Philippa Kaur

Breed differences in animal feed conversion and economic trait performanceThroughout the feed intake measurement period, summary statistics shows animals on test had an average DMI of 1.11 kg/d (SD = 0.18), ADG of 0.27 kg/d (SD = 0.1), FCR of 4.04 kg of DMI/ Kg of ADG (SD = 0.1), start weight of 29.60 kg (SD = 3.7), final live weight of 46.00 kg (SD = 2.9), carcass weight of 20.20 kg (SD = 1.6), and a KO% of 44.1% (SD = 2.3). Average daily gain (P = 0.005), FCR (P = 0.035), CW (P  More

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Bergmann’s ruleVariation in avian body size has arisen through millions of years of evolution43, and our data reflects this by showing that log body mass is strongly predicted by phylogeny (Supplementary Table 1). Yet, avian body size also shows large geographical variation (Fig. 1a), and our analysis provides strong support for Bergmann’s rule across the global community of birds. Phylogenetic linear mixed models indicated that the temperature variables explain from 9.0% to 11.8% of the variance in log-transformed body size (estimated with r-squared; Fig. 1b). These models are substantially better supported than the null model and the model with latitude alone (Fig. 1b), suggesting that the observed geographical pattern is linked to thermoregulation. All of these temperature models indicate that temperature negatively correlates with body size (Fig. 1c and Supplementary Fig. 1), as predicted by Bergmann’s rule.Fig. 1: Global test of Bergmann’s rule across 9962 (99.7%) avian species.Distribution of log-transformed body mass across species geographic ranges, shown as their geometric centroids (a). Model selection procedure for predicting log body mass (b), with six temperature measures assessed within species geographic ranges, as sole fixed effects; AIC—Akaike Information Criterion, r2—coefficient of determination. An exemplar Bergmann’s model (c), showing decreasing body size with max temperature of all months; see Supplementary Fig. 1 for surrogate models based on the other temperature measures (evaluated in b). The shaded area around the trend line is simple shading to facilitate reading. The p values refer to the significance of temperature effect and whether it differs from zero as derived from a two-tailed test. The results were obtained with phylogenetic linear regression by phylolm models on a single maximum clade credibility phylogenetic tree.Full size imageAllometry of appendagesAllen’s hypothesis3 implies that the length of animal’s appendages varies with temperature in relative (not absolute) terms, thus when asking how the appendage length vary across temperature gradient, we always need to control for body size. Phylogenetic log-log regression models revealed that body mass explains 72.7% and 72.5% of variance in beak and tarsus length (estimated with r-squared of models shown in Fig. 2a and Supplementary Fig. 3a), respectively, confirming that the evolution of absolute avian appendage size is substantially constrained by body size. These null allometric models predict that log-transformed beak length (Fig. 2a) and tarsus length (Fig. 3a) scale with log-transformed body mass in a linear manner:$${{{{rm{log }}}}}_{e}left({{{{rm{Beak}}}}},{{{{rm{Length}}}}}right)=1.4345+0.3362,{{{{rm{log }}}}}_{e}{{{{rm{Body}}}}},{{{{rm{Mass}}}}}$$
(1)
$${{{{rm{log }}}}}_{e}left({{{{rm{Tarsus}}}}},{{{{rm{Length}}}}}right)=2.1141+0.2883,{{{{rm{log }}}}}_{e}{Body},{{{{rm{Mass}}}}}$$
(2)
Fig. 2: Global test of Allen’s rule on avian beak length across 9962 (99.7%) bird species.The null allometric model (a) used to scale the absolute (log-transformed) beak length with log body size, the residuals from which were used as the relative beak length. Distribution of relative beak length across species geographic ranges (b). Model selection procedure for predicting log beak length (c), involving models with log body mass and either of six temperature measures within species geographic ranges included as fixed and interaction terms; AIC—Akaike Information Criterion, r2—coefficient of determination. An exemplar Allen’s model (d) showing increasing beak length with max temperature of all months, while controlling for body size as fixed term. An exemplar model with interaction of body size and max temperature of all months (e) illustrating how Allen’s rule operates across steeping quantiles of body size (left) and how allometry varies across quantiles of temperature (right). See Supplementary Fig. 2 for surrogate models based on the other temperature measures (evaluated in c). The p values refer to the significance of model’s fixed (d) or interaction terms (e) derived from two-tailed tests. The shaded area around the trend line is simple shading to facilitate reading. The results were obtained with phylogenetic linear regression by phylolm models on a single maximum clade credibility phylogenetic tree.Full size imageFig. 3: Global test of Allen’s rule on avian tarsus length across 9962 (99.7%) bird species.The null allometric model (a) used to scale the absolute (log-transformed) tarsus length with log body size, the residuals from which were used as the relative tarsus length. Distribution of relative tarsus length across species geographic ranges (b). Model selection procedure for predicting log tarsus length (c), involving models with log body mass and either of six temperature measures within species geographic ranges included as fixed and interaction terms; AIC—Akaike Information Criterion, r2—coefficient of determination. An exemplar Allen’s model (d) showing decreasing tarsus length with max temperature of all months, while controlling for body size as fixed term. An exemplar model with interaction of body size and max temperature of all months (e) illustrates how Allen’s rule operates across steeping quantiles of body size (left) and how allometry varies across steeping quantiles of temperature (right). See Supplementary Fig. 3 for surrogate models based on the other temperature measures (evaluated in c). The p values refer to the significance of model’s fixed (d) or interaction terms (e) derived from two-tailed tests. The shaded area around the trend line is simple shading to facilitate reading. The results were obtained with phylogenetic linear regression by phylolm models on a single maximum clade credibility phylogenetic tree.Full size imagethe normalized formulas of which give us the logarithmic equations:$${{{{rm{Beak}}}}},{{{{rm{Length}}}}}={4.1975,{{{{rm{Body}}}}},{{{{rm{Mass}}}}}}^{0.3362}$$
(3)
$${{{{rm{Tarsus}}}}},{{{{rm{Length}}}}}={8.2821,{{{{rm{Body}}}}},{{{{rm{Mass}}}}}}^{0.2883}$$
(4)
Because these allometric plots (Figs. 2a and 3a) relate the length of the appendage (one dimensional linear measure) to the body mass (three-dimensional volumetric measure) it means that the size of appendages would scale isometrically (proportionally) with the body size if the allometric coefficient was 0.3333. Thus, beak length equals to body mass to a power of 0.3362 means that the beak elongates almost exactly proportionally with body size. However, tarsus length equals to body mass to a power of 0.2883 means that the extent to which tarsus elongates with body size is slightly more pronounced in smaller species and weaker in larger species.These allometric relationships have implications for how we interpret subsequent patterns. For example, consider a species that experience a temporal increase in temperature, or invades a warmer climate. Then, if only Bergmann’s rule is operating (and in the absence of other confounds), a gradual decrease in body size should result in a proportional decrease in absolute beak length, and a gradually larger decrease in absolute tarsus length. Conversely, if species follow only Allen’s rule (and not follow Bergmann’s rule), then the increase in beak length should be similar between larger and smaller species, while the increase in tarsus length should be weaker in larger species and stronger in smaller species. Thus, Allen’s assumption that the increase in the ratio of body width to body length is steeper in larger species3, should not be a direct effect of the allometric rules, as appendages tend to increase proportionally with body size (beak) or increase milder at larger body sizes (tarsus).Allen’s ruleAfter excluding the effect of allometry, relative beak length is still tightly associated with phylogeny (Supplementary Table 1), while showing an impressive geographic variation (Fig. 2b). Our phylogenetic analysis concurs with an array of existing studies12,16,17,44,45 that found that the length of avian beak follows Allen’s rule, and is a general pattern across birds as a whole. Among our models predicting beak length, those with temperature variables among fixed terms are more informative than the null allometric model, where log body mass (allometry) is put as sole predictor (Fig. 2c). Most of the temperature variables also predict beak length better than latitude (Fig. 2c), again confirming the thermoregulatory basis of the observed pattern. Each of the temperature variables are positively associated with longer beaks (Fig. 2d and Supplementary Fig. 2a), which remains in agreement with Allen’s rule.Some studies have reported the ambiguous46 or very weak16 Allen’s pattern for avian legs. While relative tarsus length is also well conserved in avian phylogeny (Supplementary Table 1) and shows a high geographic variation (Fig. 3b), surprisingly, our global phylogenetic analysis indicates that avian tarsus length follows the inverse of Allen’s rule. Among models explaining tarsus length, those with temperature variables are better than the null allometric model (Fig. 3c). However, these models indicate a negative correlation—thereby shorter tarsi are associated with warmer temperatures (Fig. 3d).Allen’s vs Bergmann’s rule in allometryOur analyses support the hypothesis that the way in which avian appendages size varies across temperature regimes, depends on body size and vice versa. First, among models of beak length, those with an interaction of body size and the temperature consistently perform better than models without that interaction (Fig. 2c). The interaction of temperature and body size loads positively on beak length, indicating that larger-bodied species show stronger increases in beak size with temperature (Fig. 2e, left plot). Notably, beak length does not co-vary with temperature in the smallest birds (Fig. 2e, left plot), which is in agreement with Allen’s speculations3 that being smaller reduces the need to develop elongated appendages in hot climates, as effective heat exchange is already enabled through small body size (according to Bergmann’s rule). The positive interaction between body size and temperature also indicates that the higher the temperature, the steeper the allometric relationship between beak size and body size (Fig. 2e, right plot), meaning that in warmer climates beak size increases more strongly with body size than in colder climates, exactly as Allen hypothesized.An interaction between body size and temperature is also consistently supported in models of tarsus length (Fig. 3c). This interaction has strong positive effect on tarsus length, thereby reversing the trend by which tarsus shortens with temperature (Fig. 3e, left plot). This means that despite the overall decrease of tarsus size with temperature in smaller birds (the inverse of Allen’s rule), the opposite is true for larger birds that show increasing tarsus size with temperature (Fig. 3e, left plot). The interaction holds regardless of the temperature measure examined (Supplementary Fig. 3b, upper row), even if those previously did not co-vary with tarsus length when included as simple independent term with body size (Supplementary Fig. 3a). The case of larger birds thus fits Allen’s rule, and agrees with Allen’s further speculations3 that appendages are more likely to increase in larger- than in smaller-bodied animals. However, Allen did not predict the possibility of shortening appendages toward hot temperatures as seen in small birds. Given the extent of our sampling, the effect of shortening tarsi toward the equator in small-bodied species is presumably not an artefact, but relies on yet unknown mechanisms (possibly unrelated to thermoregulation). Nevertheless, if there is an evolutionary pressure to develop a smaller tarsus in hot climates, the increased thermoregulatory needs of larger-bodied species possibly overwhelm this selective process. This may be because large species acquire higher heat loads when the ambient temperature is hot, hence necessitating the development of longer legs as cooling organs. As with beak size, the interaction also indicates substantial changes in allometry, with much more millimeters of tarsus per each gram of body in warm conditions compared to cold conditions (Fig. 3e, right plot).Our analyses also support the mirror scenario, that the extent to which body size decreases with temperature (Bergmann’s rule) depends on the length of appendage. In models predicting body size, the temperature does not interact with relative beak length (Supplementary Fig. 4a), but interacts with tarsus length (Supplementary Fig. 4b). This interaction indicates that the strongest shrinkage in body size with temperature occurs in shorter-legged birds, while in longer-legged birds body size increases with temperature (inverse Bergmann’s rule). This again supports Allen’s speculations that variation in body shape allows birds to evolve body sizes less restricted (or even unrestricted) to environmental temperature. Thus, the results support the theory that Bergmann’s and Allen’s rules are two distinct, albeit analogous strategies to deal with thermoregulation.Allen’s vs Bergmann’s rule in climatic adaptationsOur analysis shows that the interactions of body size (Bergmann’s rule), beak length and tarsus length (Allen’s rule) predict the thermal environment across birds (e.g. the max temperatures of all months across species ranges, Fig. 4a). As with body size and shape, the temperatures experienced by species within their geographic ranges are finely conserved in the avian phylogeny (Supplementary Table 1), suggesting that thermal preferences of avian species have been established through evolutionary history. Evolution of these preferences then occurred when temperature changes affected their native environments (thus causing extinctions or adaptations), or when birds invaded novel environments (thus adapting to newly-encountered climates). Log-transformed body mass, relative beak length and tarsus length clearly predict the species ambient temperature (Fig. 4b), suggesting that the phenotype changes as animals adapt to suit different climates. However, of particular note is that the addition of an interaction between body size and relative beak length substantially improves model performance (Fig. 4b). This interaction shows that for longer-beaked birds, temperature associations are unrelated to body size, but the shorter the beak, the more pronounced is the shrinking in body size in warmer temperatures (Fig. 4c, left plot). In the case of smaller-bodied birds, the adaptation to different temperatures is independent of beak length, but with larger birds, the adaptation to warmer temperatures is more likely associated with elongated beaks (Fig. 4c, right plot). These results indicate that living in warmer temperatures tends to be associated either with smaller body size (Bergmann’s rule) or longer beak (Allen’s rule), rather than both rules simultaneously, thus again supporting the hypothesis of an evolutionary compromise between shifts in body size and shape as alternative adaptations to thermal environment.Fig. 4: Global test for avian adaptation to maximum temperature across all months by shifts in body size (Bergmann’s rule) and appendage size (Allen’s rule) across 9962 (99.7%) avian species.Distribution of environmental temperature across species geographic ranges (a). Model selection procedure for predicting max temperature all months (b), involving models with different combinations of log body mass, relative beak and tarsus length as fixed and interaction terms; AIC—Akaike Information Criterion, r2—coefficient of determination. Exemplar models with two-way interaction of body size and relative beak length (c) or tarsus length (d) illustrate how Bergmann’s rule operate across steeping quantiles of relative appendage length (left plots) and how Allen’s rule operate across steeping quantiles of body size (right plots). An exemplar model with two-way interaction of relative beak length and tarsus length (e) illustrates how Allen’s rule based on the relative length of one appendage operates across steeping quantiles of the relative length of second appendage. An exemplar model with three-way interaction of log body mass, relative beak and tarsus length (f) illustrates how shifts in body size and two measures of body shape depend on each other when animals adapt to novel climates; the trend lines indicate relationships between y and x1 (axes) across combinations of min and max values of x2 and x3 (colors); see also Supplementary Fig. 5 for more detailed visualization of the model f. The p values refer to significance of two-way (c–e) and three-way (f) interaction terms derived from two-tailed tests. The shaded area around the trend line is simple shading to facilitate reading. Obtained with phylogenetic linear regression by phylolm models on a single maximum clade credibility phylogenetic tree.Full size imageThe interaction of body size and relative tarsus length also substantially improves the model predicting ambient temperature of the species (Fig. 4b). This interaction indicates that living in warmer climates is associated with smaller body size (Bergmann’s rule) only in shorter-legged birds, while in longer-legged birds the environmental temperature increases with body size (inverse Bergmann’s rule; Fig. 4d, left plot). Simultaneously, the avian environmental temperature increases with tarsus length (Allen’s rule) only in larger species, while the opposite is true for smaller species (Fig. 4d, right plot). This suggests that larger-legged avian lineages may be resistant to Bergmann’s rule and become larger when habituating to warm climates, while shorter- and average-legged birds become smaller with temperature, as predicted by Bergmann’s rule. These findings again converge with Allen’s speculations on trade-off in the evolution of body size and appendage length in relation to temperature.We found that the length of the two different appendages—beak and tarsus—show independent evolutionary patterns (Fig. 4e). The environmental temperature of a species increases with beak length independently from tarsus length, and decreases with tarsus length independently from beak length (Fig. 4e). These outcomes reject the possibility of an evolutionary compromise in climatic adaptation of two types of appendages, at least when we do not control for body size (Bergmann’s rule) as additional type of climatic adaptation.Finally, the model with a three-way interaction between body size, relative beak and tarsus length predicting temperature performs the best among all considered candidate models (Fig. 4b) and this interaction is statistically significant (Fig. 4f), suggesting that evolutionary adaptation to novel climates depends on various configurations of body size, beak, and tarsus length. This model indicates various Bergmann’s rule slopes across different settings of body shape (Fig. 4f, top-left). Namely, the steepest decrease in environmental temperature with body size (i.e. strongest Bergmann’s rule) is observed in smaller-billed and smaller-legged birds (Fig. 4f, top-left, brown trend line), whereas in longer-billed and shorter-legged birds (Fig. 4f, top-left, green trend line) body size is not associated with environmental temperature. This model also indicates that in shorter-billed, longer-legged birds (Fig. 4f, top-left, purple trend line) body size increases across temperature gradient (inverse Bergmann’s pattern). This thus strengthens the support for Allen’s theory that having bodies with elongated appendages may enable species to circumvent or even reverse Bergmann’s pattern; whereas compact bodies are more prone to decrease in size with temperature in order to deal with overheating in warm climates. Counteracting this argument, however, is that longer-billed and longer-legged birds show (moderate) typical Bergmann’s pattern (Fig. 4f, leftmost plot, bluish trend line).The three-way interaction model also shows other mixtures of expected and unexpected results. For example, the strongest increase in environmental temperature with beak length occurs in larger-bodied and shorter-legged birds (Fig. 4f, top-right plot, orange trend line), which clearly suggests a trade-off in evolution of body size and beak length and a similar trade-off in the evolution of the two types of appendages, presumably reflecting different adaptive responses for thermoregulation. However, a similar increase in beak length also occurs in tiny-bodied and longer-legged birds (Fig. 4f, top-right plot, blue trend line), which stands in contrast to this trade-off hypothesis. Likewise, the steepest increase in environmental temperature with tarsus length (Allen’s rule) occurs in larger-bodied and shorter-billed birds (Fig. 4f, bottom plot, pink trend line), again suggesting a compromise scenario, with elongated tarsus evolving as thermoregulatory organ to compensate for insufficient heat exchange due to large body and small beak. It also suggests that, in large birds, having a short beak in hot climates requires longer tarsi (Fig. 4f, top-right, pinkish and orange trend lines) and vice versa (Fig. 4f, bottom plot, rose and yellowish trend lines), indicating that in large species, the summarized length of two types of appendages is important for thermoregulation. However, by contrast, it seems that in small bodied species, beak and tarsi length evolved in a correlated way (Fig. 4f, top-right and bottom plots, green and blue trend lines) across environmental temperature (occurrences in warmer temperatures are associated with simultaneously both longer beaks and tarsi, or else simultaneously shorter beaks and tarsi). This may indicate a general tendency to correlated evolution of relative beak and tarsus lengths, perhaps for functional reasons, e.g. longer beaks may allow long-legged birds to explore substrate more efficiently, as longer necks also do47.Allen’s vs Bergmann’s rules in causal modelsOur hypothesis consequently holds within phylogenetic path analysis, where the best causal models integrate Bergmann’s and Allen’s rules to explain both the size of avian appendages (Fig. 5a) and the avian thermal environment (here, maximum temperature across all months) (Fig. 5b). The best model predicting beak and tarsus length includes the causal effect of temperature on body size (Bergmann’s rule) and then body size on beak and tarsus length (allometry), as well as the direct effect of temperature on the size of appendages (Allen’s rule) (Fig. 5a). This joint Bergmann’s and Allen’s model is substantially better than the model assuming that temperature does not affect body size before scaling for the length of appendages (Fig. 5a, Allen’s rule only). The combined Bergmann’s and Allen’s model is also better than one assuming no direct effect of temperature on appendages (Fig. 5a, Bergmann’s rule only). This again indicates that how the length of avian appendages co-varies with the ambient temperature partially depends on how avian body size co-varies with temperature, yielding results aligned with the trade-off hypothesis. This notably argues against the possibility that the increase in the length of appendages (relative to body size) with temperature is an artefact of decreased body sizes at hot temperatures (see26). However, interestingly, the model including only Allen’s rule (and allometry) explains the length of appendages with similar accuracy to the model with only Bergmann’s rule (Fig. 5a).Fig. 5: Phylogenetic path analysis with responses of the length of avian appendages (beak and tarsus) (a) and the maximum temperature of all months within species range (b) across 9962 species (99.7% of global community).In both cases model candidates include different combinations of allometry (relationship between the length of appendages and body size), Bergmann’s rule (the relationship between body size and the temperature) and Allen’s rule (relationship between the length of appendages and temperature). ∆CIC—delta C statistic Information Criterion. The results were obtained with phylogenetic path analysis by using phylopath models on a single maximum clade credibility phylogenetic tree and scaled covariates (mean = 0 ± 1 SD) to compare their effect sizes (see numbers on path diagrams).Full size imageThe best model predicting the temperature associations includes the indirect effect of body size on the length of appendages (allometry), and then the length of appendages on temperature (Allen’s rule), as well as the direct effect of body size on temperature (Bergmann’s rule) (Fig. 5b). These results again demonstrate that how the temperature varies across species ranges depends on both the size of body and appendages, suggesting that Bergmann’s and Allen’s rules describe two distinct evolutionary ways to cope with thermoregulation. Moreover, the similar performance of Allen’s model compared to Bergmann’s model (Fig. 5b) again suggests that shifts in the animal’s body size and shape represent roughly equally influential in the evolution of adaptations to novel climates.Excluding possible confounding factorsTo ensure the reliability of our findings, first we show that when explaining the phenotype (Supplementary Figs. 2b and 3b), or the temperature within species geographic ranges (Supplementary Fig. 6), the main results remain consistent whichever of the five temperature measures is included. Second, despite the fact that the relationships with relative length of appendages and the experienced temperatures are strongest in resident birds, followed by partial- and full- migrants, our results still hold when accounting for these three categories of avian migratory habits (Supplementary Fig. 7); and the compromise scenario remains similar in each of these groups independently (Supplementary Figs. 8–10). It aligns with previous studies22,46, which found that ecogeographical rules are valid regardless of variation in avian migratory habits. However, it is worth to notice that the most prominent trade-offs are found in resident species (in case of explaining environmental temperature, see Supplementary Fig. 10) or in partial migrants (in case of explaining beak length, see Supplementary Fig. 8). Third, the trade-offs in thermoregulatory strategies also hold after controlling for geographic range size (Supplementary Fig. 11) and remains quantitatively (Supplementary Figs. 12–13) or qualitatively (Supplementary Fig. 14) stable across the gradient of endemic-cosmopolitan species. Thus, even if ecogeographical rules operate within widespread species (across distanced populations, as well documented9,10,11,12,13), this does not appear to influences the results of our cross-species analysis. Fourth, the predictions of temperature within species geographic ranges are also not specific to the way by which we account for the allometry of appendages (by using residual appendage length). Parallel analyses with ratios of appendage length to body mass (Supplementary Fig. 15) or principal components of all phenotypic traits (Supplementary Fig. 16) give qualitatively similar outcomes. Fifth, we also show that results of both phylogenetic regression (Supplementary Fig. 17) and phylogenetic path models (Supplementary Fig. 18) remain consistently valid across 100 randomly chosen phylogenetic trees32, mitigating concerns regarding phylogenetic uncertainty influencing our results.Notably, there is a wider list of important ecological factors constraining or favoring variation in body size and shape, e.g. tropic levels or foraging techniques21,23,47,48, although they are also themselves constrained by phylogeny to some extent, which we control for. Nevertheless, we believe that it is likely that these constraints influenced (or were influenced by) the Bergmann-Allen trade-off. Understanding of this issue would benefit from a deeper dive into the relationship between climatic, phenotypic and ecological variation across animals.Our findings in the context of eco-evolutionary processes driven by climateTo the best of our knowledge, this study is the largest (taxonomically and geographically) simultaneous test of ecogeographical rules and it provides a first empirical evidence for a trade-off in the evolution of body size (Bergmann’s rule2) and the size of appendages (Allen’s rule3) across global temperature gradients. Our results confirm what Allen3 speculated—the larger the body, the stronger the increase in appendage size with temperature; and the larger the appendages, the milder the decrease in body size with temperature. Thus, the evolution of body size under temperature regimes likely depends on the size of appendages and, on the other hand, the extent to which temperature drives the size of appendages depends on body size. This means that these two thermoregulatory adaptations are not independent of each other, but the phenotype has at least two ways to adapt to novel climates, i.e. by the shifts in body size or the shifts in the size of appendages (or both to a lesser extent).The evolution of appendages (e.g. avian beaks49) was a dynamic process believed to overtake the changes in body size across evolutionary time50. Our analyses do not indicate, however, that shifts in body size have been more frequent than shifts in appendage size (or vice versa), at least not because of thermoregulation. Rather, they indicate that shifts in body size and shape are intertwined through avian evolutionary history, agreeing with the theory that animals select the most convenient strategy of thermoregulation to maintain functional traits of its phenotype. For functional reasons animal lineages tend to increase in body size over evolutionary time (Cope’s rule43), thus it is not surprising that strategies allowing species to maintain/develop larger bodies (i.e. over-increase in appendage size) are to be expected evolutionarily. On the other hand, some lineages may be constrained in appendage size (e.g. to forage21,47 or communicate23 effectively), hence those may favor the shifts in body size to reconcile optimal thermoregulation with a desired functionality.We found that the compromise in thermoregulatory strategies may also involve two distinct types of appendages, here beak versus tarsus. However, this is true only for larger-bodied species (see Fig. 4f top-right and bottom plots, trends for large bodies), that are more likely to acquire higher heat loads in warmer environments, thus the summarized size of many appendages may be for them crucial to disperse heat loads. Both beak and legs have been confirmed to act as key regions of heat transfer on the avian body37,38,51,52, thus both may be sensitive to thermal conditions when body size is too large to deal alone with too hot temperatures. Yet, in small-bodied species both appendages seem to evolve in concerted way across temperature gradients, and this may be in a way that conforms with Allen’s rule or not (see Fig. 4f, top-right and bottom plots, trends for small bodies), indicating that the small body ensures good temperature exchange in hot climates, thus the evolution of appendages in these species may be correlated, but independent of thermoregulatory selection pressures.It is worthwhile emphasizing that apart from shifts in body size and shape, many other elements combine to help birds meet their thermoregulatory requirements53, e.g. through variation in insulation (feathers)54, coloration55,56 metabolism57, blood circulation58 or behavior59,60,61. Extrapolating our results, these thermoregulatory strategies might also co-evolve under a trade-off to ensure optimal thermoregulation along with desired functionality. This is presumably a reason for the relatively low performance of our models; e.g. physical phenotype explains up to only 20% of the variance in ambient temperature (Fig. 4b, upper model), therefore unexplained variance must be attributed to other thermoregulatory strategies.In this study, we demonstrate that Allen’s rule may be attributed to the varying allometric functions across temperature gradients. Although logical and argued elsewhere26, it has never been addressed by any empirical research. Our findings clearly indicate the importance of considering body mass as both a fixed and interaction term in studies of Allen’s rule, but also might suggest that ambient temperature should be included in other allometric studies of animals’ morphology. That said, temperature explains very little of the variance in the size of appendages compared to body size (Figs. 2c and 3c), thus thermal conditions are unlikely to be a very crucial confounding factor for allometry in comparative analyses.In this study, we empirically confirm for the first time an evolutionary compromise theory that was first proposed almost 150 years ago3– the evolution of body size and appendages are two distinct and interacting ways to cope with thermoregulation. This may explain why many studies fail to detect Allen’s or Bergmann’s rules independently which has led to questioning of the generality of these ecogeographical patterns13,24,25. Here, our findings suggest that Bergmann’s and Allen’s rules should not necessarily be considered in isolation. We believe that these thermoregulatory strategies might intertwine through the evolutionary history of animals, as the evolution of phenotype possibly interacts to confound ecogeographical rules to evolve functional traits. This explanation also highlights the diverse mechanisms that animals may employ to expand across the world’s multiple environments. It also raises the speculation that with observed and future anticipated warming of Earth’s climate, we should expect mainly large animals to elongate in appendages, while mainly compact-bodied animals to shrink in size. More

A large-scale field study finds that different bee species experience different levels of risk from pesticides, depending on how much land is farmed within their foraging range. For bumblebees and solitary bees, more seminatural habitat means less risk from pesticides, but this is not true for honeybees.In the discussion of how to protect bees from pesticides, bees are often treated as a monolith. It is assumed that what is good for one species is good for all, and that pesticides or changes to agricultural landscapes would affect all bee species equally. This is often taken one step further, with the western honeybee (Apis mellifera) being used as a surrogate species for all bees. Yet despite this simplification there are around 2,000 species of bee in Europe1 and 20,000 worldwide2 with a dazzling diversity of niches and life histories. With this in mind, the question arises of how valid the assumption is that honeybees represent a good surrogate species. In this issue of Nature Ecology & Evolution, Knapp et al.3 investigate this question by measuring how three species of bee with differing life histories respond to different agricultural land-use intensities, and find that a species’ foraging range plays a big part in pesticide exposure risk. More

Study populations and sequencing strategyDNA libraries were prepared for 1261 D. sylvestris individuals from 115 populations (5–20 individuals per population) under a modified protocol49 of the Illumina Nextera DNA library preparation kit (Supplementary Methods S1.1, Supplementary Data 1). Individuals were indexed with unique dual-indexes (IDT Illumina Nextera 10nt UDI – 384 set) from Integrated DNA Technologies Co, to avoid index-hopping50. Libraries were sequenced (150 bp paired-end sequencing) in four lanes of an Illumina NovaSeq 6000 machine at Novogene Co. This resulted in an average coverage of ca. 2x per individual. Sequenced individuals were trimmed for adapter sequences (Trimmomatic version 0.3551), mapped (BWA-MEM version 0.7.1752,53) against a reference assembly54 (ca. 440 Mb), had duplicates marked and removed (Picard Toolkit version 2.0.1; http://broadinstitute.github.io/picard), locally realigned around indels (GATK version 3.555), recalibrated for base quality scores (ATLAS version 0.956) and had overlapping read pairs clipped (bamUtil version 1.0.1457) (Supplementary Methods S1.1). Population genetic analyses were performed on the resultant BAM files via genotype likelihoods (ANGSD version 0.93358 and ATLAS versions 0.9–1.056), to accommodate the propagation of uncertainty from the raw sequence data to population genetic inference.Population genetic structure and biogeographic barriersTo investigate the genetic structure of our samples (Fig. 2A, Supplementary Fig. S2), we performed principal component analyses (PCA) on all 1261 samples (“full” dataset) via PCAngsd version 0.9859, following conversion of the mapped sequence data to ANGSD genotype likelihoods in Beagle format (Supplementary Methods S1.2). To visualise PCA results in space (Supplementary Fig. S4), individuals’ principal components were projected on a map, spatially interpolated (linear interpolation, akima R package version 0.6.260) and had the first two principal components represented as green and blue colour channels. Given that uneven sampling can bias the inference of structure in PCA, PCA was also performed on a balanced dataset comprising a common, down-sampled size of 125 individuals per geographic region (“balanced” dataset; Fig. 2B, Supplementary Fig. S3; Supplementary Methods S1.2; Supplementary Data 1). Individual admixture proportions and ancestral allele frequencies were estimated using PCAngsd (-admix model) for K = 2–6, using the balanced dataset to avoid potential biases related to imbalanced sampling22,23 and an automatic search for the optimal sparseness regularisation parameter (alpha) soft-capped to 10,000 (Supplementary Methods S1.2). To visualise ancestry proportions in space, population ancestry proportions were spatially interpolated (kriging) via code modified from Ref. 61 (Supplementary Fig. S5).To test if between-lineage admixture underlies admixture patterns inferred by PCAngsd or if the data is better explained by alternative scenarios such as recent bottlenecks, we used chromosome painting and patterns of allele sharing to construct painting palettes via the programmes MixPainter and badMIXTURE (unlinked model)28 and compared this to the PCAngsd-inferred palettes (Fig. 2B, C; Supplementary Methods S1.2). We referred to patterns of residuals between these palettes to inform of the most likely underlying demographic scenario. For assessing Alpine–Balkan palette residuals (and hence admixture), 65 individuals each from the French Alps (inferred as pure Alpine ancestry in PCAngsd), Monte Baldo (inferred with both Alpine and Balkan ancestries in PCAngsd) and Julian Alps (inferred as pure Balkan ancestry in PCAngsd) were analysed under K = 2 in PCAngsd and badMIXTURE (Fig. 2C). For assessing Apennine–Balkan admixture, 22 individuals each from the French pre-Alps (inferred as pure Apennine ancestry in PCAngsd), Tuscany (inferred with both Apennine and Balkan ancestries in PCAngsd) and Julian Alps (inferred as pure Balkan ancestry in PCAngsd) were analysed under K = 2 in PCAngsd and badMIXTURE.To construct a genetic distance tree (Supplementary Fig. S1), we first calculated pairwise genetic distances between 549 individuals (5 individuals per population for all populations) using ATLAS, employing a distance measure (weight) reflective of the number of alleles differing between the genotypes (Supplementary Methods S1.2; Supplementary Data 1). A tree was constructed from the resultant distance matrix via an initial topology defined by the BioNJ algorithm with subsequent topological moves performed via Subtree Pruning and Regrafting (SPR) in FastME version 2.1.6.162. This matrix of pairwise genetic distances was also used as input for analyses of effective migration and effective diversity surfaces in EEMS25. EEMS was run setting the number of modelled demes to 1000 (Fig. 2A, Supplementary Fig. S8). For each case, ten independent Markov chain Monte Carlo (MCMC) chains comprising 5 million iterations each were run, with a 1 million iteration burn-in, retaining every 10,000th iteration. Biogeographic barriers (Fig. 2A, Supplementary Fig. S7) were further identified via applying Monmonier’s algorithm24 on a valuated graph constructed via Delauney triangulation of population geographic coordinates, with edge values reflecting population pairwise FST; via the adegenet R package version 2.1.163. FST between all population pairs were calculated via ANGSD, employing a common sample size of 5 individuals per population (Supplementary Fig. S6; Supplementary Methods S1.2; Supplementary Data 1). 100 bootstrap runs were performed to generate a heatmap of genetic boundaries in space, from which a weighted mean line was drawn (Supplementary Fig. S7). All analyses in ANGSD were performed with the GATK (-GL 2) model, as we noticed irregularities in the site frequency spectra (SFS) with the SAMtools (-GL 1) model similar to that reported in Ref. 58 with particular BAM files. All analyses described above were performed on the full genome.Ancestral sequence reconstructionTo acquire ancestral states and polarise site-frequency spectra for use in the directionality index ψ and demographic inference, we reconstructed ancestral genome sequences at each node of the phylogenetic tree of 9 Dianthus species: D. carthusianorum, D. deltoides, D. glacialis, D. sylvestris (Apennine lineage), D. lusitanus, D. pungens, D. superbus alpestris, D. superbus superbus, and D. sylvestris (Alpine lineage). This tree topology was extracted from a detailed reconstruction of Dianthus phylogeny based on 30 taxa by Fior et al. (Fior, Luqman, Scharmann, Zemp, Zoller, Pålsson, Gargano, Wegmann & Widmer; paper in preparation) (Supplementary Methods S1.3). For ancestral sequence reconstruction, one individual per species was sequenced at medium coverage (ca. 10x), trimmed (Trimmomatic), mapped against the D. sylvestris reference assembly (BWA-MEM) and had overlapping read pairs clipped (bamUtil) (Supplementary Methods S1.3). For each species, we then generated a species-specific FASTA using GATK FastaAlternateReferenceMaker. This was achieved by replacing the reference bases at polymorphic sites with species-specific variants as identified by freebayes64 (version 1.3.1; default parameters), while masking (i.e., setting as “N”) sites (i) with zero depth and (ii) that didn’t pass the applied variant filtering criteria (i.e., that are not confidently called as polymorphic; Supplementary Methods S1.3). Species FASTA files were then combined into a multi-sample FASTA. Using this, we probabilistically reconstructed ancestral sequences at each node of the tree via PHAST (version 1.4) prequel65, using a tree model produced by PHAST phylofit under a REV substitution model and the specified tree topology (Supplementary Methods S1.3). Ancestral sequence FASTA files were then generated from the prequel results using a custom script.Expansion signalTo calculate the population pairwise directionality index ψ for the Alpine lineage, we utilised equation 1b from Peter and Slatkin (2013)31, which defines ψ in terms of the two-population site frequency spectrum (2D-SFS) (Supplementary Methods S1.4). 2D-SFS between all population pairs (10 individuals per population; Supplementary Data 1) were estimated via ANGSD and realSFS66 (Supplementary Methods S1.4), for unfolded spectra. Unfolding of spectra was achieved via polarisation with respect to the ancestral state of sites defined at the D. sylvestris (Apennine lineage) – D. sylvestris (Alpine lineage) ancestral node. Correlation of pairwise ψ and (great-circle) distance matrices was tested via a Mantel test (10,000 permutations). To infer the geographic origin of the expansion (Fig. 3), we employed a time difference of arrival (TDOA) algorithm following Peter and Slatkin (2013);31 performed via the rangeExpansion R package version 0.0.0.900031,67. We further estimated the strength of the founder of this expansion using the same package.Demographic inferenceTo evaluate the demographic history of D. sylvestris, a set of candidate demographic models was formulated. To constrain the topology of tested models, we first inferred the phylogenetic tree of the three identified evolutionary lineages of D. sylvestris (Alpine, Apennine and Balkan) as embedded within the larger phylogeny of the Eurasian Dianthus clade (note that the phylogeny from Fior et al. (Fior, Luqman, Scharmann, Zemp, Zoller, Pålsson, Gargano, Wegmann & Widmer; paper in preparation) excludes Balkan representatives of D. sylvestris). Trees were inferred based on low-coverage whole-genome sequence data of 1–2 representatives from each D. sylvestris lineage, together with whole-genome sequence data of 7 other Dianthus species, namely D. carthusianorum, D. deltoides, D. glacialis, D. lusitanus, D. pungens, D. superbus alpestris and D. superbus superbus, that were used to root the D. sylvestris clade (Supplementary Methods S1.5). We estimated distance-based phylogenies using ngsDist68 that accommodates genotype likelihoods in the estimation of genetic distances (Supplementary Methods S1.5). Genetic distances were calculated via two approaches: (i) genome-wide and (ii) along 10 kb windows. For the former, 110 bootstrap replicates were calculated by re-sampling over similar-sized genomic blocks. For the alternative strategy based on 10 kb windows, window trees were combined using ASTRAL-III version 5.6.369 to generate a genome-wide consensus tree accounting for potential gene tree discordance (Supplementary Methods S1.5). Trees were constructed from matrices of genetic distances from initial topologies defined by the BioNJ algorithm with subsequent topological moves performed via Subtree Pruning and Regrafting (SPR) in FastME version 2.1.6.162. We rooted all resultant phylogenetic trees with D. deltoides as the outgroup70. Both approaches recovered a topology with the Balkan lineage diverging prior to the Apennine and Alpine lineages (Supplementary Fig. S9). This taxon topology for D. sylvestris was supported by high ASTRAL-III posterior probabilities ( >99%), ASTRAL-III quartet scores ( >0.5) and bootstrap values ( >99%). Topologies deeper in the tree were less well-resolved (with quartet scores More

Different picocyanobacterial communities exhibit distinct gene repertoiresTo analyze the distribution of Prochlorococcus and Synechococcus reads along the Tara Oceans transect, metagenomic reads corresponding to the bacterial size fraction were recruited against 256 picocyanobacterial reference genomes, including SAGs and MAGs representative of uncultured lineages (e.g., Prochlorococcus HLIII-IV, Synechococcus EnvA or EnvB). This yielded a total of 1.07 billion recruited reads, of which 87.7% mapped onto Prochlorococcus genomes and 12.3% onto Synechococcus genomes, which were then functionally assigned by mapping them onto the manually curated Cyanorak v2.1 CLOG database [19]. In order to identify picocyanobacterial genes potentially involved in niche adaptation, we analyzed the distribution across the oceans of flexible (i.e. non-core) genes. Clustering of Tara Oceans stations according to the relative abundance of flexible genes resulted in three well-defined clusters for Prochlorococcus (Fig. 1A), which matched those obtained when stations were clustered according to the relative abundance of Prochlorococcus ESTUs, as assessed using the high-resolution marker gene petB, encoding cytochrome b6 (Fig. 1A; [24]). Only a few discrepancies can be observed between the two trees, including stations TARA-070 that displayed one of the most disparate ESTU compositions and TARA-094, dominated by the rare HLID ESTU (Fig. 1A). Similarly, for Synechococcus, most of the eight assemblages of stations discriminated based on the relative abundance of ESTUs (Fig. 1B) were also retrieved in the clustering based on flexible gene abundance, except for a few intra-assemblage switches between stations, notably those dominated by ESTU IIA (Fig. 1B). Despite these few variations, four major clusters can be clearly delineated in both Synechococcus trees, corresponding to four broadly defined ecological niches, namely (i) cold, nutrient-rich, pelagic or coastal environments (blue and light red in Fig. 1B), (ii) Fe-limited environments (purple and grey), (iii) temperate, P-depleted, Fe-replete areas (yellow) and (iv) warm, N-depleted, Fe-replete regions (dark red). This correspondence between taxonomic and functional information was also confirmed by the high congruence between distance matrices based on ESTU relative abundance and on CLOG relative abundance (p-value  0.01) are marked by a cross. Φsat: index of iron limitation derived from satellite data. PAR30: satellite-derived photosynthetically available radiation at the surface, averaged on 30 days. DCM: depth of the deep chlorophyll maximum.Full size imageIdentification of individual genes potentially involved in niche partitioningTo identify genes relevant for adaptation to a specific set of environmental conditions and enriched in specific ESTU assemblages, we selected the most representative genes from each module (Dataset 5; Figs. 3, S2). Most genes retrieved this way encode proteins of unknown or hypothetical function (85.7% of 7,485 genes). However, among the genes with a functional annotation (Dataset 6), a large fraction seems to have a function related to their realized environmental niche (Figs. 3, S2). For instance, many genes involved in the transport and assimilation of nitrite and nitrate (nirA, nirX, moaA-C, moaE, mobA, moeA, narB, M, nrtP; [6]) as well as cyanate, an organic form of nitrogen (cynA, B, D, S), are enriched in the Prochlorococcus blue module, which is correlated with the HLIIA-D ESTU and to low inorganic N, P, and silica levels and anti-correlated with Fe availability (Fig. 2A–C). This is consistent with previous studies showing that while only a few Prochlorococcus strains in culture possess the nirA gene and even less the narB gene, natural Prochlorococcus populations inhabiting N-poor areas do possess one or both of these genes [40,41,42]. Similarly, numerous genes amongst the most representative of Prochlorococcus brown, red and turquoise modules are related to adaptation of HLIIIA/IVA, HLIA and LLIA ESTUs to Fe-limited, cold P-limited, and cold, mixed waters, respectively (Fig. 3). Comparable results were obtained for Synechococcus, although the niche delineation was less clear than for Prochlorococcus since genes within each module exhibited lower correlations with the module eigenvalue (Fig. S2). These results therefore constitute a proof of concept that this network analysis was able to retrieve niche-related genes from metagenomics data.Fig. 3: Violin plots highlighting the most representative genes of each Prochlorococcus module.For each module, each gene is represented as a dot positioned according to its correlation with the eigengene for each module, the most representative genes being localized on top of each violin plot. Genes mentioned in the text and/or in Dataset 6 have been colored according to the color of the corresponding module, indicated by a colored bar above each module. The text above violin plots indicates the most significant environmental parameter(s) and/or ESTU(s) for each module, as derived from Fig. 2.Full size imageIdentification of eCAGs potentially involved in niche partitioningIn order to better understand the function of niche-related genes, notably of the numerous unknowns, we then integrated global distribution data with gene synteny in reference genomes using a network approach (Datasets 7, 8). This led us to identify clusters of adjacent genes in reference genomes, and thus potentially involved in the same metabolic pathway (Figs. 4, S3, S4; Dataset 6). These clusters were defined within each module and thus encompass genes with similar distribution and abundance in situ. Hereafter, these environmental clusters of adjacent genes will be called “eCAGs”.Fig. 4: Delineation of Prochlorococcus eCAGs, defined as a set of genes that are both adjacent in reference genomes and share a similar in situ distribution.Nodes correspond to individual genes with their gene name (or significant numbers of the CK number, e.g. 1234 for CK_00001234) and are colored according to their WGCNA module. A link between two nodes indicates that these two genes are less than five genes apart in at least one genome. The bottom insert shows the most significant environmental parameter(s) and/or ESTU(s) for each module, as derived from Fig. 2.Full size imageeCAGs related to nitrogen metabolismThe well-known nitrate/nitrite gene cluster involved in uptake and assimilation of inorganic forms of N (see above), which is present in most Synechococcus genomes (Dataset 6), was expectedly not restricted to a particular niche in natural Synechococcus populations, as shown by its quasi-absence from WGCNA modules. In Prochlorococcus, this cluster is separated into two eCAGs enriched in low-N areas (Fig. S5A, B), most genes being included in Pro-eCAG_002, present in only 13 out of 118 Prochlorococcus genomes, while nirA and nirX form an independent eCAG (Pro-eCAG_001) due to their presence in many more genomes. The quasi-core ureA-G/urtB-E genomic region was also found to form a Prochlorococcus eCAG (Pro-eCAG_003) that was impoverished in low-Fe compared to other regions (Fig. S5C, D), in agreement with its presence in only two out of six HLIII/IV genomes. We also uncovered several other Prochlorococcus and Synechococcus eCAGs that seem to be involved in the transport and/or assimilation of more unusual and/or complex forms of nitrogen, which might either be degraded into elementary N molecules or possibly directly used by cells for e.g. the biosynthesis of proteins or DNA. Indeed, we detected in both genera an eCAG (Pro-eCAG_004 and Syn-eCAG_001; Fig. S6A, B; Dataset 6) that encompasses speB2, an ortholog of Synechocystis PCC 6803 sll1077, previously annotated as encoding an agmatinase [29, 43] and which was recently characterized as a guanidinase that degrades guanidine rather than agmatine to urea and ammonium [44]. E. coli produces guanidine under nutrient-poor conditions, suggesting that guanidine metabolism is biologically significant and potentially prevalent in natural environments [44, 45]. Furthermore, the ykkC riboswitch candidate, which was shown to specifically sense guanidine and to control the expression of a variety of genes involved in either guanidine metabolism or nitrate, sulfate, or bicarbonate transport, is located immediately upstream of this eCAG in Synechococcus reference genomes, all genes of this cluster being predicted by RegPrecise 3.0 to be regulated by this riboswitch (Fig. S6C; [45, 46]). The presence of hypA and B homologs within this eCAG furthermore suggests that, in the presence of guanidine, these homologs could be involved in the insertion of Ni2+, or another metal cofactor, in the active site of guanidinase. The next three genes of this eCAG, which encode an ABC transporter similar to the TauABC taurine transporter in E. coli (Fig. S6C), could be involved in guanidine transport in low-N areas. Of note, the presence in most Synechococcus/Cyanobium genomes possessing this eCAG of a gene encoding a putative Rieske Fe-sulfur protein (CK_00002251) downstream of this gene cluster, seems to constitute a specificity compared to the homologous gene cluster in Synechocystis sp. PCC 6803. The presence of this Fe-S protein suggests that Fe is used as a cofactor in this system and might explain why this gene cluster is absent from picocyanobacteria thriving in low-Fe areas, while it is present in a large proportion of the population in most other oceanic areas (Fig. S6A, B).Another example of the use of organic N forms concerns compounds containing a cyano radical (C ≡ N). The cyanate transporter genes (cynABD) were indeed found in a Prochlorococcus eCAG (Pro-eCAG_005, also including the conserved hypothetical gene CK_00055128; Fig. S7A, B). While only a small proportion of the Prochlorococcus community possesses this eCAG in warm, Fe-replete waters, it is absent from other oceanic areas in accordance with its low frequency in Prochlorococcus genomes (present in only two HLI and five HLII genomes). In Synechococcus these genes were not included in a module, and thus are not in an eCAG (Dataset 6; Fig. S7C), but seem widely distributed despite their presence in only a few Synechococcus genomes (mostly in clade III strains; [6, 47, 48]). Interestingly, we also uncovered a 7-gene eCAG (Pro-eCAG_006 and Syn-eCAG_002), encompassing a putative nitrilase gene (nitC), which also suggests that most Synechococcus cells and a more variable fraction of the Prochlorococcus population could use nitriles or cyanides in warm, Fe-replete waters and more particularly in low-N areas such as the Indian Ocean (Fig. 5A, B). The whole operon (nitHBCDEFG; Fig. 5C), called Nit1C, was shown to be upregulated in the presence of cyanide and to trigger an increase in the rate of ammonia accumulation in the heterotrophic bacterium Pseudomonas fluorescens [49], suggesting that like cyanate, cyanide could constitute an alternative nitrogen source in marine picocyanobacteria as well. However, given the potential toxicity of these C ≡ N-containing compounds [50], we cannot exclude that these eCAGs could also be devoted to cell detoxification [45, 47]. Such an example of detoxification has been described for arsenate and chromate that, as analogs of phosphate and sulfate respectively, are toxic to marine phytoplankton and must be actively exported out of the cells [51, 52].Fig. 5: Global distribution map of the eCAG involved in nitrile or cyanide transport and assimilation.A Prochlorococcus Pro-eCAG_006. B Synechococcus Syn-eCAG_002. C The genomic region in Prochlorococcus marinus MIT9301. The size of the circle is proportional to relative abundance of each genus as estimated based on the single-copy core gene petB and this gene was also used to estimate the relative abundance of other genes in the population. Black dots represent Tara Oceans stations for which Prochlorococcus or Synechococcus read abundance was too low to reach the threshold limit.Full size imageWe detected the presence of an eCAG encompassing asnB, pyrB2, and pydC (Pro-eCAG_007, Syn-eCAG_003, Fig. S8), which could contribute to an alternative pyrimidine biosynthesis pathway and thus provide another way for cells to recycle complex nitrogen forms. While this eCAG is found in only one fifth of HLII genomes and in quite specific locations for Prochlorococcus, notably in the Red Sea, it is found in most Synechococcus cells in warm, Fe-replete, N and P-depleted niches, consistent with its phyletic pattern showing its absence only from most clade I, IV, CRD1, and EnvB genomes (Fig. S8; Dataset 6). More generally, most N-uptake and assimilation genes in both genera were specifically absent from Fe-depleted areas, including the nirA/narB eCAG for Prochlorococcus, as mentioned by Kent et al. [36] as well as guanidinase and nitrilase eCAGs. In contrast, picocyanobacterial populations present in low-Fe areas possess, in addition to the core ammonium transporter amt1, a second transporter amt2, also present in cold areas for Synechococcus (Fig. S9). Additionally, Prochlorococcus populations thriving in HNLC areas also possess two amino acid-related eCAGs that are present in most Synechococcus genomes, the first one involved in polar amino acid N-II transport (Pro-eCAG_008; natF-G-H-bgtA; [53]; Fig. S10A, B) and the second one (leuDH-soxA-CK_00001744, Pro-eCAG_009, Fig. S10C, D) that notably encompasses a leucine dehydrogenase, able to produce ammonium from branched-chain amino acids. This highlights the profound difference in N acquisition mechanisms between HNLC regions and Fe-replete, N-deprived areas: the primary nitrogen sources for picocyanobacterial populations dwelling in HNLC areas seem to be ammonium and amino acids, while N acquisition mechanisms are more diverse in N-limited, Fe-replete regions.eCAGs related to phosphorus metabolismAdaptation to P depletion has been well documented in marine picocyanobacteria showing that while in P-replete waters Prochlorococcus and Synechococcus essentially rely on inorganic phosphate acquired by core transporters (PstSABC), strains isolated from low-P regions and natural populations thriving in these areas additionally contain a number of accessory genes related to P metabolism, located in specific genomic islands [6, 14, 30,31,32, 54]. Here, we indeed found in Prochlorococcus an eCAG containing the phoBR operon (Pro-eCAG_010) that encodes a two-component system response regulator, as well as an eCAG including the alkaline phosphatase phoA (Pro-eCAG_011), both present in virtually the whole Prochlorococcus population from the Mediterranean Sea, the Gulf of Mexico and the Western North Atlantic Ocean, which are known to be P-limited [30, 55] (Fig. S11A, B). By comparison, in Synechococcus, we only identified the phoBR eCAG (Syn-eCAG_005, Fig. S11C) that is systematically present in warm waters whatever the limiting nutrient, in agreement with its phyletic pattern in reference genomes showing its specific absence from cold thermotypes (clades I and IV, Dataset 6). Furthermore, although our analysis did not retrieve them within eCAGs due to the variability of gene content and synteny in this genomic region, even within each genus, several other P-related genes were enriched in low-P areas but partially differed between Prochlorococcus and Synechococcus (Figs. 3, S2, S11; Dataset 6). While the genes putatively encoding a chromate transporter (ChrA) and an arsenate efflux pump ArsB were present in both genera in different proportions, a putative transcriptional phosphate regulator related to PtrA (CK_00056804; [56]) was specific to Prochlorococcus. Synechococcus in contrast harbors a large variety of alkaline phosphatases (PhoX, CK_00005263 and CK_00040198) as well as the phosphate transporter SphX (Fig. S11).Phosphonates, i.e. reduced organophosphorus compounds containing C–P bonds that represent up to 25% of the high-molecular-weight dissolved organic P pool in the open ocean, constitute an alternative P form for marine picocyanobacteria [57]. We indeed identified, in addition to the core phosphonate ABC transporter (phnD1-C1-E1), a second previously unreported putative phosphonate transporter phnC2-D2-E2-E3 (Pro-eCAG_012; Fig. 6A). Most of the Prochlorococcus population in strongly P-limited areas of the ocean harbored these genes, while they were absent from other areas, consistent with their presence in only a few Prochlorococcus and no Synechococcus genomes. Furthermore, as previously described [58,59,60], we found a Prochlorococcus eCAG encompassing the phnYZ operon involved in C-P bond cleavage, the putative phosphite dehydrogenase ptxD, and the phosphite and methylphosphonate transporter ptxABC (Pro-eCAG_0013, Dataset 6; Fig. 6B, [60,61,62]). Compared to these previous studies that mainly reported the presence of these genes in Prochlorococcus cells from the North Atlantic Ocean, here we show that they actually occur in a much larger geographic area, including the Mediterranean Sea, the Gulf of Mexico, and the ALOHA station (TARA_132) in the North Pacific, even though they were present in a fairly low fraction of Prochlorococcus cells. These genes occurred in an even larger proportion of the Synechococcus population, although not found in an eCAG for this genus (Fig. S12; Dataset 6). Synechococcus cells from the Mediterranean Sea, a P-limited area dominated by clade III [24], seem to lack phnYZ, in agreement with the phyletic pattern of these genes in reference genomes, showing the absence of this two-gene operon in the sole clade III strain that possesses the ptxABDC gene cluster. In contrast, the presence of the complete gene set (ptxABDC-phnYZ) in the North Atlantic, at the entrance of the Mediterranean Sea, and in several clade II reference genomes rather suggests that it is primarily attributable to this clade. Altogether, our data indicate that part of the natural populations of both Prochlorococcus and Synechococcus would be able to assimilate phosphonate and phosphite as alternative P-sources in low-P areas using the ptxABDC-phnYZ operon. Yet, the fact that no picocyanobacterial genome except P. marinus RS01 (Fig. 6C) possesses both phnC2-D2-E2-E3 and phnYZ, suggests that the phosphonate taken up by the phnC2-D2-E2-E3 transporter could be incorporated into cell surface phosphonoglycoproteins that may act to mitigate cell mortality by grazing and viral lysis, as recently suggested [63].Fig. 6: Global distribution map of eCAGs putatively involved in phosphonate and phosphite transport and assimilation.A Prochlorococcus Pro-eCAG_012 putatively involved in phosphonate transport. B Prochlorococcus Pro-eCAG_013, involved in phosphonate/phosphite uptake and assimilation and phosphonate C-P bond cleavage. C The genomic region encompassing both phnC2-D2-E2-E3 and ptxABDC-phnYZ specific to P. marinus RS01. The size of the circle is proportional to relative abundance of Prochlorococcus as estimated based on the single-copy core gene petB and this gene was also used to estimate the relative abundance of other genes in the population. Black dots represent Tara Oceans stations for which Prochlorococcus read abundance was too low to reach the threshold limit.Full size imageeCAGs related to iron metabolismAs for macronutrients, it has been hypothesized that the survival of marine picocyanobacteria in low-Fe regions was made possible through several strategies, including the loss of genes encoding proteins that contain Fe as a cofactor, the replacement of Fe by another metal cofactor, and the acquisition of genes involved in Fe uptake and storage [14, 15, 36, 39, 64]. Accordingly, several eCAGs encompassing genes encoding proteins interacting with Fe were found in modules anti-correlated to HNLC regions in both genera. These include three subunits of the (photo)respiratory complex succinate dehydrogenase (SdhABC, Pro-eCAG_014, Syn-eCAG_006, Fig. S13; [65]) and Fe-containing proteins encoded in most abovementioned eCAGs involved in N or P metabolism, such as the guanidinase (Fig. S6), the NitC1 (Fig. 5), the pyrB2 (Fig. S8), the phosphonate (Fig. 6, S12), and the urea and inorganic nitrogen eCAGs (Fig. S5). Most Synechococcus cells thriving in Fe-replete areas also possess the sodT/sodX eCAG (Syn-eCAG_007, Fig. S14A, B) involved in nickel transport and maturation of the Ni-superoxide dismutase (SodN), these three genes being in contrast core in Prochlorococcus. Additionally, Synechococcus from Fe-replete areas, notably from the Mediterranean Sea and the Indian Ocean, specifically possess two eCAGs (Syn-eCAG_008 and 009; Fig. S14C, D), involved in the biosynthesis of a polysaccharide capsule that appear to be most similar to the E. coli groups 2 and 3 kps loci [66]. These extracellular structures, known to provide protection against biotic or abiotic stress, were recently shown in Klebsiella to provide a clear fitness advantage in nutrient-poor conditions since they were associated with increased growth rates and population yields [67]. However, while these authors suggested that capsules may play a role in Fe uptake, the significant reduction in the relative abundance of kps genes in low-Fe compared to Fe-replete areas (t-test p-value  More

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Modular components of the DFTe energy functionalThe central ingredient of DFTe is an energy functional E, assembled according to Eq. (1). The methodology of DFTe can be understood by inspecting the dispersal and environmental energies in Eqs. (2) and (3) without interactions. In our first case study, illustrated in Fig. 2 and Supplementary Fig. 2, we demonstrate that equation (3), in conjunction with Eq. (2), can realistically describe the influence of the environment on species’ distributions. Mechanisms that alter the trade-off between dispersal and environment can be introduced as part of Eint. For instance, back reactions on the environment could be modelled with a bifunctional Ebr[Venv, n] that yields the equilibrated modified environment ({V}_{s}^{{{{{{{{rm{env}}}}}}}}}+delta {E}_{{{{{{{{rm{br}}}}}}}}}[{{{{{{{{bf{V}}}}}}}}}^{{{{{{{{rm{env}}}}}}}}},{{{{{{{bf{n}}}}}}}}]/delta {n}_{s}({{{{{{{bf{r}}}}}}}})), cf. Eq. (5).In the following we make explicit the interaction and resource energies that enter Eq. (1) and are used in our case studies of Figs. 2–7. We let Eint[n] include all possible bipartite interactions$${E}_{gamma }[{{{{{{{bf{n}}}}}}}}]=mathop{sum }limits_{{s,{s}^{{prime} }!=!1}atop {{s}^{{prime} }ne s}}^{S}{int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}})({{{{{{{rm{d}}}}}}}}{{{{{{{{bf{r}}}}}}}}}^{{prime} }){n}_{s}{({{{{{{{bf{r}}}}}}}})}^{{alpha }_{s}},{gamma }_{s{s}^{{prime} }}({{{{{{{bf{r}}}}}}}},, {{{{{{{{bf{r}}}}}}}}}^{{prime} }){n}_{{s}^{{prime} }}{({{{{{{{{bf{r}}}}}}}}}^{{prime} })}^{{beta }_{{s}^{{prime} }}},$$
(6)
which include amensalism, commensalism, mutualism, and so forth. Here, ({alpha }_{s},, {beta }_{{s}^{{prime} }}ge 0), and the interaction kernels ({gamma }_{s{s}^{{prime} }}) are assembled from fitness proxies of species s and ({s}^{{prime} }) (Supplementary Table 1). Higher-order interactions can be introduced, for example, through (i) terms like ({n}_{s},{gamma }_{s{s}^{{prime} }},{n}_{{s}^{{prime} }},{gamma }_{{s}^{{prime} }{s}^{{primeprime} }}^{{prime} },{n}_{{s}^{{primeprime} }}) that build on pairwise interactions or (ii) genuinely multipartite expressions like ({gamma }_{s{s}^{{prime} }{s}^{{primeprime} }}{n}_{s},,{n}_{{s}^{{prime} }},{n}_{{s}^{{primeprime} }}). Multi-partite interactions based on bipartite interactions do not seem to be an uncommon scenario48. However, there may be systems where nonzero coefficients ({gamma }_{s{s}^{{prime} }{s}^{{primeprime} }}) couple all species. This poses a challenge for mechanistic theories in general. Then, ‘simpler subsystems’ that have to be included in the DFTe workflow of Fig. 1a can only refer to situations where other energy components are absent, such as resource terms or complex environments. For example, the coefficients ({gamma }_{s{s}^{{prime} }{s}^{{primeprime} }}) could be extracted in an experiment with a controlled simple environment and then used to model the interacting species in a real-world setting. For (({alpha }_{s},, {beta }_{{s}^{{prime} }})=(1,1)) we identify the contact interaction in physics as ({gamma }_{s{s}^{{prime} }}propto delta ({{{{{{{bf{r}}}}}}}}-{{{{{{{{bf{r}}}}}}}}}^{{prime} })) with the two-dimensional delta function δ( ), while the Coulomb interaction amounts to setting ({gamma }_{s{s}^{{prime} }}propto 1/|{{{{{{{bf{r}}}}}}}}-{{{{{{{{bf{r}}}}}}}}}^{{prime} }|). The mechanistic effect of these interaction kernels on the density distributions is the same in ecology as it is in physics—a mathematical insight that inspired us to build ecological analogues to the phenomenology of quantum gases, which feature functionals of the kind in Eq. (6). Note that we do not introduce any quantum effects into ecology despite the fact that the mathematical structure of DFTe is borrowed in part from quantum physics. While the contact interaction is a suitable candidate for plants and especially microbes52, we expect long-range interactions (for example, repulsion of Coulomb type) to be more appropriate for species with long-range sensors, such as eyes. Both types of interactions feature in describing the ecosystems addressed in this work.In a natural setting the equilibrium abundances are ultimately constrained by the accessible resources. It is within these limits of resource availability that environment as well as intra- and inter-specific interactions can shape the density distributions. An energy term for penalising over- and underconsumption of resources is thus of central importance. Each species consumes resources from some of the K provided resources, indexed by k. A subset of species consumes the locally available resource density ρk(r) according to the resource requirements νks, which represent the absolute amount of resource k consumed by one individual (or aggregated constituent) of species s. The simple quadratic functional$${E}_{{{{{{{{rm{Res}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]={int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}})mathop{sum }limits_{k=1}^{K}{{{{{{{{mathcal{L}}}}}}}}}_{k}left({{{{{{{bf{n}}}}}}}},, {rho }_{k}right)equiv zeta {int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}})mathop{sum }limits_{k=1}^{K}{w}_{k}({{{{{{{bf{r}}}}}}}}){left[mathop{sum }limits_{s=1}^{S}{nu }_{ks}{n}_{s}({{{{{{{bf{r}}}}}}}})-{rho }_{k}({{{{{{{bf{r}}}}}}}})right]}^{2}$$
(7)
proves appropriate. Here, νksns is the portion of resource density ρk that is consumed by species s. That is, νks  > 0 indicates that s requires resource k. If Eq. (7) is the total energy functional, then a single-species system with a single resource equilibrates with density n1(r) = ρ1(r)/ν11 at every position r, and additional DFTe energy components would modify this equilibrium. Predator–prey relationships are introduced by making species k a resource ({rho }_{k}=left]{n}_{k}right[), where (left]nright[) declares n a constant w.r.t. the functional differentiation of E, that is, the predator tends to align with the prey, not the prey with the predator. In view of the energy minimisation, the quadratic term in Eq. (7) entails that regions of low resource density ρk are less important than regions of high ρk. The different resources k have the same ability to limit the abundances, such that the limiting resource k = l at r has to come with the largest of weights wl(r), irrespective of the absolute amounts of resources at r. For example, the weights wk have to ensure that an essential but scarce mineral has (a priori) the same ability to limit the abundances as a resource like water, which might be abundant in absolute terms. To that end, we specify the weights$${w}_{k}({{{{{{{bf{r}}}}}}}})=frac{1}{{bar{rho }}_{k}^{2}}mathop{sum}limits_{s}eta ({nu }_{ks})exp left[sigma left(frac{{lambda }_{ks}}{{lambda }_{ls}}-1right)right],$$
(8)
which are inspired by the smooth minimum function, where σ  λls irrelevant at r. Using ({E}_{{{{{{{{rm{Res}}}}}}}}}), we show that an analytically solvable minimal example of two amensalistically interacting species already exhibits a plethora of resource-dependent equilibrium states (see Supplementary Notes and Supplementary Fig. 1).We specify the DFTe energy functional in Eq. (1) by summing Eqs. (2), (3), (6), and (7) and by (optionally) constraining the abundances to N via Lagrange multipliers μ:$$E[{{{{{{{bf{n}}}}}}}},, {{{{{{{boldsymbol{mu }}}}}}}}]({{{{{{{bf{N}}}}}}}}) equiv E[{{{{{{{bf{n}}}}}}}}]+{E}_{{{{{{{{boldsymbol{mu }}}}}}}}}[{{{{{{{bf{n}}}}}}}}]({{{{{{{bf{N}}}}}}}})\ equiv {E}_{{{{{{{{rm{dis}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]+{E}_{{{{{{{{rm{env}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]+{E}_{gamma }[{{{{{{{bf{n}}}}}}}}]+{E}_{{{{{{{{rm{Res}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]+mathop{sum }limits_{s=1}^{S}{mu }_{s}left({N}_{s}-{int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}}),{n}_{s}right).$$
(9)
Uniform situations are characterised by spatially constant ingredients ns = Ns/A, ρk = Rk/A, coefficients τs, etc. for the DFTe energy, such that Eq. (9) reduces to a function E(N) with building blocks$${E}_{{{{{{{{rm{dis}}}}}}}}}longrightarrow frac{1}{2,A}mathop{sum }limits_{s=1}^{S}{tau }_{s},{N}_{s}^{2},$$
(10)
$${E}_{{{{{{{{rm{env}}}}}}}}}longrightarrow mathop{sum }limits_{s=1}^{S}{V}_{s}^{{{{{{{{rm{env}}}}}}}}},{N}_{s},$$
(11)
$${E}_{gamma }longrightarrow mathop{sum }limits_{{s,{s}^{{prime} }!=!1}atop {{s}^{{prime} }ne s}}^{S}frac{{N}_{s}^{{alpha }_{s}},{gamma }_{s{s}^{{prime} }},{N}_{{s}^{{prime} }}^{{beta }_{{s}^{{prime} }}}}{{A}^{{alpha }_{s}+{beta }_{{s}^{{prime} }}-1}},$$
(12)
$${E}_{{{{{{{{rm{Res}}}}}}}}}longrightarrow Amathop{sum }limits_{k=1}^{K}{{{{{{{{mathcal{L}}}}}}}}}_{k}left({{{{{{{bf{N}}}}}}}}/A,, {R}_{k}/Aright).$$
(13)
Ecosystem equilibria from the DFTe energy functionalThe general form of Eq. (9) gives rise to two types of minimisers (viz., equilibria): First, we term$${{{{{{{mathcal{H}}}}}}}}({{{{{{{bf{N}}}}}}}})equiv E[tilde{{{{{{{{bf{n}}}}}}}}}]equiv mathop{min }limits_{{{{{{{{bf{n}}}}}}}}}left{E[{{{{{{{bf{n}}}}}}}}],left|,{int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}}),{{{{{{{bf{n}}}}}}}}({{{{{{{bf{r}}}}}}}})={{{{{{{bf{N}}}}}}}},{{{{{{{rm{(fixed)}}}}}}}}right.right}$$
(14)
the ‘DFTe hypersurface’, with (tilde{{{{{{{{bf{n}}}}}}}}}) the energy-minimising spatial density profiles for given (fixed) N. Second, the ecosystem equilibrium is attained at the equilibrium abundances (hat{{{{{{{{bf{N}}}}}}}}}={int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}}),hat{{{{{{{{bf{n}}}}}}}}}({{{{{{{bf{r}}}}}}}})), which yield the global energy minimum$${{{{{{{mathcal{H}}}}}}}}(hat{{{{{{{{bf{N}}}}}}}}})=mathop{min }limits_{{{{{{{{bf{N}}}}}}}}},{{{{{{{mathcal{H}}}}}}}}({{{{{{{bf{N}}}}}}}}),$$
(15)
where the minimisation samples all admissible abundances, that is, ({{{{{{{bf{N}}}}}}}}in {left({{mathbb{R}}}_{0}^{+}right)}^{times S}) if no further constraints are imposed.The direct minimisation of E[n] is most practical for uniform systems, which only require us to minimise E(N) over an S-dimensional space of abundances. For the general nonuniform case, we adopt a two-step strategy that reflects Eqs. (14) and (15). First, we obtain the equilibrated density distributions on ({{{{{{{mathcal{H}}}}}}}}) for fixed N from the computational DPFT framework26,27,28,29,30,31. Second, a conjugate gradient descent searches ({{{{{{{mathcal{H}}}}}}}}({{{{{{{bf{N}}}}}}}})) for the global minimiser (hat{{{{{{{{bf{N}}}}}}}}}). Technically, we perform the computationally more efficient descent in μ-space. Local minima are frequently encountered, and we identify the best candidate for the global minimum from many individual runs that are initialised with random μ. Note that system realisations with energies close to the global minimum, especially local minima, are likely observable in reality, assuming that the system can equilibrate at all. There is always an equilibrium if the energy functional is bounded from below, together with the fact that the support (abundances/densities) of the energy functional is finite in any practical application. If some DFTe energy components are chosen (too) negative, the system can be unstable, in which case the energy functional has no minimum and is inappropriate for modelling the equilibrium in question. This means that another energy functional has to be considered, or, in the worst case, that DFTe is incapable of simulating this system. We also caution that no numerical optimisation algorithms for non-convex black-box functions can guarantee to find the global minimum, not even approximately. Without analytically available characteristics of the global minimum, all one may hope for are candidates of the minimiser, and those may not even be local minima—there is no way to be certain that an optimum proposed by a numerical optimisation algorithm is stable.Density-potential functional theory (DPFT) in Thomas–Fermi (TF) approximationDefining$${V}_{s}({{{{{{{bf{r}}}}}}}})={mu }_{s}-frac{delta {E}_{{{{{{{{rm{dis}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]}{delta {n}_{s}({{{{{{{bf{r}}}}}}}})}$$
(16)
for all s, we obtain the reversible Legendre transform$${E}_{{{{{{{{rm{dis}}}}}}}}}^{{{{{{{{rm{L}}}}}}}}}[{{{{{{{bf{V}}}}}}}}-{{{{{{{boldsymbol{mu }}}}}}}}]={E}_{{{{{{{{rm{dis}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]+mathop{sum }limits_{s=1}^{S}{int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}}),({V}_{s}-{mu }_{s}),{n}_{s}$$
(17)
of the dispersal energy and thereby supplement the total energy with the additional variables V:$$E[{{{{{{{bf{V}}}}}}}},, {{{{{{{bf{n}}}}}}}},, {{{{{{{boldsymbol{mu }}}}}}}}]({{{{{{{bf{N}}}}}}}})={E}_{{{{{{{{rm{dis}}}}}}}}}^{{{{{{{{rm{L}}}}}}}}}[{{{{{{{bf{V}}}}}}}}-{{{{{{{boldsymbol{mu }}}}}}}}]-{int}_{A}({{{{{{{rm{d}}}}}}}}{{{{{{{bf{r}}}}}}}}),{{{{{{{bf{n}}}}}}}}cdot ({{{{{{{bf{V}}}}}}}}-{{{{{{{{bf{V}}}}}}}}}^{{{{{{{{rm{env}}}}}}}}})+{E}_{{{{{{{{rm{int}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]+{{{{{{{boldsymbol{mu }}}}}}}}cdot {{{{{{{bf{N}}}}}}}}.$$
(18)
This density-potential functional is equivalent to (but more flexible than) the density-only functional E[n,  μ](N). The minimisers of E[n] are thus among the stationary points of Eq. (18) and are obtained by solving$${n}_{s}[{V}_{s}-{mu }_{s}]({{{{{{{bf{r}}}}}}}})=frac{delta {E}_{{{{{{{{rm{dis}}}}}}}}}^{{{{{{{{rm{L}}}}}}}}}[{V}_{s}-{mu }_{s}]}{delta {V}_{s}({{{{{{{bf{r}}}}}}}})}$$
(19)
and$${V}_{s}[{{{{{{{bf{n}}}}}}}}]({{{{{{{bf{r}}}}}}}})={V}_{s}^{{{{{{{{rm{env}}}}}}}}}({{{{{{{bf{r}}}}}}}})+frac{delta {E}_{{{{{{{{rm{int}}}}}}}}}[{{{{{{{bf{n}}}}}}}}]}{delta {n}_{s}({{{{{{{bf{r}}}}}}}})}$$
(20)
self-consistently for all ns while enforcing ∫A(dr) ns(r) = Ns. Specifically, starting from V(0) = Venv, such that ({n}_{s}^{(0)}={n}_{s}[{V}_{s}^{(0)}-{mu }_{s}^{(0)}]), we iterate$${n}_{s}^{(i)}mathop{longrightarrow }limits^{{{{{{{{rm{equation}}}}}}}},(20)}{V}_{s}^{(i+1)}={V}_{s}[{{{{{{{{bf{n}}}}}}}}}^{(i)}]mathop{longrightarrow }limits^{{{{{{{{rm{equation}}}}}}}},(19)}{n}_{s}^{(i+1)}=(1-{theta }_{s}),{n}_{s}^{(i)}+{theta }_{s},{n}_{s}left[{V}_{s}^{(i+1)}-{mu }_{s}^{(i+1)}right]$$
(21)
until all ns are converged sufficiently. This self-consistent loop establishes a trade-off between dispersal energy and effective environment V by forcing an initial out-of-equilibrium density distribution to equilibrate at fixed N. We adjust ({mu }_{s}^{(i)}) in each iteration i such that ({n}_{s}^{(i)}) integrates to Ns. Small enough density admixtures, with 0  More