ARTICLE https://doi.org/10.1038/s41467-021-23894-3 OPEN The long lives of primates and the ‘invariant rate of ageing’ hypothesis Fernando Colchero 1,2,46 ✉, José Manuel Aburto 2,3,4, Elizabeth A. Archie5,6, Christophe Boesch 7,8, Thomas Breuer9,10, Fernando A. Campos 11, Anthony Collins12, Dalia A. Conde 2,13,14, Marina Cords 15,16, Catherine Crockford7,8, Melissa Emery Thompson 17,18, Linda M. Fedigan19, Claudia Fichtel 20, Milou Groenenberg9,21, Catherine Hobaiter22,23, Peter M. Kappeler 20,24, Richard R. Lawler25, Rebecca J. Lewis26,27, Zarin P. Machanda18,28, Marie L. Manguette7,9, Martin N. Muller17,18, Craig Packer 29, 1234567890():,; Richard J. Parnell9, Susan Perry30, Anne E. Pusey 31, Martha M. Robbins7, Robert M. Seyfarth32, Joan B. Silk33, Johanna Staerk 2,13,14, Tara S. Stoinski34, Emma J. Stokes35, Karen B. Strier 36, Shirley C. Strum37,38,39,40, Jenny Tung 31,41,42, Francisco Villavicencio 43, Roman M. Wittig 7,8, Richard W. Wrangham18,44, Klaus Zuberbühler22,23,45, James W. Vaupel2,42 & Susan C. Alberts 31,41,42,46 ✉ Is it possible to slow the rate of ageing, or do biological constraints limit its plasticity? We test the ‘invariant rate of ageing’ hypothesis, which posits that the rate of ageing is relatively fixed within species, with a collection of 39 human and nonhuman primate datasets across seven genera. We first recapitulate, in nonhuman primates, the highly regular relationship between life expectancy and lifespan equality seen in humans. We next demonstrate that variation in the rate of ageing within genera is orders of magnitude smaller than variation in pre-adult and age-independent mortality. Finally, we demonstrate that changes in the rate of ageing, but not other mortality parameters, produce striking, species-atypical changes in mortality patterns. Our results support the invariant rate of ageing hypothesis, implying biological constraints on how much the human rate of ageing can be slowed. A full list of author affiliations appears at the end of the paper. NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 T he maximum human life expectancy has increased since for long-lived species, including nonhuman primates, the closest the mid-1800s by ~3 months per year1. These gains have relatives of humans. Nonetheless, these are precisely the species resulted from shifting the majority of deaths from early to that will shed most light on how biological constraints have later and later ages, with no evidence of slowing the rate at which shaped the evolution of ageing within the lineage leading to mortality increases with age (i.e. the ‘rate of ageing’)2–4. Further humans. substantial extensions of human longevity will depend on whe- To better understand biological constraints on ageing, here we ther it is possible to slow the rate of ageing or otherwise reduce answer two questions. First, is the highly regular linear relation- late life mortality. Consequently, the nature of biological con- ship between life expectancy and lifespan equality in humans also straints on ageing is a central problem in the health sciences and, evident in other primates? Second, if so, do biological constraints because of its implications for demographic patterns, is also of on ageing underlie this highly regular relationship? To address long-standing interest in ecology and evolutionary biology. these questions, we have assembled a large dataset on age-specific Across species, rates of ageing are strongly correlated with mortality rates in multiple populations of several different pri- other aspects of the life history—pre-adult mortality, age at first mate genera. Our combined dataset includes data from both wild reproduction, birth rate, metabolic rate and generation time—as and captive primate populations. The data from wild populations well as with morphological traits such as body size and growth consists of individual-based birth and death data on males and rate5,6. These correlations suggest that ageing evolves in concert females from 17 continuous long-term studies of wild primate with a suite of other traits, which may produce constraints on the populations representing 6 genera distributed across the order rate of ageing within species. Indeed, researchers have long Primates, and include African monkeys (2 genera), Central hypothesised that the rate of ageing is relatively fixed within and South American monkeys (1 genus), great apes (2 genera), species, not only in humans but also other animals7–9. and an indriid (1 genus, endemic to Madagascar) (Supplementary This ‘invariant rate of ageing’ hypothesis has received mixed Data 1). support. Several studies have documented a strong phylogenetic For these genera, we also obtained individual-based birth and signal in the rate of ageing across multiple species of birds and death data from 13 species in zoos from Species360’s Zoological mammals, suggesting biological constraints and little within- Information Management System (ZIMS)18 (see Methods, Sup- species variance in this rate9,10. Furthermore, Bronikowski and plementary Data 1). We also include data on a 7th primate genus, colleagues11 observed greater variation in initial adult mortality Homo, using male and female human mortality data from nine of than in the rate of ageing across several populations of baboons. the human datasets studied by Colchero et al.13. These On the other hand, across multiple mammal species, measurable nine populations had not benefited from modern advances in differences in the rate of ageing have been documented between public health, medicine, and standards of living, enabling us to populations in different environments (e.g. zoo versus wild12). carry out the most salient comparisons with nonhuman primates. Understanding the nature and extent of biological constraints We use life tables from the Human Mortality Database19 for (1) on the rate of ageing and other aspects of age-specific mortality Sweden from 1751 to 1759, (2) Sweden in 1773, (3) Sweden from patterns is critical for identifying possible targets of intervention 1850 to 1859, (4) and Iceland in 1882. We also use human life to extend human lifespans, and for understanding the evolu- tables for (5) England from 1600 to 172520, (6) Trinidad from tionary forces that have shaped lifespans within and across spe- 1813 to 181521, (7) Ukraine in 193322 and two hunter gatherer cies. Although no consensus has been reached about the invariant populations, (8) the Hadza, based on data collected between 1985 rate of ageing hypothesis, further evidence that biological con- and 200023 and (9) the Ache during the pre-contact period of straints may shape human ageing comes from the remarkably 1900–197824. In the aggregate, our 39 combined datasets (17 wild consistent relationship between life expectancy at birth (e0) and and 13 zoo nonhuman primates, and 9 human populations; lifespan equality (ε0) in a diverse set of human populations3,13,14. Supplementary Data 1) comprise a taxonomically diverse sample While life expectancy at birth (a measure of the ‘pace’ of of primates and represent considerable environmental variability mortality15) describes the average lifespan in a population, life- within genera, maximising the probability of detecting variation span equality (a measure of the ‘shape’ of mortality15) describes in ageing. the spread in the distribution of ages at death in a population (see To understand potential constraints on primate ageing, we also16,17). compare age-specific changes in the risk of death across multiple Lifespan equality is highly correlated with other measures of populations of each genus. The age-specific risk of death, often the distribution of ages at death, such as the coefficient of var- described by a hazard rate, is the basic building block of the iation and the Gini coefficient, often used to measure economic distribution of ages at death, and therefore determines both life inequality13. The distribution of ages at death tells us whether expectancy and lifespan equality for a population (see ‘Methods’). deaths are evenly distributed across the range of observed life- Among most mammal species, the risk of death is high in infancy, spans, or are concentrated around certain ages. For instance, if rapidly declines during the immature period, remains relatively deaths are evenly distributed across age classes or show multiple low until early adulthood and then rises with age as a result of modes, the result is high lifespan variation and low lifespan senescence. This pattern can be described mathematically by the equality, while if deaths are concentrated at the tail-end of the five-parameter Siler mortality function25, given by lifespan distribution (as in most developed nations), the result is     low lifespan variance and high lifespan equality. The tight positive μðxÞ ¼ exp a0  a1 x þ c þ exp b0 þ b1 x ; for x ≥ 0 ð1Þ linear relationship between life expectancy (e0) and lifespan equality (ε0) across a large range of human populations indicates where a0, a1, c, b0, b1 are mortality parameters, each of which strong but poorly understood constraints underlying variation in governs different stages of the age-specific mortality. In short, human mortality3,13. parameters a0 and a1 drive infant and juvenile mortality, c is Understanding the biological constraints on ageing requires commonly described as the age-independent mortality, and b0 mortality data for multiple populations of nonhuman species, as and b1 control senescent mortality. Parameters a0, c and b0 are well as for humans. However, comparative data across multiple scale parameters, while a1 determines the speed of decline in populations of nonhuman animals are rarely available, making it infant and juvenile mortality and b1 determines the rate of difficult to unveil the forces underlying mortality differences increase in adult and senescent mortality, analogous to the rate of within versus between species. The challenge is particularly acute senescence or rate of ageing. 2 NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 ARTICLE Here, we fit these Siler models of age-specific mortality for which have a disproportionately large effect on the slopes of the males and females for each of the 30 non-human primate regression lines. To pursue this question, we initially conducted a populations (Methods, Supplementary Data 1 and 2), and we sensitivity analysis by simulating independent changes in each of examine how each of the five Siler parameters varied within and the Siler mortality parameters (Fig. 1C) and graphically exam- between the genera (Supplementary Figs. 1 and 2). We also cal- ining the effects of these changes on the life expectancy-lifespan culate sex-specific values for life expectancy at birth (e0) and equality relationships. Specifically, we varied one Siler parameter lifespan equality (ε0) in each population, and use these values to at a time within each genus, keeping the other four Siler para- examine the relationship between life expectancy and lifespan meters constant at the value found at the midpoint of the equality within each genus (Supplementary Data 3). We conduct regression line. genus-level rather than species-level analyses because restricting This approach produced striking results: within each genus, ourselves to the species level severely limits the availability of simulated variation in pre-adult mortality (captured by Siler individual-based datasets (e.g., among guenons, only one or two parameters a0 and a1) and in age-independent mortality (Siler individual-based datasets are available for each species, while parameter c) all produced lines of similar direction to the examining the genus provides five such datasets). We show that observed regression lines (Fig. 1D). That is, within the observed the highly regular linear relationship between life expectancy and range of e0 values, changes in these three Siler parameters resulted lifespan equality in humans is indeed recapitulated in other pri- in ε0 similar to the observed range. Therefore, consistent with mates. Further, our results suggest that the regularity in this theory and with the long-understood effect of averting early relationship is driven by a combination, within each genus, of deaths, observed variation in life expectancy and lifespan equality high variability in early and age-independent mortality and low within each primate genus appears to be largely accounted for by variability in senescent mortality. variation in the pattern of early deaths, and very little by actuarial senescence. In stark contrast, simulated variation in the rate-of-ageing Results parameter (Siler parameter b1) produced lines with conspicuously Age-specific mortality across populations and life different direction from the observed regression lines. Specifically, expectancy–lifespan equality relationship. Our regression ana- changing b1 moved the life expectancy–lifespan equality values lyses yielded clear linear relationships between e0 and ε0 within away from the regression lines (Fig. 1D). each primate genus, mirroring the relationship observed within humans (Fig. 1A, B and Supplementary Fig. 3). Importantly, the linear relationship between e0 and ε0 is not a simple artefact of Sensitivity of life expectancy and lifespan equality to mortality our modelling. For instance, Stroustrup et al.26 demonstrated in parameters. These findings led us to postulate that, while varia- laboratory experiments with C. elegans that changes in life tion in early deaths is the primary cause of observed variation in expectancy occur with no change in lifespan variance. Similarly, life expectancy and lifespan equality within each genus, changes Jones et al.5 found no correlation between a measure of the length in the rate of ageing in one or more populations in a genus could of life and a measure of relative variation in lifespans, across shift those populations towards the lines of other genera. To 46 species drawn from different taxa. Colchero et al.13 found no further investigate this possibility, we derived mathematical correlation between life expectancy and lifespan equality across functions for the sensitivity of life expectancy and lifespan 15 non-primate mammal species. Aburto and van Raalte27 equality to changes in any given mortality parameter (see showed that in Eastern European countries life expectancy and ‘Methods’). These sensitivity functions allowed us to obtain pre- lifespan equality often moved independently of each other cise measures of the amount of change in life expectancy and between the 1960s and 1980s, and van Raalte et al.28 showed that lifespan equality for a unit change in any given mortality para- life expectancy and lifespan equality have a negative relationship meter at any point in the life expectancy–lifespan equality land- (i.e., inequality increases with life expectancy) in some human scape (including along each of the regression lines). populations (Finland in the 20th and 21st centuries, in their The resulting vectors of change (Fig. 2A) are consistent with example). our graphical exploration, and they also revealed the relative This linear relationship between life expectancy and lifespan magnitudes of changes that each mortality parameter produces equality emerged in our analysis despite considerable variation in the life expectancy–lifespan equality landscape (Fig. 2B). among populations of each genus in age-specific mortality, in the Specifically, a unit change in the rate of ageing parameter b1 distribution of ages at death, and in the Siler mortality parameters shifts the life expectancy and lifespan equality values in a (Supplementary Figs. 1–4, Supplementary Data 2). The slopes of direction almost perpendicular to the regression lines, and the these regression lines were statistically significant (i.e., p value < magnitude of that change is disproportionately large compared 0.05) in five of seven genus-level datasets for females and in four to the other four parameters. We then calculated the degree of of seven for males (Fig. 1A, B, Supplementary Data 4). The collinearity (how parallel versus perpendicular two vectors are) regression lines did not reach statistical significance in analyses between the seven genus-specific regression lines for females and that included relatively few populations or that included small or the vectors of change for each parameter. We found that the two heavily censored datasets. The slopes of the regression lines were parameters that govern infant mortality, a0 and a1, and the age- statistically significantly different than the slope of the line for independent parameter c, produce vectors of change that humans in female sifaka, baboons, guenons and gorillas, and in are almost parallel to the regression lines. In contrast, Siler male guenons, gorillas and chimpanzees. parameter b0 produces vectors that are intermediate between parallel and perpendicular, while the rate-of-ageing parameter, Drivers of the linear relationship between life expectancy and b1, produces vectors that are almost perpendicular to the lifespan equality. Having confirmed that the relationship regression lines (Fig. 2C). In short, changes in pre-adult between life expectancy and lifespan equality is linear and highly mortality and in age-independent mortality tend to move a regular within other primate genera, as it is in humans, we next population along the regression line typical of its genus. In sought possible causes for this regularity. Specifically, we asked contrast, changes in the ageing parameters, b0 and particularly which Siler mortality parameters best explain variation among b1, will shift a population away from this line, into the space populations in life expectancy and lifespan equality, and therefore occupied by other genera in the landscape. NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 a Females b Males Guenon on Gorilla Babo Gorilla Guenon Lifespan equality 0.5 in h e uc nze p e pa Ca ze im an Ch p im on Ch bo Ba chin pu 0.0 Ca Sifaka an a Sifak m p−val < 0.05 Hu an 0.05 < p−val < 0.1 −0.5 m Hu p−val > 0.1 0 10 20 30 40 0 10 20 30 40 Life expectancy c d 0.6 a0 a1 c (b enc ty & Siler mortality, µ(x) Lifespan equality sc ali b1 e 0.0 ne r t se t mo ) Infa mor ta a 1) 0, ul nt & lity −0.6 Ad 0 20 40 (a 0 juv Chimpanzee , eni 0.6 b0 b1 line le Age independent 0.0 mortality (c) Line of change −0.6 0 20 40 0 20 40 Age Life expectancy Fig. 1 The life expectancy–lifespan equality landscape for seven genera of primates for for both sexes. a Life expectancy and lifespan equality regression lines for females; each species is represented by a different colour. b Life expectancy and lifespan equality regression lines for males. Each genus is characterised by a relatively constrained relationship between life expectancy and lifespan equality, and thus a distinct regression line; colours as in a. The central lines are the predicted fitted values of the regression and the type of line (e.g. continuous, dashed, or dotted) depicts three levels for the p values of the slopes (how significantly different from 0 they are, two-sided t test, H0: β1 = 0, Supplementary Table 1), while the shaded polygons show the 95% confidence intervals of the regressions. c The relationship between the Siler mortality parameters and the resulting mortality function, given by the equation μ(x) = exp(a0 – a1 x) + c + exp(b0 + b1 x), where infant and juvenile mortality (blue) are controlled by parameters a0 and a1, age-independent mortality (orange) is captured by c, and senescent mortality (green) is captured by b0 (initial adult mortality) and b1 (rate of ageing). d Each box shows how gradual changes in each Siler mortality parameter modify the life expectancy and lifespan equality values (thick purple lines). The green line in each box corresponds to the regression line for female chimpanzees, shown for reference to illustrate the general trends among all genus lines. The purple curves show the changes in life expectancy and lifespan equality after varying individual Siler parameters while holding the other parameters constant. Note the striking change in life expectancy and lifespan equality that would result from changes in the ageing parameters, particularly b1. See Supplementary Fig. S3 for plots that include individual points for each population. Source data to generate the regression lines are available in Supplementary Data 3. Amount of change in each mortality parameter along the genus to quantify the total amount of change in each parameter for all lines. If variation in pre-adult and age-independent mortality seven genera. We found that, in agreement with our previous parameters account for most of the within-genus differences in results, in all cases the parameters that govern infant and age- life expectancy and lifespan equality, we expect the parameters independent mortality changed orders of magnitude more than that control infant and age-independent mortality to be much those that drive adult and senescent mortality (Fig. 3). more highly sensitive to perturbations of e0 and ε0 than the parameters that control adult and senescent mortality, particu- larly b1. To test these expectations, we quantified the relative Discussion change in each parameter along each genus line by calculating the To our knowledge, our results provide the most comprehensive partial derivatives of the log-transformed parameter with respect support to date for the idea that observed variation in mortality to changes in e0 and ε0. (see ‘Methods’). These partial derivatives patterns among populations of a given genus is driven largely by of the log-transformed parameter values represent standardised changes in pre-adult mortality: previous support for this idea measures that allow direct comparison among parameters that comes from studies of just one or a few species, typically including differed in the absolute magnitude of change. We then calculated humans or primarily captive animal populations3,7,10,12. Notably, path integrals of these sensitivities along each genus line in order recent research on human populations3 shows that increases in life 4 NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 ARTICLE c c a 1.0 0.75 Lifespan equality a0 0.8 0.50 a1 Collinearity 0.6 0.25 b0 ine 0.00 0.4 l ee nz pa im Ch −0.25 0 10 20 30 40 50 b1 b 0.2 a0 a1 c b0 b1 0.5 0.0 −0.5 0.0 0 25 50 0 25 50 0 25 50 0 25 50 0 25 50 a0 a1 c b 0 b 1 Life expectancy Parameter 1 = parallel Chimpanzee Sensitivity vector 0 = perpendicular Fig. 2 Sensitivities of life expectancy and lifespan equality to changes in mortality parameters. a Using the female chimpanzee line (bright green) as an example, vectors depict the sensitivity at the mid-point of the genus line. Each vector depicts the direction and magnitude of change in life expectancy and lifespan equality for a unit change in the corresponding Siler mortality parameter. The x- and y-axes show the life expectancy and lifespan equality values of the sensitivity vectors for a0 (light blue), a1 (dark blue), and b0 (light green); vectors for c (orange) and b1 (dark green) are particularly large, represented by broken lines (Source data are provided as a Source Data File and available in Supplementary Table 2). b Gradient field of sensitivities of life expectancy and lifespan equality to changes in each mortality parameter, showing the direction of change any population would experience for a given change in the parameter, from any starting point in the landscape. The green chimpanzee line is provided for reference. Each sensitivity vector (bright purple) can be interpreted as those in A, but calculated from different points on the landscape). c Boxplots representing the values of the seven collinearity values (one for each genus) for each of the Siler parameters for n = 7 independent genera. Collinearity is calculated between the mid-point of the genus line and the sensitivity vector for each parameter; a value of 1 would imply that the vector is parallel, a value of 0 would imply that it is perpendicular. Note the relatively large collinearity values for a0 (light blue), a1 (dark blue), and c (orange), the intermediate value for b0 (light green) and the relatively small value for b1 (dark green). The boxplots indicate median (horizontal black line), 25th and 75th percentiles (box), the whiskers are extend to 1.5 the interquartile range, and the open points are extreme values (Source data are provided as a Source Data File and available in Supplementary Table 3). expectancy can occur not just through decreases in pre-adult individuals within a population in rate of ageing. More impor- mortality but also through decreases in adult mortality. In the tantly, it implicates changes in the rate of ageing as a likely source context of the Siler model, this would most likely translate into of variation in lifespan between distantly related taxa8. reductions in the b0 parameter. This possibility is supported by Furthermore, by considering populations exposed to a wide our result that the vectors of change for Siler parameter b0 pro- range of environmental conditions—from high predation and low duced by our sensitivity analysis are markedly less colinear with resource availability, to unconstrained resources and veterinary our genus-specific regression lines than the vectors of change for care in zoos—our results have implications both for life history the pre-adult mortality parameters (Fig. 2C). theory and for conservation. Life history theory predicts that More strikingly, our results provide fresh insight into the among species with slow life histories (i.e., long lifespans, small ‘invariant rate of ageing’ hypothesis. In support of that hypoth- litters and delayed maturity), adult survival should be buffered esis, we find that, within primate genera, rates of ageing (captured from environmental variability, while juvenile survival is expected by the Siler parameter b1) do indeed vary across populations, but to vary widely in response to the environment29–32. Our findings along each genus line they generally vary orders of magnitude less support this buffering hypothesis, in that the most dramatic than other mortality parameters (an exception is sifaka, Fig. 3). observed changes in life expectancy occur because of changes in Further, our results illustrate that, within any given genus, large juvenile survival, while changes in adult or senescent survival changes in the rate of ageing would shift a population across the account for relatively little of the observed variation within life expectancy-lifespan equality landscape to a position closer to each genus. other genera. This result supports the ‘invariant rate of ageing’ Importantly, sufficient demographic information to under- hypothesis, although it does not rule out heterogeneity among stand and predict population dynamics exists for less than 1.5% NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 a) Human h) Human Females Males 10 1 Amount of change in parameter along the genus line 0.1 0 a0 a1 c b0 b1 a0 a1 c b0 b1 b) Chimpanzee c) Gorilla i) Chimpanzee j) Gorilla Females Females Males Males 10 1 0.1 0 d) Guenon e) Baboon k) Guenon l) Baboon Females Females Males Males 10 1 0.1 0 f) Capuchin g) Sifaka m) Capuchin n) Sifaka Females Females Males Males 10 1 0.1 0 a0 a1 c b 0 b 1 a0 a1 c b 0 b 1 a0 a1 c b 0 b 1 a0 a1 c b 0 b 1 Fig. 3 Relative magnitude of change of each parameter along the genus lines. Pre-adult and age-independent mortality parameters (a0 a1, and c) vary several orders of magnitude more, within each genus, than the ageing parameters (b0 and b1). Colours: a0 (light blue) a1, (dark blue), c (orange), b0 (light green) and b1 (dark green). Values were calculated by numerically solving the path integral in Eq. (9) (see ‘Material and Methods’) for each parameter along each genus line. The y-axes were scaled by the logarithm base 10 to improve interpretability. a–g depict results for females, and h–n for males (Source data are provided as a Source Data File and available in Supplementary Table 4). of extant vertebrate species33. By unravelling the interdependence Methods of mortality parameters within a species or genus, we can con- Data for non-human primates. We obtained 30 datasets for six genera of non- human primates: sifaka (Propithecus spp), gracile capuchin monkey (Cebus spp), tribute to filling these glaring demographic knowledge gaps and guenon (Cercopithecus spp), baboon (Papio spp), gorilla (Gorilla spp), and chim- further our understanding of the ecology and evolution of a wide panzee (Pan troglodytes) (Supplementary Data 1). Of these, 17 datasets correspond range of animal species, as well as the conservation of species to long-term projects in the wild, while 13 were contributed by the non-profit worldwide. Species360 from ZIMS18, which is the most extensive database of life history Finally, can we humans slow our own rate of ageing? Our information for animals under human care. findings support the idea that, in historical population when life expectancies were low, mortality improvements for infants, and Basic demographic functions. Let X be a random variable for ages at death, with observations x ≥ 0, and let μ (x|θ) be a continuous, non-negative parametric in age-independent mortality, were the central contributors to hazards rate or mortality function defined as the decades-long trend towards longer human life expectancies and greater lifespan equality3. These improvements were largely PrðxxÞ μðx; j; θÞ ¼ lim ; ð2Þ the result of environmental influences including social, eco- Δx!0 Δx nomic, and public health advances13,34,35. Since the middle of the given that the limit exists, where θ 2 Rp is a p-dimensional vector of mortality 20th century, however, declines in the baseline level of adult parameters. The cumulative hazards rate is mortality—measured in the context of the Siler model by b0— Z have very likely played an increasingly important role in x U ðxjθÞ ¼ μðtjθÞdt; ð3Þ industrialised societies3,8. As we show here, improvements in the 0 environment are unlikely to translate into a substantial reduction in the rate of ageing, b1, or in the dramatic increase in lifespan which results in the survival function that would result from such a change. It remains to be seen if future advances in medicine can overcome the biological con- SðxjθÞ ¼ exp½UðxjθÞ: ð4Þ straints that we have identified here, and achieve what evolution The Cumulative distribution function (CDF) of ages at death is F (x | θ) = 1 – S has not. (x | θ), and the probability density function (PDF) of ages at death is f (x | θ) = 6 NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 ARTICLE μ (x | θ) S (x | θ), for x ≥ 0. The remaining life expectancy after age x is calculated as π = [π1, π2] is the vector of probabilities of out-migration while the subscripts u R1 and k refer to unknown and known, respectively. tf ðtjθÞdt Following Colchero et al.13, we used published data, expert information and an eðxjθÞ ¼ x F ð1Þ  F ðxÞ agent-based model to estimate the mortality and out-migration prior parameters R1 ð5Þ SðtjθÞdt for each population. We assumed a normal (or truncated normal distribution ¼ x ; depending on the parameter’s support) for all the parameters. We used vague S ðx Þ priors for the mortality and natal out-migration parameters (sd = 10), and which yields a life expectancy at birth given by informative priors for the immigrant out-migration parameters (sd = 0.5). We ran Z 1 six MCMC parallel chains for 25 000 iterations each with a burn-in of 5000 eð0jθÞ ¼ SðxjθÞdx: ð6Þ iterations for each population, and assessed convergence using potential scale 0 reduction factor38. The lifespan inequality at birth, as proposed by Demetrius16,36 and later by For the zoo data we used a simplified version of the model described above, Keyfitz17, is given by which omitted all parts that related to out-migration. In order to produce R1 Supplementary Figs. 1 and 2, we used the same method as for the zoo data on the SðxjθÞlog½SðxjθÞdx Hð0jθÞ ¼  0 human life tables. To achieve this, we created an individual level dataset from the lx eð0jθÞ column of each population, and then fitted the Siler model to this simulated data. It R1 ð7Þ 0 SðxjθÞUðxjθÞdx is important to note that the Siler model provides a close fit to the nonhuman ¼ : primate data and to high-mortality human populations, although it does not eð0jθÞ provide the best fit to low-mortality human populations, in part due to the late life Following Colchero et al.13, we define the lifespan equality as mortality plateau common among human populations39 (Supplementary Fig. 6). It εðxjθÞ ¼ log½HðxjθÞ: ð8Þ is therefore possible that the values of the mortality parameter b1 we report in Supplementary Data 2 for the human populations are under-estimated. For simplicity, henceforth we note the life expectancy, lifespan inequality and Nonetheless, and for the purposes of our analyses, the Siler fits to the human lifespan equality at birth as e(0 | θ) = e, H (0 | θ) = H, and ε (0 | θ) = ε, populations we considered here are reasonable (Supplementary Fig. 6) and we can respectively. therefore confidently state that the limitations of the Siler model do not affect the generality of our results. Survival analysis. To estimate age-specific survival for all the wild populations of non-human primates, we modified the Bayesian model developed by Colchero Estimation of life expectancy and lifespan equality. Based on the results of the et al.13 and Barthold et al.37. This model is particularly appropriate for primate Bayesian inference models, we calculated life expectancy at birth as studies that follow individuals continuously within a study area and when indi- viduals of one or both sexes can permanently leave the study area (out-migration), Z 1   while other individuals can join the study population from other areas (in- e¼ S tjθ^ dt; ð11Þ migration). Thus, it allowed us to make inferences on age-specific survival (or 0 mortality) and on the age at out-migration. where S (x) is the cumulative survival function as defined in Eq. (4) and where ^θ is Here we use the five parameter Siler mortality function25, as in Eq. (1) where θ = the vector of mortality parameters calculated as the mean of the conditional pos- [a0, a0, c, b0, b1] is a vector of parameters to be estimated, and where a0, b0 2R and a1, terior densities from the survival analysis described above. We calculated the c, b1 ≥ 0. For all species we studied, individuals of one or both sexes often leave their lifespan inequality17,36, H, as natal groups to join other neighbouring groups in a process commonly identified as Z natal dispersal. For some species, individuals who have undergone natal dispersal can 1 1  ^ h  ^i then disperse additional times, described as secondary dispersal. Although dispersal H¼ S xjθ log S xjθ dx; ð12Þ within monitored groups (i.e. those belonging to the study area) does not affect the e 0 estimation of mortality, the fate of individuals that permanently leave the study area to from which we calculated lifespan equality, ε, as in Eq. (8). We calculated both join unmonitored groups can be mistaken for possible death. We identify this process measures for each of the study populations, and performed weighted least squares as “out-migration”, which we classify as natal or immigrant out-migration, the first for regressions for each genus, with weights given by the reciprocal of the standard natal and the second for secondary dispersals to unmonitored groups. This distinction error of the estimated life expectancies. is particularly relevant because not all out-migrations are identified as such, and therefore the fate of some individuals is unknown after their last detection. For these individuals we define a latent out-migration state at the time they were last detected, Sensitivities of life expectancy and lifespan equality to mortality parameters. given by the random variable indicator O, with observations oij ∈ {0,1}, where oij = 1 if As we mentioned above, for simplicity of notation, we will express all demographic individual i out-migrated and oij = 0 otherwise, and where j = 1 denotes natal out- functions by their variable notation (e.g. e = e (0 | θ), S = S (x | θ), etc.), while we migration and j = 2 for immigrant out-migration. For known out-migrations, we will alternatively note first partial derivatives, for instance the derivative of e with automatically assign oij = 1. The model therefore estimates the Bernoulli probability of respect to a given mortality parameter θ ∈ θ, as eθ or ∂e / ∂θ. out-migration, πj, such that Oij ~ Bern(πj). Those individuals assigned as exhibiting Proposition: If S :R ≥ 0 ! ½0; 1 is a continuous non-increasing parametric out-migration, as well as known emigrants and immigrants, contribute to the survival function with parameter vector θRp , with continuous differentiable estimation of the distribution of ages at out-migration. Here, we define a gamma- cumulative hazards function U :R ≥ 0 ! R ≥ 0 , and with life expetancy at birth, distributed random variable V for ages at out-migration, with realisations v ≥ 0, where lifespan inequality and lifespan equality as in Eqs. (4)-(6), respectively, then the Vj | Oj = 1 ~ Gam(γj1, γj2) and where γj1, γj2 > 0 are parameters to be estimated with sensitivity of life expectancy, e, to a given parameter θ ∈ θ is j defined as above. The probability density function for the gamma distribution is Z 1 ∂e gV(v | γj1, γj2) for v ≥ 0, with v = xl - αj, where xl is the age at last detection and αj is the eθ ¼ ¼ Sθ dx; ð13Þ minimum age at natal or immigrant out-migration. ∂θ 0 In addition, since not all individuals have known birth dates, the model samples while the sensitivity of lifespan equality to θ is the unknown births bi as xil = til – bi, where til is the time of last detection for individual i. The likelihood is then defined as   R1 ∂ε eθ 1 þ H 1  H 1 0 Sθ Udx ð14Þ 8 εθ ¼ ¼ ; > f ðx Þ ∂θ e < Sðx il Þ ð1  π j Þ if oij ¼ 0 pðxil ; xif ; j; θ; γ1 ; γ2 ; π j ; oij Þ ¼ ; ð9Þ if > where : ð il Þ π j g ðxil  αj Þ if oij ¼ 1 S x Sðxif Þ V ∂ where xif is the age at first detection, given by xif = tif – bi, with tif as the Sθ ¼ SðxjθÞ ð15Þ ∂θ corresponding time of first detection. The parameter vectors γ1 and γ2 are for natal and immigrant out-migration, respectively. In other words, individuals with oij = 0 is the sensitivity of the survival function at age x to changes in parameter θ. are assumed to have died shortly after the last detection, while those with oij = 1 Proof. The sensitivity of lifespan equality to changes in θ is derived from are censored and contribute to the estimation of the distribution of ages at out- Z ∂ 1 migration. The full Bayesian posterior is then given by eθ ¼ Sdx; ð16Þ     ∂θ 0 p θ;γ1 ; γ2 ; π; bu ; ou ; j; bk ; ok ; tf ; tl / p xl ; xf ; j; θ; γ1 ; γ2 ; π; d     ð10Þ which, by Leibnitz’s rule, Eq. (16) becomes ´ pðθÞp γ1 p γ2 pðπ Þ; Z 1 Z 1 ∂S where the first term on the right-hand-side of Eq. (10) is the likelihood in Eq. (9), eθ ¼ dx ¼ Sθ dx: ð17Þ 0 ∂θ 0 and the following terms are the priors for the unknown parameters. The vector NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 The sensitivity of lifespan equality to changes in θ can be calculated as integral Θj provides a measure of the relative change in parameter θ along the ∂   genus line (Fig. 3). To allow comparisons between all genera, we scaled the values εθ ¼ log H of each path integral by the length of each line. ∂θ ∂ ¼  log H Applications to the Siler mortality model. The Cumulative hazards for the Siler ∂θ 1 ∂H ð18Þ mortality model in Eq. (7) is given by ¼ H ∂θ R ea0 eb0  b1 x   1  U ðx Þ ¼ ð1  ea1 x Þ þ cx þ e 1 ; ð28Þ 1 ∂ 0 SUdx a1 b1 ¼ : H ∂θ e The sensitivities in Eqs. (13) and (14) require calculating Sθ for all θ ∈ θ. By the quotient and Leibnitz’s rules, Eq. (18) can be modified as Treating S (x) as the function composition W (V), where W = exp(x) and V = - U,  Z 1  Z 1  then Sθ is 1 ∂ ∂e εθ ¼  2 SUdx e  SUdx dW He ∂θ 0 0 ∂θ Sθ ¼ V ¼ SU θ ; ð29Þ Z 1 R1 ð19Þ dV θ 1 ∂ 1 0 SUdx ∂e ¼ ðSU Þdx þ : where Uθ is the first derivative of U(x | θ) with respect to θ. For each of the Siler He 0 ∂θ He e ∂θ mortality parameters, we then have The first term in Eq. (19) can be further decomposed by the product rule, while the second term can be modified following the equality for H in Eq. (7), which ea0 a1 x Sa0 ¼ SðxjθÞ ðe  1Þ ð30Þ yields a1 Z 1     1 ∂S ∂U 1 ea0 1 1 εθ ¼  U þS dx þ eθ Sa1 ¼ SðxjθÞ  ea1 x x þ ð31Þ He 0 ∂θ ∂θ e a1 a1 a1 Z 1 Z 1  ð20Þ 1 ∂U 1 ¼ Sθ Udx þ S dx þ eθ : He 0 0 ∂θ e Sc ¼ SðxjθÞx ð32Þ By the chain rule, we have that ∂U ∂ 1 ∂S ∂θ ¼  ∂θ log S ¼  S ∂θ, which modifies Eq. (20) eb0   as Sb0 ¼ SðxjθÞ 1  eb1 x ð33Þ Z 1 Z 1  b1 1 ∂S 1 εθ ¼  Sθ Udx  dx þ eθ    He 0 0 ∂θ e 1 1 Z 1  Sb1 ¼ SðxjθÞ eb1 x x  : ð34Þ 1 1 b1 b1 ¼  Sθ Udx  eθ þ eθ 40 He 0 e All analyses were performed in the free open source programme R . The R R1   ð21Þ functions we created for this project can be found in41. 0 Sθ Udx eθ 1 ¼  þ 1þ He e H  1  R 1 1 Reporting summary. Further information on research design is available in the Nature eθ 1 þ H H 0 Sθ Udx ¼ ; Research Reporting Summary linked to this article. e hence completing the proof. ∎ Data availability Full datasets for survival analyses on wild populations and populations under human Changes in parameters along the genus lines. From the results in Eqs.D(13) Eand care of non-human primate species supporting the findings of this study were used under (14), we calculated the vectors of change (gradient vectors) at any point ej ; εj of license for the current study and are not publicly available; specific requests for the access the life expectancy-lifespan equality landscape, as a function of each of the Siler to the wild data should be addressed to the PIs of the data. Data of animals under human mortality parameters (See Fig. 2A, B). care are however available from Species360 (https://www.species360.org/) upon To quantify the amount of change of each parameter along the genus lines, we reasonable request. Summarised data underlying the analyses here and sufficient to derived the sensitivities of a given mortality parameter θ to changes in life calculate life tables and summary statistics such as life expectancy and lifespan equality expectancy and lifespan equality, namely ∂θ ∂θ ∂e ¼ eθ for eθ ≠ 0; and ∂ε ¼ εθ for εθ ≠ 0. 1 1 are available in the Dryad data repository, URL: https://doi.org/10.5061/dryad.4b8gthtb4. Data from human populations were obtained from the Human Mortality Database With these sensitivities we calculated the gradient vector (https://www.mortality.org/) and published sources. Source data are provided with ∂θ ∂θ this paper. ∇θ ¼ ; ð22Þ ∂e ∂ε for any parameter at any point along the genus lines. Here we find a linear Code availability relationship between life expectancy and lifespan equality, given by The code showing an overview of the analyses used in the manuscript is available at   https://github.com/fercol/ColcheroEtal2021NatComm (https://doi.org/10.5281/ m eik ¼ ^εik ¼ β0k þ β1k eik ; ð23Þ zenodo.4736892). for i = 1, …, nk, where nk is the number of populations for genus k, and ^εik is the fitted value of lifespan equality for population i in genus k, and β0k and β1k are linear regresssion parameters for genus k. To estimate the amount of change in Received: 29 November 2020; Accepted: 19 May 2021; parameter θ along the line for genus k, we can solve the path integral Z Θk ¼ ∇θdr; ð24Þ Ck where path Ck is determined by the linear model for genus k and dr ¼ hde; d^εi ¼ de; d mðeÞ is the rate of change in the velocity vector r ¼ he; ^εi ¼ e; mðeÞ . References In order to compare results between the different mortality parameters in vector 1. Oeppen, J. & Vaupel, J. 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(Chapman and Hall/CRC, 2013). © The Author(s) 2021 NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications 9 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23894-3 1 Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark. 2Interdisciplinary Centre on Population Dynamics, University of Southern Denmark, Odense, Denmark. 3Department of Sociology, Leverhulme Centre for Demographic Science, Nuffield College at University of Oxford, Oxford, UK. 4Lifespan Inequalities Research Group, Max Planck Institute for Demographic Research, Rostock, Germany. 5Department of Biological Sciences, University of Notre Dame, Notre Dame, IN, USA. 6Institute of Primate Research, National Museums of Kenya, Nairobi, Kenya. 7Max Planck Institute for Evolutionary Anthropology, Leipzig, Germany. 8Taï Chimpanzee Project, CSRS, Abidjan, Côte d’Ivoire. 9Mbeli Bai Study, Wildlife Conservation Society Congo Program, Brazzaville, Congo. 10World Wide Fund for Nature - Germany, Berlin, Germany. 11Department of Anthropology, University of Texas at San Antonio, San Antonio, TX, USA. 12Gombe Stream Research Centre, Jane Goodall Institute, Kigoma, Tanzania. 13Species360 Conservation Science Alliance, Bloomington, MN, USA. 14Department of Biology, University of Southern Denmark, Odense, Denmark. 15Department of Ecology, Evolution, and Environmental Biology, Columbia University, New York, NY, USA. 16New York Consortium in Evolutionary Anthropology, New York, NY, USA. 17Department of Anthropology, University of New Mexico, Albuquerque, NM, USA. 18Kibale Chimpanzee Project, Fort Portal, Uganda. 19Department of Anthropology and Archaeology, University of Calgary, Calgary, AB, Canada. 20Behavioral Ecology & Sociobiology Unit, German Primate Center, Leibniz Institute for Primate Research, Göttingen, Germany. 21World Wide Fund for Nature- Cambodia Program, Phnom Penh, Cambodia. 22School of Psychology and Neuroscience, University of St Andrews, St Andrews, Scotland, UK. 23Budongo Conservation Field Station, Masindi, Uganda. 24Department for Sociobiology/ Anthropology, Johann-Friedrich-Blumenbach Institute of Zoology and Anthropology, University of Göttingen, Göttingen, Germany. 25Department of Sociology and Anthropology, James Madison University, Harrisonburg, VA, USA. 26Department of Anthropology, University of Texas at Austin, Austin, TX, USA. 27Ankoatsifaka Research Station, Morondava, Madagascar. 28Departments of Anthropology and Biology, Tufts University, Medford, MA, USA. 29College of Biological Sciences, Department of Ecology, Evolution and Behavior, University of Minnesota, Saint Paul, MN, USA. 30Department of Anthropology, and Behavior, Evolution & Culture Program, UCLA, Los Angeles, CA, USA. 31Department of Evolutionary Anthropology, Duke University, Durham, NC, USA. 32Department of Psychology, University of Pennsylvania, Philadelphia, PA, USA. 33School of Human Evolution and Social Change, Institute of Human Origins, Arizona State University, Tempe, AZ, USA. 34Dian Fossey Gorilla Fund International, Atlanta, GA, USA. 35Wildlife Conservation Society, Global Conservation Program, Bronx, NY, USA. 36Department of Anthropology, University of Wisconsin-Madison, Madison, WI, USA. 37Department of Anthropology, University of California, San Diego, La Jolla, CA, USA. 38 Uaso Ngiro Baboon Project, Laikipia, Kenya. 39Kenya Wildlife Service, Nairobi, Kenya. 40African Conservation Centre, Nairobi, Kenya. 41 Department of Biology, Duke University, Durham, NC, USA. 42Duke Population Research Institute, Duke University, Durham, NC, USA. 43 Department of International Health, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD, USA. 44Department of Human Evolutionary Biology, Harvard University, Cambridge, MA, USA. 45Institute of Biology, University of Neuchâtel, Neuchâtel, Switzerland. 46 These authors contributed equally: Fernando Colchero, Susan C. Alberts. ✉email: colchero@imada.sdu.dk; alberts@duke.edu 10 NATURE COMMUNICATIONS | (2021)12:3666 | https://doi.org/10.1038/s41467-021-23894-3 | www.nature.com/naturecommunications