## Introduction

Body mass impacts almost every aspect of an animal’s biology and ecology. Locomotion, behavior, diet, social organization, energy requirements, and a host of other vital biological and ecological characteristics are directly or indirectly tied to body mass. Thus, understanding the evolution of this trait is a necessary step in reconstructing the paleobiology of extinct fossil species. Though the timing, causes, and biological implications of the increase in body mass that took place during human evolution continue to inspire a wealth of research (e.g., refs. 1,2,3,4,5), the body mass of the last common ancestor (LCA) of chimpanzees and humans remains unexplored in any rigorous fashion. This omission is startling because numerous arguments over one of the most contested topics in hominin evolution—what were the selective regimes that led to the origins of bipedalism6,7,8,9 (but see ref. 10)—depend on inferences about body mass at and prior to the root of our lineage. Various classic models9, 11,12,13 proposed a body mass increase as a proximate factor in the evolution of suspensory adaptations and the transition from an arboreal to terrestrial hominid (great apes plus humans and our fossil ancestors) as larger sizes dictated a switch between locomotor modes, while models based around an arboreal quadruped ancestor (e.g., ref. 14) implicitly assumed a smaller-body mass in order to maintain balance and stability on deformable branches of different diameters15. Note that here we define body size as body mass16.

One important reason for this omission is the paucity of African fossil hominids during the period when the chimpanzee and human lineages are believed to have diverged, perhaps 4–6 Ma (million years ago)17 or earlier at 6–8 Ma18,19,20, with the notable exceptions of putative basal hominins Orrorin tugenensis (~6 Ma)21, Sahelanthropus tchadensis (6–7 Ma)22, Ardipithecus kadabba (5.5–6.4 Ma)23, and the later Ardipithecus ramidus (4.4 Ma)7. In addition, body sizes in the more well-sampled Miocene hominoid (all living and extant apes and humans) taxa (e.g., Proconsul) appear to be extremely variable (e.g., refs. 24, 25), and questions about how these species relate to one another and to crown hominoids (reviewed in ref. 26) further complicate the usefulness of these data. Issues with body mass variation in Miocene hominoids are evident in the description of Ar. ramidus, which argues that the chimpanzee-human and African hominid LCAs were likely to be “equal to or smaller than Ar. ramidus, possibly even substantially so”7—a range extending down from the fossil’s predicted body mass (~50 kg)7 and encompassing almost all primates from chimpanzees to diminutive monkeys. Another primary reason for this omission may be a consensus among many researchers that—reinforced by the wealth of molecular work showing humans and chimpanzees are sister taxa27—the mass of the chimpanzee–human LCA resembled common chimpanzees28 (~45 kg), and a chimpanzee-sized ancestor represents the LCA of African hominids (i.e., hominines)10, hominids10, and hominoids29. However, a few recent findings argue caution with acceptance of a chimpanzee-sized series of LCAs stretching back to before the divergence of hylobatids from other hominoids around 19.5 Ma. First, this hypothesis coincides with the assumption of an overall chimpanzee-like morphology for the chimpanzee–human and hominid LCA, a topic of much debate6, 7, 30,31,32,33, with some researchers suggesting that current fossil evidence and analyses point to a generalized monkey-like ancestor34. Chimpanzee-like postcranial morphology and body mass are not necessarily linked, although this is often implied by many models that suggest a chimpanzee-like LCA. Second, the description of Pliobates cataloniae, a small-bodied (4–5 kg) hominoid from the Miocene of Spain (11.6 Ma), argues for a gibbon-sized common ancestor of all crown hominoids, rather than an extant great ape-sized ancestor with hylobatids evolving as a dwarfed lineage29. Finally, a large-scale analysis of hominin body mass found earlier hominins were smaller-bodied than previously thought3 (Table 1, Supplementary Table 1), with no evidence for an orderly increase in body mass from Australopithecus to early (non-erectus) Homo to Homo erectus as has been suggested35. Average body mass for the well sampled Australopithecus afarensis was ~5 kg less than an average common chimpanzee, and many other well sampled later hominin taxa (Australopithecus africanus, possible Paranthropus boisei, Paranthropus robustus, Homo habilis sensu stricto) are ~5–10 kg below Au. afarensis (Table 1). While body mass predictions for the earliest undisputed hominin, Australopithecus anamensis (46.3 kg)3, and the earliest putative hominins O. tugenensis (35–50 kg)36, and the later Ar. ramidus (~50 kg) (but see refs. 3, 7), are all in the range of common chimpanzees, these estimates are based on single fossils3, and overall these findings argue that the pattern of body mass evolution in our own lineage may be more complicated than either stasis or a steady increase in body mass from a chimpanzee-like ancestor. Taken together, while a chimpanzee-sized LCA has been hypothesized as the phenotype from which all hominoid branches diverged, to the best of our knowledge this has not been tested in any quantitative, phylogenetically informed fashion. In addition, the data underlying these hypotheses appear to be problematic and further compounded by poor understanding of the Miocene fossil relationships, and likely heavily influenced by the view that chimpanzees provide fairly clear windows into our evolutionary past (see Supplementary Note 1 for taxonomic scheme used here).

To test alternative hypotheses on primate body mass evolution, we employ phylogenetic comparative methods37,38,39 on body mass data from a large sample of extant primates, a wide sample of fossil primates including Miocene fossil apes, and recently published data for fossil hominins3 to reconstruct the path of body mass evolution in individual branches of the primate phylogeny. Our fossil sample includes stem fossil apes Proconsul africanus (22.5 Ma) Proconsul major (22.5 Ma), Ekembo (Proconsul) nyanzae (20.0 Ma), Ekembo (Proconsul) heseloni (20.0 Ma), and Pliobates cataloniae (11.6 Ma), early hominids Dryopithecus fontani (11.8 Ma), Hispanopithecus laietanus (9.6 Ma), and Sivapithecus indicus (12.7 Ma), very early putative hominins Orrorin tugenensis (6.0 Ma) and Ardipithecus ramidus (4.4 Ma), and unequivocal hominins Australopithecus anamensis (4.17 Ma), Australopithecus afarensis (3.77 Ma), Australopithecus africanus (3.03 Ma), Australopithecus sediba (1.98 Ma), Paranthropus boisei (2.3 Ma), Paranthropus robustus (2.0 Ma), Homo habilis (2.33 Ma), Homo erectus (1.9 Ma), Homo heidelbergensis (0.609 Ma), Homo neanderthalensis (0.13 Ma), Homo floresiensis (0.1 Ma), and modern humans (0.195 Ma) (Table 1; Supplementary Table 1). Our approach here translates hypotheses on adaptation in selective regimes (also known as adaptive regimes or zones) into explicit evolutionary models, tests alternate models against comparative data (including fossils) using a maximum-likelihood model selection framework, and infers details of evolutionary processes such as estimating optimal body sizes for a given selective regime. Species in a selective regime share a common selective factor, and the underlying cause of differences in selective regimes may be differences in this factor related to environment, habitat, locomotion, etc. 37. When the selective regime changes, the fitness landscape of functional traits changes along with the optimal values for traits such as body mass, which leads to adaptation of traits towards this new optimum. Here, we test hypotheses of when shifts in selective regimes occurred along a phylogeny and estimate new optima (i.e., the optimal body mass of the new regime) that coincide with regime shifts by reconstructing the macroevolutionary adaptive landscape for primate and hominin body size. This approach provides a new and novel source of information on a trait that directly influences numerous hypotheses on the paleobiology of the human lineage.

First, we focus on body mass evolution within the hominoid clade, and show that the LCA of humans and chimpanzees, the earliest hominins, and the ancestor of African hominids lived in a selective regime that favored a modern chimpanzee-like body mass and was likely chimpanzee-sized. Further, the ancestor of all hominoids evolved in a regime that favored a gibbon-like body size, supporting recent fossil findings40. Second, we compare details of the evolutionary process within the hominoid clade, including fossil apes and hominins, to a large sample of extant and fossil primates and show that while most of primate evolution is characterized by evolution towards only two optimal body sizes, hominids are unique among primates in having a substantially greater number of adaptive optima due to distinct selective regimes across evolutionary time.

## Results

### Body mass evolution in hominoids including fossils

To reconstruct the adaptive landscape for body mass in primates including hominins and test hypotheses on when regime shifts occurred, we used the recently introduced “SURFACE” approach38 that fits a series of evolutionary hypotheses via Ornstein-Uhlenbeck or OU stabilizing selection models37 to phenotypic species data related via a phylogeny and retains the hypotheses that best fits the data (seen in the lowest corrected Akaike Information Criterion or AICc score). OU models permit the realization of Simpson’s41 adaptive zones by allowing for the placement of selective regimes along different branches of a phylogeny, where species in each regime evolve toward a distinct trait optimum—in this case the optimal body mass for a given regime. In the OU model, species have their own local adaptive optima, the position of which is influenced by numerous selective factors. Species within the same selective regime are also pulled toward a “primary” adaptive optima that is influenced by the selective factor or factors that define that regime (e.g., arboreality as a factor could define a selective regime with a smaller bodied optimal body size). Evolutionary constraints (due to ancestry, genetic correlations, functional constraints, and so on) reduce the rate of adaptation given a change in the selective regime, and can lead to species' trait values that are quite distant from the primary optimum of a given selective regime37. Here, “optima” or “adaptive optima” always refer to primary adaptive optima, not local adaptive optima. OU models differ from the more commonly used Brownian motion (BM)-based models of evolution as they model the evolutionary process as the combination a deterministic pull toward adaptive optima and fluctuations around this optimum that result from unmeasured forces (e.g., other selective factors, genetic drift). As such, OU models are consistent with current views on evolutionary change39, 42, 43. Importantly, this approach also identifies convergence in regimes—whether distantly related species found similar optima. Here, because the selective regimes are not assigned a priori, distantly related species in convergent regimes are not necessarily under the same set of selective factors, though they share the same estimated optima.

Our results show that the earliest putative hominins (O. tugenensis, Ar. ramidus), and the early australopith Au. anamensis shared an selective regime with both species of Pan (Fig. 1a, b; Regime “m” in Table 2), and along with evidence that these fossil taxa were the mass of a chimpanzee (see above and Table 1), argue that the LCA of chimpanzees and humans was indeed chimpanzee-sized. This selective regime begins after the divergence of Hispanopithecus and includes the African hominid LCA—while geographically and temporally distinct, this result suggests that all of these species lived in an environment/or environments that favored the same optimal body size. While a selective regime favoring a slightly smaller body mass shared by Sivapithecus (with both species of Pongo sharing a derived larger body mass regime) and Hispanopithecus (Fig. 1a, b; Regime “b” in Table 2) was present after the divergence of hylobatids until after the LCA of hominids, the LCA of hominoids lived in a regime that was similar to extant hylobatids, stem fossil apes Ekembo heseloni, Proconsul africanus, and Pliobates cataloniae, and was plesiomorphic (primitive) and shared with a large number of Old World Monkey species and converged on by Atelidae (Fig. 1a, b; Regime “h” in Table 2). Focusing on hominins, a regime shift towards a smaller body mass (Fig. 1a, b; Regime “b” in Table 2) occurred with the arrival of Au. afarensis and persisted through all early hominins (Au. africanus, Au. sediba, P. boisei, P. robustus, H. habilis, H. floresiensis), before a regime shift to a larger optimal body mass near the origins of H. erectus shared with modern humans (Fig. 1a, b; Regime “e” in Table 2). There was also evidence that a few hominins and fossil apes converged on selective regimes shared with other hominoids—Pr. africanus converged on the body mass optima shared with both species of Pan and the earliest hominins, Pr. major with the large bodied regime of both Gorilla species, E. nyanzae and D. fontani with the larger bodied regime shared by both species of Pongo, which is also converged on by H. erectus and modern humans (Table 2).

To test how different estimates of body mass for O. tugenensis and Ar. ramidus 7, 30 affect our results, we reran our analysis using smaller body mass estimates that depend on the fossils sharing a modern human rather than a great ape pattern of scaling (Table 1). Our results (Supplementary Fig. 2A, B) show that a smaller body mass for these two taxa has no major impact on our overall findings—the LCA of chimpanzees and humans now shares a selective regime with the slightly smaller bodied pygmy chimpanzee (Pa. paniscus) as well as two of the earliest putative hominins (O. tugenensis, Ar. ramidus) but this regime now includes the slightly smaller Au. afarensis while Au. anamensis converges on the slightly larger-bodied regime that contains common chimpanzees. As the phylogenetic placement of D. fontani and other European and Asian hominids as closely related to African hominids is debated (reviewed in ref. 26), we tested how including these taxa affected our results by rerunning our analyses after removing these species. Overall results are consistent and robust (Supplementary Fig. 3A, B), with the only major change being that the larger-bodied regime that includes the Pongo lineage now includes both the hominid and the African hominid LCA, from which the smaller-bodied chimpanzee-like and larger bodied gorilla-like optima diverge. To test how inclusion of fossil taxa where average body mass estimates are based on only a few individuals (e.g., n = 1) affects our results, we reran our analyses using only those taxa where average body mass is based on more than two individuals (leading to the exclusion of O. tugenensis, Ar. ramidus, Au. anamensis, Au. sediba, H. habilis, and H. floresiensis) and are well attributed to particular taxa (leading to the exclusion of “probable” P. boisei 3). We also excluded the data from Miocene apes and other fossil primates because of uncertainties about their phylogenetic relatedness. Overall results (Fig. 2a; Supplementary Fig. 4) are extremely similar to the results using the smaller bodied estimates of Ar. ramidus and O. tugenensisAu. afarensis shares a selective regime with Pan (here both Pa. troglodytes and Pa. paniscus) with a shift to a smaller-bodied regime early in australopiths (encompassing Au. africanus and P. robustus) and increase at the time of H. erectus, but now the Pan-sized selective regime extends back to a shift after the LCA of hominoids (Node “1”). Finally, to test how using species averages for extremely sexually dimorphic species affects our findings, we used only average female body mass for the extant primates and estimated female body mass averages for our well-sampled reliably attributed hominins3, leading to the removal of a number of hominin taxa (O. tugenensis, Ar. ramidus, Au. anamensis, Au. sediba, P. boisei, and H. habilis). Here (Supplementary Fig. 5), the LCA of chimpanzees and humans shares a selective regime with the smaller bodied pygmy chimpanzee (Pa. paniscus), which stretches back to shortly after the LCA of hominoids (Node “1”) and includes Sivapithecus, Hispanopithecus, Pongo pygmaeus, and Au. afarensis. It is important to note that in some lineages, female body masses may have evolved in a different manner than average body mass, as discussed below.

We tested the relative support of our results for each iteration of our data set compared to three other alternative evolutionary hypotheses (Fig. 2b–d) using relative AICc values including a hypothesis of evolution by genetic drift (a Brownian motion model of evolution), a model where adaptive evolution is toward one body mass optimum (a single-peak OU model), and a (multi-peak) model based on the SURFACE model where the LCA of all hominoids was chimpanzee-sized and the hylobatid lineage evolved by dwarfism. This last hypothesis assumes that the increase in body mass occurred prior to the divergence of the hylobatids and proconsuloids. Our results suggest that the adaptive landscape shown above (Fig. 1) and returned by SURFACE is the most likely model as seen in its smallest AICc values, regardless of the data set used (Table 3; Supplementary Table 2).

### Patterns of body mass evolution in primates

To test how including fossil data affects our results, we reran the analysis with a phylogeny and data from only extant species (Supplementary Fig. 6). Though there are now only seven hominid species, this group again shows high variation in the number of body mass regimes, evolving towards four optima, or more than half of the total number of regimes for primates. One important change is that the LCA for hominids and African hominids as well as chimpanzees and humans now shares a selective regime with Po. pygmaeus, a regime that appears constant at nodes within the phylogeny up to both Po. pygmaeus and modern humans (Nodes “2”–“4”). Notably, the LCA of all hominoids (Node “1”) remains in a regime shared with hylobatids and a large number of other primates, suggesting that this body mass was optimal at this time period, though this could be driven by the presence of the deeply rooted hylobatids. Another important change is the half-life is now 0.21 Ma (Table 3), a significant decrease from previous runs and suggests that the addition of fossil data could provide a more accurate estimate of evolutionary parameters than analyses based purely on extant data (see ref. 50 for more on this point). Additionally, comparing exceedingly long half-lives of the extant females-only data set to the extant species averages data set (Supplementary Fig. 7), 10.13 vs. 0.21 Ma (Supplementary Table 2 vs. Table 3), provides further support that average female primate body mass may be evolving in a substantially different fashion than species averages.

## Discussion

The results of our novel comparative phylogenetic analysis of body mass evolution in primates have large consequences for the paleobiology of hominoid and hominin origins. First, our results suggest that the LCA of chimpanzees and humans lived in an environment that favored a body mass similar to modern chimpanzees (either Pa. troglodytes, Pa. paniscus, or both depending on the data set used), and this optimal body mass was shared with the earliest hominins. Consistent with fossil evidence of large body sizes, our results support earlier suggestions9, 10, 28, 29 that this LCA had a body mass close to that of modern chimpanzees. It should be noted that this regime persisted in the earliest hominins until shifting to a smaller-bodied regime near or following (depending on the data set used here) the origins of Au. afarensis at 3.77 M. While this reduction in the optimal average body mass could be due to a reduction in female body mass resulting from differential effects of ecological stresses47—such as caused by climate variability at Hadar 3.4–2.9 million years ago51—recent findings3 suggest that later early hominins from South Africa (Au. africanus, P. robustus) were smaller bodied both on average than earlier Au. afarensis, and purported males may have had a slightly larger decrease in body mass than females (~4%). Thus, if an increase in sexual dimorphism in Au. afarensis was the result of ecological stresses affecting females to a greater extent than males (e.g., refs. 47,48,49), it appears that these stresses affected the sexes in a more similar manner in later australopiths. In fact, the estimated optimal body mass for the later smaller-bodied early hominin regime was slightly below 30 kg (Table 2), which is about 10 kg smaller than the earlier Au. afarensis, and later early hominins (starting with the 30.5 kg Au. africanus at 3.03 Ma), including the small-bodied H. habilis, appear to be evolving towards this new smaller optimal body mass. The origin of Au. africanus coincides with the shift towards more open environments after 3 Ma. in South African sites52, as well as apparently increased greater climatic variability in East Africa35, which likely imposed new selective pressures on early hominins and led to a regime shift at this time. The regime shift to larger optimal body sizes near the origins of H. erectus (Fig. 1b, c) as well as larger body mass in this taxon (Table 1) could signify the combination of environmental changes to more favorable conditions or behavioral differences leading to shifts in the ability to use available resources (e.g., a greater reliance on high-quality sources such as meat53). Of course, this sequence of body mass evolution (and the results of this analysis) depends on the relationships among taxa, but at the very least there appears to be a substantial decrease in both the species average as well as average male and female body mass for hominins between 3.0 and 2.0 Ma with the extinction of Au. afarensis 3. It is also suggestive that the optimal body mass for the regime that contains H. heidelbergensis and H. neanderthalensis is slightly below the average mass of G. gorilla and G. beringei in almost all iterations here, and these hominins may in fact be evolving toward a selective regime that favored increasingly large body sizes due to factors such as colder climates54, 55 or hunting larger-bodied prey56. No doubt hominin body mass was constrained and influenced by a wealth of factors, such as sexual selection, food availability and other ecological influences, tool-use, and physiological constraints that are not tested in the current model. Taking a step back, it is also notable that we found primate female body mass adapting to new optima at a substantially slower rate than species data, seen in much larger half-life values (Supplementary Table 2), which argues that sexual dimorphism in body mass is the result of males and females responding to selection pressures on different time scales and/or selection pressures come from different sources. One possible reason that may be partially supported our findings for hominins is females appear to be more heavily influenced by the environment than males because female reproductive success depends on acquiring energetic resources for birth and lactation48, 49, 57, 58. Thus, one would expect the rate of adaptation for females to follow broad shifts in ecology—i.e., an adaptive landscape that is dominated by peaks defined by ecological factors—that likely occur on a much slower time scale than competing selective pressures on males (e.g., intraspecific sexual selection, along with ecological factors).

Finally, our results provide evidence of a complex and changing adaptive landscape in the hominin and hominid clades—while almost all other primates are evolving toward two body mass optima in our sample (e.g., 1.4 and 7.0 kg; Fig. 1a), hominids (including proconsuloids) had a substantially greater number of adaptive optima due to distinct regime shifts than any other group (Fig. 1c). While these results are preliminary, they suggest that most of primate evolution has taken place within a small number of ecological niches—one small-bodied regime principally based around arboreal quadrupedalism and leaping, one larger bodied regime, members of which evolved toward suspensory behavior (Hylobatidae and Atelidae) and continued arboreal quadrupedalism and leaping (most Cercopithecidae). Larger-bodied species adapted to terrestrial locomotion—Pa. cynocephalus and Pa. anubis here, are near their own body mass optima (Regime “k”—17.7 kg; Table 2). Within each group is variation in locomotor behavior, as well as diet, social structure, and so on—differing local selective pressures that likely led to the variation around the optimal body mass within a given regime. Together, the greater number and greater complexity of body mass optima for hominids and hominins supports the hypothesis that dramatic and uncommon shifts in the adaptive landscape drove human evolution.

## Methods

### Body mass averages

Extant primate body mass averages were taken from Isler et al.63, which focused on wild-caught primates, supplemented by data for modern humans from Ruff et al.2. Early fossil hominin body mass species averages were taken from Grabowski et al.3, supplemented by previously published data on hominins and Miocene fossil apes (Table 1; see Supplementary Table 1 for complete references). Body mass of Ar. ramidus is uncertain3, 7 based on whether this species scaled more like a modern human or chimpanzee. For ARA-VP-6/500 we used the modern human-based estimate of 32.1 kg3 and the chimpanzee-based estimate 50.8 kg30. We used a similar approach for the body mass of O. tugenensis—the modern human-based estimate of 35.8 kg was taken from Grabowski et al.3 and we calculated a chimpanzee-based estimate using the supero-inferior femoral head diameter (FHD) of 33.2 mm for BAR 1002’00 and the regression of FHD on body mass for 28 chimpanzees from Almécija et al.30.

### Phylogeny

We used a composite phylogeny based on the dated consensus tree from version 3 of 10Ktrees64, the latest phylogeny for fossil hominins from Dembo et al.65, and the addition of the stem fossil ape species Pr. africanus (22.5 Ma), Pr. major (22.5 Ma), E. nyanzae (20.0 Ma), E. heseloni (20.0 Ma)66, and P. cataloniae (11.6 Ma), European fossil hominids D. fontani (11.8 Ma), and Hi. laietanus (9.6 Ma), and Asian fossil hominid S. indicus (12.7 Ma) based on their first appearance and proposed phylogenetic relationships. Both species of Ekembo were formerly placed in Proconsul but united into their own genus based on morphology66. Here Ekembo and Proconsul form two clades66 and diverge separately from the main trunk prior to the divergence of hylobatids. Proconsul (and by extension Ekembo) is accepted as a stem hominoid by the majority of researchers67 although some argue that it is a stem catarrhine68. We placed all five early Miocene apes as originating shortly before the divergence of hylobatids, later Miocene great ape Dryopithecus originating before the Gorilla lineage, followed by Hispanopithecus, with Sivapithecus on the branch that led to Pongo. For these fossil species, ghost lineages of 1 Ma were added to the published age of the fossil as a criterion for standardization. We also updated the divergence date for H. floresiensis based on recent findings69. We added a wide array of non-hominoid fossil primates (Table 1) to our phylogeny based on their first appearance dates and proposed relationships from a time scaled phylogeny of living and extinct primates19. As most of these non-hominoid fossil taxa averages are based on rare single individuals, we assumed that average body size and female body size were the same for our females-only analyses above. See Supplementary Fig. 1 for complete phylogeny with species names, Supplementary Data 1 for phylogeny with branch lengths. To focus on hominoid evolution within a larger primate context, as well as decrease computation time and redundancy in our phylogeny, we reduced the number of species in our extant data set from 176 in Isler et al.63 to 87 for our main analysis. To do this, we first took the subset of species averages from Isler et al.63 that are based on at least one male and one female individual and therefore are the more reliable estimates of species means70—totaling 161 species. After matching these species with those in our phylogeny (see below) we used a routine where given sister species of roughly similar mass (sister species where the second sister species was > 80% the mass of the first), the second species in each comparison was dropped from the analysis, leaving 111 non-hominid species. We then combined in our fossil and extant hominid data set (Table 1; Supplementary Table 1). Finally, the tree was pruned to 87 species for our complete data set by removing random non-hominid extant branches.

### Comparative approach

Our phylogenetic comparative approach is based around the model of adaptive evolution introduced by Hansen37 and implemented in the R package “SURFACE” (“SURFACE Uses Regime with Akaike Information Criterion (AIC) to model Convergent Evolution”)38, fitted in R 3.2.371. While a purely neutral model of evolution—Brownian motion—underlies most comparative approaches for quantitative characters72, the expected value of a trait remains equal to its ancestral value, and thus this model does not make allowance for the tendency of traits to adapt in response to selection. In Hansen’s37 adaptive model, trait change is modeled according to an Ornstein-Uhlenbeck (OU) process, where the change in a quantitative trait X is the sum of a deterministic part (interpreted as the force of selection acting on the trait) and a stochastic part (interpreted fluctuations in the local fitness optima due to unmeasured factors). This can be expressed as the stochastic differential equation39:

$${\rm{d}}X\left( t \right) = \alpha [ {\theta - X\left( t \right)} ]{\rm{d}}t + \sigma {\rm{d}}B\left( t \right)$$
(1)