Evolution of brain ontogenetic growth under ecological 1 challenges

9 Large brains are metabolically expensive but support skills (or cognitive abilities, knowledge, 10 information, etc.) that allow overcoming ecological and social challenges, with social challenges 11 being thought to strongly promote large-brain evolution by causing evolutionary arms races in 12 cognition yielding exaggerated brain sizes. We formulate a mathematical model that yields quan- 13 titative predictions of brain and body mass throughout ontogeny when individuals evolve facing 14 ecological but no social challenges. We ﬁnd that ecological challenges alone can generate adult 15 brain and body mass of ancient human scale, showing that evolutionary arms races in cognition 16 are not necessary for extreme brain sizes. We show that large brains are favored by intermediately 17 challenging ecological environments where skills are moderately effective and metabolically ex- 18 pensive for the brain to maintain. We further show that observed correlations of cognitive abili- 19 ties and brain mass can result from saturation with skill maintenance of the brain metabolic rate 20 allocated to skills. 21 Here we study the possible causal contribution of ecological challenges alone to large-brain evo- lution by means of a mathematical model. We formulate a metabolically explicit model for the evo- lution of brain ontogenetic growth when individuals face ecological but no social challenges. We use the model to determine how much energy should be allocated to brain growth at each age as a re- sult of natural selection given that overcoming ecological challenges provides energetic returns (e.g., through food procurement). By excluding social challenges, the model deliberately eliminates the possibilities of evolutionary arms races in allocation to brain growth, and thus serves as a baseline for understanding brain growth evolution. We derive the model in terms of measurable parame- ters using the approach of West et al. In particular, the model incorporates parameters mea- suring the mass-speciﬁc metabolic costs of brain growth and maintenance, which capture the rela- tively large metabolic expense of the brain. These parameters can be measured empirically, and are likely to differ among species given different brain structures and efﬁciencies. Once parameterized with values obtained from data, the model yields quantitative predictions for brain and body mass throughout ontogeny under the assumption that individuals evolved under ecological challenges alone.


Introduction
Large brains use copious amounts of resources that could otherwise be allocated to reproductive simplicity we also assume constant with respect to age. Building on the metabolic model of West whereẋ i (a) denotes the derivative of x i (a) with respect to age. Equation (1) is a general equation 126 describing how growth schedules u i (a) specify tissue growth rates.

127
Skill 128 We let the individual have a number x k (a) of energy-extraction skills at age a. We assume that a frac- 129 tion v k of the brain metabolic rate is due to the energetic expense incurred by the brain for acquiring 130 (learning) and maintaining (memory) energy-extraction skills. We also assume that the brain releases 131 as heat an amount of energy E k for gaining an average skill (learning cost) and an amount B k per unit 132 time for maintaining an average skill (memory cost). We also assume v k , E k , and B k to be constant.

133
The growth rate of energy-extraction skills (see SI §1.4 for derivation) is then is the brain metabolic rate at age a (i.e., the energy released as heat by the brain per unit time with  (2) is also a general equation capturing the link of brain with skill; it is 138 general in that, for example, (2) is not restricted to energy-extraction skills (given that v k is accord-139 ingly reinterpreted). In analogy with (1), the first term in the numerator of (2) gives the heat released 140 due to energetic input for skill growth whereas the second term gives the heat released for skill main-141 tenance.

142
Skill function 143 Finally, we specify how skills allow for energy extraction. We denote the probability of energy extrac-144 tion at age a as p(x k (a)), defined as the ratio of the amount of energy extracted per unit time at age 145 a over that extracted if the individual is maximally successful at energy extraction. We assume that 146 p(x k (a)) depends on skill number but is independent of body mass. Given the empirical relationship 147 of resting metabolic rate and body mass as a power law (Kleiber, 1961, Peters, 1983, Sears et al., 2012, 148 which for humans also holds ontogenetically to a good approximation (Fig. S4), we show in the SI ( §1.5) that resting metabolic rate takes the form 150 B rest (a) = K p(x k (a)) x T (a) β , where β is a scaling coefficient, K is a constant independent of body mass, and body mass is x T (a) = 151 i ∈{b,r,s} x i (a). Equation (4) captures the notion that energy extraction gives the individual energy 152 that it can use to grow or maintain its different tissues. 153 We consider energy extraction at age a as a contest against the environment. We thus let the 154 probability of energy extraction p(x k (a)) take the form of a contest success function (Hirshleifer, 155 1995, Skaperdas, 1996): which we assume increases with the number x k (a) of energy-extraction skills, and depends on two Whereas predicted body growth patterns are qualitatively similar with either power or exponen-246 tial competence, they differ quantitatively (Fig. 1b,f). With power competence, the predicted body 247 mass is quantitatively nearly identical to that observed in modern humans throughout life (Fig. 1b). 248 In contrast, with exponential competence, the predicted body mass is larger throughout life than that 249 of modern human females (Fig. 1f). 250 Regarding brain mass, the model predicts it to have the following growth pattern. During early 251 childhood, brain mass remains static, in contrast to the observed pattern (Fig. 1d). During mid 252 childhood, brain mass initially grows quickly, then it slows down slightly, and finally grows quickly 253 again before brain growth arrest at the onset of preadolescence (Fig. 1d). Predicted brain growth is 254 thus delayed by the obtained early-childhood period relative to the observed brain growth in modern 255 humans (Fig. 1d). As previously stated, this delay in predicted brain growth may be an inaccuracy 256 arising from the underestimation of resting metabolic rate during early childhood by the power law 257 of body mass.

258
Predicted brain growth patterns are also qualitatively similar but quantitatively different with 259 power and exponential competence (Fig. 1d,h). Adult brain mass is predicted to be smaller or larger 260 than that observed in modern human females depending on whether competence is respectively a 261 power or an exponential function (Fig. 1d,h). Remarkably, considering body and brain mass together, 262 the predicted adult body and brain mass can match those observed in late H. erectus if competence 263 is a power function (Fig. 1b,d). In contrast, the predicted adult body and brain mass can match those 264 of Neanderthals if competence is an exponential function (Fig. 1f,h). Consequently, the encephaliza-265 tion quotient (EQ, which is the ratio of observed adult brain mass over expected adult brain mass for 266 a given body mass) is larger with exponential competence for the parameter values used (Table 1).

267
Skills through ontogeny 268 The obtained ESGS predict the following patterns for energy-extraction skills throughout ontogeny. 269 For the scenario in Fig. 1, the individual gains most skills during childhood and adolescence, skill 270 number continues to increase after brain growth arrest, and skill number plateaus in adulthood ( Fig.   271 2). That is, skill growth is determinate, in agreement with empirical observations (Fig. 2). Yet, if mem-272 ory cost B k is substantially lower, skill number can continue to increase throughout the individual's 273 reproductive career (i.e., skill growth is then indeterminate; Fig. S8e) [see equation (2)]. Neverthe- 274 less, in that case, the agreement between predicted and observed body and brain mass throughout 275 ontogeny is substantially reduced (Fig. S8b,c).
cally achieved [from equation (2) settingẋ k (a) = 0 and u * b (a) = 0] is wherex k is the asymptotic skill number, x * b (a a ) is the adult brain mass, v k is the fraction of brain 280 metabolic rate allocated to energy-extraction skills, and B b is the brain mass-specific maintenance 281 cost. The requirement for skill growth to be determinate is that the brain metabolic rate allocated terpreted), this result provides an explanation for these correlations in terms of saturation of brain 291 metabolic rate with skill (cognitive ability) maintenance. 292 We now vary parameter values to assess what factors favor a large brain at adulthood.

293
A large brain is favored by intermediate environmental difficulty, moderate skill effec-294 tiveness, and costly memory 295 A larger adult brain mass is favored by an increasingly challenging environment [increasing α; equa-296 tion (5)], but is disfavored by an exceedingly challenging environment (Fig. 3a). Environmental dif-297 ficulty favors a larger brain because more skills are needed for energy extraction [equation (5)], and 298 from equation (2) more skills can be gained by increasing brain metabolic rate in turn by increasing 299 brain mass. Thus, a large brain is favored to energetically support skill growth in a challenging en-300 vironment. However, with exceedingly challenging environments, the individual is favored to repro-301 duce early without substantial body or brain growth because it fails to gain enough skills to maintain 302 its body mass as (allo)parental care decreases with age (Fig. S12).

303
A larger adult brain is favored by moderately effective skills. When skills are ineffective at energy 304 extraction [γ → 0; equation (5)], the brain entails little fitness benefit and fails to grow in which case 305 the individual also reproduces without substantially growing (Fig. 3b). When skill effectiveness (γ) 306 crosses a threshold value, the fitness effect of brain becomes large enough that the brain becomes 307 favored to grow. Yet, as skill effectiveness increases further and thus fewer skills are needed for energy 308 extraction, a smaller brain supports enough skill growth, so the optimal adult brain mass decreases 309 with skill effectiveness (Fig. 3b). Hence, adult brain mass is largest with moderately effective skills.

310
10 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint (costly memory, increasing B k ), but exceedingly costly memory prevents body and brain growth (Fig.   312 3c). Costly memory favors a large brain because then a larger brain mass is required to energetically 313 support skill growth [equation (2)]. If memory is exceedingly costly, skills fail to grow and energy 314 extraction is unsuccessful, causing the individual to reproduce without substantial growth (Fig. 3c).

315
Factors favoring a large EQ and high skill 316 A large EQ and high adult skill number are generally favored by the same factors that favor a large 317 adult brain. However, the memory cost has a particularly strong effect favoring a large EQ because it 318 simultaneously favors increased brain and reduced body mass (Fig. 3c,f). In contrast to its effect on 319 EQ, increasing memory cost disfavors a high adult skill number (Fig. 3f). That is, a higher EQ attained 320 by increasing memory costs is accompained by a decrease in skill number (Fig. 3c,f). The factors that 321 favor a large brain, large EQ, and high skill are similar with either power or exponential competence 322 ( Fig. 3 and Figs. S15,S16). Importantly, although with the estimated parameter values the model 323 can recover modern human growth patterns yielding adult body and brain mass of ancient humans, 324 our exploration of the parameters that were not estimated from data suggests that the model cannot 325 recover modern human growth patterns yielding adult body and brain mass of modern humans.

327
Our model shows that ecological challenges alone can be sufficient, and that evolutionary arms races 328 in cognition are not necessary, to generate major human life history stages as well as adult brain 329 and body mass of ancient human scale. We find that the brain is favored to grow to energetically 330 support skill growth, and thus a large brain is favored when simultaneously (1) competence at energy 331 extraction has a steep dependence on skill number, (2) many skills are needed for energy extraction 332 due to environmental difficulty and moderate skill effectiveness, and (3) skills are expensive for the 333 brain to maintain but are still necessary for energy extraction.

334
While the model considers ecological challenges alone and so evolutionary arms races in cogni-335 tion do not take place, the model can recover body and brain mass of ancient human scale. Predicted rameter values for non-human taxa would allow to determine the model's ability to predict diverse 341 life histories and brain growth patterns (Moses et al., 2008), offering a means to assess the explanatory potential of ecological challenges for large-brain evolution across taxa.

343
The model also provides an explanation for observed inter-and intraspecific correlations be- explanation is the saturation with skill maintenance of the brain metabolic rate allocated to skills 347 during the individual's lifespan [equation (6)]. The proportionality arises because the adult brain 348 metabolic rate is found to be proportional to brain mass. This explanation follows from a general 349 equation for the learning rate of skills [equation (2)] that is based on metabolic considerations (West 350 et al., 2001) without making assumptions about skill function; yet, this equation assumes that the 351 fraction of brain metabolic rate allocated to the skills of interest (v k ) is independent of brain mass 352 (and similarly for B b and B k ). The model further predicts that additional variation in correlations be-353 tween cognitive ability and brain mass can be explained by variation in maintenance costs of brain 354 and skill, and by variation in brain metabolic rate allocation to skill [equation (6)]. However, the 355 model indicates that adult skill number and brain mass need not be correlated since saturation with 356 skill maintenance of the brain metabolic rate allocated to skills may not occur during the individual's 357 lifespan, for example if memory is inexpensive, so skill number increases throughout life (Fig. S8e).

358
Predicted adult brain mass and skill have non-monotonic relationships with their predictor vari-359 ables ( Fig. 3 and Figs. S15,S16). Consequently, conflicting inferences can be drawn if predictor vari- the finding that moderately effective skills are most conducive to a large brain and high skill is sim-369 ply a consequence of the need of more skills when their effectiveness decreases (Fig. 3b). Regarding 370 memory cost, the strong effect of memory cost on favoring a high EQ at first glance suggests that a 371 larger EQ than the observed in modern humans is possible if memory were costlier (see dashed lines 372 in Fig. 3e). However, such larger memory costs cause a substantial delay in body and brain growth, 373 and the resulting growth patterns are inconsistent with those of modern humans (Figs. S9-S11). lenges and social learning, our results are relevant for a set of hypotheses for human-brain evolution.

376
In particular, food processing (e.g., mechanically with stone tools or by cooking) has previously been 377 advanced as a determinant factor in human-brain evolution as it increases energy and nutrient avail-378 ability from otherwise relatively inaccessible sources (Wrangham, 2009, Zink andLieberman, 2016 tutes an ecological rather than a social challenge, but also in that it may help satisfy at least two of 389 the three key conditions identified for large-brain evolution listed in the first paragraph of the Dis-390 cussion. First, a shift in food-processing technology (e.g., from primarily mechanical to cooking) 391 may create a steeper relationship between energy-extraction skills and competence by substantially 392 facilitating energy extraction (relating to condition 1). Second, food processing (e.g., by building the 393 required tools or lighting a fire) is a challenging feat to learn and may often fail (relating to condi-394 tion 2). Yet, there are scant data allowing to judge the metabolic expense for the brain to maintain 395 tool-making or fire-control skills (condition 3). Our results thus indicate that food processing may 396 well have been a key causal factor in human brain expansion. Also, although we did not consider so-       CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made                         CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint selection on relative brain size in the guppy reveals costs and benefits of evolving a larger brain.

21
. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint  and maintenance of different tissues, in particular the brain. individual is its assimilation rate. Part of this energy per unit time is stored in the body (S) and the rest is the 54 total metabolic rate which is the energy released as heat per unit time after use. Part of the total metabolic rate 55 is the resting metabolic rate B rest and the rest is the energy released as heat per unit time due to activity B act . In 56 turn, part of the resting metabolic rate is due to maintenance of existing biomass B maint , and the rest is due to 57 production of new biomass B syn . We refer to B syn as the growth metabolic rate. This partitioning is illustrated 58 in Fig. S1. We formulate our model in terms of allocation of resting metabolic rate B rest to maintenance and 59 growth of the different tissues.
which gives the part of resting metabolic rate due to body mass maintenance (Hou et al., 2008).
which gives the rate of heat release in biosynthesis (Hou et al., 2008), and we call it the growth metabolic rate.

Tissue mass 77
Let the mass of an average cell of type i be x ci for i ∈ {b, r, s}. Then, the mass of tissue i at age a is and hence, using (S3), we have that Defining E i = E ci /x ci , this gives 4 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint and B i = B ci /x ci . We will denote body mass at age a as x T (a) = x b (a) + x r (a) + x s (a). 83 We consider that some of the brain metabolic rate is to acquiring and maintaining energy-extraction skills. We 84 assume that the individual at age a has a number x k (a) of energy-extraction skills. From energy conservation 85 and (S1) and (S2), the brain metabolic rate must equal M brain (a) = x b (a)B b +ẋ b (a)E b . We thus let v k be the 86 fraction of brain metabolic rate that is due to acquiring and maintaining energy-extraction skills (or brain's 87 allocation to energy-extraction skills). Suppose that the brain releases as heat an amount of energy E k for 88 acquiring an average energy-extraction skill (learning cost). Similarly, assume that the brain releases as heat 89 an amount of energy B k per unit time for maintaining an average energy-extraction skill (memory cost). Hence, 90 from energy conservation,

Skills
(S9) which is equation (2)  also possibly depends on body mass). Let us use x ≡ y to denote that x is defined as y. Then, we define the 101 probability of energy extraction at age t as the normalized production per unit time at age a: We also define the ratio of resting metabolic rate and energy obtained per unit time as and, motivated by (S12), the quantity From (S13b), we have that 105 B rest (a) = p(x k (a)) B rest,max (a).
Consequently, B rest,max (a) gives the resting metabolic rate when the individual is maximally successful at en-106 ergy extraction. Adult resting metabolic rate typically scales with adult body mass as a power law across all 107 5 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint to a good approximation ( Fig. S4; see also Sears et al. (2012)). Hence, assuming that p(x k (a)) is independent of body mass, we assume that where K is a constant independent of body mass. Equation (S14) then becomes equation (4) in the main text. 111 1.6 Fitness and evolving traits 112 We consider the growth schedules u i (a) for i ∈ {b, r, s} as evolving traits, and we make assumptions (see below) 113 that imply that these schedules attain evolutionarily stable values (Lande, 1982; Mylius and Diekmann, 1995).

114
To obtain evolutionarily stable growth schedules we need a fitness measure. To obtain this measure, we con- where µ is the mortality rate. For simplicity, we take mortality rate as constant. 119 We obtain a measure of fertility as follows. We partition the mass-specific resting metabolic rate of repro-120 ductive tissue B r into a component due to maintenance of reproductive tissue itself B ra and a component due 121 to production of offspring cells B ro . That is, B r = B ra +B ro (note that B ro is not part of E r because the latter refers 122 to the production of mother's cells). LetṄ o (a) be the number of offspring cells produced by the individual per where C 2 , C 3 , and f 0 are proportionality constants defined in the absence of density dependence competition. 127 We also assume that costs of parental or alloparental care are included in f 0 . Fertility is then proportional to 128 the mass of reproductive tissue (King and Roughgarden, 1982).

129
From (S16)-(S17), the individual's lifetime number of offspring produced in the absence of 130 density-dependent competition (Mylius and Diekmann, 1995) is then given by where τ is an age after which the individual no longer reproduces. With additional standard assumptions, 132 evolutionarily stable growth schedules in the population of constant size regulated through fertility must max-133 imize R 0 (Mylius and Diekmann, 1995), and so we take R 0 as a fitness (objective) function that is maximized 134 by the evolving growth schedules u i (a) at an evolutionary equilibrium. . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint ables is expressed in terms of the growth schedules u i (a) that we take as evolving traits and of 22 parameters: 139 namely, 11 tissue-and skill-metabolism parameters (K , β, v k , and B i and E i for i ∈ {b, r, s, k}); 3 demographic 140 parameters ( f 0 , µ, and τ); 2 contest success parameters (α and γ); 2 (allo)parental care parameters (ϕ 0 and ϕ r ); 141 and 4 newborn tissue mass and newborn skill parameters [x i (0) for i ∈ {b, r, s, k}]. Parameter f 0 only displaces 142 the objective vertically and thus has no effect on the optimal growth schedules. 143 We now formulate the optimal control problem posed by our evolutionary model and later describe how 144 we estimated parameter values from empirical data. For readability, we will suppress the argument in u(a) and x(a), and write u and x. 152 We then have the optimal control problem where from (S16)-(S18) subject to the dynamic constraints with 156 g i (u, x, a) = e i u i B syn (x, a) for i ∈ {b, r, s} (S19f) which are obtained from (S7) and (S10), where e i = 1/E i , d 1 = v k /E k , and d 2 = B k /E k . From (S4), (S8), (S14), 157 and (S15), we have that growth metabolic rate is where body mass is and, from (5) in the main text, the probability of energy extraction at age a is where competence at energy extraction is e γx k (exponential competence). (S19k) Finally, the initial conditions of (S19e) are 162 x i (0) = x i 0 for all i (S19l) and we do not consider any terminal conditions for (S19e) .
where λ i is the costate variable associated to state variable i and λ is the vector of costates. Here we also drop responding state variable (Dorfman, 1969). Thus, we now proceed to maximize the Hamiltonian to obtain 173 candidate optimal controls u * that satisfy these necessary conditions for optimality.

174
Due to the constraint u b +u r +u s = 1, we set u r = 1−u b −u s and only two controls must be determined: u * b 175 and u * s . Using (S19f) and (S19g), collecting for B syn in (S20), and evaluating at x = x * we have We thus seek to maximize (S21) with respect to u = (u b , u s ).
evaluated at (x * , u * ), where we define for i ∈ {b, r, s, k}. Note that the marginal returns on energy extraction from increasing skill and skill synergy are Hence, 10 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint We present the analytical results for the candidate optimal controls in this section, and their derivations in 193 section 4. In these two sections, we assume that growth metabolic rate is positive; that is, B syn (x * , t ) > 0. 194 The Hamiltonian of the optimal control problem (S19) is affine (or, less rigorously, linear) in the controls Together, these cases show that there are seven possible growth regimes (Table S1). Regimes B, R, and S 207 involve pure growth of one of the three tissues, whereas regimes BS, BR, RS, and BRS are singular arcs where at 208 least two tissues grow simultaneously. These regimes occur as indicated in Table S1 depending on the sign of 209 both the switching functions and their difference. Numerical illustration of these regimes is given in Fig. S2.
The "·" means any sign.
For simplicity of presentation in the remainder of section 3 and 4, we will explicitly write the arguments of 216 a function only when defining the function and will suppress their writing elsewhere, except in a few places 217 where it is useful to recall them.

11
. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  (S32e) Here we have for i , j , k, l , m ∈ {b, r, s, k}, and a subscript "/" in χ l m i j k in (S32) denotes a removed subscript. In turn, functions 13 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Finally, to complete the specification of (S32), we have The analytical solutions for the candidate optimal controls given by Table S1 and (S32) are functions of the 226 candidate optimal states x * and costates λ, which we have not specified analytically. To assess if these analyti-227 cal candidate optimal controls are indeed optimal, we compare them to optimal controls found numerically by which are part of the output given by GPOPS (Fig. S2b-e). Feeding these numerically obtained optimal states 234 and costates to the expressions for the analytical candidate optimal control, we plot in Fig. S3 the analytical 235 solutions for the candidate optimal controls given by Table S1 and (S32). Comparison with Fig. 1a,e shows that 236 the analytical candidate optimal controls closely follow the controls found numerically by GPOPS. Figure S3: Plots of the analytically found candidate optimal controls. (a) is for the power competence case in Fig. 1a-d. (b) is for the exponential competence case in Fig. 1e- , a b , a m , a a ), the analytically found controls can be greater than one or smaller than zero, possibly due to negligible numerical error in the location of the switching points. 14 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Here we derive the expressions forû b (x * , λ, t ) andû s (x * , λ, t ) during the singular arcs given by (S32). To do so, 244 we make use of the well-known result, according to whichû b andû s can be obtained from the age derivatives 245 of the switching functions up to some even, but not odd, order (Kelley et al., 1967). Note that during singular 246 arcs, either σ i = 0 for some i or the difference σ s −σ b = 0, and hence their age derivatives also equal zero during 247 the singular arcs. We thus obtain the singular controls by taking second age derivatives of these functions, but 248 before doing so, we obtain expressions that will be useful.

249
By differentiating (S31d) and (S31c) with respect to age, we obtain From (S27), taking the second age derivatives for the costates and noting thatψ i =ψ for i ∈ {b, r, s}, we find

Singular controls for regime BS
We now obtain the singular controls for growth regime BS. The procedure is essentially the same for growth From (S37), we also have the simplifications Since σ s − σ b = 0, we have thatσ s −σ b = 0, which using (S25), (S38), and (S39) becomes 15 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Hereλ k during the singular arc BS is similarly not an explicit function of the controls.

260
In (S40c), onlyψ andψ k are functions of u * . Expanding these terms in (S40c), we obtain an affine equation 261 in the singular controlû b : Therefore, assuming that ζ bsk = 0, the singular control for regime BS is From (S37), we have the simplifications From σ b = 0, we have thatσ b = 0, which becomes 269 e bλb − e rλr + d 1λk = 0 (S46a) 16 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Again, in (S46c), onlyψ andψ k are functions of u * . Expanding these terms in (S46c), we similarly obtain 271 an affine equation in the singular controlû b : Therefore, assuming that ζ brk = 0, the singular control for regime BR is For regime RS, we have that (u * b , u * s ) = (0,û s ). Hence, during regime RS the variable φ is again no longer an 276 explicit function of the controls: We have the simplifications From σ s = 0, we have thatσ s = 0, which becomes 17 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint ρ rs (x * , λ, a) =φ e r ψ r − e s ψ s − e r f 0 µe −µa = −e r λ r e r ψ r e r ψ r − e s ψ s − e r f 0 e −µa µ + e r ψ r − e s ψ s . (S53) Once again, onlyψ is a function of u * in (S52c). Expanding this term in (S52c), we obtain an affine equation 281 in the singular controlû s : where we define Therefore, assuming that ζ sr = 0, the singular control for regime RS is For regime BRS, we have that (u * b , u * s ) = (û b ,û s ). As before, the variable φ is no longer an explicit function of 286 the controls: Similarly, we have the simplifications From σ s = 0, we have thatσ s = 0, which is 289 e sλs − e rλr = 0 (S58a) −e s φψ +φψ s + e r φψ +φψ r − f 0 µe −µa = 0 (S58b) where as before 18 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint ψ x T (û s ω sr +û b ω br + ω r ) φ(e r − e s ) + ρ rs = 0 (S60a) Now, from σ b = 0, we have thatσ b = 0, which is where as before Expandingψ andψ k in (S62c), we obtain another affine equation in the two controlsû s andû b : . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Therefore, solving (S60b) and (S64b) and assuming that ζ sr ζ brk −ζ br ζ srk = 0, the singular controls for regime 297 BRS are 298û s (x * , λ, a) = ζ r ζ brk − ζ br ζ rk ζ sr ζ brk − ζ br ζ srk (S66a) 20 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Here we summarize the values of the 22 parameters used in numerical solutions. From these, 13 parameters 300 are estimated as described in section 6 and they refer to newborn mass, tissue metabolism, and demography 301 (Table S2). The estimates of E i for are less accurate than those of B i for i ∈ {b, s, r} as they require stronger 302 assumptions given the available data (see Moses et al. (2008)). Since the parameter f 0 only displaces the ob-303 jective vertically and thus has no effect on the solution, we choose its value to scale the objective R 0 (Table S2).

304
The remaining 8 parameters refer to skill metabolism, contest success, and (allo)parental care, for which we 305 use values that produce body and brain mass that closely approach ontogenetic modern human data. Hence, 306 we use different benchmark values with either power (Table S3) or exponential (Table S4)

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Here we describe how we obtained the parameter values in Table S2. We use ontogenetic data for modern human females published in Table S2   Let c 1 (a) be the ratio of glucose uptake by the brain per unit time at age a divided by the resting metabolic 325 rate at that age. Let c 2 (a) be the fraction of brain glucose metabolism that is oxidative. Then, the empirically 326 estimated brain metabolic rate at age a is the product B rest (a)c 1 (a)c 2 (a). c 1 (a) is obtained from Table S2   . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint E b = M brain (0)/Ẋ b (0) = 123.7584 MJ/kg. 350 E r : We have that B syn (a) = i ∈{b,r,s}Ẋi (a)E i . We assume that shortly before adulthood most growth is repro-351 ductive. So assumingẊ r (A a − 1) = 0 whileẊ i =r (A a − 1) ≈ 0, we have that assuming that at birth most resting metabolic rate is due to growth so B rest (0) − B maint (0) ≈ B rest (0). We have  Using the ontogenetic (averaged) data in Table S2   pectancy is 1/µ. We thus let µ = 1 29 y = 0.034 1 y .

373
For Hadza and Gainj hunter-gatherers, the average age at menopause is about 47 years (Eaton et al., 1994).

375
24 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint 7.1 Brain metabolic rate through ontogeny 377 With the obtained ESGS, brain metabolic rate is predicted to peak at the age of brain growth arrest, which is 378 qualitatively consistent with recent findings for brain glucose intake (Figs. S5a,b  2014), a peak in brain metabolic rate is predicted during mid childhood. The predicted small peak in brain 383 metabolic rate results from brain growth arrest (Figs. S5b and S6b) and is enhanced by a peak in allocation to 384 brain growth just before brain growth arrest (Figs. 1a,e). The predicted ratio of brain metabolic rate and resting 385 metabolic rate is also qualitatively consistent with brain glucose intake in modern humans (Figs. S5c and S6c).  Figure S5: Predicted and observed brain metabolic patterns in humans qualitatively agree. Plots are for the scenario in Fig. 1a-d (power competence). (a) Maintenance (blue; x * b B b ), growth (green;ẋ * b E b ), and total (red; M brain ) brain metabolic rates. (b) Brain metabolic rate peaks at the age of brain growth arrest. (c) Ratio of brain metabolic rate to resting metabolic rate vs. age. Dots are (a) the energy-equivalent brain glucose intake observed in modern human females or (c) the ratio of the latter to resting metabolic rate (Kuzawa et al., 2014).
A similar pattern is predicted with exponential competence (Fig. S6).

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S7: Effect of the absence of (allo)parental care with exponential competence. Parameters are as in Fig.   1e-h, except that here (allo)parental is absent; i.e., ϕ 0 = 0.

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S8: Indeterminate skill growth with inexpensive memory and exponential competence. Parameters are as in Fig. 1e-h, except that here B k = 1 MJ/y/skill rather than B k = 50 MJ/y/skill.

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S9: Larger EQ than that in Fig. 1 with exponential competence, but predicted body mass is less consistent with observation. Parameters are as in Fig. 1e-h, except that here B k = 60 MJ/y/skill rather than B k = 50 MJ/y/skill. Jitter in the controls indicates that the optimal control problem is computationally challenging for GPOPS (this applies to all plots in the main paper and SI).  Figure S10: Larger EQ than that in Fig. 1 with exponential competence, but predicted body mass is less consistent with observation. Parameters are as in Fig. 1e-h, except that here B k = 70 MJ/y/skill rather than B k = 50 MJ/y/skill. 28 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S11: Larger EQ than that in Fig. 1 with exponential competence, but predicted body mass is less consistent with observation. Parameters are as in Fig. 1e-h, except that here B k = 80 MJ/y/skill rather than B k = 50 MJ/y/skill. 29 . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S12: Reproduction without substantial growth with exponential competence when the environment is exceedingly challenging. Parameters are as in Fig. 1e-h, except that here α = 1.5 rather than 1.15. The mass of reproductive tissue grows from 0 kg at birth, to 0.77 g at the age of a m ≈ 6 months, and reaches a peak of 4.64 g at a b ≈ 8 months. Jitter in the controls indicates that the optimal control problem is computationally challenging for GPOPS (this applies to all plots in the main paper and SI).   Figure S13: Brain and body collapse in adulthood with exponential competence when learning is exceedingly inexpensive. Parameters are as in Fig. 1e-h, except that here and E k = 100 MJ/skill rather than 250 MJ/skill.

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S14: Brain and body collapse with exponential competence when the newborn has overly many skills.
Parameters are as in Fig. 1e-h, except that here x k (0) = 4 skills rather than 0.

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint Figure S15: Predicted adult body and brain mass, EQ, and skill vs. other parameter values with exponential competence. See legend of Fig. 3. In d, jitter in EQ is due to increasing jittering in the controls when body and brain mass collapse. When varying newborn skills, a larger adult brain mass is predicted when the newborn has fewer skills 451 (Fig. S15b). If the newborn has overly many skills, the individual grows more during the (allo)parental care 452 period than what it can maintain when (allo)parental care is absent, causing brain and body collapse during 453 adulthood (Figs. S15b and S14).

454
Regarding allocation of brain metabolic rate to energy-extraction skills, brain mass is predicted to be larger 455 with a decreasing, but not exceedingly, small brain allocation to skills (Fig. S15c). With an exceedingly small 456 brain allocation to skills, the individual reproduces without substantial growth because skills grow little and 457 the individual is unable to support itself when (allo)parental care becomes absent. Above a threshold, an 458 increasing brain allocation to skills predicts a decreasing adult brain mass because the energetic input to skill 459 growth is larger without the brain having to be as large [equation (A2)]. In contrast to brain mass and EQ, the 460 predicted adult skill number increases with brain allocation to skills (Fig. S15f).

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. CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made  Figure S16: Predicted comparative patterns with power competence. See legend of Fig. 3. Jitter in EQ is due to increasing jittering in the controls when body and brain mass collapse. . CC-BY-NC 4.0 International license available under a not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which was this version posted April 27, 2016. ; https://doi.org/10.1101/050534 doi: bioRxiv preprint