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Our experimental data reveals a scaling law between the mass of a layer along the vertical coordinate, M(z), and the weight that it supports, W(z), namely: (W(z) sim M(z)^a) with (a approx 1.5). To better understand the physical mechanism that yields this scaling law, we derive the force balance equation of a layer of the swarm and solve for W(z). We then equate the analytical expression for W(z) with the experimentally determined scaling law, (W(z) sim M(z)^a), to connect the swarm mass distribution to the exponent a and formulate the expressions for M(z) and W(z) in terms of a. We then consider a dimensional analysis of the strength of each layer of the swarm, S, or the maximum weight that it can support before the grip of the bees on one another breaks. As will be described in detail below, we find that (S sim M^{1.5}), which is close to the experimentally determined (a = 1.53). Deviation from this value increases the fraction of maximum strength exerted by different parts of the swarm.Force balance model of the weight distribution in the swarmWe assume that the swarm is at quasi-equilibrium (the shape does not change although individual bees may move), that all of the bees in each layer contribute equally to supporting the weight of the bees underneath that layer, that the layer thickness is very small, and that the swarm is radially symmetrical about the z-axis. We use a cylindrical coordinate system with a vertical coordinate z, as shown in Fig. 1e, and we consider layers of the swarm along the z-axis of thickness dz. Variables labeled with a tilde, as in (tilde{W}(z)), represent analytically derived expressions; variables without a tilde, as in W(z), represent values determined with power law fits to experimental data.We begin our analysis by applying the force balance principle to each layer of a swarm. As shown by the free body diagram in Fig. 1f, the force with which each layer of bees has to grasp the layer above it is equal to the weight of that layer and all of the layers underneath it: (tilde{F} = tilde{W}(z)). We express (tilde{W}(z)) using the force balance equation (a continuous version of the discrete definition in Eq. (5).):$$begin{aligned} tilde{W}(z) = g int _z^L tilde{M}(z) dz, end{aligned}$$
(8)
where the mass of bees per layer is (tilde{M}(z)), the swarm length is L, and g is the gravitational constant. Inspired by our experimental observation that the mass of the layers near the base is highest and the mass of the layers at the tip of the swarm is lowest in Fig. 3a, we model (tilde{M}(z)) as a monotonically decreasing function of z. To keep the units consistent, we normalize the z coordiante by the length of the swarm:$$begin{aligned} tilde{M}(z) = c left( 1-frac{z}{L}right) ^{tilde{b}}, end{aligned}$$
(9)
where the c factor in this expression ensures that the units of the mass per layer are mass/length, and (tilde{b}) is an unknown exponent. Choosing this function form allows us to easily integrate the expression for (tilde{W}(z)) when we substitute (tilde{M}(z)) into it, set this force balance derivation for (tilde{W}(z)) equal to the experimentally determined expression (W(z) = C M(z)^a), and compare the exponents a and (tilde{b}).To solve the expression for (tilde{W}(z)), we substitute the expression for (tilde{M}(z)), Eq. (9), into Eq. (8) and integrate. We then express (tilde{b}) in terms of the experimentally determined a by equating this expression for (tilde{W}(z)) to the scaling law we observe in our experiments, Eq. (7), (W(z) = C tilde{M}(z)^a). The exponent in the expression for (tilde{M(z)}), Eq. (9), is$$begin{aligned} tilde{b} = frac{1}{a-1}. end{aligned}$$
(10)
The weight supported by each layer is then:$$begin{aligned} tilde{W}(z) = cLg left( 1 – frac{1}{a}right) left( 1-frac{z}{L}right) ^{frac{a}{a-1}}. end{aligned}$$
(11)
Next, we test how well our force balance model predicts the data by comparing the predicted value of (tilde{b}) using the force balance to the value of b calculated using experimental fits. We first separate the expression for the layer mass, Eq. (9) into the product of the layer area, (tilde{A}(z)) and the layer density, (tilde{rho }(z)):$$begin{aligned} tilde{M}(z) sim tilde{A}(z) tilde{rho }(z). end{aligned}$$
(12)
To simplify our analysis, we model (tilde{A}(z)) and (tilde{rho }(z)) with a similar monotonically decreasing function to that in Eq. (9):$$begin{aligned} tilde{A}(z) = c_1 left( 1-frac{z}{L}right) ^{tilde{b}_1}, end{aligned}$$
(13)
and$$begin{aligned} tilde{rho }(z) =c_2 left( 1-frac{z}{L}right) ^{tilde{b}_2} end{aligned}$$
(14)
we can then separately measure the effect of the changes in area and density on the exponent in the mass per layer expression in Eq. (9), (tilde{b} = tilde{b}_1 + tilde{b}_2).We first calculate (tilde{b}) using the expression derived from the force balance, Eq. (10), and our experimental result for a, which yields (tilde{b} = 2 pm 0.47). Second, we calculate b by separately calculating power law fits to the data for A(z) in Fig. 2e according to Eq. (13) and (rho (z)) in Fig. 2d according to Eq. (14), which yields (b_1 = 1.38 pm 0.2) and (b_2 = 0.51 pm 0.09). Thus, (b = b_1 + b_2 = 1.89 pm 0.25). See Supplementary Fig. S5(a–c) for log-log plots of M(z), A(z) and (rho (z)), and Supplementary Fig. S5(d–f) for plots of the resulting b, (b_1), and (b_2).We calculate the deviation of (tilde{b}) from b, (frac{tilde{b} – b}{tilde{b}} = 0.03 pm 0.11), and plot the deviation of b from (tilde{b}) in Supplementary Fig. S5(g) as a comparison for the individual CT scans. The values of b and (tilde{b}) being on the same order of magnitude validates the model and allows us to compare (tilde{W}(z)) to a maximum strength of each layer, which we find with dimensional analysis in the following section.Strength of a swarm layer and individual beesThe strength of the layer, (tilde{S}(z)), or the maximum weight that it could support, can be greater than or equal to (tilde{W}(z)): (tilde{S}(z) ge tilde{W}(z)). If the weight of the bees underneath a layer were to exceed its strength (tilde{S}(z)), the layer would not be able to support the weight of those bees, and the swarm would break apart. We perform a dimensional analysis on the strength of each layer to find the relationship between the mass of a layer and its maximum strength, (tilde{S}(z) sim tilde{M}(z)^{alpha }). Force is proportional to mass, which is proprtional to volume, or a length cubed, so a layer’s strength scales with length cubed, (tilde{S}(z) propto L^3). The mass of each layer, with units of mass/length, is proportional to an area, or a length squared, so (tilde{M}(z)) scales with length squared, (tilde{M}(z) propto L^2). Thus, (alpha) must be 1.5 for (tilde{S}(z) sim tilde{M}(z)^{alpha }) to be dimensionally correct. This is similar to the relationship between weightifting capacity and body weight in Ref.16.Estimating (tilde{W}(z)/tilde{S}(z)) gives a measure of how much of its maximum strength each layer uses to hold up the rest of the swarm:$$begin{aligned} frac{tilde{W}(z)}{tilde{S}(z)} sim left( 1-frac{1}{a}right) left( 1-frac{z}{L}right) ^frac{2a-3}{2a-2} end{aligned}$$
(15)
The average number of bees that a bee in a swarm layer supports, (tilde{F}_{bee}(z)), is equal to the mass of bees supported by a layer divided by the sum of the mass of bees in a layer of bees that has the thickness of the length of a bee, (l approx 1.5), as a continuous version of the discrete equation in Eq. (6):$$begin{aligned} tilde{F}_{bee}(z) =frac{int _z^L tilde{M}(z) dz}{int _z^{z+l} tilde{M}(z) dz}. end{aligned}$$
(16)
After integrating, we get an expression for (tilde{F}_{bee} (z)):$$begin{aligned} tilde{F}_{bee}(z)= frac{left( 1-frac{z}{L}right) ^{frac{a}{a-1}}}{left( 1-frac{z}{L}right) ^{frac{a}{a-1}} – left( 1-frac{z + l}{L}right) ^{frac{a}{a-1}}}. end{aligned}$$
(17)
We use the expression for (frac{tilde{W}(z)}{tilde{S}(z)}), Eq. (15), and (tilde{F}_{bee}(z)), Eq. (17), in the next section to evaluate how the force distribution in the swarm would change for swarms with different values of a.Effect of a on the mass of each layer, the fraction of its maximum stregnth it uses, and the average force per beeWe now consider the effect of varying a on the mass and force distribution inside the swarm. To visualize the effect of a on the distribution of bees, we plot the mass per layer of a 1000-g, 12.5 cm long swarm, (tilde{M}(z)) vs. z/L, with (a = 1.5, 1.01, 1000), and (-0.2) in Fig. 3c and the corresponding average force per bee, (F_{bee}(z)) vs. z/L in Fig. 3d. These values of a are example values for the four possible cases of mass distribution in the swarm. We then evaluate how these values of a affect the fraction of maximum strength each layer uses to support the layers underneath it using Eq. (15).If (a approx alpha), as we found in our experiments, layers with higher mass near the attachment surface support the less massive layers under them, as in the solid black line in Fig. 3c. Correspondingly, Fig. 3d shows (tilde{F}_{bee}(z=0) approx 3) at the top of the swarm, and decreases towards the tip. The strength of each layer and the weight it supports are proportional to one another, (tilde{W}(z)/tilde{S}(z) sim 1/3), meaning that the fraction of maximum strength used by a layer is the same for all z. If (1< a < alpha), the swarm approaches one massive layer of bees, as in the dashed purple line in Fig. 3c. The dimensional analysis results in a very small fraction of the total strength used by this layer, (tilde{W}(z)/tilde{S}(z) rightarrow 0 (1-frac{z}{L})^{-infty }). The force supported by each bee in Fig. 3d shows (tilde{F}_{bee}(z) = 1) for the entire swarm, meaning that each bee only supports its own weight. This configuration would either require packing a large number of bees into one very dense or one very wide layer. A swarm with one very dense layer at the top would compress all of the bees; a swarm with one very wide layer would require a large surface area, which would put the swarm in danger from predators and changes in weather. Thus, despite a potentially lower fraction of strength used by the largest layer of bees, this configuration would put the swarm in danger by requiring a large surface area.For values of (a > alpha), as (a rightarrow infty), all the layers of the swarm have the same mass, as in the dash-dot red line in Fig. 3c. The force per bee in Fig. 3d shows (tilde{F}_{bee}(z=0) approx 8) at the top of the swarm, 2.5 times that of the (a = alpha) configuration. In this configuration, the top layers use a higher percentage of their available strength than the lower layers, (tilde{W}(z)/tilde{S}(z) rightarrow (1-frac{z}{L})). Thus, for large swarms, the bees that support the swarm would be under more strain, and the swarm would be more likely to break under external perturbation.Finally, (a < 0) ((0 le a le 1) results in negative values for (tilde{W}(z))) would suggest that the top layers of the swarm have a lower mass than the bottom layers, as in the dotted orange line in Fig. 3c. This is not a realistic range of values for a, but we include it here as a demonstration of a potential mass distribution with the largest layers being on the bottom of the swarm. This configuration would put even more strain on the layers of bees at the top of the swarm, as smaller layers near the attachment surface have a smaller maximum strength. As (a rightarrow 0) on the (a < 0) side, (tilde{W}(z)/tilde{S}(z) rightarrow infty (1-z/L)^{1.5}), and bees in the top layers use a much greater fraction of their strength than bees in the bottom layers. Accordingly, the mean force per bee in Fig. 3d exceeds the maximum bee grip strength of 35 bee weights, and the swarm could not support itself in this configuration.The swarm configuration with (a approx 1.5) uses the full strength of each layer and puts a lower strain on the bees than most other values of a, and avoids weight distributions that could expose a large number of bees to external danger. More

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All procedures accorded to administrative provision of animal welfare of the Fisheries Research Education Agency Japan. All statistical tests used in this study are two-sided.Otolith samplesFrom the western North Pacific, age-0 JP sardine were collected from samples taken during acoustic and sub-surface trawl surveys in the offshore Oyashio region conducted during 2006–2010 and 2014–2015. The surveys were conducted by Japan Fisheries Research and Education Agency every autumn since 2005 which aim to estimate the abundance of small pelagic species. The abundance of young-of-the-year sardine in the region in the season, approximately 10–15 cm in standard length (SL), is considered a proxy for the abundance of recruits of the Pacific stock and used to tune the cohort analysis in stock assessment4. As representatives of the young-of-the-year population in the region, 2–6 trawl stations each year that had relatively larger catch-per-unit-effort were selected (Supplementary Fig. 1), and 9–20 individuals were randomly selected from each station for otolith analyses (Supplementary Table 1). Age of fish was initially judged by SL (10–15 cm) and later confirmed by the counts of otolith daily increments.From the eastern North Pacific, archived otoliths of CA sardine captured in cruise surveys and in the pelagic fishery of the Southern California Bight during 1987, 1991–1998, and 2005–2007 were collected. Fish in the size range of 10–16 cm SL were regarded as age-1 individuals born in the previous year, following Takahashi and Checkley56. The number of individuals varied between year classes in the range of 4–20 (Supplementary Table 2).Otolith processing, microstructure and somatic growth analysisSagittal otoliths were cleaned to remove the attached tissue in freshwater and then air-dried. Otoliths of JP sardine were embedded in epoxy resin (Petropoxy 154, Burnham Petrographics LLC) on slide-glass, while those of CA were glued to slide-glass using enamel resin and then ground and polished with sandpaper to expose the core. For some otoliths of CA sardine, the polished surface was coated with additional resin to facilitate identification of the daily increment width. Using an otolith measurement system (RATOC System Engineering Co. Ltd.), the number and location of daily increments were examined along the axis in the postrostrum from the core. Although daily increments were clearly observed until the otolith edge for JP sardine, it was difficult to do this for CA sardine probably because they had experienced winter when otolith growth slowed down. Therefore, the rings were counted as far as possible for CA sardine, which typically resulted in more than 150 counts. The first daily increment was assumed to form after 3 days post hatch (dph) for JP and 8 dph for CA sardine following Takahashi et al.26 and Takahashi and Checkley56. The otolith radius at each age was calculated by adding all the increment widths up to that age. Standard lengths at each age were back-calculated assuming a linear relationship between otolith radius and standard length using the biological intercept method34 as follows:$${SL}_{n}=left({{SL}}_{{catch}}-{{SL}}_{{first}}right)times left({{OR}}_{n}-{{OR}}_{{first}}right)/left({OR}_{catch}-{{OR}}_{{first}}right)+{{SL}}_{{first}}$$
(1)
where SLn is the standard length at age n, SLcatch is the standard length at catch, SLfirst is the standard length at the age of first daily increment deposition fixed at 5.9 mm for JP sardine and 5.5 mm for CA sardine following the previous studies26,56, ORn is the otolith radius at age n, ORfirst is the otolith radius at the age of first daily increment deposition, and ORcatch is the otolith radius at catch. Based on rearing experiments of field collected eggs, Lasker57 showed the SL of CA sardine at 6–8 dph ranged between 3.8 to 6.5 mm, and Matsuoka and Mitani58 showed the total length at 2–4 dph ranged between 4.8 to 6.2 mm, corresponding to 4.7 to 6.1 mm in SL. To deal with these uncertainties regarding the size at the age of first daily increment deposition, we conducted Monte Carlo simulations (10,000 times) to estimate the uncertainties of back-calculated SL, assuming that the initial SLs fall between 3.8 to 6.5 mm for both sardines. Standard deviations of the temporal back-calculated SL at each age were presented as the uncertainty of each SLn estimation, which varied between 0.51 and 0.73 at the end of larval stage (JP: 45 dph, CA: 60 dph), between 0.34 and 0.64 at the end of early juvenile stage (JP: 75 dph, CA: 90 dph) and between 0.20 and 0.53 at the end of late juvenile stage (JP: 105 dph, CA: 120 dph). These values were significantly smaller than the variability of estimated SL among individuals assuming initial sizes of 5.9 and 5.5 mm for JP and CA sardine, respectively (standard deviations: 4.2, 8.1 and 8.3 in JP sardine and 5.5, 9.1 and 10.3 in CA sardine for the end of larval, early juvenile and late juvenile stages, respectively), suggesting that the back-calculated SL is robust to variations of initial size. Nevertheless, the biological intercept method assumes a constant linear relationship between fish and otolith size within individual59, which can vary depending on physiological or environmental conditions60,61. Therefore, to examine the relationships between temperature and growth, we used both otolith growth, which contains fewer assumptions, and back-calculated somatic growth as growth proxies. Since the use of the two proxies did not show remarkable differences in the relationships between temperature and growth (Supplementary Figs. 11, 12), we mainly used the back-calculated SL in the discussion, which has a more direct ecological implication.To more generally test whether growth trajectories are different between the western and eastern boundary current systems, otolith growth data of JP and CA sardines were compared with those of sardines in the east to south and west coasts of South Africa. The biological intercept method to back-calculate standard length could not be used in sardine from South Africa because the size at catch was large, some over 20 cm, and otolith radius and standard length were not linearly correlated for fish of this size. Therefore, the otolith radius and increment width were directly used as proxy for size and growth in this comparison, respectively. For visualisation (Fig. 2a), the means of year class mean otolith radii were estimated for JP and CA sardines. For CA sardine, otolith radii at ages were simply averaged within each year class. For JP sardine, to account for the variation in the number of individuals captured at the same station, otolith radii were first averaged within each station, and the station means were averaged within each year, weighted by catch-per-unit-effort. For South African sardine, data of otolith daily increment widths from hatch to 100 dph of 67 adults captured at six stations on the east to south coast ( >22oE), and 51 individuals captured at six stations on the west coast ( 0.05). Theoretically, the relationship between metabolism and temperature tends to show a linear trend after the metabolic rate is log-transformed79. Thus, we applied “identity (data without transformed)” and “log (data transformed)” links to evaluate if model shows a better linearity with data transformation. Based on AIC, however, the result showed Moto have a better linearity without data transformation (Supplementary Table 7). We, therefore, used “identity” links for the further model selection. Model selection base on AIC was performed for models including temperature, region (JP and CA sardines), life history stages (larvae, early juvenile and late juvenile) and interactions of these factors. The full model including all the interactions had the lowest AIC (Supplementary Table 7). As the diagnostic for the full model showed normality and homogeneity of residuals (Supplementary Fig. 9), we selected this model for interpretation. The CA sardine at the larval stage as the baseline, we found only JP sardine at early and late juvenile stages has relatively higher Moto values, and the temperature-dependent slope is significantly gentler in JP sardine at early and late juvenile stages (Supplementary Table 8).Next, the diversity of Moto across temperature range was assessed to estimate the optimal temperature in each stage. The relationship between the maximum metabolic rate and temperature is known to be parabolic, while that between the standard metabolic rate and temperature is logarithmic28,79. As the highest field metabolic rate would be constrained by maximum metabolic rate and the lowest field metabolic rate would be close to resting metabolic rate43, fish would have the most diverse metabolic performance at the optimal temperature with the widest aerobic scope. Thus, we modelled the highest and lowest Moto values in each 1 °C bin using a polynomial regression and a generalised linear model with Gaussian distribution and a log link for the 95th and 5th percentile values of each bin, respectively (Supplementary Fig. 10). The values of the bin that included less than four values were excluded from the regression analyses to reduce the uncertainty caused by under-sampled temperature bins. The gap between the two regression lines was considered as a proxy for the aerobic scope, and the temperature at which the gap reached the maximum was regarded as the optimal temperature.Statistical analyses for the relationships between temperature and growthTo understand how variation in ambient water temperature affects early life growth of sardines, we compared back-calculated standard length at around the end of the larval stage (hatch–35 mm; JP: 45 dph, CA: 60 dph), the end of the early juvenile stage (35–60 mm; JP: 75 dph, CA: 90 dph), and the end of the late juvenile stage (60–85 mm; JP: 105 dph, CA: 120 dph) and the mean seawater temperature from hatch to the ages. Median of each sampling batch were used as minimal data unit. Pearson’s r and p-values were first calculated for each comparison (Supplementary Table 9). As the relationship between mean temperature and standard length of JP at 75 dph seemed to be dome-shaped rather than linear, we introduced quadratic term of temperature and tested whether the term increased explanatory power using a linear model and stepwise model selection based on AIC. The model selection showed that the full model (Standard length ∼ Temperature2 + Temperature) was the best model, and the coefficients of the quadratic and linear terms were both significant (Supplementary Table 10). To account for these multiple tests, we corrected the p-values of the coefficients of the quadratic term in the linear model for JP sardine at 75 dph and of the Pearson’s r for the rest using the Benjamini-Hochberg procedure with α = 0.05, and selected the null hypotheses that could be rejected (Supplementary Table 9). To compare the temperature that allow maximisation of growth rate and optimal temperature derived from the analysis of Moto for each stage, median somatic growth rate and otolith increment width in each 1 °C bin was calculated together with its 3-window running mean (Supplementary Figs. 11, 12).Statistical analyses for the relationships between sea surface temperature and survival indexTo test whether habitat temperatures during the first 4 months after hatch affect the survival of sardines in the first year of life on a multidecadal scale, satellite-derived sea surface temperature (SST) since 1982 and survival of JP and CA sardines were compared. The log recruitment residuals from Ricker recruitment models (LNRR)13, representing early life survivals taking into account the effect of population density, were calculated based on the stock assessment data for JP and CA sardines as follows:$${LNR}{R}_{t}={ln}({R}_{t}/{S}_{t}) , – , (a+btimes {S}_{t})$$
(6)
where LNRRt is the LNRR at year t, Rt is the recruitment of year-class t, St is the spawning stock biomass in year t, and a and b are the coefficients of linear regression of ln(Rt/St) on St. Pearson’s r between the LNRR and the mean SST values from March to June for JP and from April to July for CA sardine was calculated for each grid points in the western and eastern boundaries of the North Pacific basin, derived from a SST product based on satellite and in situ observations80 (Global Ocean OSTIA Sea Surface Temperature and Sea Ice Reprocessed (https://resources.marine.copernicus.eu/product-detail/SST_GLO_SST_L4_REP_OBSERVATIONS_010_011/INFORMATION), accessed on 11th August and 28th October 2021). The correlations were generally negative and positive in the western and eastern regions, respectively (Supplementary Fig 13a, b). In particular, mean SST values in the area where eggs, larvae and juveniles of JP or CA sardines are mainly found in the months26,39,49,56,78,81,82 (dotted areas in Supplementary Fig 13a, b) were compared with LNRR values to test the relationship between habitat temperature and survival in the early life stages (Supplementary Fig 13c). It should be noted that the mean SST values were not significantly correlated with otolith-derived year-class mean temperatures of JP and CA sardines during the larval to late juvenile stages (JP: r = 0.01, p = 0.98, n = 7, CA: r = 0.29, p = 0.38, n = 11), likely due to the short periods analysed, patchy distribution and inter annual variation in larval and juvenile dispersal and migration patterns. Nevertheless, the regions included areas where SST showed weak to significant (p  More