Ulrich Brose

Phillippe Archambault

Andrew D Barnes

Louis-Felix Bersier

Thomas Boy

João Canning-Clode

Erminia Conti

Marta Dias

Christoph Digel

Awantha Dissanayake

Augusto A V Flores

Katarina Fussmann

Benoit Gauzens

Clare Gray

Johanna Häussler

Myriam R Hirt

Ute Jacob

Malte Jochum

Sonia Kéfi

Orla Mclaughlin

Muriel M Macpherson

Ellen Latz

Katrin Layer-Dobra

Pierre Legagneux

Yuanheng Li

Carolina Madeira

Neo D Martinez

Vanessa Mendonça

Christian Mulder

Sergio A Navarrete

Eoin J O'gorman

David Ott

José Paula

Daniel Perkins

Denise Piechnik

Ivan Pokrovsky

David Raffaelli

Björn C Rall

Benjamin Rosenbaum

Remo Ryser

Ana Silva

Esra H Sohlström

Natalia Sokolova

Murray S A Thompson

Ross M Thompson

Fanny Vermandele

Catarina Vinagre

Shaopeng Wang

Jori M Wefer

Richard J Williams

Evie Wieters

Guy Woodward

Alison C Iles

Predator traits and food-web architecture

Predator-prey interactions in natural ecosystems generate complex food webs that have a simple universal body-size architecture where predators are systematically larger than their prey. Food-web theory shows that the highest predator-prey body-mass ratios found in natural food webs may be especially important as they create weak interactions with slow dynamics that stabilize communities against perturbations and maintain ecosystem functioning. Identifying these vital interactions in real communities typically requires arduous identification of interactions in complex food webs. Here, we overcome this obstacle by developing predator-trait models to predict average body-mass ratios based on a database comprising 290 food webs from freshwater, marine and terrestrial ecosystems across all continents. We analyzed how species traits constrain body-size architecture by changing the slope of the predator-prey body-mass scaling. Across ecosystems, we found high bodymass ratios for predator groups with specific trait combinations including (1) small vertebrates and (2) large swimming or flying predators. Including the metabolic and movement types of predators increased the accuracy of predicting which species are engaged in high body-mass ratio interactions. We demonstrate that species traits explain striking patterns in the body-size architecture of natural food webs that underpin the stability and functioning of ecosystems, paving the way for community-level management of the most complex natural ecosystems.

Introduction

Prey rarely if ever give up their lives willingly to their predators. Predators overcome their prey's resistance by being, on average, larger than their prey, yielding a systematic pattern in communities where the non-cannibalistic average ratio of predator-to-prey body mass (hereafter: body-mass ratio) is higher than unity [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF] . The notable exceptions to this include when animals cooperate to overcome larger prey (e.g. pack hunters) and where consumers are parasites or parasitoids. The variation in body-mass ratios within food webs typically spans several orders of magnitude and includes some predators that are smaller than their prey [START_REF] Nakazawa | Predator-prey body size relationships when predators can consume prey larger than themselves[END_REF] , but is dominated by situations of the larger feeding on the smaller. The varying body-mass ratios limit which trophic interactions are realized in a community [START_REF] Woodward | Body size in ecological networks[END_REF][START_REF] Petchey | foraging, and food web structure[END_REF][START_REF] Eklof | The dimensionality of ecological networks[END_REF] , and the strength of these interactions [START_REF] Rall | Taxonomic versus allometric constraints on non-linear interaction strengths[END_REF][START_REF] Emmerson | Predator-prey body size, interaction strength and the stability of a real food web[END_REF][START_REF] Reuman | Estimating relative energy fluxes using the food web, species abundance, and body size[END_REF] . Predators typically exert the strongest feeding pressure on prey that are one to two orders of magnitude smaller 1 , while weaker interaction strengths are realized with prey that are smaller or larger than this size [START_REF] Schneider | Body mass constraints on feeding rates determine the consequences of predator loss[END_REF][START_REF] Brose | Foraging theory predicts predator-prey energy fluxes[END_REF] . Specifically, interactions of predators with small prey are characterized by high body-mass ratios that yield weak interactions with slow dynamics, which play a central role in maintaining food-web stability 12- 16 and ecosystem functioning [START_REF] Schneider | Body mass constraints on feeding rates determine the consequences of predator loss[END_REF][START_REF] Schneider | Animal diversity and ecosystem functioning in dynamic food webs[END_REF][START_REF] Wang | Biodiversity and ecosystem functioning in food webs: the vertical diversity hypothesis[END_REF] . Moreover, they also buffer natural communities against perturbations from global warming [START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF] , eutrophication [START_REF] Rall | Food-web connectance and predator interference dampen the paradox of enrichment[END_REF] and secondary extinction waves [START_REF] Brose | Predicting the consequences of species loss using size-structured biodiversity approaches[END_REF] . Therefore, identifying these unique interactions is paramount to determining the stability of natural food webs to perturbations and functioning, but applications of this concept to natural communities have been hampered by the difficulty of describing the myriads of interactions present in natural food webs. Using traits of predator species as proxies of body-mass ratios and the resulting interaction strengths that they govern could provide the means to understand which species are drivers of community stability and functioning without having to perform the often logistically and economically impossible task of quantifying entire interaction networks. This approach could ultimately help predict how extinctions, invasions and other anthropogenic environmental changes affect community stability and functioning through shifts in community trait structure.

Despite the importance of understanding how species traits affect body-mass ratios, there is much uncertainty about these relationships. A pioneering study [START_REF] Cohen | Body sizes of animal predators and animal prey in food webs[END_REF] showed that vertebrate predators exhibit systematically higher body-mass ratios than invertebrates and this has been supported by subsequent analyses also documenting higher body-mass ratios in aquatic versus terrestrial communities [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF] . These studies also found that predator-prey bodymass scaling is superlinear with slopes higher than unity, meaning that body-mass ratios increase with body mass (see Supplementary Figure 1, red line). However, other studies have proposed that the scaling relationship is either sublinear (decreasing body-mass ratios with body mass; Supplementary Figure 1, yellow line) [START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF][START_REF] Cohen | Body sizes of animal predators and animal prey in food webs[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF] or superlinear [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF][START_REF] Carbone | Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments[END_REF] , depending on the ecosystem type [START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF] , predator metabolic group [START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF][START_REF] Carbone | Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments[END_REF][START_REF] Naisbit | Phylogenetic signal in predatorprey body-size relationships[END_REF] or resource supply [START_REF] Costa-Pereira | Prey limitation drives variation in allometric scaling of predator-prey interactions[END_REF] . Additionally, besides body mass and metabolic type, little is known about how body-mass ratios vary with other species traits and across different ecosystem types. Predator and prey movement types and feeding behavior are likely to influence scaling relationships by limiting maximum achievable attack speeds [START_REF] Carbone | Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments[END_REF][START_REF] Hirt | A general scaling law reveals why the largest animals are not the fastest[END_REF] . Interaction dimensionality, which describes whether predators forage in three dimensions (e.g. the water column of lakes and oceans) or on twodimensional surfaces (e.g. epigeic terrestrial or benthic aquatic predators), also influences predator-prey attack rates [START_REF] Carbone | Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments[END_REF][START_REF] Pawar | Dimensionality of consumer search space drives trophic interaction strengths[END_REF] . As these variables affect the likelihood and strength of predator attack rates and scale with individual body mass, we expected that they should also modify the scaling relationship between predator and prey body masses.

Insert Fig. 1 here Here, we provide a comprehensive assessment of how species traits modulate predator-prey body-mass scaling relationships and body-mass ratios in natural, complex food webs across an unprecedented range of ecosystems. To do so, we collated the most extensive global food-web database (GlobAl daTabasE of traits and food Web Architecture, GATEWAy version 1.0) to date, comprising 290 food webs (with 222,151 feeding links between 5736 species; see Supplementary Table 1) distributed across the globe (Fig. 1), and including information on four different species traits and five ecosystem types (see Supplementary Table 2). First, we analyzed the scaling of predator and prey body masses over 17 orders of magnitude (fresh masses ranging from the 2 10 -9 g protozoan Bodo saltans, to the 275 10 6 g sperm whale Physeter microcephalus). Some prior studies advocated the use of major axis regressions to account for the bidirectional causality between predator and prey body mass 1 , whereas others used mixed-effects models to include random effects of the study 2 . As these two types of analyses are mutually exclusive in traditional statistics, we used Bayesian modelling to implement a combination of major axis regressions with mixed effects. Second, we tested for the importance of co-factors in this scaling relationship (ecosystem type, predator and prey metabolic types, interaction dimensionality, predator and prey movement types). These analyses address relationships between species traits and food-web architecture across ecosystems. Third, we developed predictions of average body-mass ratios of predators by their traits, which identifies ecological attributes that broadly predict ecological perturbation stability and functioning in natural communities without requiring detailed knowledge of complex food-web structure.

Results

In our first analysis, we addressed the scaling of predator and prey body masses.

Ordinary least squares (OLS) regressions relating these two variables generate different slopes depending on which variable is chosen as the independent variable (Fig. 2, magenta and blue lines). This discrepancy arises because there is no unidirectional causal relationship between the two variables and because both have measurement errors of the same magnitude, which renders major axis regression the appropriate tool for analyzing these data [START_REF] Warton | Bivariate line-fitting methods for allometry[END_REF] . Thus, we used Bayesian modelling to fit a major axis regression, which makes no assumptions about a causal relationship between the variables. The major axis regression showed that the overall allometric scaling relationship between predator and prey body mass was superlinear, exhibiting a slope higher than unity (Fig. 2, black solid line, Bayesian major axis regression, slope = 1.315, 95% CI: 1.307-1.323). This suggests that the body-mass ratios between predators and their prey increase with the masses of prey and predators (i.e. the distance between the regression line and the dashed diagonal indicating equally sized predator-prey pairs). On average, interactions between relatively large predator and prey species are characterized by higher body-mass ratios than interactions between smaller species. The Bayesian approach also allowed us to fit mixed-effects models to the data, particularly the random effects on the intercept of the different studies (database variable: link.citation; see Supplementary Table GATEWAy metadata). This yielded a very similar scaling relationship as the non-mixed major axis regression (Fig. 2, green line). Based on the similarity of the results and statistical arguments (see Methods), we have based the following analyses on Bayesian major axis regressions, while results of the analyses with Bayesian mixed-effects major axis regressions are shown in the supplement.

Insert Fig. 2 here

In our second analysis, we used major axis regressions to fit six models of predatorprey body mass scaling that each contained one co-variable (ecosystem type, predator or prey metabolic type, predator or prey movement type, interaction dimensionality; see Supplementary Table 2 for variable description). Model comparisons demonstrated that adding any of these co-variables improves the fit substantially over the simple scaling model (Table 1, lower WAIC scores indicate higher model adequacy). According to these WAIC ranks, the best-performing models included predator metabolic type (rank 1) or predator movement type (rank 2) as co-variables, whereas models including the same trait variables of the prey led to lower ranks (Table 1, ranks 4 and 6). Hence, WAIC values suggested that predator traits were more important for determining body-mass scaling than prey traits.

Insert Table 1 here

While the overall relationship was superlinear (Fig. 3, black solid lines), the relationships for ectotherm and endotherm vertebrate predators exhibited strong sublinear scaling, implying that the body-mass ratios of vertebrate predators decrease with their body mass (Fig. 3a). As vertebrate prey often have vertebrate predators, a similar pattern might be expected for the scaling relationship within the prey metabolic groups. Surprisingly, we found superlinear scaling for all vertebrate prey groups (Fig. 3b). Together, these results suggest that the sublinear scaling characterizes vertebrate predators irrespective of whether their prey are vertebrate or invertebrate species.

The second most important co-variable in our analyses was predator movement type.

Interestingly, we found that swimming, flying and sessile predators exhibit superlinear scaling relationships that are similar to the overall model, whereas walking predators exhibit sublinear scaling (Fig. 3c). Although many walking predators feed on walking prey, our analyses of the prey movement type show superlinear scaling across groups (Fig. 3d).

Similar to the metabolic groups, this implies that changes in predator-prey body-mass ratios are mainly driven by predator movement type, irrespective of prey movement type. Together, our analyses of species' traits suggest that the traits of predators have stronger implications for scaling relationships and body-mass ratios than the traits of their prey.

Insert Fig. 3 here

Comparing the two environmental characteristics showed that ecosystem type (WAIC rank 3) improved the model substantially more than interaction dimensionality (2D vs. 3D; rank 5, Table 1). Among ecosystem types, marine, stream and terrestrial aboveground ecosystems follow superlinear scaling similarly to the overall relationship (although streams followed steeper scaling relationships), whereas lake and terrestrial belowground ecosystems exhibit sublinear scaling, parallel to each other (Fig. 4a). Both 2D and 3D interaction dimensionality demonstrate superlinear scaling with a slope similar to the overall pattern. However, 3D interactions tend to involve predator-prey pairs with greater body-mass ratios compared to 2D interactions (Fig. 4b). Although many marine or lake interactions occur in the pelagic 3D part of the ecosystem, ecosystem type does not completely overlap with interaction dimensionality as these aquatic ecosystems also include benthic 2D interactions.

Insert Fig. 4 here Finally, we addressed how well we can predict which predators in a food web have the highest average body-mass ratios compared with their prey in the absence of information on food-web structure and traits of the prey species. The statistical models thus included the predator traits (body mass, metabolic and movement type) and ecosystem type as independent variables ("predator-trait model", see Methods for details and Supplementary Table 4 for parameters). The overall predator-trait model fitted the data well (Fig. 5a). We found that predictive accuracy varied across ecosystem types and with the fraction of target predators (Fig. 5b). We anticipated that typically a low fraction of predators will be chosen for applied population management and used a fraction of target predators of 25% as an arbitrary example to illustrate our results (Fig. 5b, grey area; note that qualitatively similar results could be obtained for any fraction of 30% or lower). At this fraction of target predators, the accuracy of the predator-trait model predictions is almost always higher than the prediction accuracy when the same fraction of predators is chosen at random (Fig. 5b, diagonal line). An exception to this pattern were the terrestrial belowground systems (Fig. 5b), potentially as a consequence of the substantially higher degree of omnivory in soil communities [START_REF] Digel | Unravelling the complex structure of forest soil food webs: higher omnivory and more trophic levels[END_REF] or the widespread use of poison by soil predators [START_REF] Laigle | Species traits as drivers of food web structure[END_REF] . In contrast, the predatortrait model had high accuracy in streams (89%), marine (61%), terrestrial aboveground (64%) and lake ecosystems (61%), exceeding the 25% accuracy of random predictions (Fig. 5b, diagonal line in the grey area). This implies that for these ecosystems the predator-trait model improves the predictions by a factor between 2.44 (marine and lake ecosystems) and 3.56 (streams), which is close to the maximum improvement factor of 4 (occurring with 100% prediction accuracy relative to the 25% random prediction accuracy at a fraction of target predators of 25%, grey shaded area in Fig. 5b).

Insert Fig. 5 here

Discussion

Using a global database of 290 food webs we show that (1) the overall allometric scaling relationship between predator and prey body mass is superlinear, implying that the largest species have the highest body-mass ratios and that (2) predator traits (metabolic and movement type) are more important than prey traits in determining these scaling relationships. Subsequently, we developed a predator-trait model that successfully predicted the predators with the highest average body-mass ratio. Food-web theory has shown that these high body-mass ratios yield weak interactions with slow dynamics that are critically important for buffering communities against external perturbations and maintaining ecosystem functioning [START_REF] Schneider | Body mass constraints on feeding rates determine the consequences of predator loss[END_REF][START_REF] Mccann | Weak trophic interactions and the balance of nature[END_REF][START_REF] Brose | Allometric scaling enhances stability in complex food webs[END_REF][START_REF] Rooney | Structural asymmetry and the stability of diverse food webs[END_REF][START_REF] Otto | Allometric degree distributions facilitate food-web stability[END_REF][START_REF] Schneider | Animal diversity and ecosystem functioning in dynamic food webs[END_REF][START_REF] Wang | Biodiversity and ecosystem functioning in food webs: the vertical diversity hypothesis[END_REF] . Historically, these theoretical results have had little realworld application, as they require the logistically challenging task of assessing all or at least a large fraction of the food-web links. By focusing on predator traits and ecosystem type while discarding prey traits and the specific links of the food-webs, our predator-trait model provides a generalizable and feasible solution that can bridge the gap between food-web theory and applied ecosystem conservation. For instance, our results suggest that population protection of small vertebrates (e.g. mustelids) and large swimming (e.g. sharks)

or flying predators (e.g. birds of prey) might be most effective at buffering natural communities against external perturbations such as extinctions, invasions, pollution, eutrophication and warming. This trait-based approach enables the management of perturbation vulnerability in natural communities without detailed knowledge of the food-web structure.

Within the debate over the allometric scaling relationships of predator and prey bodymasses in natural food webs, the superlinear relationship presented here is consistent with some prior studies [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF] , while deviating from others that demonstrate sublinear scaling [START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF][START_REF] Cohen | Body sizes of animal predators and animal prey in food webs[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF] .

Our comparison of regression methods suggests that this discrepancy could be partially attributed to the alternative use of major axis regressions 1 (consistently yielding superlinear scaling) or ordinary least square (OLS) regressions [START_REF] Barnes | Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency[END_REF][START_REF] Cohen | Body sizes of animal predators and animal prey in food webs[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF] (suggesting superlinear or sublinear scaling depending on which is the independent variable). Our comparison of the two OLS regressions with either predator or prey mass as the independent variable reveals substantial uncertainty as they make opposite predictions on how body-mass ratios scale with predator and prey mass, and there is no a priori argument over which OLS regression should be preferred. Hence, major axis regressions are the most appropriate statistical method because: (1) there is no a priori expectation for a causal relationship between predator and prey mass, and (2) both body masses are quantified with the same measurement error [START_REF] Warton | Bivariate line-fitting methods for allometry[END_REF] . Our results show that major axis regression is not only statistically more appropriate but also that the choice of statistical approach has important biological implications for interpretation of the allometric scaling relationship. This approach, combined with our newly compiled food-web database, has enabled refining our understanding of how the scaling relationship between predator and prey body mass varies across ecosystems and between predator-prey combinations of different movement type and metabolic group.

Despite the overall superlinear relationship between predator and prey body mass, our analyses identified several species' traits and ecosystem characteristics that are associated with a sublinear scaling relationship. Most notably, both ectotherm and endotherm vertebrate predators demonstrate strong sublinear scaling, making predator metabolic type the most important factor among those we considered for predicting predator-prey body-mass scaling relationships. Consistent with previous research [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Cohen | Body sizes of animal predators and animal prey in food webs[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF] , we found that large vertebrate predators tend, on average, to feed on prey that are more equally sized (e.g. orcas feeding on minke whales), whereas small vertebrate predators consume relatively smaller prey (e.g. arctic foxes preying on lemmings). This result suggests that large and small vertebrate predators may be constrained by different factors, such as the limitations of maximum attack speed which are only experienced by the largest species [START_REF] Hirt | A general scaling law reveals why the largest animals are not the fastest[END_REF] . Interestingly, some (often large) vertebrate predators hunt in groups to attack larger prey to improve their attacking success and overcome the body mass and speed constraints. Indeed, the next most important factor in our analysis was predator movement type, which separates species categories of different speeds (e.g. flying predators are faster than walking predators). In our analysis, walking predators demonstrate sublinear scaling in contrast to all other movement types. The highest body-mass ratios were observed for the largest swimming and flying predators. Further investigations of the physiological constraints related to predator movement type, metabolic type and relative predator-prey body masses on predator feeding rates would help illuminate the processes behind these observed patterns. Generally, our model selection results suggest that predator metabolic and movement traits had much stronger effects on the scaling relationship than the equivalent prey traits. This is partially supported by the greater similarity between the major axis regression (accounting for bi-directional causalities) and the OLS regression with prey body mass as the dependent variable. We therefore conclude that top-down prey selection by predators has a stronger effect on prey mass than does the bottom-up influence of prey mass on predator masses. It is likely that both top-down and bottom-up influences are important, but our results indicate the dominance of the former, which stimulated the development of the predator-trait models of our third analyses predicting which predators have the highest average body-mass ratios across food webs.

Our results also identify ecosystem type as an important co-factor of the predator-prey body-mass scaling relationship, which is generally consistent with prior studies [START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF][START_REF] Carbone | Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments[END_REF][START_REF] Tucker | Examining predator-prey body size, trophic level and body mass across marine and terrestrial mammals[END_REF] . We expected this effect to be partially explained by the habitat dimensionality of the interaction (2D or 3D), which has important consequences for the strength of predator attack rates [START_REF] Carbone | Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments[END_REF][START_REF] Pawar | Dimensionality of consumer search space drives trophic interaction strengths[END_REF] .

Although we found an effect of interaction dimensionality with overall higher body-mass ratios in 3D than in 2D habitats, surprisingly it did not explain the different scaling relationships between different ecosystem types. This may be explained by the fact that the ecosystem type varies across food webs, whereas variance in interaction dimensionality plays an important role across the different predator-prey pairs with food webs. The superior explanatory power of the model including ecosystem type compared to that including interaction habitat dimensionality suggests that there are ecosystem characteristics not related to dimensionality, such as laminar viscosity, that may have a stronger effect on predator-prey interactions. We found relatively high body-mass ratios and a very steep bodymass scaling relationship in stream ecosystems. The streams exhibit some differences to the other ecosystem types of our database: (1) the higher physical drag force of the water, (2) the higher dependence on allochthonous resources, (3) the dendritic environmental structure, and (4) the relatively narrower range of body masses included in our data. While each of these points could be responsible for the difference in scaling relationships, the last point calls for additional data on stream interactions between larger species such as fish to see if the steep increase in the scaling relationship holds. While terrestrial aboveground and marine interactions exhibited superlinear scaling relationships as the overall relationship, those of lake and terrestrial belowground systems were sublinear. Furthermore, the lack of vertebrate predators with high body masses and high body-mass ratios may at least partially explain this for soil communities, but this surprising result requires more mechanistic investigation of the so far untested similarity between lake and belowground interactions.

Our approach to characterize predator-prey body-mass ratios in natural food webs has some limitations. First, in order to encompass a wide range of body masses, taxonomy and ecosystem types, we assume that interacting individuals have population-average body masses [START_REF] Ings | Ecological networks -beyond food webs[END_REF] . As in prior studies [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Cohen | Body sizes of animal predators and animal prey in food webs[END_REF][START_REF] Riede | Stepping in Elton's footprints: a general scaling model for body masses and trophic levels across ecosystems[END_REF][START_REF] Tucker | Examining predator-prey body size, trophic level and body mass across marine and terrestrial mammals[END_REF] , we rely on population-averaged body masses, since we do rarely have measurements for the actual body masses of the interacting individuals.

Thus, for many predator species, particularly those with ontogenetic diet shifts, actual bodymass ratios are likely to have a lower variation than body-mass ratios calculated from population averages. Unfortunately, the lack of individual data for entire food webs across ecosystems hampers any alternative approach. As prior comparisons of individual-based versus population-based food webs have shown [START_REF] Ings | Ecological networks -beyond food webs[END_REF][START_REF] Nakazawa | Scale dependence of predator-prey mass ratio: Determinants and applications[END_REF] , our population-based approach likely underestimates the intercepts of the scaling relationships. Second, the study sampling design, environmental factors such as temperature and the species' phylogeny may also affect the scaling relationship [START_REF] Naisbit | Phylogenetic signal in predatorprey body-size relationships[END_REF][START_REF] Wood | Effects of spatial scale of sampling on food web structure[END_REF][START_REF] Dobashi | Decomposing the effects of ocean environments on predator-prey body-size relationships in food webs[END_REF][START_REF] Gibert | Temperature alters food web body-size structure[END_REF] , and these would ideally be included as co-variables in the analyses. As these data were not systematically available for the data sets included, we accounted for them by random effects in mixed models 2 , which leaves the need for more detailed analyses for future studies. As major axis regressions with random effects are not generally available, we addressed this issue by using Bayesian models throughout the study, which allowed comparisons with hierarchical models including random effects (i.e. mixed-effects models). Although the mixed effects model results do not change our findings substantially (see Supplement for a comparison between mixed and non-mixed Bayesian models) and, due to potentially confounding clustering effects (see Methods), we have focused our analysis on the model without mixed effects. Third, our analyses were restricted to predator-prey interactions, whereas interactions of other consumer types such as parasites, parasitoids or herbivores were excluded. As these interaction types are typically characterized by different body-mass ratios [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Lafferty | Parasites dominate food web links[END_REF][START_REF] Marcogliese | Parasites in food webs: the ultimate missing links[END_REF] , future studies should address their scaling relationships in our GATEWAy database. Fourth, we employed simple scaling relationships with up to one single co-variable to gain an in-depth mechanistic understanding, whereas models with interactions between multiple co-variables were omitted from our analyses of the predator-prey body-mass scaling. These more complex relationships with higher order interactive effects, however, could be addressed by black box approaches such as machine learning algorithms, which could provide accurate predictions of food-web structures [START_REF] Laigle | Species traits as drivers of food web structure[END_REF] . Fifth, our study illustrates systematic differences in body-mass ratios across ecosystem types and species' traits, whereas explanations for these differences remain to be revealed by studies integrating mechanistic models with our data.

Our analyses provide insights into how predator and prey body masses scale with each other in natural food webs. The discovery that predator traits are more important than prey traits in predicting body-mass scaling and that ecosystem type has a greater effect than interaction dimensionality offers new possibilities for understanding and predicting differences in food-web structure, community stability and ecosystem functioning across community and ecosystem types. Specifically, our results highlight that critically important high body-mass ratios occur in interactions with predators that are (1) small vertebrates or (2) large swimming or flying species. With only three species traits (body mass, metabolic and movement type), our models were able to predict which 25% of the predators possess the highest average body-mass ratios with surprisingly high accuracy in most ecosystem types (58-89%). We anticipate that this accuracy will be increased by additional species traits (e.g. predation strategy, use of poison, sub-habitat association) that compose the multiple dimensions of natural food webs [START_REF] Eklof | The dimensionality of ecological networks[END_REF][START_REF] Laigle | Species traits as drivers of food web structure[END_REF] . Our trait-based food-web analyses enable generalizations of food-web theory from the food webs studied to the vast majority of communities for which only species and trait information is available. Updated with additional traits, this approach has great potential for managing ecosystem functioning and stability against external perturbations such as pollution, eutrophication and warming without full knowledge of food-web structure. The trait-based body-mass ratio approach therefore presents an important integration of food-web theory with applied ecosystem management that provides a theoretical foundation for the community-level conservation of the most complex natural ecosystems.

Methods

We compiled a global database of traits and food-web architecture (GATEWAy version 1.0, see Supplement), where each link is characterized by the taxonomy and trait variables of both the consumer and the resource (see Supplement metadata for variables). We included food webs with (1) a sufficient quality in terms of taxonomic resolution, which prevents nodes aggregating species with very different trophic interactions; (2) a reasonable completeness integrating all trophic levels and community compartments; (3) trait information for the trophic species including at least their population-averaged body mass, their metabolic type and their movement type (see Supplementary metadata table for definitions); (4) information for each trophic link such as the type (e.g. predacious), the dimensionality (2D and 3D) and the classification (individual-based and non-individualbased) (see Supplement metadata table for definitions); (5) descriptors for the ecosystems such as the ecosystem type and the geographic location.

In our analyses, we focused on predatory (variable: interaction.type) and individualbased (variable: interaction.classification) interactions. The former excludes interactions of other types (e.g. herbivorous, detritivorous, parasitic, parasitoid), whereas the latter discards interactions of consumers attacking groups, swarms or films of resources (e.g. filter feeding, grazing). Some of the studies included in our database sampled the same ecosystem at different locations, resulting in replicated predator-prey species pairs [START_REF] Digel | Unravelling the complex structure of forest soil food webs: higher omnivory and more trophic levels[END_REF] . To avoid pseudoreplication, each unique combination of taxonomy, life stage, and individual body mass for predator and prey species was included only once. After exclusion of interactions with missing variables, the resulting data included 88,197 unique predator-prey interactions among the original 222,151 feeding links.

First, we analyzed the reduced data for the relationship between the base-10 logarithms (log10) of predator and prey body masses [gram fresh mass]. We compared the fit of two ordinary least squares (OLS) regressions (either predator mass or prey mass as the dependent variable) to that of a major axis regression and a mixed-effects major axis regression including random effects on the intercept of the different studies (variable: link.citation). Traditional methods only allow to fit either major axis regressions or mixed models with random effects. Hence, our aim of comparing major axis regressions with and without random effects (i.e. random intercepts for each study) could only be achieved by realizing models that were fitted by Bayesian methods using the RStan package [START_REF] Development | RStan: the R interface to Stan[END_REF] (see supplementary statistical methods for details). Consistent with traditional major axis regressions, we minimized the sum of squared orthogonal distances of the observations (x,y) to the regression line [START_REF] Warton | Bivariate line-fitting methods for allometry[END_REF] instead of the vertical distance (y) as in OLS (model I) regressions.

Second, we used Bayesian major axis models to compare the fit of the simple scaling model to six models, whereby each included one co-variable: ecosystem type, predator or prey metabolic type, predator or prey movement type or interaction dimensionality (see Supplementary Table 2 for variables). Overall, the results were mostly consistent between the mixed-effects and non-mixed models. The mixed-effects models fit the relationships separately for each study. As the body-mass ranges within studies do not cover the entire body-mass gradient and the number of data points within studies is much lower than in the entire database, some of the fitted scaling relationships can become arbitrary as single points can strongly affect the slope. Averaging across all slopes and all intercepts using hierarchical approaches can lead to clusters of such arbitrary slopes, which can exert substantial leverage on the average relationship across all studies. In our data, the clustering remained even when using random intercepts and a fixed slope across all studies.

Therefore, the mixed-effects modelling of our data suffered from two limitations: (1) it loses information about the overall trend across the whole database (i.e. none of the study-specific scaling relationships spans the entire body-mass gradient), and (2) the joint mean slope and intercept are affected by partially arbitrary slopes (data sets with few points). As both regressions also yielded qualitatively similar results, we report the results of the non-mixed major axis regressions in the manuscript (Figs. 3,4) with comparisons to the fits of the mixed major axis regressions in the supplement (Supplementary Figures 234567). Model comparison (based on their WAIC values, Watanabe-Akaike Information Criterion) of these seven models (the simple model without co-variable and the six models with one co-variable each) provided a ranking of their performance, and we used the model parameters to gain an understanding of how they modify the relationship. In the analysis of predator-prey bodymass scaling, we refrained from analyzing more complex models with interactions between these co-variales for three reasons: (1) they imply impossible combinations (e.g. swimming predators in terrestrial ecosystems), (2) their higher order interactions hamper the mechanistic understanding of individual effects, and (3) their strong collinearity causes interference between factors. Third, we analyzed our database for the dependence of the predators' average predator-prey body-mass ratios on predator traits (body mass, metabolic and movement type) and ecosystem type. To avoid circularity in the statistical model (predator body mass in both the dependent and independent variables), we fitted Bayesian major axis regressions with log10 prey mass as the dependent and log10 predator mass as the independent variable with the co-variables predator metabolic type, predator movement type, and ecosystem type.

We restructured the resulting predator-trait model equation to calculate the effect of the independent and co-variables on predator-prey body-mass ratios. By discarding prey species traits, these analyses allow prediction of which predators in a community have the highest average body-mass ratios without knowledge of the predator-prey links.

The accuracy of this approach was determined in a five-step cross-validation process.

First, we chose one of the food webs ("test data") and ranked its predators according to their empirical average body-mass ratios. Second, we ran the predator-trait regression model described above in the remaining database containing the other 289 food webs ("training data") to predict the predators' average body-mass ratios depending on their traits. Third, we calculated the proportion of predators that were correctly predicted by this "predator-trait model" (hereafter: accuracy) for a fraction x of the highest ranked predators of the test-data food web (hereafter: fraction of target predators). For example, a fraction of target predators of 0.1 implies that the 10% highest ranked predators (i.e. those with the highest average body-mass ratios) of the empirical "test data" are compared to the 10% highest ranked predators as predicted by the predator-trait model of the "training data". An exemplary accuracy of 0.8 would indicate an 80% overlap between the two species lists. Fourth, this assessment of prediction accuracy was systematically replicated across a gradient in the fraction of target predators x between 5% and 95% (steps of 5%). Finally, these four steps were repeated for each of the 290 food webs independently to calculate the average accuracy across food webs depending on the fraction of target predators. See Supplementary Table 3 for model parameters. Our extensive statistical considerations hold that the Bayesian major axis regression (black line) is the most appropriate model. 3 for model parameters. 3 for model parameters.

Fig. 1 :

 1 Fig. 1: The global distribution of food webs in GATEWAy (GlobAl daTabasE of traits and food Web Architecture, GATEWAy version 1.0, see Supplement).

Fig. 2 :

 2 Fig. 2: Overall scaling of predator and prey body mass assessed by four regression methods (n=88,197). Ordinary least squares regression (OLS) of prey mass depending on predator mass (blue line), ordinary least squares regression of predator mass depending on prey mass (magenta line), Bayesian major axis regression (black line), mixed Bayesian major axis (ma) regression with random intercepts (green line). Bayesian regression parameters are the means of the posterior distributions. The dashed line indicates equal body masses of predator and prey for comparison. All body masses are gram fresh masses.

Fig. 3 :

 3 Fig. 3: Species' traits constrain the scaling of log10 predator body mass with log10 prey body mass (n=88,197): (a) predator metabolic type, (b) prey metabolic type, (c) predator movement type, (d) prey movement type. Solid black lines represent the overall scaling relationship, and the colored lines show the relationships for subgroups. Bayesian regression parameters are the means of the posterior distributions. Dashed lines indicate equal body masses of predator and prey for comparisons. See Supplementary Table3 for

Fig. 4 :

 4 Fig. 4: Ecosystem characteristics constrain the scaling of log10 predator body mass with log10 prey body mass (n=88,197): (a) ecosystem type, (b) interaction dimensionality. Solid black lines represent the overall scaling relationship, and the coloured lines show the relationships for the subgroups. Bayesian regression parameters are the means of the posterior distributions. Dashed lines indicate equal body masses of predator and prey for comparisons. See Supplementary Table3for model parameters.

Fig. 5 :

 5 Fig. 5: The predator-trait model predicts the target predators with the highest body-mass ratios across different ecosystem types (color code) (n=7296). (a) Observed versus predicted average body-mass ratios characterize the goodness of fit (R²=0.633, RMSE=0.914). Dashed diagonal line shows where observations and predictions are identical. (b) Accuracy (proportion of correct predictions) in an out-of-sample food web depending on the fraction of target predators to be predicted. The diagonal line characterizes predictions when predators are chosen at random. Grey area corresponds to an exemplary

Acknowledgements

This study was supported by the German Centre for integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig funded by the German Research Foundation (FZT 118). RT was supported by an Australian Research Council Future Fellowship (FT110100957). ACI was supported by the Alexander von Humboldt Foundation. C.V. acknowledges a researcher position and the strategic project UID/MAR/04292/2013, funded by the Portuguese Science Foundation. We thank Lena Rohde, Florian Schwarzmüller and Anthony Dell for help in organizing a prior version of the database.

Data availability

The data supporting the findings of this study (GATEWAy 1.0) are available at the iDiv data repository [START_REF] Brose | GlobAL daTabasE of traits and food Web Architecture (GATEWAy) version 1.0. iDiv Data Repository[END_REF] .

Code availability

The R code of the statistical analyses is available as a supplement.

Competing interests

The authors declare no competing financial interests.

Table 1:

Comparison (Watanabe-Akaike Information Criterion, WAIC) of six predator-prey body-mass scaling models with one co-variable. Bayesian major axis models (ma) as in Fig. 3 and Fig. 4 of the main manuscript and mixed Bayesian major axis models as in Supplementary Figures 234567.

Bayesian ma