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Drag coefficients of plesiosaurs, ichthyosaurs and modern cetaceansAt equal Reynolds numbers (same body length and same flow velocity), the total drag coefficients of plesiosaurs (Cd) are higher than the estimated values for ichthyosaurs and modern cetaceans (Fig. 1a). The limbless bodies, however, display similar Cd in all three groups and are even lower-than-average in the long-necked plesiosaurs, indicating that the limbs are responsible for the observed high Cd. The limbs of plesiosaurs contribute to more than 20% of their total drag coefficient: up to 32.2% in the basal Meyerasaurus and averaging 25% in derived plesiosaurs, with no major differences between plesiosaur morphotypes. In parvipelvian ichthyosaurs the contribution of the limbs to Cd is 11.2–15.6%, compared to 8.7–14.3% in modern cetaceans. Some of the living taxa we include provide a functional reference for this analysis. Our computed drag coefficient for the bottlenose dolphin model (Cd = 0.00413 at Re = 107) for example, is consistent with the estimates from a gliding living dolphin33 (Cd = 0.0034 at Re = 9.1 × 106) and other static CFD simulations34 (Cd = 0.00413 at Re = 107). It is worth noting that these values are, as expected, lower than estimates obtained from kinematic models, as motion is not accounted for35. In a former study, drag coefficients for a plesiosaur (Cryptoclidus), two ichthyosaurs and various cetaceans were obtained from rigid models in water tanks36. However, the pressure drag component (Cp) was likely overestimated due to the proximity of the models to the air–water interface, and thus are not directly comparable to ours.Fig. 1: Comparison of the drag coefficient of derived plesiosaurs, ichthyosaurs and cetaceans.a Total drag coefficient computed for the full models including the limbs (‘body + limbs’, circles) and the limbless models (‘body’, squares). Average (point) and range (bar) shown for calculations at Re = 5 × 106–107. The derived short-necked plesiosaurs are highlighted in orange; the parvipelvian ichthyosaurs in blue and the extant cetaceans in red. A basal plesiosaur included as a reference is highlighted in purple. b Representative two-dimensional plots of the flow velocity magnitude at Re = 5 × 106 (inlet velocity of 5 ms−1) in lateral view. For dorsal view see Supplementary Fig. 1. Images of Tursiops and the three ichthyosaurs modified from Gutarra et al.29.Full size imageIn all models across the various clades, velocity plots display a stagnation point at the anterior tip of the model, a thin velocity gradient along the body corresponding to the boundary layer, an area of higher velocity around the greatest diameter and a low velocity wake behind the body, characteristic features of a fully developed external flow (Fig. 1b, Supplementary Fig. 1). The acceleration of flow results in areas of low pressure (Supplementary Fig. 2), while high pressure areas are observed where stagnation occurs. Our CFD methodology has been previously validated against experimental data from slender torpedo-like shapes26 and has been shown to provide a reliable distribution of internal drag components29 essential when dealing with streamlined bodies35. In all our simulations, the proportion of frictional and pressure drag was consistent with the expected values for slender geometries31: most of the drag originated from skin friction with a minor pressure drag component (Supplementary Fig. 2). The relatively larger limbs of plesiosaurs (Supplementary Table 1) produce a small increase in skin friction (Supplementary Fig. 2a), but a large increase in the pressure drag coefficient (Supplementary Fig. 2b), indicating that the latter largely explains differences in total drag coefficient between the groups. These effects might be explained by the low local Reynolds number of the flippers (resulting from a small chord length) producing high local Cd relative to the rest of the body31, alongside interference drag (i.e. drag caused by the interaction of flow fields where limbs and body meet), which might be higher for larger flippers.Effect of body shape and body size on drag-related costs of steady swimmingWhen comparing morphologies at the same volume (proxy for body mass) and the same velocity, to focus on the effect of shape alone, derived plesiosaurs produce on average 30% more drag than parvipelvian ichthyosaurs and modern cetaceans (Fig. 2a, Supplementary Table 3; two-sample t-tests p  0.05). In these conditions, the drag-related costs of steady swimming of plesiosaurs fall within the range observed in both modern cetaceans and ichthyosaurs. Normalised against a 2.85 m-long Tursiops, the COTdrag for derived plesiosaurs ranges from 0.42, estimated for the large elasmosaur Thalassomedon, to 1.41 in the medium-sized Dolichorhynchops. In the parvipelvians, COTdrag spans from 0.33 estimated for the large Temnodontosaurus, to 1.76 in a 2.5 m-long Stenopterygius. Cetaceans show a smaller lower limit, because they include the largest animal in our sample, a 16 m-long humpback whale, with a COTdrag of 0.13 compared to Tursiops. The estimated cetacean upper COTdrag limit is 1.54 for a 1.9 m Tursiops. On the other hand, comparisons of the total drag power (Pdrag, i.e., the non-mass normalised version of COTdrag) for the same speed of 1 ms−1 (Fig. 3), show a different trend. Pdrag is highest for Megaptera, higher than in any fossil taxa included in this study, and is lowest in Tursiops. Thalassomedon is comparable both in total drag power and COTdrag to the killer whale. Similarly, the thalassophonean pliosaurid Liopleurodon matches the elasmosaurian Hydrotherosaurus in having a similarly low mass-normalised COTdrag but requiring about 4× more total drag power than Tursiops. Smaller forms like the polycotylid Dolichorhynchops and the thunnosaurian Ophthalmosaurus resemble the extant bottlenose dolphin in having a relatively high COTdrag and low total power.Fig. 3: Comparative plot of mass-normalised drag power and total drag power.Values of mass-normalised drag power (i.e., drag per unit of volume or COTdrag calculated as in Fig. 2b) in grey, and non-mass-normalised total drag power, in black, for an array of derived plesiosaurs, parvipelvian ichthyosaurs and modern cetaceans compared at the same inlet velocity of 1 ms−1. Error bars represent minimum and maximum values accounting for taxon body size variation (see Supplementary Data). Values are normalised to the results for Tursiops.Full size imageThus, in contrast to the volume-normalised simulations, differences between animals at their life-size scale are mainly influenced by size. For example, medium-sized plesiosaurs and ichthyosaurs, such as Dolichorhynchops and Ophthalmosaurus, have values of COTdrag close to that of a dolphin, while large plesiosaurs like Thalassomedon are more like the parvipelvian ichthyosaur Temnodontosaurus and a modern Orcinus. It is worth noting that the inflow velocity of 1 ms−1, is a reference velocity used for comparative purposes, and is not equivalent to the optimal cruising speed (i.e. speed at which COT is minimum16). This parameter is known to vary little in nature, with most vertebrates displaying values of preferred speed between 1–2 ms−1 regardless of body size40,41,42, which means it is reasonable to assume all tested taxa, regardless of their size, were able to swim at this velocity. Using a different reference velocity (2 ms−1) has no effect on the relative values of drag per unit of volume and the mass-normalised drag power (Supplementary Fig. 3; Supplementary Data). A reduction of mass-normalised drag-related costs of cruising as body size increases is selectively advantageous, as energy savings can be used to extend foraging and mating range, increase swimming speed and fuel other activities42,43.Our analysis shows that for highly aquatic tetrapods, size dominates over shape in affecting the drag-related costs of steady locomotion. This is because COTdrag (i.e., the balance of drag to volume) is highly sensitive to surface/volume proportion (Fig. 2f), and so is much influenced by isometry in streamlined animals.Interplay between neck anatomy and body size in plesiosaur dragSimulations at constant Reynolds number (i.e., comparing models at same total length and same flow velocity), show that necks up to 5× the length of the trunk do not increase substantially the total drag coefficient. Longer neck ratios up to 7× were found to impact the drag coefficient by as little as 3% (Fig. 4a). We estimated a 4–10% increase in skin friction drag coefficient for neck ratios of 3–7×, but also a comparable reduction in pressure drag resulting in almost no change in the total drag coefficient. A previous CFD-based study also found no differences in drag coefficient between plesiosaur models with variable neck proportions20, but further comparison is not possible because of great differences in the order of magnitude of Cd, the use of a different scaling reference area and the lack of information on skin and pressure drag20. Here, we have shown that long necks produce only a small increase in skin friction, although not as great as previously speculated25,30, and this is nullified by reduced pressure drag.Fig. 4: Influence of neck length and its interaction with body size on the drag-related costs of swimming in plesiosaurs.a Total drag coefficient and skin friction drag coefficient for an array of hypothetical plesiosaurs with varying neck ratios computed at Re = 5 × 106 (same total length and inflow velocity). b Drag per unit of trunk volume computed for the same array of models scaled at the same trunk length and tested at the same speed of 1 ms−1. The hypothetical models were created by modifying the length in the model of the basal plesiosaur Meyerasaurus victor which has a neck ratio of 0.87×. The limits of the trunk (which extends along the torso and includes the edges of the pectoral and pelvic girdles) are shown in red in the rendered models. c Three-dimensional models of a wide array of plesiosaurs, in dorsal view, at their life-size dimensions, showing the differences in body proportions and sizes. The limits of the trunk in the models (defined as in b) are coloured by group. Basal plesiosaurs are highlighted in purple. Among the derived groups, thalassophonean plesiosaurs (derived pliosaurid plesiosaurs) are highlighted in light orange, polycotylid plesiosaurs in dark orange and elasmosaurid plesiosaurs in green. d Scatterplot of trunk length (cm) and neck ratio showing the relative drag per unit of trunk volume as a gradient of colour for each taxon analysed and for the plot area in between (contour lines represent the interpolated values of drag per unit of volume). e Plot of the relative drag per unit of trunk volume versus the trunk length showing results highlighted by group. Line plots at the right-hand side show the range for each group. The D/Vtr and the trunk length show a significant negative correlation (Pearson’s correlation coefficient calculated with log-transformed variables, p = 2.28 × 10−7, R2 = −0.92). A small version of the fitted power curve (regression equation (y=69.76{x}^{-0.94})) is shown on the right upper corner. The grey area around the curve represents a confidence interval of 95%. All values in b, d and e are normalized to the results for the Meyerasaurus model.Full size imageNext, we explored the impact of neck proportions on drag-related costs of swimming in simulations where the size factor is removed. We found that if trunk dimensions are kept constant while the neck is enlarged, the drag per unit of trunk volume does not change appreciably for neck ratios up to 2×. However, longer neck proportions did impact resistive forces. This was moderate for a 3× ratio, with 12% more drag per unit of trunk volume, but became more substantial for longer necks, with 22%, 35% and 59% excess drag for necks of 4×, 5× and 7× respectively (Fig. 4b). This means that elasmosaurine elasmosaurs, with necks commonly 3–4× the length of the trunk23 might have experienced higher drag than other plesiosaurs of similar trunk dimensions.To test if the ‘long neck effect’ remains when body size is accounted for, we compared the relative amount of drag-per-unit-trunk-volume (D/Vtr) in a wide sample of plesiosaurs (Fig. 4c) at life-size scale for a constant velocity of 1 ms−1, including three species with neck ratios above 2×: Styxosaurus (2.76×), Hydrotherosaurus (3.18×) and Albertonectes (3.72×), the last being the elasmosaur with the longest reported neck44. Our results show great variability in D/Vtr. Small-bodied plesiosaurs such as Plesiosaurus, Meyerasaurus and Dolichorhynchops generated up to six times more D/Vtr than the largest plesiosaurs, Kronosaurus and Aristonectes (Fig. 4d, e). Comparisons per group show that both basal plesiosaurs and derived polycotylids, the groups with the smallest specimens, produced generally higher D/Vtr. Moreover, we did not find substantial differences between elasmosaurs and thalassophonean pliosauroids (Fig. 4e, Supplementary Table 4; all two-sample t-tests p  > 0.05). Both groups had similarly low ranges of D/Vtr regardless of neck length, lower on average than in polycotylids. These results stand even if we exclude Aristonectes, which belongs to the aristonectines, an elasmosaur subfamily with reduced neck length23,45. Further comparisons by morphotype show no significant differences between short-necked pliosauromorphs (here arbitrarily including plesiosaurs with neck ratios below 2×) and long-necked plesiosauromorphs (Supplementary Table 4, all two-sample t-tests p  > 0.05). The highest values of D/Vtr occur in animals with trunk lengths of 100 cm or less, followed by a steep decrease between 100–150 cm and a steadier decrease in longer trunks. This indicates a strong negative correlation between trunk dimensions and D/Vtr (Pearson’s product-moment correlation between the log-transformed variables, adjusted r2 = −0.92, p = 2.28 × 10−7). The curve that best describes this relationship is the power equation, D/Vtr = 69.76 × Trunk length−0.944 (Fig. 4e), an almost inversely proportional relationship, consistent with the streamlined nature of these animals for which skin friction drag is dominant.Polycotylids and thalassophonean pliosaurs, both derived pliosauromorph plesiosaurs9,21, share the same general body proportions9,21,46, but the latter had larger bodies and therefore needed less power in relation to their muscles to move at the same speed. Elasmosaurs on the other hand, despite their disparate morphologies, were no different from thalassophonean pliosaurs in their drag-related costs of forward swimming (Fig. 4c–e) and therefore they were likely to have been equally efficient cruisers.Earlier research suggested that, even if long necks did not add extra drag during forward swimming, speed in elasmosaurs would have been limited to avoid added drag when their necks bent20. However, when the neck is bent in living forms, the course of swimming changes, as does the flow direction, but the body remains streamlined in the direction of incoming flow. For example, sea lions perform non-powered turns initiated by the head in which the body glides smoothly in a curved position, limiting deceleration47. Further biomechanical research is needed to understand the role of plesiosaur necks in manoeuvrability and other aspects of swimming performance, as well as how these were influenced by shape and flexibility. The well-established idea that long-necked plesiosaurs were sluggish, slow swimmers7,30 is thus not supported here, not because long necks did not increase drag20, but because body size overrode this drag excess.Long necks evolved in large-bodied plesiosaurs: implications for dragWe analysed trends of body size and neck proportion in a wider sample of sauropterygians, including plesiosaurian and non-plesiosaurian Triassic sauropterygians. Long necks (neck ratio > 3×) occur in taxa with trunk lengths > 150 cm, whereas most sauropterygians had neck ratios of ≤ 2× (Fig. 5a). The great plasticity of body proportions of sauropterygians before and after their transition to a pelagic lifestyle after the Triassic has been well documented21,23,46, but this is the first time that neck and body size have been explored in the context of swimming performance for such a wide sample. We show that overall, sauropterygians and particularly plesiosaurs, mainly explored neck morphologies with little or no effect on drag costs and did not enter morphospaces that were suboptimal for aquatic locomotion (i.e., corresponding to small trunks with long necks; Fig. 5a). In fact, ancestral state reconstruction for trunk length shows that the ancestor of elasmosaurs was likely around 180 cm long and had a relatively short neck with a ratio smaller than 2× (Fig. 5b, c). This indicates that large trunks preceded neck elongation in elasmosaurs and suggests that extreme proportions might have been favoured by a release of hydrodynamic constraints.Fig. 5: Evolutionary trends of neck proportions and body size in Sauropterygia and their implications for the drag-related costs of swimming.a Bivariate plot of the length of trunk and the neck ratio of 79 sauropterygian taxa. Polygons in different colours show area occupied by the main sauropterygian groups. The functional trends describing the effect of each axis are based on results from flow simulations. On the top of this graph, a univariate plot shows the distribution and mean values of trunk length for each group. b, c Phenograms showing the disparity of trunk length (b) and neck ratio (c) in sauropterygians through time. The branches corresponding to basal Plesiosauria (including Rhomaleosauridae and Plesiosauridae), thalassophonean pliosaurs, polycotylids and elasmosaurs are highlighted (colour coding as in a). d, e Sauropterygian trees showing the evolutionary rates for trunk length (d) and neck ratio (e) represented by colour gradient (see Supplementary Fig. 5 for an alternative analysis to 5d using the log10-transformed trunk length). Consensus trees show average results from analyses of 20 cal3-dated trees (see Supplementary Figs. 4 and 6 for analysis on Hedman-dated trees). Rates are based on the mean scalar evolutionary rate parameter.Full size imageWe next explored evolutionary rates of relative neck length and trunk length in sauropterygians. The pattern of trunk length evolution is consistent with a heterogeneous rates model, not a homogeneous Brownian motion model (log Bayes Factor48 (BF)  > 5 in 100% of the sampled trees and > 10 in 92.5%, Supplementary Table 5). Analysis of non-transformed trunk data shows that through the evolution of Sauropterygia, there was a general increase in trunk length with some higher rates, in Triassic nothosauroids, Jurassic rhomaleosaurids and Cretaceous aristonectine elasmosaurs (Fig. 5d; Supplementary Fig. 4a). Additionally, analysis of the log10-transformed trunk data highlights variation in the small-to-medium size ranges and reveals high rates in Triassic eosauropterygians (Supplementary Figs. 5 and 6). The largest trunks evolved independently in two groups, thalassophonean pliosaurids and elasmosaurid plesiosauroids, with no evidence of high rates in the former. In the plesiosauroids, rates are not particularly high in the basal branches, but they are very high in derived aristonectines, and rates for the whole clade were significantly higher than the background rate in 40% of randomisation tests (Supplementary Fig. 7 and Table 6). A progressive increase in body mass over evolutionary time has been described for various clades of aquatic mammals49 and seems to be a common hallmark of the aquatic adaptation to marine pelagic lifestyles in secondarily aquatic tetrapods44. Whether body size reaches a plateau as is the case in cetaceans49 and what constraints influence the evolutionary patterns of size in plesiosaurs remains unexplored. Against this general trend, some derived plesiosaurs, such as polycotylids, saw a reduction in body size, which might have been related to pressures on niche selection, such as adaptation to specific prey, the need for higher manoeuvrability or other ecological factors. As shown earlier, small sizes require lower amounts of total power for a given speed, and therefore would be favoured if for example food resources were limited. This suggests that, in spite of the energy advantages of large size in terms of reduced mass-specific drag29 and metabolic rates49,50, which make it a common adaptation to the pelagic mode of life, other constraints limiting very large sizes were also at work50,51.A heterogeneous evolutionary rates model for neck proportion is also strongly supported (log BF  > 5 in 100% of the sampled trees and > 10 in 45%, Supplementary Table 5). Fast rates are consistently seen at the base of Pistosauroidea (including some Triassic forms and plesiosaurs) and, interestingly, also within elasmosaurs (Fig. 5e; Supplementary Fig. 4b). The neck proportions of elasmosaurs were found to evolve at a faster pace than the background rate in 90% of analyses (randomisation test p-value < 0.001 in 80% and < 0.01 in 10% of the sampled trees; Supplementary Fig. 7 and Table 6). Very fast rates in elasmosaurs are concentrated in the most derived branches (i.e., Euelasmosauridia from the late Upper Cretaceous52) and represent both rapid neck elongation in elasmosaurines and rapid neck shortening in weddellonectians (i.e., aristonectines and closely related taxa52). Additionally, various other independent instances of relative shortening of the neck occurred during the evolution of Sauropterygia, most notably in placodonts, pliosaurs and polycotylids, but these are not associated with high rates.Our findings contrast with a previous study23 which did not identify any significant evolutionary rate shifts in the neck ratio across Sauropterygia. Here we use a larger number of taxa and a different model fitting approach, which might account for these discrepancies. The association between very long necks and large trunks, along with our flow simulations results and the evidence of high rates in the elongation of necks in elasmosaurines (Fig. 5e), suggests that neck elongation was facilitated by large body sizes. The question remains why neck ratios did not evolve longer than 4×. According to our data, hydrodynamic constraints might have operated against the selection of such long necks. However, it is possible that the primary function for which they were selected, which is still debated30,53, did not require necks with those characteristics. Neck anatomy is likely to be the result of a compromise between different functions/constraints, one of them being hydrodynamic, as shown by the results presented herein. More

Minimum distance to weak efficient frontierBriec and Charnes et al. first proposed the Minimum distance to weak efficient frontier (MinDW) model39,40, which can be expressed as (m + n) linear programming ((m) is the number of input indicators and (n) is the number of output indicators), assuming that the input variable is (x) and the output variable is (y). The specific formula is shown in Eq. (1):$$ begin{aligned} & max beta_{z} ,z = 1,2, ldots ,m + n \ & s.t.left{ begin{gathered} sumnolimits_{j = 1}^{q} {alpha_{j} x_{rj} + beta_{z} e_{r} le x_{rk} ,r = 1,2, ldots ,m} hfill \ sumnolimits_{j = 1}^{q} {alpha_{j} x_{ij} + beta_{z} e_{i} ge y_{ik} ,i = 1,2, ldots ,n} hfill \ alpha_{j} ge 0 hfill \ end{gathered} right. \ end{aligned} $$
(1)
(e_{r}) and (e_{i}) are constants. In the programming formula, only one (e) is equal to 1, and the others are 0, that is shown in Eq. (2):$$ begin{aligned} & e_{r} = 1;{text{ if}}; , r = z; , e_{r} = 0 , ;{text{if}}; , r ne z \ & e_{i} = 1 , ;{text{if}}; , i = z – m; , e_{r} = 0 , ;{text{if}}; , i ne z – m \ end{aligned} $$
(2)
The efficiency value of model is expressed as Eq. (3):$$ theta_{z}^{*} = frac{{1 – frac{1}{m}sumnolimits_{r = 1}^{m} {frac{{beta_{z}^{*} e_{r} }}{{x_{rk} }}} }}{{1 + frac{1}{n}sumnolimits_{i = 1}^{n} {frac{{beta_{z}^{*} e_{i} }}{{y_{ik} }}} }} $$
(3)
The efficiency value of MinDW model is expressed as (theta_{max }^{*} = max (theta_{z}^{*} ,z = 1,2, cdots ,m + n)), and the maximum efficiency value corresponds to the minimum (beta^{*}), that is the nearest distance to the frontier.This paper uses the MinDW model with negative output to conduct empirical analysis. The method can be expressed as (m + n + d) linear programming ((m) is the number of inputs, (n) is the number of desirable output, (d) is the number of unexpected output), assuming that the input variable is (x), the desirable output variable is (y), and the undesirable output variable is (f). The specific formula is shown in Eq. (4):$$ begin{aligned} & max beta_{z} ,z = 1,2, ldots ,m + n + d \ & s.t.left{ begin{gathered} sumnolimits_{j = 1}^{q} {alpha_{j} x_{rj} + beta_{z} e_{r} le x_{rk} ,r = 1,2, ldots ,m} hfill \ sumnolimits_{j = 1}^{q} {alpha_{j} x_{ij} – beta_{z} e_{i} ge y_{ik} ,i = 1,2, ldots ,n} hfill \ sumnolimits_{j = 1}^{q} {alpha_{j} x_{lj} + beta_{z} e_{l} le f_{lk} ,l = 1,2, ldots ,d} hfill \ alpha_{j} ge 0 hfill \ end{gathered} right. \ end{aligned} $$
(4)
(e_{r}), (e_{i}) and (e_{l}) are constants. In the programming formula, only one (e) is equal to 1, and the others are 0, that is shown in Eq. (5):$$ begin{aligned} & e_{r} = 1;{text{ if}}; , r = z; , e_{r} = 0 , ;{text{if}}; , r ne z \ & e_{i} = 1 , ;{text{if }};i = z – m; , e_{r} = 0 , ;{text{if}}; , i ne z – m \ & e_{l} = 1 , ;{text{if}}; , l = z – m – n; , e_{l} = 0 , ;{text{if}}; , l ne z – m – n \ end{aligned} $$
(5)
The efficiency value of model is expressed as Eq. (6):$$ theta_{z}^{*} = frac{{1 – frac{1}{m}sumnolimits_{r = 1}^{m} {frac{{beta_{z}^{*} e_{r} }}{{x_{rk} }}} }}{{1 + frac{1}{n + d}left( {sumnolimits_{i = 1}^{n} {frac{{beta_{z}^{*} e_{i} }}{{y_{ik} }}} + sumnolimits_{l = 1}^{d} {frac{{beta_{z}^{*} e_{l} }}{{f_{lk} }}} } right)}} $$
(6)
The efficiency value of MinDW model is expressed as (theta_{max }^{*} = max (theta_{z}^{*} ,z = 1,2, cdots ,m + n + d)), and the maximum efficiency value corresponds to the minimum (beta^{*}), which means the nearest distance to the frontier.The efficiency value of MinDW model will not be less than the efficiency value of directional distance function model with any direction vector or other distance types (such as radial model and SBM model). In other words, the efficiency value of MinDW model is the largest. Combined with the above process, we can define the common boundary ((beta^{meta*})) and the model is as Eq. (7):$$ begin{aligned} & beta^{meta*} = max frac{{1 – frac{1}{m}sumnolimits_{r = 1}^{m} {frac{{beta_{z} e_{r} }}{{x_{rk} }}} }}{{1 + frac{1}{n + d}left( {sumnolimits_{i = 1}^{n} {frac{{beta_{z} e_{i} }}{{y_{ik} }}} + sumnolimits_{l = 1}^{d} {frac{{beta_{z} e_{l} }}{{f_{lk} }}} } right)}} \ & s.t.left{ begin{gathered} sumnolimits_{j = 1}^{{q_{m} }} {alpha_{j} x_{rj} + beta_{z} e_{r} le x_{rk} ,r = 1,2, cdots ,m} hfill \ sumnolimits_{j = 1}^{{q_{m} }} {alpha_{j} x_{ij} – beta_{z} e_{i} ge y_{ik} ,i = 1,2, cdots ,n} hfill \ sumnolimits_{j = 1}^{{q_{m} }} {alpha_{j} x_{lj} + beta_{z} e_{l} le f_{lk} ,l = 1,2, cdots ,d} hfill \ alpha_{j} ge 0 hfill \ end{gathered} right. \ end{aligned} $$
(7)
Similarly, the efficiency value of DMU relative to the scale frontier ((beta^{scale*})) can be obtained by the Eq. (8):$$ begin{aligned} & beta^{scale*} = max frac{{1 – frac{1}{m}sumnolimits_{r = 1}^{m} {frac{{beta_{z} e_{r} }}{{x_{rk} }}} }}{{1 + frac{1}{n + d}left( {sumnolimits_{i = 1}^{n} {frac{{beta_{z} e_{i} }}{{y_{ik} }}} + sumnolimits_{l = 1}^{d} {frac{{beta_{z} e_{l} }}{{f_{lk} }}} } right)}} \ & s.t.left{ begin{gathered} sumnolimits_{j = 1}^{{q_{s} }} {alpha_{j} x_{rj} + beta_{z} e_{r} le x_{rk} ,r = 1,2, ldots ,m} hfill \ sumnolimits_{j = 1}^{{q_{s} }} {alpha_{j} x_{ij} – beta_{z} e_{i} ge y_{ik} ,i = 1,2, ldots ,n} hfill \ sumnolimits_{j = 1}^{{q_{s} }} {alpha_{j} x_{lj} + beta_{z} e_{l} le f_{lk} ,l = 1,2, ldots ,d} hfill \ alpha_{j} ge 0 hfill \ end{gathered} right. \ end{aligned} $$
(8)
Finally, in the common frontier model, the technology gap ratio (TGR) is equal to the ratio of the efficiency value of the common frontier to the scale frontier41. The formula is as Eq. (9):$$ TGR^{MinDW} = frac{{beta^{meta*} }}{{beta^{scale*} }} $$
(9)
(beta^{meta*}) and (beta^{scale*}) represent the optimal solution of formula (7) and formula (8), respectively. Obviously, (0 le TGR le 1). TGR is used to measure the distance between the optimal production technology and the potential optimal technology of a group, and identify whether there are any differences in LHG under different groups. The closer the TGR is to 1, the closer the technology level is to the optimal potential technology level. Conversely, it shows the larger gap between the technology level and the potential optimal technology level.Metafrontier-Malmquist–Luenberger indexMalmquist productivity index is widely used in the study of dynamic efficiency change trend, and has good adaptability to multiple input–output data and panel data analysis. The actual production process often contains unexpected output. After Chung et al. proposed Malmquist–Luenberger (ML) index, any Malmquist index with undesired output can be called ML index42. Oh constructed the Global-Malmquist–Luenberger index43. All the evaluated DMUs are included in the global reference set, which avoids the phenomenon of infeasible solution in VRS. The global reference set constructed in this paper is as Eqs. (10)–(11):$$ Q^{G} left( x right) = Q^{1} left( {x^{1} } right) cup Q^{2} left( {x^{2} } right) cup cdots cup Q^{T} left( {x^{T} } right) $$
(10)
$$ Q^{t} left( {x^{t} } right) = left{ {left( {y^{t} ,f^{t} } right)left| {x^{t} ;can;produce} right.;left( {y^{t} ,f^{t} } right)} right} $$
(11)
This paper takes MML index as the LHG.$$ begin{aligned} MML_{t – 1}^{t} & = sqrt {frac{{1 – D_{t – 1} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{t – 1} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} times frac{{1 – D_{t} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{t} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}}} \ & = sqrt {frac{{1 – D_{t – 1} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}}{{1 – D_{t} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} times frac{{1 – D_{t – 1} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{t} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}} \ & ;;;;; times frac{{1 – D_{t} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{t – 1} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} \ end{aligned} $$
(12)
Next, it further decompose the MML index into efficiency change (EC) and technology change (TC). The specific formula is shown in Eqs. (13)–(14):$$ TC_{t – 1}^{t} = sqrt {frac{{1 – D_{t – 1} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}}{{1 – D_{t} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} times frac{{1 – D_{t – 1} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{t} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}} $$
(13)
$$ EC_{t – 1}^{t} = frac{{1 – D_{t} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{t – 1} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} $$
(14)
where (left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} } right)) and (left( {x^{t} ,y^{t} ,f^{t} } right)) represent the input, expected output and unexpected output of t-1 and t, respectively. (TC_{t – 1}^{t}) is the devotion to LHG raise of DMU’s technical progress from (t – 1) to (t). And (EC_{t – 1}^{t}) represents the devotion to LHG raise of DMU’s efficiency improvement from (t – 1) to (t). The higher the value is, the larger the devotion is. The (MML) index is recorded as (MI). The value of (MI) is the LHG. The green total factor productivity index of laying hens breeding under the common frontier and scale frontier are as Eqs. (15)–(16):$$ metaMI_{t – 1}^{t} = sqrt {frac{{1 – D_{{_{t – 1} }}^{m} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{{_{t – 1} }}^{m} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} times frac{{1 – D_{{_{t} }}^{m} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{{_{t} }}^{m} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}}} $$
(15)
$$ groupMI_{t – 1}^{t} = sqrt {frac{{1 – D_{{_{t – 1} }}^{g} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{{_{t – 1} }}^{g} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}} times frac{{1 – D_{{_{t} }}^{g} left( {x^{t} ,y^{t} ,f^{t} ;y^{t} , – f^{t} } right)}}{{1 – D_{{_{t} }}^{g} left( {x^{t – 1} ,y^{t – 1} ,f^{t – 1} ;y^{t – 1} , – f^{t – 1} } right)}}} $$
(16)
For the DMUs with scale heterogeneity, we can measure the technology gap between the group frontier and the common frontier, which is caused by the specific group structure.Data and variablesBased on the research of the existing literature36, this paper selects five indexes to build the input–output indicator system. Details are as below:

1.

Input variables:

(1)

Quantity of concentrated forage. Mainly includes seeds of crops and their by-products.

(2)

Quantity of grain consumption. Quantity of grain consumed is the quantity of grain consumed by laying hens when they are raised. For example: corn, sorghum, broken rice, wheat, barley, wheat bran, etc.

(3)

Material expenses. The sum of water and fuel power costs, labor costs, and medical epidemic prevention fees. Water and fuel power costs include water, electricity, coal and other fuel power costs; labor costs mean the human management cost of each laying hen from the brood stage to the laying stage; medical and epidemic prevention costs include the cost of disease prevention and control.

2.

Positive output Main product production, which is the egg production per layer.

3.

Negative output Total discharge. According to the calculation method of The Manual of Pollutant Discharge Coefficient, Eq. (17) is used to calculate the COD, TN, and the TP of each layer. Then, according to the calculation method of class GB3838-2002 water quality standard in V, Eq. (18) is used to calculate the total discharge.

$$ POLLUTANTS = FP(FD) times Days $$
(17)
$$ TOTAL , POLLUTANTS = frac{COD}{{40}} + frac{TN}{2} + frac{TP}{{0.4}} $$
(18)
where, (FP(FD)) is the pollution discharge coefficient and the (Days) is the average raising days. Descriptive statistics of input and output indicators are shown in Table 1.Table 1 Descriptive statistics of input and output indicators.Full size tableThe quantity of concentrate, the quantity of food consumed, the cost of labor, the cost of medical treatment all come from “National Agricultural Product Cost and Benefit Data Compilation”. The pollutant discharge coefficient of laying hens is derived from “The Manual of Pollutant Discharge Coefficient”. According to the definition of scale in above two materials, a small scale 300–1000 laying hens, a medium scale 1000–10,000 laying hens, and a large scale greater than 10,000 laying hens are grouped to calculate cost efficiency.From 2004 to 2018, this paper selects 24 major egg-producing provinces (municipalities) in China as samples, after eliminating singular data in the three scales and averaging the missing data, the final small-scale group is left with 7 provinces including Liaoning, Shandong, Henan, Heilongjiang, Jilin, Shanxi, and Shaanxi; the medium-scale group is the remaining 21 provinces of Beijing, Hebei, Jiangsu, Liaoning, Shandong, Tianjin, Zhejiang, Anhui, Henan, Heilongjiang, Jilin, Hubei, Inner Mongolia, Shanxi, Yunnan, Gansu, Ningxia, Shaanxi, Sichuan, Xinjiang, Chongqing; the large-scale group has 18 provinces, including Beijing, Fujian, Guangdong, Henan, Jiangsu, Liaoning, Shandong, Tianjin, Anhui, Henan, Heilongjiang, Hubei, Jilin, Shanxi, Yunnan, Gansu, Sichuan and Chongqing.As is shown in Table 2, after dividing the provinces by region, the eastern region has 10 provinces (municipalities): Liaoning, Shandong, Beijing, Hebei, Jiangsu, Tianjin, Zhejiang, Fujian, Guangdong, Henan. The central region has 7 provinces (autonomous region): Henan, Heilongjiang, Jilin, Shanxi, Anhui, Hubei, Inner Mongolia. The western region has 7 provinces (municipalities): Shaanxi, Gansu, Ningxia, Sichuan, Xinjiang, Chongqing, Yunnan.Table 2 Samples selected from 2004–2018.Full size table More

Animals such as this orangutan in Indonesia are endangered because of illegal deforestation.Credit: Jami Tarris/Future Publishing via Getty

Funding battles stymie plan to protect global biodiversityScientists are frustrated with slow progress towards a new deal to protect the natural world. Government officials from around the globe met in Geneva, Switzerland, on 14–29 March to find common ground on a draft of the deal, known as the post-2020 global biodiversity framework, but discussions stalled.The framework so far sets out 4 broad goals, including slowing species extinction, and 21 mostly quantitative targets, such as protecting at least 30% of the world’s land and seas. It is part of an international treaty known as the United Nations Convention on Biological Diversity, and aims to address the global biodiversity crisis, which could see one million plant and animal species go extinct in the next few decades.Many who were at the meeting say that disagreements over funding for biodiversity conservation were the main hold-up in negotiations. For example, the draft deal proposed that US$10 billion of funding per year should flow from developed nations to low- and middle-income countries to help them to implement the biodiversity framework. But many think this is not enough.Negotiators say they will now have to meet again before a highly anticipated UN biodiversity summit later this year, where the deal was to be signed.‘Collegiality’ influences researchers’ promotion prospectsUniversities in North America often consider how well researchers interact with each other when making decisions about who gets promoted, a study has found, even though these factors are not formally acknowledged in review guidelines.A researcher’s performance is usually assessed according to three pillars: research, teaching and service. But in recent years, there has been a push from some academics to add another pillar: collegiality. Many say that the concepts of cooperation, collaboration and respect, which broadly fall under the definition of collegiality, are important to the functioning of laboratories and research teams.DeDe Dawson, an academic librarian at the University of Saskatchewan in Saskatoon, Canada, and colleagues analysed more than 860 review, promotion and tenure documents from different departments at 129 universities in the United States and Canada to get a sense of how often collegiality is taken into account.The study, published on 6 April (D. Dawson et al. PLoS ONE 17, e0265506; 2022), found that the concept of collegiality was widespread: the word ‘collegiality’ and related terms, such as ‘citizenship’ or ‘professionalism’, appeared 507 times in 213 of the documents, suggesting that it was often taken into account in evaluations. But just 85 documents included a definition of the term, and fewer still explained how it was measured or used in assessments.

Source: D. Dawson et al. PLoS ONE 17, e0265506 (2022)

Collegiality was mentioned most often in research-intensive institutions (see ‘Academia’s fourth pillar’). The authors say that this could be because the behaviour involved is valued in research groups.Dawson and her colleagues warn that relying on collegiality in performance reviews without adequate guidance could introduce bias, as those in charge fill in the blanks with their own definitions.“We need to make sure that we don’t use collegiality to exclude others that may communicate or interact differently,” says Sujay Kaushal, a geologist at the University of Maryland in College Park, who has previously studied collegiality. More

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