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Well-mixed populationAs shown in the Methods section, in a well-mixed population, the model can be described in terms of the replicator-mutator dynamics. I begin, by a case where the quality of the two resources are similar, r1 = r2 = r, and plot the frequency (solid blue), the average payoffs from the game (dashed red), and the amplitude of fluctuations (dotted blue) for different strategies in Fig. 1a–d. Here, the replicator dynamic is solved starting from a uniform initial condition in which all the strategies’ initial frequency is equal. The results of simulations in finite populations are in good agreement with the replicator dynamics results (see Supplementary Note 2 and Supplementary Figs. 1, 2, 3, and 4 for comparison to simulations). Throughout this manuscript I fix c = 1.Fig. 1: The frequency, game payoffs, and the amplitude of fluctuations for different strategies.The frequency (solid blue), game payoffs (dashed red), and amplitude of fluctuations (dotted blue) of costly cooperators (a), costly defectors (b), non-costly cooperators (c), and non-costly defectors (d), as a function of the enhancement factor, r. As r increases, above a first threshold (r* = 1 + cg), cooperation in the costly institution evolves, and above a second threshold (approximately r = 2) cooperation in both the costly and the free institutions evolves. For medium r, the system shows periodic fluctuations. Parameter values: g = 5, nu = 10−3, π0 = 2, cg = 0.398. The replicator dynamic, derived in the Methods Section, is solved for 9000 time steps, and the time averages are taken over the last 2000 time steps.Full size imageFor small enhancement factors r, the dynamics settle in a fixed point where only defectors in the free institution survive. The advantage of the costly institution becomes apparent as r increases beyond r* = 1 + cg, such that the maximum possible payoff of the costly institution, which is achieved when nobody defects and is equal to r − 1 − cg becomes positive. As shown in the Methods, a focal defector in a group composed of ({n}_{C}^{1}) costly cooperators and ({n}_{D}^{1}) other costly defectors (and ({n}_{C}^{2}+{n}_{D}^{2}) individuals who prefer the free resource), obtains a payoff of ({n}_{C}^{1}r/({n}_{D}^{1}+{n}_{C}^{1}+1)-{c}_{g}). This payoff becomes negative for a small enough value of ({n}_{C}^{1}) (or a large enough value of ({n}_{D}^{1})). Since groups with a small number of costly cooperators are drawn with a high probability when ({rho }_{C}^{1}) is small (these probabilities can be derived in terms of multinational coefficients, see the Methods), the average payoff of a costly defector remains negative in this regime (for instance, given at the transition ({rho }_{C}^{1}approx nu), a mutant costly defector finds itself in a group with no costly cooperator with probability ({(1-{rho }_{C}^{1})}^{g-1}approx {(1-nu )}^{g-1}), which is close to 1 for low mutation rates, and pays a pure cost of −cg). On the other hand, a costly cooperator’s payoff is equal to (({n}_{C}^{1}+1)r/({n}_{D}^{1}+{n}_{C}^{1}+1)-{c}_{g}-1), which for small enough ({n}_{D}^{1}) becomes positive. As in this region, the frequency of costly defectors, ({rho }_{D}^{1}), is small, such group compositions occur with a high probability (at the transition, ({rho }_{D}^{1}approx nu), and thus the probability that a costly cooperator joins a group with no costly defectors is ({(1-{rho }_{D}^{1})}^{g-1}approx {(1-nu )}^{g-1}), which is close to 1 for low mutation rates). Consequently, the average payoff of costly cooperators from the game becomes positive, and thus, larger than the dominant non-costly defectors’ payoff (who receive a payoff of zero). Consequently, the frequency of costly cooperators rapidly increases at r*. However, due to the rapid increase in the frequency of costly cooperators at r*, the probability of formation of such mixed groups increases, and costly defectors start to appear in the system. Further increasing r in this region, the frequency of costly cooperators and costly defectors increases at the expense of non-costly defectors.As r increases above a second threshold, cyclic fluctuations set in, and the dynamics settle in a periodic orbit. An example of this periodic orbit is presented in Fig. 2a, b. Interestingly, the average payoff of costly cooperators, costly defectors, and non-costly defectors in this region remains close to zero despite the evolution of cooperation. Although individuals constantly update their strategy to overcome others, no strategy wins in the evolution. Instead, individuals engage in a winnerless red queen dynamic. The game payoffs of costly cooperators and costly defectors fluctuate around zero (which is equal to the game payoff of non-costly defectors). The dynamics of the system in this regime resembles the frequency-dependent selection in the host-parasite evolution, coined the red queen dynamic based on the fact that no matter how much they run, all end up in the same place53,54. On this basis, I call this periodic orbit the red queen periodic orbit.Fig. 2: Red queen and black queen orbits.The frequency of different strategies (a) and the game payoffs (b) in the red queen, and the black queen (c, d) periodic orbits. In the red queen orbit, cooperators in the costly institution survive. However, the payoff of the surviving strategies fluctuates around zero, and none dominate others. In contrast, cooperators in both institutions evolve in the black queen orbit, and cooperators of each type suppress defection in their opposite institution. Consequently, the payoff of all the strategies starts to deviate from zero. Parameter values: g = 5, nu = 10−3, π0 = 2, and cg = 0.398. In (a, b) r = 1.7, and in (c, d) r = 2.2.Full size imageThe existence of a costly institution can facilitate the evolution of cooperation in its competing free institution too. As the amplitude of fluctuations increases, episodes where most of the individuals prefer the costly institution occur. During these episodes, ({rho }_{D}^{2}) drops to a small value. Consequently, the probability that a mutant non-costly cooperator finds itself in a group devoid of non-costly defectors ({(1-{rho }_{D}^{2})}^{g-1}), increases. In such groups, non-costly cooperators receive a payoff of r−1, which is larger than the payoff of all the other strategies and outcompete other strategies. At this point, a second periodic orbit emerges in which cooperation in both the costly and free institutions evolves. The evolution of cooperation in the free institution can, in turn, have a positive impact on cooperation in the costly institution. This is the case because above the point where cooperation in the free institution evolves, the frequency of individuals who prefer the free institution starts to increase by increasing r. This effect decreases the frequency of those who prefer the costly institution and its effective size. This decreases the mixing probability between costly cooperators and costly defectors and increases the costly cooperators’ payoffs. Consequently, a functional complementation between cooperators with different game preferences emerges, which is reminiscent of a black queen dynamics in which different types crucially depend on each other for performing vital functions36,55. While vulnerable to defectors in their own institution, cooperators complement each other by beating defectors in their opposite institution. Synergistically thus, they can suppress defection in the population and alternately dominate the population (see Fig. 2c, d). At this stage, the game payoff of all the strategies starts to increase beyond zero. I call this periodic orbit the black queen orbit.The picture depicted above is the typical behavior of the model for large enough values of the cost. To see this, in Fig. 3a, I plot the phase diagram of the model in the cg − r plane. Here, the frequency of cooperators in the population, ({rho }_{C}={rho }_{C}^{1}+{rho }_{C}^{2}), is color plotted as well (see Supplementary Fig. 1 for the frequency of different strategies). Red dashed lines show the boundary of the region where the system settles in a periodic orbit. For high costs, as r increases, the system shows a series of successive cross-overs from a defective fixed point to the red queen periodic orbit, black queen periodic orbit, and finally a cooperative fixed point. On the other hand, for small costs, the system possesses a bistable region where both the red queen and black queen periodic orbits (or a partially cooperative fixed point and black queen periodic orbit to the left of the red dashed line in the bistable region) are stable, and the system shows a discontinuous transition between these two orbits. Orange circles show the lower boundary of the bistable region, below which the black queen orbit is unstable. Its upper boundary, above which the red queen orbit becomes unstable, is plotted by red squares, in Fig. 3. The transition between the two periodic orbits becomes a continuous transition at a single critical point (see Supplementary Fig. 4).Fig. 3: Evolution of cooperation.a Time average total frequency of cooperators, ({rho }_{C}={rho }_{C}^{1}+{rho }_{C}^{2}) in the r − cg plane is color plotted. The dynamics can settle in fixed point (FP) (small and large enhancement factors), or two different periodic orbits, red queen periodic orbit (RPO) where cooperation only in the costly institution evolve and black queen periodic orbit (BPO) where cooperation in both institutions evolve. b Time average difference between the probability that an individual in the costly institution is a cooperator from the probability that an individual in the free institution is a cooperator, (gamma ={rho }_{C}^{1}/({rho }_{C}^{1}+{rho }_{D}^{1})-{rho }_{C}^{2}/({rho }_{C}^{2}+{rho }_{D}^{2})). Individuals are more likely to be cooperators in a costly institution. c The time average total frequency of cooperators in the r − cg plane under pure selection dynamic (ν = 0). Red queen and black queen periodic orbit can occur for, respectively, small and large enhancement factors. In other regions, the dynamics settle in a fixed point where either non-costly defectors (small enhancement factors), costly cooperators (inside the region marked with dashed black line), or non-costly cooperators survive. Parameter values: g = 5, and π0 = 2. In (a, b) ν = 10−3, and in (c) ν = 0. In (a, b) the replicator dynamic is solved for 8000 time steps, and the time average is taken over the last 2000 steps. In (c) the replicator dynamic is solved for 200,000 time steps, and the time average is taken over the last 150,000 time steps.Full size imageExamination of the overall cooperation in the population shows that an entrance cost has a contrasting effect on population cooperation for large and small enhancement factors. An entrance cost keeps free-riders away from a costly institution. This fact makes the relative frequency of cooperators to defectors higher in the costly institution than that in the free institution. To see that defectors are less likely to join the costly institution, I plot the difference between the probabilities that an individual in the costly institution is a cooperator and the probability that an individual in the free institution is a cooperator, (gamma ={rho }_{C}^{1}/({rho }_{C}^{1}+{rho }_{D}^{1})-{rho }_{C}^{2}/({rho }_{C}^{2}+{rho }_{D}^{2})) in Fig. 3b, where it can be seen it is always positive. Intuitively, as a costly defector’s payoff in a group with ({n}_{C}^{1}) cooperators and ({n}_{D}^{1}) other defectors is equal to (r{n}_{C}^{1}/({n}_{C}^{1}+{n}_{D}^{1})-{c}_{g}), a costly defector can reach a positive payoff only when ({n}_{C}^{1}) is large. Otherwise, costly defectors are better off hedging the risk of obtaining a negative payoff by joining the free institution, where their payoff is necessarily non-negative. Consequently, the expected number of cooperators in the costly institution, ({rho }_{C}^{1}g), sets a bound for the frequency of costly defectors. This fact increases a costly institution’s profitability, especially for small enhancement factors, and positively impacts cooperation in the population. On the other hand, for high enhancement factors, a large entrance cost is detrimental to cooperation. This is because, although the frequency of defectors in the costly institution remains close to zero, fewer individuals are willing to choose a costly institution with a high cost. This increases the effective size of the free institution and the mixing between cooperators and defectors in the free institution. Since defectors can better exploit cooperators in well-mixed groups, the increased mixing between cooperators and defectors in the free institution hinders cooperation.As shown in the Supplementary Note 3, while the phenomenology of the model remains the same for lower mutation rates, lower mutation rates increase the size of the region where the dynamics settle in a periodic orbit (see Supplementary Fig. 5). Regarding the dependence of the dynamics on the mutation rate, an interesting case is the zero mutation rate, where selection is the sole driver of the dynamics. The time average cooperation for zero mutation rate, starting from an initial condition where all the strategies are equal, is plotted in Fig. 3c (See Supplementary Fig. 6 for the frequency of different strategies). Both the red queen (for small enhancement factors) and the black queen (for large enhancement factors) periodic orbits are observed in this case. However, for zero mutation rate, both the amplitude and period of fluctuations increase: The fluctuating dynamics go through periods where one of the surviving strategies reaches a frequency close to 1 only to be later replaced by another strategy (see Supplementary Fig. 7). The dynamics can also settle in different fixed points. For cg = 0, depending on the enhancement factor, either cooperators or defectors in both institutions survive in equal densities. For nonzero cg, however, only one of the strategies survives. For small enhancement factors, non-costly defectors dominate the population. For larger enhancement factors, either costly cooperators (the region marked with a dashed black line) or non-costly cooperators dominate the population.In the Supplementary Note 2, I consider a case where the two institutions have different productivities, i.e., different enhancement factors, and show that similar phases are at work in this case (see the Supplementary Figs. 2 and 3). For instance, I show that a large entrance cost destabilizes full defection, removes the system’s bistability, and ensures the evolution of cooperation starting from all the initial compositions of the population. In addition, I study the continuous replicator dynamics and show similar phenomenology is at work in this case (see Supplementary Notes 1.4 and 4, and Supplementary Figs. 8 and 9).Finally, I note that a similar phenomenology is at work in a context where instead of a costly and a cost-free institution, two costly institutions interact. To see this, assume institution 1 has a cost cg and institution 2 has a cost ({c}_{g}^{0}). Without loss of generality, assume ({c}_{g} , > , {c}_{g}^{0}). Writing ({c}_{g}=({c}_{g}-{c}_{g}^{0})+{c}_{g}^{0}), it is easy to see that it is possible to absorb ({c}_{g}^{0}) in the base payoff b (as all the individual pay a cost ({c}_{g}^{0}) irrespective of their institution choice). Thus, the model is equivalent to a context where resource 2 has zero cost, resource 1 has a cost of ({c}_{g}-{c}_{g}^{0}), and all the individuals receive a shifted base payoff of (b-{c}_{g}^{0}) (see Supplementary Note 5 and Supplementary Fig. 10).Structured populationIn contrast to the well-mixed population, the model shows no bistability in a structured population, and the fate of the dynamics is independent of the initial condition. To see why this is the case, I note that in a well-mixed population, a situation where all the individuals are defectors, and randomly prefer one of the two institutions, is the worst case for the evolution of cooperation, as in this case, mutant cooperators are in a disadvantage in both institutions. However, in a structured population, starting from such an initial condition, blocks of defectors, most of whom prefer the same institution, form due to spatial fluctuations. A mutant cooperator who prefers the minority institution in these blocks obtains a high payoff and proliferates. This removes the bistability of the dynamics in a structured population.To study the model’s behavior in a structured population, I perform simulations starting from an initial condition in which all the individuals are defectors and prefer one of the two institutions at random. The model shows similar behavior in a structured population to that in a well-mixed population. This can be seen in Fig. 4a–d, where the densities of different strategies are color plotted in the cg − r plane (see Supplementary Note 6 and Supplementary Figs. 11 and 12 for further details). As was the case in a well-mixed population, cooperation does not evolve for too small values of r. As r increases beyond a threshold, cooperation does evolve in the costly institution but not in the free institution. In this region, for a fixed enhancement factor, an optimal cost, approximately equal to cg = r − 1, optimizes the cooperation in the population. On the other hand, cooperation in both the costly and the free institutions evolves for large enhancement factors. In this region, increasing the cost can slightly increases defection in the free institution and have a detrimental effect on the evolution of cooperation, but not as much as it does in a well-mixed population.Fig. 4: The frequency of different strategies in the cg − r plane in a structured population.The time average frequencies of costly cooperators (a), costly defectors (b), non-costly cooperators (c), and non-costly defectors (d) in the cg − r plane are color plotted. The system shows a red queen dynamic in which cooperators only in the costly institution survive in large numbers (for smaller enhancement factors), or a black queen dynamic, where cooperators in both institutions survive and help each other to suppress defection (for larger enhancement factors). Parameter values: g = 5, nu = 10−3, and π0 = 2. The population resides on a 200 × 200 first nearest neighbor square lattice with von Neumann connectivity and periodic boundaries. The simulation is performed for 5000 time steps starting from an initial condition in which all the individuals are defectors and prefer one of the two institutions at random. The time average is taken over the last 2000 steps.Full size imageInstead of periodic orbits observed in the well-mixed population, on a spatial structure the model’s dynamic is governed by the cyclic dominance of different strategies through spatiotemporal fluctuations manifested by traveling waves. In Fig. 5, I present snapshots of the population’s stationary state in different phases. In this figure, I consider a model in which individuals reproduce with a probability proportional to the exponential of their payoff, π, times a selection parameter, β, (exp (beta pi )) (see the Supplementary Note 1.3), with β = 5. The situation in the model where individuals reproduce with a probability proportional to their payoff is similar. In Fig. 5a, I have set r1 = r2 = 1.7, and cg = 0.6. This phase corresponds to the red queen periodic orbit in the well-mixed population case. Here, the majority of the population are non-costly defectors. Costly cooperators experience an advantage over the former and can proliferate in the sea of non-costly defectors. However, costly cooperators are vulnerable to both costly defectors and non-costly cooperators. The former can only survive in small bands around costly cooperators, as they rapidly get replaced by non-costly defectors once they eliminate costly cooperators. This phenomenon shows that spatial competition between defectors with differing institution preferences can positively impact the evolution of cooperation. Non-costly cooperators, in turn, can survive by forming compact domains where they reap the benefit of cooperation among themselves. However, as the effect of network reciprocity is too small to promote cooperation in this region, non-costly cooperators get eliminated by non-costly defectors once costly cooperators are out of the picture. Consequently, the system’s dynamic is governed by traveling waves of costly cooperators followed by small trails of costly defectors and non-costly cooperators in a sea of non-costly defectors (see the Supplementary Video, SV.156, and Supplementary Note 7 for an illustration of the dynamics in this regime).Fig. 5: Snapshots of the population in the stationary state for different parameter values.Different strategies are color codded (legend). In (a,) r1 = 1.7, r2 = 1.7, in (b,) r1 = 3.5 and r2 = 3.5, and in c, r1 = 3, and r2 = 1.8. In all the cases cg = 0.6. For small enhancement factors (a), the red queen dynamics in which cooperators only in the costly institute survive in large numbers occur. For larger enhancement factors (b), the black queen dynamics in which cooperators in both institutions survive and help each other suppress defection occur. By increasing the enhancement factors (c), non-costly cooperators dominate. However, a small frequency of costly cooperators survives and purge the population from defectors by moving along the bands of non-costly defectors. Here, individuals reproduce with a probability proportional to the exponential of their payoff with a selection parameter equal to β = 5. The population resides on a 400 × 400 square lattice with von Neumann connectivity and periodic boundaries. Parameter values: g = 5 and ν = 10−3.Full size imageFigure 5b shows a snapshot of the population for r1 = r2 = 2.2. This phase corresponds to the black queen periodic orbit in a well-mixed population. In this phase, cooperators in both the costly and free institutions evolve. Cooperators are vulnerable to defectors in their institution and lose their territory to defectors of similar type. Defectors are in turn vulnerable to cooperators in their opposite institution and are replaced by them. Consequently, the dynamic of the model is governed by traveling waves of cooperators, chased by defectors of similar type, who are in turn extincted by cooperators of the opposite type. Thus, while cooperators of different types on their own either can not survive (in the case of non-costly cooperators) or are doomed to a winnerless competition with defectors (in the case of costly cooperators), they complement each other to efficiently suppress defection in the population (see the Supplementary Video, SV.256, for an illustration of the dynamics in this regime).Another manifestation of functional complementation between cooperators of different types can be seen in the regime of large enhancement factors. An example of this situation is plotted in Fig. 5c. Here, r1 = r2 = 3.5 and cg = 0.6. In this region, non-costly cooperators dominate the population. However, non-costly defectors can survive in small bands in the sea of non-costly cooperators. While at a disadvantage in the sea of non-costly cooperators, costly cooperators beat non-costly defectors. Consequently, small blocks of costly cooperators are formed within the bands of non-costly defectors. These blocks of costly cooperators move along the bands of non-costly defectors and purge the population from non-costly defectors. In this way, although costly cooperators exist only in small frequency, they play a constructive role in helping non-costly cooperators to dominate the population.In summary, the analysis of the spatial patterns reveals that competition or synergistic relation between individuals with different institution preferences plays an essential role in the evolution of cooperation in the system. Defectors with different institution preferences always appear as competitors who compete over space. By eliminating each other, they play a surprisingly constructive role in the evolution of cooperation. Cooperators, on the other hand, while having direct competition over scarce sites, can also act synergistically and help the evolution of cooperation in their opposite institution since they can eliminate defectors in their opposite institution. In this way, by purging defectors with an opposite game preference, cooperators help fellow cooperators with an opposite game preference. Consequently, cooperators with different game preferences can engage in a mutualistic relation to efficiently suppress defection in the population.Finally, as shown in the Supplementary Note 6, the spatial model shows similar phases in the case where the two public resources have heterogeneous profitability, that is, when r1 ≠ r2 (see the Supplementary Fig. 12). More

Data for all analysed radionuclides are presented in the “Supplementary Material”. Cryoconite samples were collected on Nalli Glacier (Supplementary Fig. S1) on Sept. 25, 2017 (samples 1701–1714) and on Sept. 10, 2018 (samples 1801–1814) at 28 spots (Fig. 2, Supplementary Table S1). Gamma spectrometric analysis of samples showed the presence of anthropogenic radionuclides 137Cs, 241Am, and 207Bi. All quoted radioactivity values were recalculated for the sampling date, except those for 241Am since the concentration of the parent 241Pu isotope is unknown. However, for this isotope, the correction for decay is negligible. The activity of 137Cs reached 8093 (± 69) Bq kg−1 of dry weight, that of 241Am reached 58.3 (± 2.3) Bq kg−1 and that of 207Bi reached 6.3 (± 0.6) Bq kg−1. The natural radionuclides 210Pb and 7Be were also present in all samples. The activity of 210Pb varied in the range of 1280–9750 Bq kg−1. In addition, in the investigated samples, a significant amount of short-lived cosmogenic radionuclide 7Be was found, whose specific activity reached 2418 (± 76) Bq kg−1 (Fig. 3, Supplementary Table S2). To evaluate the contribution of atmospheric components to the total 210Pb activity, 226Ra activity was determined and found to be 17–27 Bq kg−1 (Supplementary Table S2). Based on the 210Pb/226Ra ratio, we conclude that more than 98% of 210Pb was of atmospheric provenance.Figure 2Location of sampling points on Nalli Glacier. A—137Cs activity zone  95%) of corresponding rocks and numerous outcrops likely promoted entrapment of these elements into explosion clouds, and their subsequent precipitation with radionuclides. This feature of the geological structure of the area explains the extremely high enrichment of surface waters in elements such as Zn, Pb, Sr, Ni, As, Cr, Co, Se, Te, Cd, W, Cu, Sb, and Sn; for many of them, the excess reaches 10-fold with respect to the Clrake values51. This hypothesis is supported by obvious correlations between the concentrations of Bi, Ag, Sn, Sb, Pb, Cd, W, and Cu and the activity of anthropogenic radionuclides 137Cs, 241Am and 207Bi. This relationship is obviously related to the simultaneous release of elements and radionuclides from the contaminated ice layer and their entrapment in cryoconite holes. More

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ParticipantsThe Ministry of Health, Labour, and Welfare of Japan has been conducting The Longitudinal Survey of Newborns in the 21st Century since 2001 to establish strategies to counter the declining birthrate in Japan. The survey targeted all babies born in Japan between January 10 and 17 or between July 10 and 17 of 2001. Baseline questionnaires were sent to a total of 53,575 families when eligible babies reached the age of 6 months and 47,015 families initially completed the baseline questionnaire (88% response rate). These respondents were mailed follow-up questionnaires to investigate medical conditions and behaviors when children reached the ages of 1.5, 2.5, 3.5, 4.5, 5.5, 7, 8, 9, 10, 11, 12, 13, 14, and 15 years20,21,22,23. Birth record data from Vital Statistics of Japan are also linked for each child participating in the study. The current study included data for children/families who responded both to the baseline questionnaire and the fifteenth questionnaire at age 15 years.The baseline survey at age 6 months included questions regarding children’s perinatal status as well as household and socioeconomic factors such as parental academic attainment, parental smoking status, and daycare attendance. The subsequent annual surveys starting at age 1.5 years included questions regarding each child’s height, weight and health status. We excluded 2382 children born before 37 weeks of pregnancy and one child with responses only for the baseline survey and the survey at age 15 years. A total of 26,778 children (315,581 data points) were included in the final analysis. A total of 11,141 children (41.61%) had responses to all 15 questionnaires between the ages of 6 months and 15 years, and responses to more than 12 questionnaires were available for the majority (91.94%) of children (Fig. 1, Table S1).Figure 1Flowchart of study participants.Full size imageMeasuresWe calculated BMI based on each participant’s reported annual height and weight. Each participant’s annual BMI was converted to a BMI Z-score using smoothed L, M, and S values for BMI standards from a representative population of Japanese children24. Briefly, the LMS (lambda–mu–sigma) method is a method proposed by Cole et al. to monitor changes in the skewness of the distribution during childhood as a way of constructing normalized growth standards25. Participants were then classified into four BMI categories based on the World Health Organization (WHO) criteria26: underweight (BMI standard deviation [SD] score of − 5 or more but less than − 2), normal weight (BMI SD score of − 2 or more but less than 1), overweight (BMI SD score of 1 or more but less than 2), and obese (BMI SD score of 2 or more but less than 5). The definitions of overweight and obesity were different for children under 5 years of age: a BMI Z-score of 2 SD or more was categorized as overweight and a BMI Z-score of 3 SD or more was categorized as obese. BMI category at age 15 years was the main outcome of interest in the current study.We also calculated annual height growth for each participant by subtracting the height reported at the previous survey from that reported in the current survey. For annual height growth between 5.5 and 7 years of age, this value was multiplied by 2/3 because of the 1.5-year interval between surveys.Statistical analysesWe first compared baseline characteristics among the four BMI categories (underweight, normal weight, overweight and obese) at age 15 years. To evaluate potential selection bias resulting from losses to follow-up, we also compared the baseline characteristics of children included in the analysis and those of children lost to follow-up through to the fifteenth survey (at age 15 years).We retrospectively examined annual aggregate categorical changes in individuals of the four BMI categories (groups) at age 15 years. For each group, the proportion of each BMI category at each survey between the ages of 1.5 and 14 years was calculated. In addition, we prospectively calculated the proportion of children in each BMI category at each survey between the ages of 1.5 and 14 years who eventually became underweight, normal weight, overweight, or obese at age 15 years. Note that these analyses were based on aggregate data and do not describe individual BMI changes and were performed using only the data obtained without imputation of missing values.Under the assumption that missing data were missing at random, mixed effect models with natural cubic regression splines were applied to calculate the trajectories of BMI Z-scores and annual BMI Z-score changes through age 15 years for participants of each BMI category at age 15 years. Knots at seven locations were placed in percentiles of age to yield a sufficient number of measurements between each consecutive knot (age 1.5, 3.5, 5.5, 8.5, 11, 13 and 15 years), as recommended by Harrell27. The mixed effect model is useful for describing population average growth trajectories and individual growth trajectories even when data are not available for all children at all ages28,29,30,31. Briefly, the population average growth trajectory was modeled with fixed effects, while the individual variability is represented as random effects.After fitting individual BMI trajectories using a mixed-effects model with natural cubic spline function, we estimated individual adiposity rebound timing as the age where the first derivative of the trajectory reached its minimum and the second derivative was positive32. Children were then classified into five categories (1.5–2.5 years, 3.5–4.5 years, 5.5–7 years, 8–10 years, and 11 years or older) for analysis of adiposity rebound timing33,34. The distribution of adiposity rebound timing was calculated for individuals of each BMI status at age 15 years overall and by gender.Finally, we modelled annual height change and its associations with BMI status at age 15 years separately for each gender using mixed-effects models with natural cubic regression splines.All statistical analyses were performed using Stata version 16 (StataCorp LLC, College Station, TX, USA). This study was approved by the Institutional Review Board at Okayama University Graduate School of Medicine, Dentistry, and Pharmaceutical Sciences (No.1506-073) and was conducted in accordance with the 1964 Helsinki Declaration and Ethical Guidelines for Medical and Health Research Involving Human Subjects. Informed consent was obtained by the opt-out method on the university’s website. More

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