Guisan, 2019). It was built based on expert knowledge and phytosociological field experience of botanists and ecologists, and contains information regarding plants’ environmental preferences and ecological strategies (Landolt et al., 2010). Notice that this sort of information represents prior knowledge that we have on plants’ differences, but that it is rarely accounted for in most distribution modelling approaches.

Species’ environmental preferences in this database are additional independent data that can be used to inform distribution models—e.g. as prior information in a Bayesian framework. These are characterized following the ecological indicator values developed by Landolt et al. (2010), providing both an estimate of the average conditions in which a species can be found as well as a broad description of their range of variation. These values are provided for a range of 8 environmental variables, including temperature, continentality, light conditions, as well as moisture, acidity and nutrient content of the soil (see a full list and description of the ecological indicators in the Supplementary Table S1; Landolt et al. 2010). In addition to species’ environmental preferences, the attribute data also contain information on species’ introduction status, classifying species based on whether these are indigenous, archaeophytes or neophytes (Landolt et al., 2010). For simplicity, we consider those species that spread into the region before and after 1500 AD as historical and recent range expanders, respectively, and all other species as indigenous. Finally, in the attribute database, one can also find information regarding species’ change tendency, indicating species that have shown decline or increase in their populations over the last five decades. We summarize this information in more detail in Supplementary Table S1.

Baseline model

There is a long list of model structures well suited to characterizing species’ distributions (see Norberg et al. 2019). As a baseline model, however, we were interested in a hierarchical model—a multi-level model to jointly study all species together—that does not make any assumptions regarding the shape of the distributions, and yet explicitly incorporates all information that we have regarding plant’s environmental preferences. More specifically, we wanted to account not only for plant distribution data, but also for information provided by the climatic indicator values and range of variation registered in the attribute database for all plants in our dataset. These two values provide basic information regarding a plants’ optimal environmental conditions and the width of its distributions.

Response curve

To choose an appropriate response curve, we first need to agree on what we truly know about the system. Given the prior information that we have about the system, we know that species occupy specific geographic ranges; therefore, we know that their distributions have finite variance. While we could also assume that many other factors might influence a species’ presence in a given site—e.g. the biotic interactions among species in the site—we do not necessarily have an a priori expectation of how exactly these factors will influence the shape of its distributions. Therefore, for this baseline model, we choose the maximum entropy distribution with finite variance: a Gaussian distribution (Supplementary Fig S2a). That is, given the presence/absence or abundance y_ij of any species i in any given site j, and an environmental variable x_j, we can define species’ responses to the environment as

$$\begin{array}{c}{y}_{ij}\sim F\left(p_{ij}\right)\\ \log \left(p_{ij}\right)=-{\alpha }_i-{\gamma }_i^{\prime }{\left(x_j-{\beta }_i\right)}^2,\end{array}$$ (1)

where F is the likelihood function, ${\gamma }_i^{\prime }={\gamma }_i/2$, and α_i, β_i, and ${\gamma }_i^{-1}$ describe the amplitude of the probability p_ij, species’ average climatic suitability and range of variation along the environmental gradient, respectively. Notice that F characterizes a Binomial distribution when considering binary data, and it characterizes an ordered categorical likelihood function when we consider Braun-Blanquet abundance-dominance classes as response variables (see the full description of both models in the Supplementary Methods; Guisan and Harrell 2000). For the sake of simplicity, we use only one environmental variable to characterize the species’ probability distribution. That said, this model can easily be generalized to account for multiple predictors (see Supplementary Methods).

Model priors

The model structure described above allows us to explicitly incorporate all prior knowledge that we have regarding species’ distributions contained in the attribute database. To do so, we define the prior distributions for the parameters in model (1) as:

$$\begin{array}{cc}\hfill \log \left({\alpha }_i\right)& \sim \text{Normal}\left(\widehat{\alpha },{\sigma }_{\alpha }\right)\hfill \\ \hfill {\beta }_i& \sim \text{MVNormal}\left(\widehat{\beta },{\text{Σ}}^{\beta }\right)\hfill \\ \hfill \log \left({\gamma }_i\right)& \sim \text{MVNormal}\left(\widehat{\gamma },{\text{Σ}}^{\gamma }\right)\hfill \\ \hfill \widehat{\alpha },\widehat{\beta },\widehat{\gamma }& \sim \text{Normal}(0,1)\hfill \\ \hfill {\sigma }_{\alpha }& \sim \text{Exponential}(1)\hfill \end{array}$$ (2)

where parameters γ_i and β_i are expressed as multivariate normal distributions—i.e. Gaussian processes—such that Σ^β and Σ^γ are variance-covariance matrices describing species’ similarity in terms of their average climatic suitability and range of variation along the different environmental gradients, respectively. We define these variance-covariance matrices as follows:

$${\text{Σ}}_{ij}=\eta \exp \left(-\rho D_{ij}^2\right)+{\delta }_{ij}\sigma ,$$ (3)

where Σ_ij characterizes the covariance between any pair of species i and j, and δ_ij is the Kronecker delta. Such a covariance structure declines exponentially with the square of a distance matrix D_ij, which characterizes differences between species computed using our prior information. In the attribute database, this information is represented by the set of ordinal traits specified for the different species. While there are many different ways to turn ordinal data into distance matrices, we choose to use a mixed-membership stochastic block model because it allows us to deal with cases of missing data (see Supplementary Methods for extended details; Godoy-Lorite et al. 2016). In each covariance matrix, the hyperparameter ρ determines the rate of decline of the covariance between any two species, and η defines its maximum value. The hyperparameter σ describes the additional covariance between the different observations for any given species. For all these hyperparameters, we choose weakly informative priors such that

$$\begin{array}{cc}\hfill \sigma ,\eta & \sim \text{Exponential}(1)\hfill \\ \hfill \rho & \sim \text{Exponential}(0.5).\hfill \end{array}$$ (4)

Notice that other structures can be used to define the covariance matrices of the different Gaussian processes (McElreath, 2020), including structures that account for multiple distance matrix D_ij for any given parameter (Supplementary Methods).

Sampling the posterior

We generated the posterior samples for the Bayesian models with the Hamiltonian Monte Carlo algorithm implementation provided by the R packages ‘rstan’ and ‘cmdstanr’ (Stan Developent Team, 2021). Sampling models like the ones described above can be computationally very intensive. This is especially true when using ordered categorical likelihood functions (see Stan Development Team 2021). Therefore, we focus on those species for which we have at least 20 occurrences when modelling both binary data and ordinal data (251 species in total), excluding any rare species (Michaelis and Diekmann, 2017).

To test the performance of the model as well as our choice of prior distributions, we modelled simulated data and compared the sampled posterior distributions to the data-generating parameters (e.g. Supplementary Fig. S3; see Code Availability section). Notice that using the link function in Eq. (1) could cause problems when sampling the model, and some adjustments need to be made when specifying the model (see Code Availability section). To perform the data analysis and generate the figures, we used some of the functions available with the R package ‘rethinking’ (McElreath, 2020). A discussion on the effect of unequal sampling and prior information can be found in Supplementary Notes S1–S3.

Modifying the baseline model

We proposed a baseline model that is naive regarding how the data is distributed, and yet accounts for all prior information that we have about the system. Now, we want to modify this model to test the extent to which empirical species’ distributions showcase different shapes. We focused on two properties: fat-tailed and skewed responses. While there are several model structures that could account for these properties, we propose new species’ response curves following three criteria. First, the probability distribution of a species along an environmental gradient must have a defined mean and variance. This is important because we know that species naturally have different environmental preferences as well as finite geographic ranges. Second, the Gaussian shape must be a special case of the probability distribution, allowing species to showcase variation regarding the presence (or lack thereof) of any given pattern. Finally, there must be a re-parametrization of the model that allows us to keep the same prior information and interpretable parameters.

Fat-tailed response curve

Fat-tailed distributions represent populations with relatively high representation of individuals at the extremes. While many different distributions exhibit this property, we decided to accommodate this feature into our baseline model by considering a response curve that follows a generalized error distribution. Such a distribution is useful because the Gaussian shape is a special case of it, and it contains a parameter that regulates the level of kurtosis—ranging from longer to shorter tails than the Gaussian case (Supplementary Fig S2b). In particular, we can adapt Eq. (1) to present this non-linear form as follows:

$$\log \left(p_{ij}\right)=-{\alpha }_i-{({\gamma }_i^{\prime }\left|x_j-{\beta }_i\right|)}^{v_i},$$ (5)

where ${\gamma }_i^{\prime }=g\left({\gamma }_i,v_i\right)$ describes the relationship between the parameters controlling for the variance γ_i and kurtosis ν_i ∈ (1, ∞) of distributions. Following this, we choose an adaptive prior for this set of new parameters such that

$$\begin{array}{cc}\hfill \log \left(v_i-1\right)& \sim \text{Normal}\left(\widehat{v},{\sigma }_v\right)\hfill \\ \hfill \widehat{v}& \sim \text{Normal}(0,1)\hfill \\ \hfill {\sigma }_v& \sim \text{Exponential}(2).\hfill \end{array}$$ (6)

Given the relationship between ${\gamma }_i^{\prime }$ and γ_i, we can re-parametrize the model and follow Eq. (2) to define the prior distributions (see Supplementary Table S2; Nadarajah 2005). Notice that the Gaussian distribution will naturally emerge when ν_i = 2. For a generalized error distribution, the level of kurtosis can be directly computed using ν_i (Supplementary Methods; Kerman and McDonald 2013).

Alternatively, we could have used other distributions that present fat tails and fulfil the selection criteria described above. For example, the non-standardized Student’s t-distributions is an interesting distribution because, as opposed to the generalized error distribution, it allows for fat tails without generating a cusp at the center (see Supplementary Fig S2b). However, we avoided using the non-standardized Student’s t-distributions because it does not allow for tails that are lighter than normal (e.g. ν_i > 2 in Eq. 5; Supplementary Fig S2b), and the sampling of the model can be somewhat more challenging.

Skewed response curve

As opposed to bell-shaped, skewed responses characterize populations that present steeper declines towards either side of the distribution. One way to accommodate this feature in our models is by considering a skewed normal distribution. As for the case described above, the Gaussian is a special case of this distribution, and it contains a parameter that controls for the level and direction of ‘skewness’ (Supplementary Fig S2c). Importantly, this distribution presents normal-like tails; therefore, the added skewness does not make additional assumptions regarding how species are distributed along the gradient. To test for the existence of this feature, we modified Eq. (1) as

$$\log \left(p_{ij}\right)=-{\alpha }_i-{\gamma }_i^{\prime }{\left(\frac{x_j-{\beta }_i^{\prime }}{1+{\lambda }_i\mathrm{sgn}\left(x_j-{\beta }_i^{\prime }\right)}\right)}^2,$$ (7)

where ${\gamma }_i^{\prime }=q_1\left({\gamma }_i,{\lambda }_i\right)$ and ${\beta }_i^{\prime }=q_2\left({\gamma }_i,{\beta }_i,{\lambda }_i\right)$ describe the relationship between the parameters controlling for the variance γ_i, mean β_i and skewness λ_i ∈ (−1, 1) of distributions. The function sgn (x) characterizes the sign function. We chose λ_i to have an adaptive prior such that

$$\begin{array}{cc}\hfill \text{logit}\left(\frac{{\lambda }_i+1}{2}\right)& \sim \text{Normal}\left(\widehat{\lambda },{\sigma }_{\lambda }\right)\hfill \\ \hfill \widehat{\lambda }& \sim \text{Normal}(0,1)\hfill \\ \hfill {\sigma }_{\lambda }& \sim \text{Exponential}(1).\hfill \end{array}$$ (8)

Notice that this model can be re-parametrized following q₁ and q₂, allowing us to set the rest of the prior distributions as described for the baseline model (see Supplementary Table S2; Code Availability section). In this case, the Gaussian distribution is a special case of Eq. (7) when λ_i = 0 (Ashour and Abdel-hameed, 2010). For a skewed normal distribution, the level of skewness can be directly computed using λ_i (Supplementary Methods; Kerman and McDonald 2013).

Fat-tailed and skewed response curve

Finally, one could consider a response curve with both kurtosis and skewness. A convenient way to achieve this is by using a response curve that follows a skewed generalized error distribution. This is a combination of the two distributions described above, containing two parameters that control for both the level and direction of kurtosis and skewness (Supplementary Fig S2d). The skewed generalized error distribution can be considered by modifying the species’ response curve in Eq. (1) as

$$\log \left(p_{ij}\right)=-{\alpha }_i-{\left(\frac{{\gamma }_i^{\prime }\left|x_j-{\beta }_i^{\prime }\right|}{1+{\lambda }_i\mathrm{sgn}\left(x_j-{\beta }_i^{\prime }\right)}\right)}^{v_i},$$ (9)

where ${\gamma }_i^{\prime }=f_1\left({\gamma }_i,v_i,{\lambda }_i\right)$ and ${\beta }_i^{\prime }=f_2\left({\gamma }_i,{\beta }_i,v_i,{\lambda }_i\right)$ describe the relationship between the parameters controlling for the variance γ_i, mean β_i, skewness λ_i and kurtosis ν_i of distributions (Fig. 1). We define ν_i, λ_i and their prior distributions as in Eq. (6) and (8), respectively. Again, we can re-parametrize the model following f₁ and f₂, and set the rest of the prior distributions as in the baseline model (see Supplementary Table S2; Code Availability section). Notice that the generalized error distribution (i.e. Eq. 5) and the skew normal distribution (i.e. Eq. 7) are special cases of Eq. (9) when λ_i = 0 and ν_i = 2, respectively. For a skewed generalized error distribution, the level of skewness and kurtosis can both be directly computed using λ_i and ν_i (Supplementary Methods; Kerman and McDonald 2013).

Figure 1.

Parameters controlling the different properties of distributions. For each property, the table presents an example of a distribution where all parameters are fixed except for the one controlling the property in question.

Evaluating the log-likelihood

The log-likelihood values measure the goodness of fit of a statistical model to any data point, for a given sample of the posterior distributions. These values can be used to understand where the models fail to capture the variation of our empirical data. For example, high log-likelihood values indicate those data points that are well captured by a given distribution model, while low values signal those points that are instead unexpected by the model. Therefore, one could use these values to understand what aspects of the shape of distributions are missing. To do so, for every sample of the model, we computed the log-likelihood values and the normalized probability distribution. This normalized probability is defined such that its maximum is set to 1 for all species in our dataset. In particular, for a heavy-tailed and skewed response, the normalized probability distribution was calculated for every sample of the Bayesian model using Eq. (9), where α_i was set to 0 for any value of x_j. Notice that the normalized probability distribution is interesting when comparing the log-likelihood values across species because it can be used to understand whether the model errors are at the tails of the distributions or their center.

Results

The mean-variance relationship

We studied the distribution data to characterize species’ realized niches along the main axis of variation of all environmental variables. Using the presence and absence of species across sites as the response variable, we sampled the posterior distributions of the baseline model, accounting for the information in the attribute database regarding species’ indicator values and range of variation. This allowed us to map the center and variance of species’ distributions along the environmental gradient (Fig. 2). Studying the relationship between these properties, we found these to be negatively correlated (i.e. β_i and γ_i in the baseline model were positively correlated; Fig. 2). This means that species found at the lower end of the gradient (i.e. higher temperature and lower elevation) have generally wider distributions than those found at the higher end (i.e. lower temperature and higher elevation). The same relationship was found when independently using elevation or mean temperature as explanatory variables (Supplementary Fig. S4) as well as when using Braun-Blanquet abundance classes (Supplementary Fig. S5); however, the pattern was not present along the second main axis of variation of our environmental variables (i.e. correlated to precipitation seasonality; Supplementary Figs. S1 and S6).

Figure 2.

Relationship between β_i and γ_i from Eq. (1) across species. Panel (a) describes the relationship between the mean (β_i) and variance $\left({\gamma }_i^{-1}\right)$ of distributions. Each point represents the average value of the corresponding posterior distributions for any given species. The value in the bottom-left corner displays the Pearson’s correlation coefficient between parameters across all model samples. Panel (b) and (c) display the estimates for the center (sorted along the x axis) and variance (sorted along the y axis) of species’ distributions, respectively. The points and lines represent the mean and 89% confidence intervals (McElreath, 2020) of the posterior distributions.

The comparison between the other parameter estimates revealed additional, somewhat more expected, relationships. In particular, we found the amplitude of distributions (i.e. height of the probability distributions) to be positively and negatively correlated with their variance and mean, respectively (i.e. α_i is positively correlated with γ_i and β_i; Supplementary Fig. S7). This implies that, at the higher end of the gradient (i.e. lower temperature and higher elevation), species’ distributions generally have lower amplitudes (i.e. lower overall probability of occurrence, indicating rare species in the dataset). A study of how the prior knowledge we had regarding species’ environmental preferences and range of variation helped us estimate the different parameters of the model is presented in the Supplementary Note S4 and Supplementary Fig. S8, where we describe and interpret the model hyperparameters.

The skewness and kurtosis of distributions

Maintaining the symmetry of species’ distributions, we then allowed the kurtosis—or shape of the tails—of these to vary in different ways. To do so, we changed the response curve of our Bayesian model to follow a generalized error distribution (i.e. Eq. 5). A comparison of the WAIC values showed this non-linear regression to outperform the baseline model (Supplementary Fig. S9). Studying the resulting posterior distributions, we found the average excess kurtosis of the distributions to be slightly greater than zero, which corresponds to distributions with longer tails than the Gaussian case (Fig. 3). However, the parameter controlling for the kurtosis ν_i displayed a lot of variation among species (Supplementary Fig. S10), which might indicate that the shape of the tails is species-specific and potentially explained by species’ ecological traits.

Figure 3.

Average excess kurtosis and skewness of species’ distributions. Panel (a) displays the posterior distributions for the average excess kurtosis obtained using a generalized error distribution (black line) and a skewed generalized error distribution (yellow line), ranging from distributions with “short tails” and “long tails”. Panel (b) displays the posterior distributions for the average skewness obtained using a skewed normal distribution (black line) and a skewed generalized error distribution (yellow line), ranging from distributions that are skewed “towards low” and “towards high” elevations. The gray dotted lines indicate the conditions by which species are normally distributed along the environmental axis.

Using Eq. (7), we next studied the skewness of species’ distributions. Based on the estimates for the WAIC values, this model outperformed the first two (Supplementary Fig. S9), which sheds light on the naturally skewed nature of species’ distributions. Perhaps most importantly, studying the mean value of the skewness across species, we found this to be consistently below zero (Fig. 3). This indicates that species’ distributions generally present steeper declines towards colder environments (i.e. $\widehat{\lambda }<0$), although there is variation among species (Supplementary Fig. S11). The same was true when using a model that allowed for both fat-tailed and skewed response curves (Eq. 9). This model outperformed the rest, presenting Akaike weights close to 1 (Supplementary Fig. S9), suggesting that both the kurtosis and skewness are useful properties to describe empirical distributions (Fig. 3). Notice that these properties are also not fully independent, as the consideration of both fat-tails and skewed responses substantially changes the results obtained using the previous two models (i.e. Eqs. 5 and 7; Fig. 3). Importantly, the projection of different distributions onto a map highlights that small changes in ν_i and λ_i can lead to significant changes over gradients (Supplementary Figs. S12 and S13).

The shape along gradients and across species

The model characterizing fat-tailed and skewed distributions allowed us to study the posterior distributions for the parameters describing the mean, variance, amplitude, kurtosis and skewness of species’ realized niches, simultaneously. We observed that different types of species seem to present characteristically different distributions (Fig. 4 and Supplementary Fig. S14). Focussing on the negative correlation between the mean and variance of species’ distributions, we found some species to escape such constraints, as those correlations are not detectable (Supplementary Fig. S15). Moreover, focussing on recent and historical range expanders (Supplementary Table S1), we often found these species to be at the lower end of the environmental axis (i.e. higher temperature and lower elevation), presenting higher amplitudes, and distributions that appear to showcase steeper declines towards warmer sites (Fig. 4)—potentially showing these species to spread towards the higher elevations. Notice the nature of these results does not depend on the presence or absence of a species at the edge of the sampling area, as the same model produced comparable results when using simulated and subsampled data (Supplementary Note S3 and Supplementary Fig. S16). Moreover, these results did not substantially change when using ordinal data (Supplementary Fig. S17).

Figure 4.

Relationship between distribution properties across indigenous, historical range expanding and recent range expanding species. The main panels (a-d) characterize the relationship between the mean (from “low elevation” to “high elevation”), and the variance (from “narrow” to “wide”), amplitude (from “low” to “high”), skewness (from skewed “towards low” to “towards high” elevations) and excess kurtosis (from “short tails” to “long tails”) of the species’ distributions. The points describe average value across all samples of the model, and the values in the bottom corners display the Pearson’s correlation coefficient between parameters. The margin plots compare the distribution of mean values for the different groups of species, where confidence intervals are calculated aggregating the 89% confidence intervals for each species.

The missing piece

Finally, we wanted to identify what aspects of the shape of distributions were still missing. We used the computed log-likelihood values and normalized probabilities to understand where our best performing model—the one with fat-tailed and skewed response curve—fails to capture the variation in empirical plant distributions. We found most data points to be located at the tails of distributions (normalized probability ≈ 0) and to present high log-likelihood values (Supplementary Fig. S18). This is not surprising as the study area spans an extensive elevation gradient, and species’ distributions are generally narrow relative to it; therefore, the model accurately predicts that species are usually absent in those sampling sites that fall relatively far from the peak of their distributions. However, studying instead only those points for which the model did not perform well (with a likelihood ≤ 0.5), we found these to generally be associated with high normalized probabilities (Fig. 5). This indicates that the unexplained variation is often found at the peak of species’ distributions. Similar results were found when using ordinal data (Supplementary Fig. S19). This is important because it potentially suggests that species are not always more likely to be found at the center of their distributions. The underlying shape assumed by the model—a distribution with the highest probability values at the center—might fail to capture the variation of the data along our environmental gradient.

Figure 5.

Mapping of the log-likelihood and normalized probability values for all species across all samples. The shading characterizes the overall percentage of points falling within a given hexagon. For any given species, a normalized probability close to zero corresponds to data points found at the edge of its realized niche, while higher normalized probabilities describe those points found at the center of its realized niche. Notice that there are only displayed those points that present a likelihood smaller than 0.5, indicating points where the model did not perform well. The mapping of all log-likelihood values is presented in Supplementary Fig. S18.

Discussion

In this work, we use a set of Bayesian non-linear models to characterize the shape of species’ distributions and range shape changes along gradients and across species. We found both the amplitude and variance of distributions to be negatively correlated with their mean, and we found species’ distributions to generally be characterized by fat tails and steeper declines towards higher elevations (i.e. λ < 0 and ν < 2 in Fig. 1). Moreover, some species might present different distribution patterns. For example, rapid and historical range expanders seem to be more prevalent in warmer environments and have distributions with relatively higher amplitudes. Studying the variation in the data that remained unexplained, we found this to be generally located at the peak of distributions, identifying potential general properties of empirical distributions that were missed by our model. Putting this all together, our results uncovered several aspects of the shape of empirical plant distributions and revealed differences in the way species—such as indigenous and range expanding species—are assembled along environmental gradients.

Our approach allowed us to parsimoniously compare the shape of the species’ realized niches along an altitude gradient, which proved difficult using traditional niche and distribution models, often requiring post-hoc estimations of species response curves (Elith et al., 2005). With this comparison, we found correlations across the different distribution properties measured by the model (i.e. mean, variance, amplitude, skewness, and kurtosis; Fig.4). These correlations could suggest the existence of macroecological constraints that dictate the way different species assemble across environments, where the different properties of species’ distributions are not independent from each other. Some of these correlations are somewhat expected, such as the negative correlation between a species’ position along the gradient (i.e. the mean) and the over-all height of its probability distribution (i.e. the amplitude), as plant species might be limited to sparse habitats at higher elevations. Other correlations, however, are direct tests of classic macroecological hypotheses. For example, the Rapoport’s rule predicts wider ranges of species at higher latitudes and altitudes (Stevens, 1992); and therefore, one would expect a positive correlation between the mean and variance of species distributions. A common explanation for the Rapoport’s rule is that climatic variability selects for species with greater climatic tolerances. But while this pattern has been largely studied for multiple systems and across gradients (McCain and Knight, 2013), contrasting evidence suggests that this rule is not pervasive across species (Bhattarai and Vetaas, 2006; McCain and Knight, 2013; Ribas and Schoereder, 2006). A potential explanation for such uncertainty is that, in contrast to our approach, tests for such a pattern often rely on the prior definition of metrics for range size, and they are often post-hoc tests of the relationship between such metrics and elevation (McCain and Knight, 2013). Indeed, our results seem to contradict the predictions of the Rapoport’s rule in our dataset, as we observed a negative correlation between species’ range width and elevation. That is, plant species seems to generally present narrower distributions at higher elevations. Our results also suggest that species such as recent and historical range expanders might not obey this same constraints, as these showcase variability in the relationship between the mean, variance and amplitude of distributions (Supplementary Fig. S15). With that being said, we should interpret these results with caution, as correlations do not imply an underlying biological mechanism, and these patterns simply represent descriptions of the nature of the data studied here.

The level of skewness of species’ distribution as well as the variability in the shape of their tails diverged from traditionally assumed bell-shaped curves. This allowed us to focus on other interesting macroecological hypotheses. For instance, the abiotic stress limitation hypothesis predicts species’ distributions to present steeper declines towards conditions that are generally limiting for plant growth (Austin, 1990). Normand et al. (2009) and Dvorský et al. (2017) tested this for vegetation data using the statistical models by Huisman et al. (1993) for several independent species, finding that a large minority—around 19% of species along an altitude gradient—support the hypothesis (but see Ziffer-Berger et al. 2014). Our results, however, showcased species’ distributions to generally present steeper declines towards higher elevations, providing clear evidence of this geographical pattern. Moreover, we were able to highlight the degree to which different species might present varying levels of decline towards stressful conditions. Notably, plants found at low elevations—including recent and historical range expanding species—displayed contrasting levels of skewness. While it is important to acknowledge that these differences between species might be tenuous due to the limited number of range expanding species and their restricted elevational range, our findings provide glimpses of the different stages of species’ assembly processes. That is, plants found at lower elevations can have distributions that are positively skewed, potentially spreading towards higher elevations as a result of abiotic and biotic stress induced by changes in climate (Fig. 4).

There are many other properties characterizing empirical distributions that might not have been captured by the different models presented here. One possible way to untangle these properties is by studying the unexplained variation in the empirical data. We observed that this variation is often located at the peak of distributions, which suggests that the aspects of their shape not picked up by the models involve those points at the peak of the distributions. This observation is directly linked to another macroecological pattern: the so called abundant-center hypothesis (Sagarin and Gaines, 2002). This hypothesis predicts species to be most abundant at the center of their distributions, and it is an implicit assumption at the core of most modelling approaches. Namely, if one is only willing to assume that species have finite geographic ranges, the abundant-center hypothesis is a consequence of our state of ignorance (i.e. the maximum entropy distribution). That said, several studies have pointed out that the abundant-center hypothesis is not pervasive in empirical distributions (Chevalier et al., 2021; Dallas et al., 2017; Pironon et al., 2017; Wagner et al., 2011), suggesting that population abundance could often be more strongly driven by interactions and community structure than the environment (Dallas et al., 2017). Our results, for both binary and ordinal data, support these observations, suggesting that species’ probability of appearance—as well as likelihood of presenting high abundance at a given site—might not ubiquitously be highest at the center of their distributions. While there are other abiotic and biotic factors that could play a role in defining the shape of distributions that were not accounted for in our study, the models’ inability to explain some of the points at the peak of distributions across species suggests that their realized niches might not always follow the traditionally pictured bell-shaped distributions. Allowing species to showcase other distribution shapes, such as truncated, multimodal or plateaued distributions, could potentially resolve some of the unexplained variation. The prevalence of multimodal distributions, in particular, is an interesting perspective, as this hypothesis could be empirically tested using direct extensions to the approaches used here (Chakraborty et al., 2015). Indeed, studying the tails of species’ distributions, we observed several species presenting low kurtosis levels. While this implied that these distributions had shorter tails than normal, it could also reflect multimodal or plateau-shaped response curves.

The different hypotheses regarding the shape of species’ distributions address central topics in ecology and evolution (Sagarin and Gaines, 2002). Distributions are the result of environmental variability (Butterfield, 2015; Helmuth et al., 2002), biotic interactions (Hastings et al., 1997; Wisz et al., 2013) and historical contingencies (Frick et al., 2010), and their shape determines gene flow (Haldane and Ford, 1956; Lesica and Allendorf, 1995; Pironon et al., 2017) and energy balances along gradients (Hall et al., 1992). Perhaps most importantly, the shape of species’ distributions will influence their responses to environmental changes (Channell and Lomolino, 2000a), and it could therefore be used as an ecological compass to inform conservation and management decisions (Channell and Lomolino, 2000b; Maggini et al., 2011). In this context, we identify two areas we feel represent key steps from which to move forward. First, trait data could crucially inform the different parameters controlling the shape of distributions (Scherrer and Guisan, 2019). For example, if the skewness of species’ distributions is the result of uneven environmental tolerances along the gradient (Sunday et al., 2011), this information should be accounted for analogously to the way we used the expert knowledge on plants’ environmental preferences. The same is true for species’ ecological strategies, with aspects regarding their competitive ability potentially informing the shape of distributions (Heegaard and Vandvik, 2004). Second, our models have clear interpretable parameters, and can be used to directly compare the shape of species’ realized niches. These comparisons could be used to generate hypotheses regarding where and when different species might strongly interact with one another along an environmental gradient (Louthan et al., 2015), making ecologically-informed predictions regarding the presence and absence of these relationships (Callaway et al., 2002; He et al., 2013).

Data and Code Availability

The primary data associated with this manuscript is publicly available (see ‘Empirical data’ section of the Methods). The method presented here is available online (Bramon Mora, 2023).