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    Nutritional value and bioaccumulation of heavy metals in nine commercial fish species from Dachen Fishing Ground, East China Sea

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    Climate change increases cross-species viral transmission risk

    At least 10,000 virus species have the capacity to infect humans, but at present, the vast majority are circulating silently in wild mammals1,2. However, climate and land use change will produce novel opportunities for viral sharing among previously geographically-isolated species of wildlife3,4. In some cases, this will facilitate zoonotic spillover—a mechanistic link between global environmental change and disease emergence. Here, we simulate potential hotspots of future viral sharing, using a phylogeographic model of the mammal-virus network, and projections of geographic range shifts for 3,139 mammal species under climate change and land use scenarios for the year 2070. We predict that species will aggregate in new combinations at high elevations, in biodiversity hotspots, and in areas of high human population density in Asia and Africa, driving the novel cross-species transmission of their viruses an estimated 4,000 times. Because of their unique dispersal capacity, bats account for the majority of novel viral sharing, and are likely to share viruses along evolutionary pathways that will facilitate future emergence in humans. Surprisingly, we find that this ecological transition may already be underway, and holding warming under 2 °C within the century will not reduce future viral sharing. Our findings highlight an urgent need to pair viral surveillance and discovery efforts with biodiversity surveys tracking species’ range shifts, especially in tropical regions that harbor the most zoonoses and are experiencing rapid warming. More

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    The rising moon promotes mate finding in moths

    The moon increases mate finding in mothsTo investigate the impact of natural and artificial light sources on mate finding, we analyzed flight behavior in male moths, which were reliably attracted by caged virgin females (see Materials and Methods for details). Since we used these females specifically to exploit their attraction effect, we refer to them as ‘traps’ in the following. To establish a choice scenario (see below), males were released equidistantly from the traps, which were located north and south of the core release site in central Germany. Besides the stars, the moon creates the natural light environment that moths might use for visual orientation. We therefore first tested if the moon affects mate finding. We found that the percentage of males arriving within the experimental time (8 min from release, 58.6% of flights) at a trap increased significantly with the appearance of the moon (logistic regression: z = −2.06, p = 0.04, n = 58) and did not depend on the presence of clouds in front of the moon (z = −0.83, p = 0.406, n = 58). A few males reached the females later during the experimental night (13.8% of flights) and were released again on the next day. Some males never reached a trap and could therefore not be tested again in the next days (27.6% of flights). Furthermore, the time that successful males needed to reach a trap was significantly influenced by the height of the moon above or below the horizon (Fig.1; Cox PH survival model, z = 2.46, p = 0.014, n = 34): the higher the moon was above the horizon, the faster males were able to locate and reach the females. The presence of clouds in front of the moon did not play a significant role in this context either (z = −0.65, p = 0.519, n = 34), leading to the conclusion that the moon was equally well perceived if covered partly by clouds and used for effective orientation towards the females. Although the lunar phase changed during the period of the experiment from full moon to new moon, flight duration was not significantly affected by the percentage of the lit moon disk (z = 0.44, p = 0.66, n = 34). Thus, the properties of the moon that affected the flight duration of males were independent of the lunar phase.Fig. 1: Expected flight duration of a moth.Flight duration (black line) was calculated as the median flight duration predicted by the Cox PH model (p = 0.014, n = 34) for arrivals within 8 minutes after release and averaged over all individuals. Circles represent the actual measured values. Dashed lines indicate the confidence interval of the predicted duration at α = 5% level estimated by bootstrapping (5000 replicates).Full size imageIt is important to emphasize that the results were not significantly affected by traits on the individual level like body size or origin of the animal (see Supplementary Results and Discussion for details). Furthermore, a possible learning effect of animals that were released more than once was not detectable since flight duration did not decrease depending on ‘experience’ but only with the elevation of the moon (Fig. S1). Thus, the moon as an easily perceivable orientation cue increased mate finding in general but also depended on its elevation. Despite two exceptions of long flight durations at moon elevations > 20° that go back to the same animal probably for individual reasons (Fig. S1), the variance in flight duration was highest at low moon elevations (Fig. 1). This relatively high variance at low moon elevations emphasizes the question if artificial lights affected mate finding, particularly whenever the moon as a natural light cue was not yet prominent.Linking flight behavior to the light environmentWe used a calibrated digital all-sky camera to track changes in the natural and artificial components of the night sky brightness24 (Fig. 2 a–c). A similar camera system was recently used to study dung beetle behavior21. Although the impact of light pollution on the site was not strong, the night sky was also not completely pristine. Luminance (LVv) values were about 0.34 mcd/m² at zenith and 1.6 mcd/m² near the horizon under clear sky conditions when the moon was not visible. A natural (unpolluted) sky brightness is 0.25 mcd/m² at zenith and can be used as the reference value “Natural Sky Unit” (NSU) for easy comparison (see also Materials and Methods). The analysis of specific sky sectors revealed that the moon was the strongest factor determining the ambient brightness, brightening every sector of the sky as soon as it appeared above the horizon (Fig. 2d). During observation times, the course of the moon mainly progressed through the eastern part of the sky, affecting particularly the LvV values in the corresponding sectors (Fig. 2d). Furthermore, light conditions never corresponded to a non-light polluted sky, as NSU values were always greater than one. Most sectors in the south, west and north (sectors seven to 12 and one) were hardly subjected to fluctuations. Nevertheless, it is recognizable that the moon made a decisive contribution to the light environment in all directions since images with the moon above the horizon were always brighter than those with the moon below the horizon (Fig. 2d).Fig. 2: Quantification of the light environment with all-sky imagery and its impact on flight behavior of moths.a Raw RGB all-sky image with clear sky and a visible moon 26° above the horizon at 119° azimuth angle, South-east (24 July 2019, 03:23). b Same image as in a with processed luminance values. c Processed all-sky image in luminance with clear sky, a visible milky way (green patches in a ‘ribbon-shape’ across the (blue) night sky), skyglow near the horizon, and a non-visible moon 0° above the horizon at 87° azimuth angle, East (24 July 2019, 0:25). The colors of the processed image correspond to the legend in b. The black lines mark the sky segments used to quantify the light environment. The outer ring covers 5° above the horizon (85°−90° zenith angle), the inner ring 20° above the outer ring (65°−85° zenith angle). Furthermore, the sky was divided into 12 sectors of 30° width along the azimuth direction (extension by dashed line), starting with the sector marked with the small circle (counting clockwise). d Luminance in natural sky units (NSU) for each full sector of 30°. The moon icons indicate sectors in which the moon was visible, regardless of its phase. The size of each symbol encodes the rank of the frequency (n = 33). e Trap choice of arrived males depending on the position of the moon at the moment of release on the north-south axis (north = 0°). The y-axis displays choice of the southern trap at 0.0 and of the northern trap at 1.0. p = 0.022, n = 42. f Male moth affinity to northern trap in response to the direction of maximum luminance measured in the outer ring of 5°. Each circle indicates an observed arrival, p = 0.753, n = 41. g Male moth affinity to northern trap as in f but with luminance measured in the inner ring of 20°, p = 0.065, n = 41. e–g The line represents the prediction of the logistic model, providing a probability value for arriving at the northern trap (north prone = 1; south prone = 0). Dashed lines indicate the confidence interval of the prediction at α = 5% level estimated by bootstrapping (5000 replicates).Full size imageDue to the design of the experiment with one trap located in the north and the other in the south of a central release site, we were able to investigate the choice behavior of males, especially in respect of the possible influence of the cardinal position of the moon as it was almost exclusively visible in the southern hemisphere of the sky (Fig. 2d). Although the moon continued to move south during the night, the moon’s cardinal position never overlapped with the exact direction of the southern trap. The only parameter that had a significant effect on choice behavior was indeed the cardinal position of the moon (Fig. 2e, logistic regression, z = −2.3, p = 0.022, n = 42). The more southern the moon’s position was, the more likely males flew to the southern trap. However, while some clouds in front of the moon had no significant effect on choice behavior (z = 0, p = 1, n = 42), moon above the horizon showed a tendency to affect males (z = −1.82, p = 0.069, n = 42). The results indicate that despite the general increase of ambient brightness by the moon, it is its position that mainly influenced the flight direction of males. Thus, moths preferred a flight direction with the prominent compass cue ahead to steer their flight towards the females but it is important to emphasize that moon and trap had an angular difference of at least 23° (80.8° to the moon’s mean cardinal direction). Therefore, males that chose to fly towards the southern trap did not fly directly towards the direction of the moon.As the moon represents a natural distant light source, we tested whether distant artificial light sources or skyglow might elicit a comparable effect on the behavior of male moths and if such light sources might mask the moon. To evaluate the light environment with regards to these aspects, we defined sky segments of particular interest that occurred due to the location of the experimental field (Fig. 2c). For each arrival at a trap, the brightest sector of the environment was determined and placed on a north-south axis of maximum 180 degrees (Fig. 2f, g). If we look at the brightest sector of the environment and distinguish between the area close to the horizon, i.e. “outer ring” (Fig. 2f) and the one above, i.e. “inner ring” (Fig. 2g), we can observe differences in trap choice. The line indicates the logistic regression model and provides the probability of arriving at the northern trap. For the Lv in the area close to the horizon no effect of maximum Lv on trap choice could be found (logistic regression, z = 0.31, p = 0.753, n = 41). For the segment further above the horizon the probability of flying to the southern trap increased with maximum Lv but the results are marginally not significant (z = −1.85, p = 0.065, n = 41). Our results for trap selection indicate that distant artificial lights of the surroundings did not attract males and support the hypothesis that the moon, once it appears above the horizon and stands out from the general light (pollution) near the horizon (above five degrees), is used as an effective visual cue with moths rather flying towards than away from.Digital cameras are suitable to measure the dynamics of night-time lighting conditions25,26, and allow researchers to track changes in artificial lighting conditions and brightness of the sky simultaneously27. However, it is not straightforward to distinguish between ALAN and natural light sources like the moon with luminance images when the moon is close to the horizon and thus in the section of the sky where most light pollution occurred. Yet, once the moon rose higher than 5° and thus stood out distinctly from the light-polluted horizon, it could be clearly identified on the images (Fig. 2b). In this context, it is particularly remarkable that the speed at which the females were reached increased reliably only above a similar threshold (Fig. 1), with the only exceptions of two flights with long durations at a moon elevation greater than 20° (Fig. 1); both flights originated from the same individual (Fig. S1). Thus, the high variance of flight durations at low moon elevations (Fig. 1) supports our hypothesis that the moon, as an orientation cue, can be masked by artificial light for the animals as well. Yet, this hypothesis needs to be explicitly tested in future experiments. In general, the possible consequences of light pollution are still uncertain28, especially because the amount of artificial light emitted during the night continues to increase exponentially worldwide18. But regardless of this, the moon is the decisive orientation cue as soon as it is visibly silhouetted against the horizon despite possible diffuse light pollution.Another interesting next research project would be to investigate the relevance of polarized light, as this could provide an explanation for the occasional fast flights at times of low lunar elevations (cf. Figure 1). Furthermore, it might explain why flight duration was not significantly affected by clouds in front of the moon since the polarization pattern extends over the whole sky and is therefore not shielded completely by scattered clouds29. For dung beetles it has been already shown that they are capable of using the polarization signal for navigation16,30,31 and it has been proposed that moths might be capable of utilizing the same signal32. At the same time, it has already been demonstrated that urban skyglow can diminish the lunar polarization signal33, making a detailed investigation of the interplay between these two factors and the significance for moth orientation particularly exciting to understand underlying mechanisms.Our results confirm that moths use the moon as an orientation cue, supporting the notion of Vickers & Baker34 that pheromones alone are not sufficient for successful (and fast) orientation. Since flight duration decreased as a function of lunar elevation, we conclude that the moon contributes to mating success, especially when it can be easily perceived. Since nocturnal landscapes around the world have been drastically restructured in terms of light intensity and light spectrum due to the rapid spread and increase of electrical lighting18, a deeper understanding of orientation mechanisms even in the absence of the moon as an easily perceivable cue could provide a valuable contribution to counteract insect decline. More

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    Sustainable human population density in Western Europe between 560.000 and 360.000 years ago

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    A perspective of scale differences for studying the green total factor productivity of Chinese laying hens

    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