Philippa Kaur

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To calculate total root biomass C colonized by AM and EcM fungi, we developed a workflow that combines multiple publicly available datasets to ultimately link fine root stocks to mycorrhizal colonization estimates (Fig. 1). These estimates were individually derived for 881 different spatial units that were constructed by combining 28 different ecoregions, 15 land cover types and six continents. In a given spatial unit, the relationship between the proportion of AM- and EcM-plants aboveground biomass and the proportion of AM- and EcM-associated root biomass depends on the prevalence of distinct growth forms. Therefore, to increase the accuracy of our estimates, calculations were made separately for woody and herbaceous vegetation and combined in the final step and subsequently mapped. Below we detail the specific methodologies we followed within the workflow and the main assumptions and uncertainties associated.Fig. 1Workflow used to create maps of mycorrhizal fine root biomass carbon. The workflow consists of two main steps: (1) Estimation of total fine root stock capable to form mycorrhizal associations with AM and EcM fungi and (2) estimation of the proportion of fine roots colonized by AM and EcM fungi.Full size imageDefinition of spatial unitsAs a basis for mapping mycorrhizal root abundances at a global scale, we defined spatial units based on a coarse division of Bailey’s ecoregions23 After removing regions of permanent ice and water bodies, we included 28 ecoregions defined according to differences in climatic regimes and elevation (deposited at Dryad-Table S1). A map of Bailey’s ecoregions was provided by the Oak Ridge National Laboratory Distributed Active Archive Center24 at 10 arcmin spatial resolution. Due to potential considerable differences in plant species identities, ecoregions that extended across multiple continents were split for each continent. The continent division was based upon the FAO Global Administrative Unit Layers (http://www.fao.org/geonetwork/srv/en/). Finally, each ecoregion-continent combination was further divided according to differences in land cover types using the 2015 Land Cover Initiative map developed by the European Space Agency at 300 m spatial resolution (https://www.esa-landcover-cci.org/). To ensure reliability, non-natural areas (croplands and urban areas), bare areas and water bodies were discarded (Table 1). In summary, a combination of 28 ecoregions, 15 land cover types and six continents were combined to define a total of 881 different spatial units (deposited at Dryad-Table S2). The use of ecoregion/land cover/continent combination provided a much greater resolution than using a traditional biome classification and allowed to account for human-driven transformations of vegetation, the latter based on the land cover data.Table 1 List of land cover categories within the ESA CCI Land Cover dataset, used to assemble maps of mycorrhizal root biomass.Full size tableMycorrhizal fine root stocksTotal root C stocksEstimation of the total root C stock in each of the spatial units was obtained from the harmonized belowground biomass C density maps of Spawn et al.20. These maps are based on continental-to-global scale remote sensing data of aboveground biomass C density and land cover-specific root-to-shoot relationships to generate matching belowground biomass C maps. This product is the best up-to-date estimation of live root stock available. For subsequent steps in our workflow, we distinguished woody and herbaceous belowground biomass C as provided by Spawn et al.20. As the tundra belowground biomass C map was provided without growth form distinction, it was assessed following a slightly different workflow (see Section 2.2.3 for more details). To match the resolution of other input maps in the workflow, all three belowground biomass C maps were scaled up from the original spatial resolution of 10-arc seconds (approximately 300 m at the equator) to 10 arc‐minutes resolution (approximately 18.5 km at the equator) using the mean location of the raster cells as aggregation criterion.As the root biomass C maps do not distinguish between fine and coarse roots and mycorrhizal fungi colonize only the fine fractions of the roots, we considered the fine root fraction to be 88,5% and 14,1% of the total root biomass for herbaceous and woody plants, respectively. These constants represent the mean value of coarse/fine root mass ratios of herbaceous and woody plants provided by the Fine-Root Ecology Database (FRED) (https://roots.ornl.gov/)25 (deposited at Dryad-Table S3). Due to the non-normality of coarse/fine root mass ratios, mean values were obtained from log-transformed data and then back-transformed for inclusion into the workflow.Finally, the belowground biomass C maps consider the whole root system, but mycorrhizal colonization occurs mainly in the upper 30 cm of the soil18. Therefore, we estimated the total fine root stocks in the upper 30 cm by applying the asymptotic equation of vertical root distribution developed by Gale & Grigal26:$$y=1-{beta }^{d}$$where y is the cumulative root fraction from the soil surface to depth d (cm), and β is the fitted coefficient of extension. β values of trees (β = 0.970), shrubs (β = 0.978) and herbs (β = 0.952) were obtained from Jackson et al.27. A mean value was then calculated for trees and shrubs to obtain a woody vegetation β value of 0.974. As a result, we estimated that 54.6% of the total live root of woody vegetation and 77.1% of herbaceous vegetation is stored in the upper 30 cm of the soil. In combination, this allowed deriving fine root C stocks in the upper 30 cm of woody and herbaceous vegetation.The proportion of root stocks colonized by AM and EcMThe proportion of root stock that forms associations with AM or EcM fungi was obtained from the global maps of aboveground biomass distribution of dominant mycorrhizal types published by Soudzilovskaia et al.14. These maps provide the relative abundance of EcM and AM plants based on information about the biomass of grass, shrub and tree vegetation at 10arcmin resolution. To match with belowground root woody plants biomass data, proportions of AM trees and shrubs underlying the maps of Soudzilovskaia et al.14 were summed up to obtain the proportion of AM woody vegetation. The same was done for EcM trees and shrubs.Our calculations are subjected to the main assumption that, within each growth form, the proportion of aboveground biomass associated with AM and EcM fungi reflects the proportional association of AM and EM fungi to belowground biomass. We tested whether root:shoot ratios were significantly different between AM and EcM woody plants (the number of EcM herbaceous plants is extremely small17). Genera were linked to growth form based on the TRY database (https://www.try-db.org/)19 and the mycorrhizal type association based on the FungalRoots database17. Subsequently, it was tested whether root:shoot ratios of genera from the TRY database (https://www.try-db.org/)19 were significantly different for AM vs EcM woody plants. No statistically significant differences (ANOVA-tests p-value = 0.595) were found (Fig. 2).Fig. 2Mean and standar error of root to shoot ratios of AM and EcM woody plant species.Full size imageEstimation of mycorrhizal fine root stocksWe calculated the total biomass C of fine roots that can potentially be colonized by AM or EcM fungi by multiplying the total woody and herbaceous fine root C biomass in the upper 30 cm of the soil by the proportion of AM and EcM of woody and herbaceous vegetation. In the case of tundra vegetation, fine root C stocks were multiplied by the relative abundance of AM and EcM vegetation without distinction of growth forms (for simplicity, this path was not included in Fig. 1, but can be seen in Fig. 3. As tundra vegetation consists mainly of herbs and small shrubs, the distinction between woody and herbaceous vegetation is not essential in this case.Fig. 3Workflow used to create mycorrhizal fine root biomass C maps specific for tundra areas.Full size imageFinally, we obtained the mean value of mycorrhiza growth form fine root C stocks in each of the defined spatial units. These resulted in six independent estimations: AM woody, AM herbaceous, EcM woody, EcM herbaceous, AM tundra and EcM tundra total fine root biomass C (Fig. 4).Fig. 4Fine root biomass stocks capable to form association with AM (a) and EcM (b) fungi for woody, herbaceous and tundra vegetation. Final AM and EcM stock result from the sum of the growth form individual maps. There were no records of fine root biomass of EcM herbaceous vegetation.Full size imageThe intensity of root colonization by mycorrhizal fungiColonization databaseThe FungalRoot database is the largest up-to-date compilation of intensity of root colonization data, providing 36303 species observations for 14870 plant species. Colonization data was filtered to remove occurrences from non-natural conditions (i.e., from plantations, nurseries, greenhouses, pots, etc.) and data collected outside growing seasons. Records without explicit information about habitat naturalness and growing season were maintained as colonization intensity is generally recorded in the growing season of natural habitats. When the intensity of colonization occurrences was expressed in categorical levels, they were converted to percentages following the transformation methods stated in the original publications. Finally, plant species were distinguished between woody and herbaceous species using the publicly available data from TRY (https://www.trydb.org/)19. As a result, 9905 AM colonization observations of 4494 species and 521 EcM colonization observations of 201 species were used for the final calculations (Fig. 5).Fig. 5Number of AM (a) and EcM (b) herbaceous and woody plant species and total observations obtained from FungalRoot database.Full size imageThe use of the mean of mycorrhizal colonization intensity per plant species is based on two main assumptions:

1)

The intensity of root colonization is a plant trait: It is known that the intensity of mycorrhizal infections of a given plant species varies under different climatic and soil conditions28,29, plant age30 and the identity of colonizing fungal species31. However, Soudzilovskaia et al.9 showed that under natural growth conditions the intraspecific variation of root mycorrhizal colonization is lower than interspecific variation, and is within the range of variations in other plant eco-physiological traits. Moreover, recent literature reported a positive correlation between root morphological traits and mycorrhizal colonization, with a strong phylogenetic signature of these correlations32,33. These findings provide support for the use of mycorrhizal root colonization of plants grown in natural conditions as a species-specific trait.

2)

The percentage of root length or root tips colonized can be translated to the percentage of biomass colonized: intensity of root colonization is generally expressed as the proportion of root length colonized by AM fungi or proportion of root tips colonized by EcM fungi (as EcM infection is restricted to fine root tips). Coupling this data with total root biomass C stocks requires assuming that the proportion of root length or proportion of root tips colonized is equivalent to the proportion of root biomass colonized. While for AM colonization this equivalence can be straightforward, EcM colonization can be more problematic as the number of root tips varies between tree species. However, given that root tips represent the terminal ends of a root network34, the proportion of root tips colonized by EcM fungi can be seen as a measurement of mycorrhizal infection of the root system and translated to biomass independently of the number of root tips of each individual. Yet, it is important to stress that estimations of fine root biomass colonized by AM and EcM as provided in this paper might not be directly comparable.

sPlot databaseThe sPlotOpen database21 holds information about the relative abundance of vascular plant species in 95104 different vegetation plots spanning 114 countries. In addition, sPlotOpen provides three partially overlapping resampled subset of 50000 plots each that has been geographically and environmentally balanced to cover the highest plant species variability while avoiding rare communities. From these three available subsets, we selected the one that maximizes the number of spatial units that have at least one vegetation plot. We further checked if any empty spatial unit could be filled by including sPlot data from other resampling subsets.Plant species in the selected subset were classified as AM and EcM according to genus-based mycorrhizal types assignments, provided in the FungalRoot database17. Plant species that could not be assigned to any mycorrhizal type were excluded. Facultative AM species were not distinguished from obligated AM species, and all were considered AM species. The relative abundance of species with dual colonization was treated as 50% AM and 50% ECM. Plant species were further classified into woody and herbaceous species using the TRY database.Estimation of the intensity of mycorrhizal colonizationThe percentage of AM and EcM root biomass colonized per plant species was spatially upscaled by inferring the relative abundance of AM and EcM plant species in each plot. For each mycorrhizal-growth form and each vegetation plot, the relative abundance of plant species was determined to include only the plant species for which information on the intensity of root colonization was available. Then, a weighted mean intensity of colonization per mycorrhizal-growth form was calculated according to the relative abundance of the species featuring that mycorrhizal-growth form in the vegetation plot. Lastly, the final intensity of colonization per spatial unit was calculated by taking the mean value of colonization across all plots within that spatial unit. These calculations are based on 38127 vegetation plots that hold colonization information, spanning 384 spatial units.The use of vegetation plots as the main entity to estimate the relative abundance of AM and EcM plant species in each spatial unit assumes that the plant species occurrences and their relative abundances in the selected plots are representative of the total spatial unit. This is likely to be true for spatial units that are represented by a high number of plots. However, in those spatial units where the number of plots is low, certain vegetation types or plant species may be misrepresented. We addressed this issue in our uncertainty analysis. Details are provided in the Quality index maps section.Final calculation and maps assemblyThe fraction of total fine root C stocks that is colonized by AM and EcM fungi was estimated by multiplying fine root C stocks by the mean root colonization intensity in each spatial unit. This calculation was made separately for tundra, woody and herbaceous vegetation.To generate raster maps based on the resulting AM and EcM fine root biomass C data, we first created a 10 arcmin raster map of the spatial units. To do this, we overlaid the raster map of Bailey ecoregions (10 arcmin resolution)24, the raster of ESA CCI land cover data at 300 m resolution aggregated to 10 arcmin using a nearest neighbour approach (https://www.esa-landcover-cci.org/) and the FAO polygon map of continents (http://www.fao.org/geonetwork/srv/en/), rasterized at 10 arcmin. Finally, we assigned to each pixel the corresponding biomass of fine root colonized by mycorrhiza, considering the prevailing spatial unit. Those spatial units that remained empty due to lack of vegetation plots or colonization data were filled with the mean value of the ecoregion x continent combination.Quality index mapsAs our workflow comprises many different data sources and the extracted data acts in distinct hierarchical levels (i.e plant species, plots or spatial unit level), providing a unified uncertainty estimation for our maps is particularly challenging. Estimates of mycorrhizal fine root C stocks are related mainly to belowground biomass C density maps and mycorrhizal aboveground biomass maps, which have associated uncertainties maps provided by the original publications. In contrast, estimates of the intensity of root colonization in each spatial unit have been associated with three main sources of uncertainties:

1)

The number of observations in the FungalRoot database. The mean species-level intensity of mycorrhizal colonization in the vegetation plots has been associated with a number of independent observations of root colonization for each plant species. We calculated the mean number of observations of each plant species for each of the vegetation plots and, subsequently the mean number of observations (per plant species) from all vegetation plots in each spatial unit. These spatial unit averaged number of observations ranged from 1 to 14 in AM and from 1 to 26 in EcM. A higher number of observations would indicate that the intraspecific variation in the intensity of colonization is better captured and, therefore, the species-specific colonization estimates are more robust.

2)

The relative plant coverage that was associated with colonization data. From the selected vegetation plots, only a certain proportion of plant species could be associated with the intensity of colonization data in FungalRoot database. The relative abundance of the plant species with colonization data was summed up in each vegetation plot. Then, we calculated the average values for each spatial unit. Mean abundance values ranged from 0.3 to 100% in both AM and EcM spatial units. A high number indicates that the dominant plant species of the vegetation plots have colonization data associated and, consequently, the community-averaged intensity of colonization estimates are more robust.

3)

The number of vegetation plots in each spatial unit. Each of the spatial units differs in the number of plots used to calculate the mean intensity of colonization, ranging from 1 to 1583 and from 1 to 768 plots in AM and EcM estimations, respectively. A higher number of plots is associated with a better representation of the vegetation variability in the spatial units, although this will ultimately depend on plot size and intrinsic heterogeneity (i.e., a big but homogeneous spatial unit may need fewer vegetation plots for a good representation than a small but very heterogeneous spatial unit).

We provide independent quality index maps of the spatial unit average of these three sources of uncertainty. These quality index maps can be used to locate areas where our estimates have higher or lower robustness. More

Real-world dataThe dataset consisted of 2986 observations from 902 freshwater shallow lakes in Denmark and North America (data extracted from the LAGOSNE database on 22 February 2022 via R LAGOSNE package version 2.0.2)56 (Supplementary Fig. 9). The Danish lakes were sampled for one or several years from 1984 to 2020 (data extracted in October 2021 from https://odaforalle.au.dk/main.aspx) (Supplementary Fig. 10). Prerequisites for inclusion in the analysis were that lakes had been sampled for physical and chemical variables at least four times or at least three times over the growing season (May to September) for the Danish or North American lakes, respectively, had a mean depth of less than 3 m and were freshwater. Water chemistry samples were analysed using standard methods and data for total phosphorus (TP), total nitrogen (TN) and chlorophyll-a are included here57. The mean and range of TP, TN and chlorophyll-a for the combined sites is given in Table 1, along with the values for each region separately.To gain a longer-term perspective on the relationship between nutrients and chlorophyll-a, we calculated the across-year averages of the summer means of TP, TN and chlorophyll-a, sequentially increasing numbers of years included in the mean up to a total of a five-year mean, at which point there were only 99 lakes left in the dataset. In calculating the multi-year means we allowed a maximum gap of 2 years between observations (i.e. two observations could cover 3 years) to avoid including time series with too many missing years in between. Hence, only lakes with sufficient numbers of sequential data were included, resulting in a large drop in lake numbers as the length of the multi-year mean increased (Table 2).Numerical methodsDiagnostic tests or proxies of alternative equilibriaWe modelled the response of chlorophyll-a to TP and TN using generalised-linear models58 with Gamma distribution and an identity link on untransformed data for single-year and multiple-year means up to 5-year means. We used the Gamma distribution, as chlorophyll fit this significantly better than a normal or log-normal distribution. We used psuedo R2 of the model along with the patterns of residuals, and finally, we plotted the kernel density of the chlorophyll-a values as diagnostics of the presence, absence or prevalence of alternative equilibria in the simulated and real work data.To test how appropriate these diagnostics or proxies of alternative stable states in terms of how well they identify the existence of alternative stable states in randomly sampled multi-year data, we

1.

Simulated two scenarios for the main manuscript, with and without alternative stable states in the data, which were as close to the real-life data as possible. The results of these scenarios appear in the main text (please see details below in the “Data Simulation” section).

2.

We provide multiple scenarios with different degrees, or prevalence, of alternative stable states in the data, see simulations of alternative stable state scenarios. The results of these scenarios appear in Supplementary note 2.

Hierarchical bootstrap approachThere are a large number of permutations of data, both real-world and simulated, that can provide a mean of the two to five sequential years from each lake in the time series data. It was vital to have a method that selects the data for analysis that provides a valid and comparable representation of both real work and simulated data and the models’ errors. In order to provide this we used a non-parametric hierarchical bootstrap procedure38. The flowchart shows the data preparation and data analysis steps of the hierarchical bootstrap procedure (Fig. 4). In the first step (step 1 in Fig. 4), all possible longer-term means are calculated for each lake. To keep as much data as possible, we decided to allow for up to 2 years of gap in the data between years. Taking the 5-year mean data as an example, if data from a lake existed for the years 1991 and 1994−1997, a 5-year mean would be calculated for the years 1991, 1994, 1995, 1996 and 1997. However, if the time series would contain a larger gap, e.g. data would only exist for the years 1991 and 1995–1998, no 5-year mean could be calculated. After the selection procedure, all the 2-year, 3-year and 5-year means are transferred into a new table (step 2 in Fig. 4).Fig. 4Data preparation and analysis steps of the hierarchical bootstrap procedure.Full size imageThe procedure is the same for each temporal scale from 2-year means to 5-year means. For the example of 5 mean years, lakes are randomly sampled from the full 5-year mean dataset in step 2 (Fig. 4) with replacement up to the number of lakes as in the original dataset, for the 5-year means 99 (step 3a). Here, the same lake can appear multiple times or not at all. This step is common for every bootstrap procedure59. However, since we have nested data (5-year means within lakes), we need a second step, in which for every resampled lake in step 3a, one 5-year mean is chosen (step 3b in Fig. 4). Then the three GLM models are produced from the randomly selected data in step 3c (Fig. 4). These steps are then repeated 1000 times to get a good representation of the uncertainties of the model. To ensure a fair comparison between single-year data and their equivalent multi-year mean data, we repeated the bootstrap procedure with single years only using only the lakes for which we also calculated multi-year means. To take the five-year mean as an example, there were 99 lakes where we could calculate at least one 5-year mean observation. First, we ran the bootstrap procedure to calculate 5-year mean values of TP, TN and chlorophyll-a (1000 times) and then took single years’ values of TP, TN and chlorophyll-a (1000 times) from exactly the same 99 lakes. With this approach, exactly the same datasets with the same lakes and observations within lakes are used for the calculation of the multi-year means and their single-year counterparts, making for a robust analysis. The GLM models did not always converge. If either the TP, TN or TP*TN model with interaction did not converge, the iteration was not used in further analysis. The number of converging models equal for each iteration of random samples is given in the results.The described hierarchical approach is the best way to reflect the structure of the original data. A simple, non-hierarchical bootstrap would favour lakes with more five-year means over lakes with fewer five-year means, simply because these make up a larger part of the data. Furthermore, sampling without replacement at the lake level would result in five-year means from lakes with few data dominating the produced random dataset, as every lake would be sampled every time, which then would result in high model leverage of 5-year means from lakes with less data. In contrast, the hierarchical procedure ensures that every lake has the same chance to end up in the randomly sampled bootstrap, in the second step, it ensures that of each sampled lake, every 5-year mean has the same chance to end up in the random dataset. These notions are in agreement with the findings of an assessment on how to properly resample hierarchical data by non-parametric bootstrap38.Data simulationGeneral approach of simulation assumptions and proceduresWe generated random scatter for the generalised-linear model based on Gamma distributions for two different “populations” of lakes with two different intercepts and slopes. At first, we calculated the linear equations for the two populations:For each population i and j, 99 samples (equalling the number of lakes in real-life data with 5-year means, nlake) were generated with a specific number of data points depending on the scenario (nyear) each, hence nlake = 1−99 for each population of lakes, e.g. with 20 years (nyear = 20) each.We found the real nutrient data to be normally distributed, with total nitrogen (TN) having a range between 0.33 and 4.93 mg/L and a constant coefficient of variation (CV, with a mean CV of 0.35) across this range (the same is true for total phosphorus (TP) at a shorter range). Hence, for each nlake, the x for the nyear = 20 were generated based on the mean range (mean per lake of the real-life data) and CV (0.35) from the real-life TN concentration data, hence with a range of 0.33 to 4.33 mg/L. Therefore the values and random variability of x in the simulations are close to the true values of the TN concentrations. The x is then fed into the linear equations above.To the resulting yi and yj we added random noise based on the Gamma distribution (using the rgamma function in R). We used a Gamma distribution because the Chlorophyll-a concentration also follows a Gamma distribution. The variability of a Gamma distribution is expressed by the shape variable. The variability of chlorophyll-a, its shape value, equals 2.63. This shape value was used in the Gamma distribution of yi and yj. The final calculated yi and yj had therefore a random rate calculated as shape/yi or shape/yj. Hence, their variability in the y dimension was close to the true chlorophyll-a variability.The data from both lake populations were then pooled and randomly sampled using the same hierarchical bootstrap procedure with 500 iterations for the scenarios in the supplementary materials and with 1000 iterations for main text simulation scenarios, which is identical to what was done for the real-world data.Simulation scenarios based on characteristics of real-world dataThe real-world 5-year mean data consisted of 99 lakes with 5–20 years of data for each lake. For the simulation scenario in the main text, we therefore randomly sampled between 5 and 20 data points for each of the 99 simulated lakes based on the x distribution described above. Intercepts and slopes of the simulation, resembled the range of the true data (see scatter plots in Fig. 2 of the main manuscript).In the alternative stable state scenario, we chose two slopes and intercepts for different populations of lakes:

Population i: ai = 0, bi = 40

Population j: aj = 50, bj = 120

We based the slopes and intercepts of the ASS scenarios on the diagnostic combination defined by Scheffer and Carpenter7 which propose an abrupt shift in (a) the time series, (b) the multimodal distribution of states and (c) the dual relationship to a controlling factor. Here, the idea is that an ecosystem will jump from one state to the next at the same (nutrient) conditions (different intercept and/or slope, condition a within ref. 7), where any change in the nutrient will have different effects on algae or macrophytes (best represented by different slope, condition c), resulting in a multimodal distribution of the response (condition b). Hence, simulations are in line with what is predicted for ASS, but we took great lengths to also show other possibilities with the simulations in the Supplementary information to ensure we did not overlook any occasional occurrence of alternative equilibria.Here, the appearance of alternative stable states in the data could happen at any point in the time series of a single lake, or the entire time series could include only one of the two alternative stable states. To accommodate these alternative stable state constellations (for each of which we made a separate simulation scenario, (see Supplementary Note 2, “Simulations of alternative stable state constellations”), we forced the alternative stable state scenario to be constructed of 1/3 of data with one state, 1/3 of data with the second state and 1/3 of data where both alternative states could occur. In the latter case, the alternative stable state appeared after the first 20% but before the last 20% of the time series. Since the variability and range of x (nutrient) and y (chlorophyll -a response) is simulated as close as possible to the real-world data in all scenarios, the measures taken here (variable time series and combination of different alternative stable state scenario constellations) produce a simulation as close to the real-world data as possible. Specifically, we found the real-world nutrient data to be normally distributed, with total nitrogen (TN) having a range between 0.33 and 4.93 mg/L and a constant coefficient of variation (CV, with a mean CV of 0.35) across this range (the same is true for total phosphorus (TP) at a shorter range). Hence, for each simulated lake, the x were generated based on this mean range and CV. Furthermore, the resulting yi and yj were randomised by using a Gamma distribution (using the rgamma function in R). We used a Gamma distribution because the chlorophyll-a concentration also follows a Gamma distribution. The variability of a Gamma distribution is expressed by the shape variable. The variability of chlorophyll-a, its shape value, equals 2.63. This shape value was used in the Gamma distribution of yi and y. The final calculated yi and yj had, therefore a random rate calculated as shape/yi or shape/y. Hence, their variability in the y dimension was close to the true chlorophyll-a variability.For the scenario without alternative stable states, both populations of data had the same intercept and slope:

Population i: ai = 0, bi = 40

Population j: aj = 0, bj = 40.

Please see Supplementary Note 2 for further simulations of different potential constellations of alternative states. There we show that our approach finds alternative stable states in response to nutrient concentration, even if they appear in time series from different lakes.Assessment of diagnostic tests or proxies of alternative equilibriaWe modelled the response of chlorophyll-a to TP and TN using generalised-linear models3 with Gamma distribution and an identity link on untransformed data for single-year and multiple-year means up to 5-year means. We used the Gamma distribution, as chlorophyll fit this significantly better than a normal or log-normal distribution. We used R2 of the model along with the patterns of residuals, and finally, we plotted the kernel density of the chlorophyll-a values as diagnostics of the presence, absence or prevalence of alternative equilibria in the simulated and real work data.The comparison of how the diagnostics/proxies of alternative stable states respond to the variation in the prevalence of alternative equilibria in the simulated datasets provides a robust assessment of their ability to identify both the presence and absence of alternative equilibria. It is the response of these diagnostic tests over time, with the increase in the temporal perspective as more years are added to the mean values of TP, TN and chlorophyll-a, that are key to the identification of the presence and or absence of alternative equilibria in a given dataset. The simulations show that a dataset which contains alternative equilibria will show (1) no improvement in R2 as the temporal perspective of the data increases (more years in the multi-year mean); (2) an increased bimodality in the residuals of the models of nutrients predicting chlorophyll-a will increase as more years are added to the multi-year mean and (3) the kernel density function of chlorophyll-a will display increasingly bimodality as more years are added to the mean. In the absence of alternative equilibria, the patterns differ with an R2, and increase in unimodality of residuals and a consistent unimodal pattern in the kernel density function. Thus, the diagnostic tests provide a robust test of both the presence and absence of alternative equilibria in a given dataset.Alternative stable state assessment for real data with limited data rangeIt could be the case that alternative stable states do not appear in the full dataset but only in a limited TN and TP concentration range. We filtered and re-analyzed the data, only keeping data points within the following two ranges: – TN concentration = 0.5−2 mg/L–TP concentration = 0.05−0.4 mg/L. In the filtered data, 1329 out of the original 2876 single-year data points, 289 out of 1028 3-year mean data points and 212 out of the 864 five-mean year data points remained. The filtered data consisted of data points from 550, 48 and 27 lakes for the single-year data, 3-year means and 5-year means, respectively. The smaller range resulted in lower R² of the models, yet the pattern that multi-year means result in higher R² compared to single-year data was largely consistent, apart from the 5-year mean TN models for which both, the single-year and mean data resulted in very low R² (Supplementary Fig. 6). Furthermore, due to the lower number of samples, the errors of all proxies are higher, making conclusions more difficult than for the full data. Still, we do not see any clear indication of alternative stable states in the scatter plots (two groups of dots are not appearing (Supplementary Fig. 5), the Kernel density plots (or model residuals (Supplementary Fig. 6)). i.e. no signs of bimodality in residuals or Kernel density plots. Please see details on this analysis in the supplementary material.Details and the R code for the steps for the random multi-year sampling can be found in the supplementary materials.Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article. More

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DatasetsImage acquisitionFor this study, we assembled three datasets: one for training of the RootDetector Convolutional Neural Network (Training-Set), one for a performance comparison between humans and RootDetector in segmenting roots in minirhizotron images (Comparison-Set), and one for the validation of the algorithm (Validation-Set). The Training-Set contained 129 images comprised of 17 randomly selected minirhizotron images sampled in a mesocosm experiment (see “Mesocosm sampling” Section), 47 randomly selected minirhizotron images sampled in a field study (see “Field sampling” Section) as well as the 65 minirhizotron images of soy roots published by Wang et al.15. The Comparison-Set contained 25 randomly selected minirhizotron images from the field-study which all were not part of the images included in the Training- and Validation-Sets. The Validation-Set contained 10 randomly selected minirhizotron images from the same field study, which had not been used in the Training-Set. All images were recorded with 2550 ✕ 2273 pixels at 300 dpi with a CI-600 In-Situ Root Imager (CID Bio-Science Inc., Camas, WA, USA) and stored as .tiff files to reduce compression loss. For all training and evaluation purposes we used raw, unprocessed output images from the CI-600.Mesocosm samplingThe mesocosm experiment was established in 2018 on the premises of the Institute for Botany and Landscape Ecology of the University of Greifswald (Fig. S1). It features 108 heavy duty plastic buckets of 100 l each, filled to two thirds of their height with moderately decomposed sedge fen peat. Each mesocosm contained one minirhizotron (inner diameter: 64 mm, outer diameter: 70 mm, length: 650 mm) installed at a 45°angle and capped in order to avoid penetration by light. The mesocosms were planted with varying compositions of plant species that typically occur in north-east German sedge fens (Carex rostrata, Carex acutiformis, Glyceria maxima, Equisetum fluviatile, Juncus inflexus, Mentha aquatica, Acorus calamus and Lycopus europaeus). The mesocosms were subjected to three different water table regimes: stable at soil surface level, stable at 20 cm below soil surface and fluctuating between the two levels every two weeks. The minirhizotrons were scanned weekly at two levels of soil depth (0–20 cm and 15–35 cm) between April 2019 and December 2021, resulting in roughly 9500 minirhizotron images of 216 × 196 mm. Manual quantification of root length would, based on own experience, take approximately three hours per image, resulting in approximately 28,500 h of manual processing for the complete dataset. Specimens planted were identified by author Dr. Blume-Werry, however no voucher specimen were deposited. All methods were carried out in accordance with relevant institutional, national, and international guidelines and legislation.Field samplingThe field study was established as part of the Wetscapes project in 201716. The study sites were located in Mecklenburg-Vorpommern, Germany, in three of the most common wetland types of the region: alder forest, percolation fen and coastal fen (Fig. S2). For each wetland type, a pair of drained versus rewetted study sites was established. A detailed description of the study sites and the experimental setup can be found in Jurasinski et al.16. At each site, 15 minirhizotrons (same diameter as above, length: 1500 mm) were installed at 45° angle along a central boardwalk. The minirhizotrons have been scanned biweekly since April 2018, then monthly since January 2019 at two to four levels of soil depth (0–20 cm, 20–40 cm, 40–60 cm and 60–80 cm), resulting in roughly 12,000 minirhizotron images of 216 × 196 cm, i.e. an estimated 36,000 h of manual processing for the complete dataset. Permission for the study was obtained from the all field owners. Figure 1Overview of the RootDetector system. The main component is a semantic segmentation network based on the U-Net architecture. The root length is estimated by skeletonizing the segmentation output and applying the formula introduced by Kimura et al.17. During training only, a weight map puts more emphasis on fine roots.Full size imageThe CNN RootDetectorImage annotationFor the generation of training data for the CNN, human analysts manually masked all root pixels in the 74 images of the Training-Set using GIMP 2.10.12. The resulting ground truth data are binary, black-and-white images in Portable Network Graphics (.png) format, where white pixels represent root structures and black pixels represent non-root objects and soil (Fig. 2b). All training data were checked and, if required, corrected by an expert (see “Selection of participants” for definition). The Validation-Set was created in the same way but exclusively by experts.Figure 2Example of segmentation and result of skeletonization. A 1000 by 1000 pixel input image (a), the manually annotated ground truth image (b), the RootDetector estimation image (c), the combined representation image (error map, d with green indicating true positives, red indicating false positive, blue indicating false negatives), the skeletonized RootDetector estimation image (e), and the skeletonized ground truth image (f).Full size imageArchitectureRootDetector’s core consists of a Deep Neural Network (DNN) based on the U-Net image segmentation architecture[27]nd is implemented in TensorFlow and Keras frameworks18. Although U-Net was originally developed for biomedical applications, it has since been successfully applied to other domains due to its generic design.RootDetector is built up of four down-sampling blocks, four up-sampling blocks and a final output block (Fig. 1). Every block contains two 3 × 3 convolutional layers, each followed by rectified linear units (ReLU). The last output layer instead utilizes Sigmoid activation. Starting from initial 64 feature channels, this number is doubled in every down-block and the resolution is halved via 2 × 2 max-pooling. Every up-block again doubles the resolution via bilinear interpolation and a 1 × 1 convolution which halves the number of channels. Importantly, after each up-sampling step, the feature map is concatenated with the corresponding feature map from the down-sampling path. This is crucial to preserve fine spatial details.Our modifications from the original architecture include BatchNormalization19 after each convolutional layer which significantly helps to speed up the training process and zero-padding instead of cropping as suggested by Ronneberger, Fischer, & Brox20 to preserve the original image size.In addition to the root segmentation network, we trained a second network to detect foreign objects, specifically the adhesive tape that is used as a light barrier on the aboveground part of the minirhizotrons. We used the same network architecture as above and trained in a supervised fashion with the binary cross-entropy loss. During inference, the result is thresholded (predefined threshold value: 0.5) and used without post-processing.TrainingWe pre-trained RootDetector on the COCO dataset21 to generate a starting point. Although the COCO dataset contains a wide variety of image types and classes not specifically related to minirhizotron images, Majurski et al.22 showed, that for small annotation counts, transfer-learning even from unrelated datasets may improve a CNNs performance by up to 20%. We fine-tuned for our dataset with the Adam optimizer23 for 15 epochs and trained on a total of 129 images from the Training-Set (17 mesocosm images, 47 field-experiment images, 65 soy root images). To enhance the dataset size and reduce over-fitting effects, we performed a series of augmentation operations as described by Shorten & Khoshgoftaar24. In many images, relatively coarse roots ( > 3 mm) occupied a major part of the positive (white) pixel space, which might have caused RootDetector to underestimate fine root details overall. Similarly, negative space (black pixels) between tightly packed, parallel roots was often very small and might have impacted the training process to a lesser extent when compared to large areas with few or no roots (Fig. 2). To mitigate both effects, we multiplied the result of the cross-entropy loss map with a weight map which emphasizes positive–negative transitions. This weight map is generated by applying the following formula to the annotated ground truth images:$$omega left( x right) = 1 – left( {tanh left( {2tilde{x} – 1} right)} right)^{2}$$
(1)
where ω(x) is the average pixel value of the annotated weight map in a 5 × 5 neighborhood around pixel x. Ronneberger, Fischer, & Brox20 implemented a similar weight map, however with stronger emphasis on space between objects. As this requires computation of distances between two comparatively large sets of points, we adapted and simplified their formula to be computable in a single 5 × 5 convolution.For the loss function we applied a combination of cross-entropy and Dice loss 25:$${mathcal{L}} = {mathcal{L}}_{CE} + lambda {mathcal{L}}_{Dice} = – frac{1}{N}sumnolimits_{i} {wleft( {x_{i} } right)y_{i} log left( {x_{i} } right) + lambda frac{{2sumnolimits_{i} {x_{i} y_{i} } }}{{sumnolimits_{i} {x_{i}^{2} sumnolimits_{i} {y_{i}^{2} } } }}}$$
(2)

where x are the predicted pixels, y the corresponding ground truth labels, N the number of pixels in an image and λ a balancing factor which we set to 0.01. This value was derived empirically. The Dice loss is applied per-image to counteract the usually high positive-to-negative pixel imbalance. Since this may produce overly confident outputs and restrict the application of weight maps, we used a relatively low value for λ.Output and post-processingRootDetector generates two types of output. The first type of output are greyscale .png files in which white pixels represent pixels associated with root structures and black pixels represent non-root structures and soil (Fig. 2c). The advantage of .png images is their lossless ad artifact-free compression at relatively small file sizes. RootDetector further skeletonizes the output images and reduces root-structures to single-pixel representations using the skeletonize function of scikit-image v. 0.17.1 (26; Fig. 2e,f). This helps to reduce the impact of large diameter roots or root-like structures such as rhizomes in subsequent analyses and is directly comparable to estimations of root length. The second type of output is a Comma-separated values (.csv) file, with numerical values indicating the number of identified root pixels, the number of root pixels after skeletonization, the number of orthogonal and diagonal connections between pixels after skeletonization and an estimation of the physical combined length of all roots for each processed image. The latter is a metric commonly used in root research as in many species, fine roots provide most vital functions such as nutrient and water transport3. Therefore, the combined length of all roots in a given space puts an emphasis on fine roots as they typically occupy a relatively smaller fraction of the area in a 2D image compared to often much thicker coarse roots. To derive physical length estimates from skeletonized images, RootDetector counts orthogonal- and diagonal connections between pixels of skeletonized images and employs the formula proposed by Kimura et al.17 (Eq. 3).$$L = left[ {N_{d}^{2} + left( {N_{d} + N_{o} /2} right)^{2} } right]^{{1/2}} + N_{o} /2$$
(3)
where Nd is the number of diagonally connected and No the number of orthogonally connected skeleton pixels. To compute Nd we convolve the skeletonized image with two 2 × 2 binary kernels, one for top-left-to-bottom-right connections and another for bottom-left-to-top-right connections and count the number of pixels with maximum response in the convolution result. Similarly, No is computed with a 1 × 2 and a 2 × 1 convolutional kernels.Performance comparisonSelection of participantsFor the performance comparison, we selected 10 human analysts and divided them into three groups of different expertise levels in plant physiology and with the usage of digital root measuring tools. The novice group consisted of 3 ecology students (2 bachelor’s, 1 master’s) who had taken or were taking courses in plant physiology but had no prior experience with minirhizotron images or digital root measuring tools. This group represents undergraduate students producing data for a Bachelor thesis or student assistants employed to process data. The advanced group consisted of 3 ecology students (1 bachelor’s, 2 master’s) who had already taken courses in plant physiology and had at least 100 h of experience with minirhizotron images and digital root measuring tools. The expert group consisted of 4 scientists (2 PhD, 2 PhD candidates) who had extensive experience in root science and at least 250 h of experience with digital root measuring tools. All methods were carried out in accordance with relevant institutional, national, and international guidelines and legislation and informed consent was obtained from all participants.Instruction and root tracingAll three groups were instructed by showing them a 60 min live demo of an expert tracing roots in minirhizotron images, during which commonly encountered challenges and pitfalls were thoroughly discussed. Additionally, all participants were provided with a previously generated, in-depth manual containing guidelines on the identification of root structures, the correct operation of the root tracing program and examples of often encountered challenges and suggested solutions. Before working on the Comparison-Set, all participants traced roots in one smaller-size sample image and received feedback from one expert.Image preparation and root tracingBecause the minirhizotron images acquired in the field covered a variety of different substrates, roots of different plant species, variance in image quality, and because tracing roots is very time consuming, we decided to maximize the number of images by tracing roots only in small sections, in order to cover the largest number of cases possible. To do this, we placed a box of 1000 × 1000 pixels (8.47 × 8.47 cm) at a random location in each of the images in the Comparison-Set and instructed participants to trace only roots within that box. Similarly, we provided RootDetector images where the parts of the image outside the rectangle were occluded. All groups used RootSnap! 1.3.2.25 (CID Bio-Science Inc., Camas, WA, USA;27), a vector based tool to manually trace roots in each of the 25 images in the comparison set. We decided on RootSnap! due to our previous good experience with the software and its’ relative ease of use. The combined length of all roots was then exported as a csv file for each person and image and compared to RootDetector’s output of the Kimura root length.ValidationWe tested the accuracy of RootDetector on a set of 10 image segments of 1000 by 1000 pixels cropped from random locations of the 10 images of the Validation-Set. These images were annotated by a human expert without knowledge of the estimations by the algorithm and were exempted from the training process. As commonly applied in binary classification, we use the F1 score as a metric to evaluate the performance RootDetector. F1 is calculated from precision (Eq. 4) and recall (Eq. 5) and represents their harmonic mean (Eq. 6). Ranging from 0 to 1, higher values indicate high classification (segmentation) performance. As one of the 10 image sections contained no roots and thus no F1 Score was calculable, it was excluded from the validation. We calculated the F1 score for each of the nine remaining image sections and averaged the values as a metric for overall segmentation performance.$$Precision;(P) = frac{{tp}}{{tp + fp}}$$
(4)
$$Recall;(R) = frac{{tp}}{{tp + fn}}$$
(5)
$$F1 = 2*frac{{P*R}}{{P + R}}$$
(6)
where P = precision, R = recall, tp = true positives; fp = false positives, fn = false negatives.Statistical analysisWe used R Version 4.1.2 (R Core Team, 2021) for all statistical analyses and R package ggplot2 Version 3.2.128 for visualizations. Pixel identification-performance comparisons were based on least-squares fit and the Pearson method. Root length estimation-performance comparisons between groups of human analysts (novice, advanced, expert) and RootDetector were based on the respective estimates of total root length plotted over the minirhizotron images in increasing order of total root length. Linear models were calculated using the lm function for each group of analysts. To determine significant differences between the groups and the algorithm, 95% CIs as well as 83% CIs were displayed and RootDetector root length outside the 95% CI were considered significantly different from the group estimate at α = 0.0529. The groups of human analysts were considered significantly different if their 83% CIs did not overlap, as the comparison of two 83% CIs approximates an alpha level of 5%30,31.This study is approved by Ethikkommission der Universitätsmedizin Greifswald, University of Greifswald, Germany. More

Our study is focussed on population trends of large carnivores; a culturally important group32, essential for regulating ecosystem function33. Large carnivores represent an important study group as their population status is unclear, with reports of devastating declines33 contrasted with remarkable recoveries23. Further, as a well-studied taxa with abundant trend and trait datasets, large carnivores present a good system to evaluate important drivers of trends without being impacted by poor inference from missing data34. Finally, as large carnivores are considered indicator species of the overall status of biodiversity within an area35, our inference may provide insight beyond our focal taxa.Population trendsWe sourced population (defined by the authors of the original studies, who reported on population trends for one or more studied groups of individuals) trend information for species in the families Canidae, Felidae, Hyaenidae, and Ursidae of the order Carnivora from two large trend datasets: CaPTrends12 and the Living Planet Database13. The CaPTrends database is the product of a semi-systematic literature search for population trends of large carnivore species (from the families listed above); the dataset possesses trend information for 50 species from locations around the world, and trends are reported in a variety of ways. The Living Planet Database contains population abundance time-series for vertebrates from thousands of sites around the world and is one of the larger population trend datasets. Combined, these datasets produce a cumulative 1123 trends (after removing duplicates and records we deemed unreliable or unsuitable), derived from >10,000 individual population estimates. In the Living Planet Database, and for most records in CaPTrends, trends are reported as a time-series of abundance (or density) estimates. We modelled these time-series with log-linear regressions, where abundance (the response) was loge transformed, and year of abundance estimates was selected as the predictor. We included a continuous Ornstein-Uhlenbeck (OU) autoregressive process to control for temporal autocorrelation in these models. The OU process estimates covariance between abundance values, under the assumption that abundances in time point 1 will be more similar to abundances in time point 2, than time-point 3, 4, 5, etc. Accounting for covariance resolves non-independence within time-series. We extracted the slope coefficient which represents the annual instantaneous rate of change, sometimes called the population growth rate (rt). Alongside the abundance time-series, CaPTrends also has three other quantitative datatypes, all of which we converted into an annual instantaneous rate of change (rt): (1) a mean finite rate of change; (2) estimates of percentage abundance change between two points in time; and (3) time-series’ of population change estimates (e.g. in year 1 the population doubled and in year 2 it halved). We converted all annual instantaneous rates of change into an annual rate of change percentage to improve interpretability. These annual rates of change ranged from −75 to 68%, but the majority of values fell within −10 to 10% (Supplementary Fig. 1a).Alongside the quantitative records, 138 populations in the CaPTrends dataset were only described qualitatively with categories: Increase, Stable, and Decrease. These qualitative records were more common for populations located in traditionally poorer-sampled countries (e.g. with lower human development), so whilst they are less informative (only describing the direction and not the magnitude), we deem them important to reduce bias (Fig. 1). As a result, we used a combination of percentage annual rates of change (N = 985) and qualitative categories (N = 138) as our response in our model (see below), representing 50 large carnivore species.CovariatesFor each population, we extracted sixteen covariates (each z-transformed) that fell into four categories: land-use, climate, governance, and traits. Our covariates were designed to cover a diverse array of factors that could influence population trends in large carnivores (Supplementary Table 1). Each covariate is described briefly in Fig. 2 with full descriptions of how variables were derived in the Supplementary material: Covariates.One of the challenges in identifying how covariates—which can vary in space and time—impact population trends is matching the spatial and temporal scale of the covariate with the population i.e. how much of the population is affected by the covariate at a given point in time. To tackle the spatial element of this problem, we used data on the area of extent of each population (e.g. how large is the spatial extent of the population or monitoring zone) to generate a circular distribution zone around the population’s coordinate centroid. We refer to this as the ‘population area’ hereafter. We then sampled covariate values within each population area, with more sampling points in larger areas (range: 13–295 sampling points, Supplementary Fig. 2b). For covariates which varied over time, we extracted the covariates across the ‘population monitoring period’, which refers to the period (from start to end year) the population was monitored for. However, as evidence suggests there can be a lag period between impact or change and any detectable changes in population abundance3, we tested how 0-, 5-, and 10-year lags in covariates changed model fits and effect sizes. We implemented these lags by extending the start of the population monitoring period backwards for each given lag e.g. for a 10-year lag, a normal population monitoring period of 1990–2000, would then capture covariates between 1980–2000. Sensitivity analysis showed a 10-year lag had the greatest balance of improved model fit, with high taxonomic and spatial coverage (see Supplementary: Sensitivity analysis).ModellingAt its core, our model is a linear mixed effects model, regressing annual rates of change against a combined 23 covariates and interactions, using random intercepts to account for phylogenetic and spatial nesting. The model was written in BUGS language and implemented in JAGS 4.3.036 via R 4.0.337. The model structure is summarised below, with a full description in Supplementary: Modelling.ResponseWe modelled our annual rate of change response with a normal error prior. However, to allow the two different types of population trend data (quantitative rates of change and qualitative descriptions of change) to be included in the same model, we treated the qualitative records as partially known. Specifically, we censored the qualitative records to indicate that the true value is unknown, but it occurs within a specified range, with annual rate changes ranging from −50 to 0%, −5 to 5%, and 0 to 50% within the decrease, stable and increase categories, respectively. The overlapping nature of these thresholds is by design, as we wanted to acknowledge that there is likely a grey area between the different categories. For instance, in one study, a 2% trend could be called stable, whilst a different study would consider this as increasing, our overlapping thresholds address this grey area. Admittedly, our category thresholds were arbitrarily selected—this is as a consequence of there being no strict rules on what population change is needed to be assigned a given category. However, despite being arbitrary, they were still carefully selected. For instance, our censoring range thresholds are similar to the range of the observed change (−75 to 68%). Further, whilst we don’t have a clear definition for what an increasing or decreasing population looks like (is it 1% or 10%), we can be confident that increasing and decreasing populations will fall above and below 0%, respectively. The stable category is most vulnerable to subjectivity, and so without clear definitions, we set a large range e.g. the maximum and minimum value we considered could be plausibly called stable was 5% and −5%, respectively.Many of the qualitative and short-term (brief monitoring period) quantitative records address known data biases as they occur in less-well represented regions, species, and time-periods (Fig. 1). However, these lower quality records can be more prone to error. As a result, we developed a weighting term within the model to inflate uncertainty around trends derived over a short timeframe, with few abundance observations, and less robust methods—see Supplementary: Modelling—Weighted error.CovariatesPrior to modelling, we identified missing values in some covariates (e.g. some species were missing Maximum longevity values), which can be problematic for inference if ignored34. We used imputation approaches38,39 to predict these missing values and recorded the associated imputation uncertainty alongside these predictions. Within our model, we accounted for uncertainty in the imputed estimates by treating imputed values of the covariates as distributions instead of point estimates. Specifically, for each imputed value we assigned a normal distribution defined by the mean and standard deviation of the imputed estimates. This approach allowed us to capture imputation uncertainty and improve inference robustness.With 16 covariates and a further seven interactive effects (23 effects in total), we were conscious of overparameterizing the model. As a result, we split these parameters into three groups: (1) core parameters—which included main effects that were considered likely drivers of population change; (2) optional parameters—which included main effects we considered interesting but with little evidence to-date of any influence on trends; and (3) interaction parameters—which includes the seven proposed interaction terms. We included our core parameters (Change in human density, Primary land loss, Population area, Body mass, Change in extreme heat, Governance, and Protected area coverage) in every model, but used Kuo and Mallick variable selection40 to identify parameters from the optional and interaction groups that improve model fit whilst balancing the risk of overfitting.Random interceptsWe used a hierarchical model structure to account for phylogenetic and spatial non-independence in the data, including species as a random intercept nested with genus, and country as a random intercept nested within sub-regions, as defined by the United Nations (https://www.un.org/about-us/member-states).Model runningWe ran the full model through three chains, each with 150,000 iterations. The first 50,000 iterations in each chain were discarded, and we only stored every 10th iteration along the chain (thinning factor of 10). We opted for a large chain and burn-in due to the model complexity, and to allow a broad selection of parameter combinations to be tested under variable selection. We assessed convergence of the full model on all parameters monitored in the sensitivity analysis, as well as the model intercept, and all 23 main and interactive effect slope coefficients. We checked the standard assumptions of a mixed effect linear model (normal residuals and heterogeneity of variance), and tested the residuals to ensure there was no spatial (Moran’s test) or phylogenetic (Pagel’s lambda) autocorrelation. We also conducted posterior predictive checks to ensure independently simulated values were broadly reminiscent of model predicted values.We report the median slope coefficient and associated credible intervals for each of the main and interactive effects, and produce marginal effect plots for a selection of important parameters. These marginal effects hold all other covariates at zero (which is the equivalent of the mean, as covariates were z-transformed).LimitationsDeveloping macro-scale models of population change is challenging as response data are biased41 and hard to summarise42, and response-covariate relationships are likely complex and numerous2. Within our workflow, we attempted to address these challenges, and overall, this allowed us to achieve a moderate model fit (conditional R2 ~ 0.4). We minimised biases in the trend data by integrating qualitative trends with quantitative estimates, which allowed us to increase the taxonomic and spatial scale of the work. However, biases are likely still present to some extent. For instance, whilst we have population trend data covering the full parameter space of our most influential variable (change in human development), we have more population trends in high human development countries (Supplementary Fig. 20)—given these biases, caution should be used when interpreting results. While we could not avoid some biases, we found inference was similar across different fragments of the data and model structures (Supplementary results: Sensitivity analysis). We also attempted to capture complexity by covering a more comprehensive array of covariates than many previous analyses, but we still lack data on likely important aspects that are cryptic and difficult to measure (e.g. poaching, persecution, culling, and the conservation benefits of being flagship species). Further, there are temporal lags between disturbance-events and observable changes in the population10 and we tested several to incorporate the lag that maximised model fit. However, it is possible that responses to different types of disturbance (e.g. habitat loss and climate change) have different lags, although this has not been quantified. Long lags (the maximum lag we explored was 10-years) may also occur and be associated with slow recoveries, but an absence of longer temporal extents in the response and covariate data largely prohibits this analysis at global scales (long temporal extent data is less available outside of the global north).Counterfactual scenariosTo explore how observed changes in land-use, climate and human development have influenced population trends, we developed three counterfactual scenarios, where we compared observed population change to predicted population change if habitat, climate, and human development remained static. For instance, in the climate change counterfactual scenario, we predicted each population trend using the global model (all covariate parameters) with available covariate data (e.g. land-use, governance and trait covariates), as well as taxa and location data (to provide sensitivity to the models varying random intercepts), but set the climate change covariate data to zero (in this case, change in extreme heat and change in drought). We then subtracted these counterfactual predictions from the observed trends to define ‘Difference in annual rate of change (%)’, whereby a positive value indicates carnivore populations would be in better shape (fewer declines) under the counterfactual scenario, and vice-versa. We summarise counterfactual scenarios by reporting the median Difference in annual rate of change and 95% quantiles across the observed 1123 populations.Socioeconomic development and non-linearity in carnivore trendsGiven the large effect of human development change on carnivore population trends within our counterfactual scenarios, we further explore the potential impacts of human development change (i.e. changes in the socioeconomic standards of society) on the dynamics of potential carnivore abundance change. Specifically, we test how changing the rate of human development growth of a hypothetical low human development country could impact carnivore abundances. We test this by simulating time series of human development change between the years 1960 and 2020 along three common development pathways for low human development countries, each given: (1) a mean rate of change in human development (%) defined as Slow (1.25%), Moderate (1.5%) and Fast (1.75%); (2) a shared deceleration rate set to −0.02% per year—a key feature of the human development data is that as human development grows, its growth rate decreases; and (3) a shared initial human development value which we set as 0.2 (a hypothetical low human development country) at year 1960 (Fig. 4a). All our selected parameter values are representative of the human development data (Supplementary Fig. 2), with the Moderate pathway being largely typical for a country with an initial human development value of 0.2, while Slow and Fast represent plausible extremes.We then used our fitted model (Fig. 2) to evaluate how the three pathways of Change in Human development would affect annual abundance of a hypothetical carnivore. This involved predicting the annual rate of change in abundance using the Change in human development pathways and the marginal effect of the Change in human development parameter from the fitted model—setting all other covariates in the model to zero, which in our z-transformed variables represents the mean. We then used the predicted annual rates of change in abundance to project carnivore abundance up to the year 2020, from an arbitrary baseline abundance of 100 in the year 1960 (Fig. 4c). These projections capture the 95% credible intervals around the human development change model coefficient, and assume constant and average values for all other effects (e.g. primary habitat loss or climate change).Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article. More