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Evolutionary Ecology Research, 2003, 5: 1199­1221

The impact of environmental factors on human life-history evolution: an optimization modelling and data analysis study
A.T. Teriokhin,1,2* F. Thomas,1 E.V. Budilova1,2 and J.F. GuÈgan1
1

Centre d'Etudes sur le Polymorphisme des Micro-Organismes, UMR 9926 IRD-CNRS, 911 Avenue Agropolis, BP 6451, 34394 Montpellier Cedex 5, France and 2 Department of Biology, Section of General Ecology, Moscow Lomonosov State University, Moscow 119992, Russia

ABSTRACT
A model of human life-history evolution based on the optimization of resource partition by an individual between its growth, reproduction and survival is used for searching evolutionarily optimal state-dependent strategies of energy allocation under different combinations of the model parameters representing food availability and environmental stresses. Using these strategies, we compute the corresponding optimal life histories and determine the dependency of their characteristics on both environmental parameters. Then, using a statistical analysis of global social and demographic data for 131 countries, we examine relationships between human life-history traits and environmental characteristics. Finally, we compare the dependencies obtained by modelling with those derived from data analysis. We show that such observed phenomena as a decrease in fertility with an increase of wealth (known as demographic transition), an increase in birth weight, age at maturity, size at maturity and life expectancy with a decrease of infection and an increase in food availability can be viewed as consequences of evolutionary optimization of the human life-history strategy of resource allocation. Keywords: age at marriage, age at maturity, availability of food, birth weight, demographic transition, dynamic programming, fertility, human life-history evolution, infection stresses, life expectancy, optimal energy allocation, size at maturity

INTRODUCTION Evolutionary optimization models of life histories (e.g. Hamilton, 1966; Perrin and Sibly, 1993; McNamara and Houston, 1996; GuÈgan et al., 2000; Teriokhin and Budilova, 2001) have been used successfully to explain and predict many general relationships between environmental conditions and life-history traits in populations evolving in such given conditions. In many studies, it has been demonstrated that some life-history traits (e.g. age at
* Author to whom all correspondence should be addressed. e-mail: ter@mat.bio.msu.su Consult the copyright statement on the inside front cover for non-commercial copying policies. © 2003 A.T. Teriokhin


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maturity, body size at maturity, life expectancy) depend on different environmental factors, in particular food availability and environmental mortality (e.g. Stearns, 1992; Teriokhin, 1998; Kozlowski and Teriokhin, 1999). The present study uses a similar approach based on the optimization of state-dependent partitioning of the individual's resources between its basic needs, such as growth, reproduction and survival, by applying a standard technique of stochastic dynamic programming (e.g. Mangel and Clark, 1988). The study consists of three stages. First, using evolutionary modelling, we evaluate the joint effects caused by changing two model parameters representing resource availability and environmental mortality stress on six human life-history characteristics (age at maturity, age at release from parental care, body size at maturity, body size at release, number of children, life expectancy). Second, using global social and demographic data, we evaluate by multiple regression analysis the joint effects of two environmental factors (food availability and infection stress) on several real human life-history characteristics (age at menarche, age at marriage, adult female body size, weight at birth, fertility, female life expectancy). Third, we compare the results obtained by optimization modelling with the results obtained by statistical analysis in order to understand to what extent the empirical dependencies of human life-history traits on environmental conditions may be explained by the process of evolutionary adaptation of human populations to their environments and whether this adaptation is ensured by biological or cultural means. EVOLUTIONARY MODEL We use a discrete time dynamic model to describe the life history of an individual (e.g. Mangel and Clark, 1988). We assume that at time 0 (the moment of birth), the individual's body size (mass) is X0 and at each time step t from 1 to Tmax it increases by the value wtEt. That is, Xt = Xt
-1

+ wtE

t

(1)

where Xt and Xt - 1 are the individual's body sizes at the beginning and the end of step t, Et is the amount of energy (measured in units of mass) produced by the individual during this step, and wt is the fraction of this energy allocated to growth. In our computations, we use the values X0 = 3, which is close to the typical human weight at birth (kg), and Tmax = 100, the age (years) that represents the limit not attained by most people. If at some step t no energy is invested in body growth (i.e. wt = 0) and hence body size remains equal to its value Xt - 1 at the beginning of the step, we use the commonly accepted allometric equation (e.g. West et al., 1999) to compute the amount of energy produced during this step: Et = aX
b t-1

(2)

The multiplier a is varied from 1.6 to 2.1 to roughly mimic the variability of food availability and the exponent b is set equal to 0.25. The range of variation of a and the value of b were so chosen to ensure that adult body weight would lie approximately within its typical limits of 40­80 kg (e.g. Eveleth and Tanner, 1976). If wt > 0 (i.e. individual body size increases in the time interval from t - 1 to t ), then we used the following more precise equation: Et = {[X
1-b t-1

+ a(1 - b)wt]

1/(1 - b)

-X

t-1

}/wt

(3)


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This equation takes into account the increase in body size in the time interval from t - 1 to t. To obtain this equation, we note that from (1) it follows that Et = (Xt - Xt - 1)/w and from (1) and (2) it follows that dXt = wtaX bdt t (3 ) Integrating (3) through the interval (t - 1, t ) (with wt assumed constant) and substituting the found expression for Xt into (3 ), we obtain (3). Still one other fraction of energy ut is allocated to reproduction. We assume that an individual can accumulate the energy allocated to reproduction during several time-steps (maybe only one) to release it in bigger portions (the moment of release can be interpreted as the cessation of parental care for a child). The reproductive energy Yt accumulated by the end of step t is obtained by adding the value utEt to the energy Yt - 1 accumulated by the beginning of this step: Yt = Y If the accumulated reproductive equal to 0 at the beginning of the We assume, in addition, that amount of released reproductive Teriokhin and Budilova, 2001):
t-1 t

(3 )

+ utE

t

(4)

energy is released at the end of some step t, then it is set next step t + 1 (we set Y0 = 0). the effective reproductive output Ft is not equal to the energy Yt, but that it depends sigmoidally on it (see also Ft = Ft(Yt) (5)

This was done because only under such an assumption may it be advantageous to accumulate energy instead of releasing it at each time-step. Here we used a sigmoid function of the following form: 0 Ymax (Yt - Ymin) Ft = Ymax - Ymin Ymax if Yt < Y if Y
min min max

< Yt < Y
max

(6)

if Yt > Y

The parameters Ymin and Ymax are set equal to 0.45 and 90, respectively, because these values ensure that the number of children is not significantly greater than 8, the maximum average fertility observed in our data. The remaining fraction of energy vt = 1 - wt - ut is assumed to be allocated to survival. We take into account two sources of mortality for an individual. The first, `environmental mortality', cannot be reduced by the individual, whereas the second, `individual mortality', can be reduced by allocating some energy to survival. More exactly, we assumed that the probability Pt for an individual to survive at step t is given by the equation Pt = exp(- m) (vtEt)d c + (vtEt)
d

(7)

The first multiplier, exp(- m), gives the probability to survive irremovable environmental stresses. The second multiplier gives the probability to survive the reducible causes of mortality. The closer to 1 is vt, the greater is vtEt, the amount of energy allocated to


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individual survival. We varied m from 0.001 to 0.011 to roughly mimic the variability of real environmental mortality (e.g. Thomas et al., 2000) and we use the values 0.0025 and 2 for parameters c and d to obtain a realistic mean life span range from 40 to 85 years observed across countries. So, the values and ranges of the model parameters a, b, c, d, m, X0, Ymin, Ymax and Tmax were chosen to make the resulting ranges of life-history traits obtained in optimization modelling as close to those observed in the data as possible. But when the values of these parameters are fixed, we can compute the dynamics of the state variables of the model, Xt and Yt, if there is a rule to determine at each time step t and for each combination of values of Xt and Yt the values of the control variables ut, vt and wt, mentioned above, and of one additional control variable zt necessary to identify the moments of releasing the accumulated reproductive energy (it can take only two values, `yes' or `no'). We searched for the rule which maximizes the expected lifetime reproductive output using stochastic dynamic programming (e.g. Mangel and Clark, 1988). It can be shown (Mylius and Diekmann, 1995; Teriokhin, 2002) that the strategy maximizing the reproductive output is evolutionarily optimal when the population is stable and this stability is attained by exercising density pressure only on offspring or only on adults. The dynamic programming searches for an optimal strategy in the form of a rule matching a set of values of control variables to each admissible set of values of state variables. It operates by iterating backwards from Tmax - 1 to 0 and at each age step and for each set of values of state variables Xt and Yt it searches for a set of values of ut, vt, wt and zt which maximizes the following gain function: Gt(Xt,Yt) = [G
t+1

(X

t+1

(ut,vt,wt,zt), Y

t+1

(ut,vt,wt,zt)) + F(Yt(ut,vt,wt,zt))]Pt(ut,vt,wt,zt)

(8)

(To find the optimum numerically, we discretize both the state and control variables.) It is natural to assume that GTmax(XTmax, YTmax) = 0 for all pairs of values of XTmax and YTmax; that is, to assume that there is no reproductive output at the maximum age Tmax. Knowing GTmax(XTmax, YTmax) and using (8), we can calculate GTmax - 1(XTmax - 1, YTmax - 1) for all pairs of values of XTmax - 1 and YTmax - 1, searching for each pair the optimal combination of control values of ut, vt, wt and zt (i.e. those which maximize the right-hand side of (8)). Then, knowing GTmax - 1(XTmax - 1, YTmax - 1), we can calculate GTmax - 2(XTmax - 2, YTmax - 2), and so on to G0(X0, Y0). What is remarkable is that during these calculations we find for each t not only Gt(Xt, Yt), the expected future fitness of an individual in state Xt, Yt at time t, but also the optimal values of control variables ut, vt, wt and zt. DATA The data for the analysis were compiled for 131 different countries using mainly Internet global databases (WHO, UNICEF, World Bank, GIS). Variables representing the environmental conditions were `food' and `infection'. For each country, the first variable was based on the average number of calories for a person per day and the second was approximated by the parasite species richness per country for the 16 most dangerous infectious diseases (typhoid, hepatitis A, hepatitis B, malaria, schistosomiasis, filariosis, meningococcosis, yellow fever, dengue fever, cholera, trypanosomiasis, dracunculosis, Chagas' disease, Lyme disease, cutaneous leishmaniosis and visceral leishmaniosis) (GuÈgan et al., 2001). Work in progress indicates that this environmental parasitic pressure index for the 16 most potent


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killers correlates strongly with a new index based on the 350 parasitic and infectious diseases infecting human populations in the world, thus indicating that the 16 pathogens estimate is a good indicator of overall parasitic stress. Other variables were used to describe the average human life-history traits of females in the population for each country. Only female characteristics were used to simplify the analysis of data and modelling. Questions pertaining to human sexual life-history dimorphism are treated elsewhere (GuÈgan et al., 2000; Teriokhin et al., 2000). Six life-history characteristics were analysed: average age at menarche, average female age at marriage, average adult female stature, weight at birth, average lifetime number of children per female, and female life expectancy at 1 year old. The dependency of each of these life-history characteristics on the two environmental characteristics (i.e. food and infection) were analysed using a two-factor linear regression (e.g. Sokal and Rohlf, 1994). The regression equations obtained were illustrated by two-dimensional contour graphs with contour lines corresponding approximately to the mean value of the analysed life-history trait and to the values distant by a half and by a full standard deviation from the mean. RESULTS OF EVOLUTIONARY MODELLING The evolutionary modelling results present the dynamics of the state and control variables and of survival in the course of an individual life history. We computed such life histories for 36 combinations of values of environmental parameters obtained by setting a = 1.6, 1.7, 1.8, 1.9, 2.0, 2.1 and m = 0.001, 0.003, 0.005, 0.007, 0.009, 0.011. An example of evolutionarily optimal life-history results (for a = 1.7 and m = 0.9) is presented in Appendix 1. Based on Appendix 1, it is possible to compute several derivative characteristics of life history. Age at maturity, Tmat, can be viewed as the age reached by the beginning of investment of energy in reproduction (i.e. before the first value of t when ut > 0). Using Appendix 1, we obtain Tmat = 20. Body size at maturity, Xmat, is the body size attained at the age at maturity. Using Appendix 1, we obtain Xmat = 53.6. Life expectancy, Texp, is calculated as the sum of probabilities, St, to survive from 0 to 1, 2, . . . , 100 years:
T
max

Texp =
t=1

St

(9)

where St is computed as the product of probabilities to survive the periods from 0 to 1, from 1 to 2, . . . , and from 99 to 100:
t

St =
i=1

P

i

(10)

Using Appendix 1, we obtain Texp = 62. The number of children, Nchi, is calculated as the sum of reproductive releases occurring before the age at menopause, which we set equal to Tmnp = 50 (Thomas et al., 2001). Using Appendix 1, we obtain Nchi = 5. The body size at release, Xrel, is defined as the average amount of energy at reproductive releases before age t = 50. Using Appendix 1, we obtain Xrel = 13.6. The period of parental care, Trel, is defined as the average period between consecutive releases of reproductive energy. Using Appendix 1, we obtain Trel = 4.


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Some irregularities (in particular, the non-monotonicity) in the dependencies of modelled life-history traits on the parameters a and m observed in several tables accompanying the figures are due to the discretizing of time and of the state and control variables. Age at maturity We obtained the following values for age at maturity, Tmat, for different combinations of values of a and m:
a m 0.011 0.009 0.007 0.005 0.003 0.001 1.6 17 19 19 21 23 23 1.7 18 19 20 22 23 24 1.8 17 18 20 22 23 24 1.9 17 19 20 21 22 24 2.0 17 18 19 21 22 24 2.1 17 18 19 21 22 24

The corresponding dependency of Tmat on a and m is illustrated in Fig. 1. We see that the age at maturity increases with a decrease in environmental mortality (i.e. the part due to parasitic pressure) and it depends very weakly on resources.

Fig. 1. A linear approximation of modelled dependency of Tmat, the age at maturity, on a and m presented as a contour plot of Tmat on a and m. The levels of Tmat are shown by the different hatching and by the numbers on the right-hand side.


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Age at release We obtained the following values for age at release, Trel, for different combinations of values of a and m:

a m 0.011 0.009 0.007 0.005 0.003 0.001 1.6 4.5 4.5 4.5 7 6.5 10 1.7 4 4 4 6 7.5 8 1.8 4 4 6 5 5 9 1.9 4 4 5 5 7 8 2.0 4 4 5 5 6 8 2.1 4 5 5 5 6 8

The corresponding dependency of Trel on a and m is illustrated in Fig. 2. We see that age at release, like age at maturity, increases with a decrease in environmental mortality and it depends again very weakly on resource availability.

Fig. 2. The linear approximation of modelled dependency of Trel, the age at release, on a and m presented as a contour plot of Trel on a and m. The levels of Trel are shown by the different hatching and by the numbers on the right-hand side.


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Body size at maturity We obtained the following values of body size at maturity, Xmat, for different combinations of values of a and m:

a m 0.011 0.009 0.007 0.005 0.003 0.001 1.6 41 46 46 52 58 57 1.7 48 51 54 59 62 65 1.8 49 52 59 66 69 71 1.9 53 60 64 68 71 78 2.0 57 61 65 73 76 85 2.1 61 65 70 78 83 90

The corresponding dependency of Xmat on a and m is illustrated in Fig. 3. We see that body size at maturity increases as environmental mortality decreases and as resource availability increases.

Fig. 3. The linear approximation of modelled dependency of Xmat, the body size at maturity, on a and m presented as a contour plot of Xmat on a and m. The levels of Xmat are shown by the different hatching and by the numbers at the top and on the right-hand side.


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Body size at release We obtained the following values of body size at release, Xrel, for different combinations of values of a and m:

a m 0.011 0.009 0.007 0.005 0.003 0.001 1.6 13.8 12.4 13.3 20.4 19.5 32.8 1.7 13.3 13.4 13.6 20.8 21.1 32.6 1.8 14.7 14.3 22.1 18.8 19.1 34.6 1.9 15.2 15.6 19.8 20 28.4 33.2 2.0 16.2 16.5 20.9 21.4 26 35.6 2.1 17.3 22 22.3 22.9 27.8 37.8

The corresponding dependency of Xrel on a and m is illustrated in Fig. 4. We see that body size at release, like body size at maturity, increases with a decrease in environmental mortality and with an increase in resource availability in the environment.

Fig. 4. The linear approximation of modelled dependency of Xrel, the body size at release, on a and m presented as a contour plot of Xrel on a and m. The levels of Xrel are shown by the different hatching and by the numbers at the top and on the right-hand side.


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Number of children We obtained the following values of number of children for a woman, Nchi, for different combinations of values of a and m:

a m 0.011 0.009 0.007 0.005 0.003 0.001 1.6 6 6 6 4 4 2 1.7 7 7 7 4 4 3 1.8 7 7 4 5 5 2 1.9 8 7 5 5 3 3 2.0 8 7 6 5 4 3 2.1 8 6 6 5 4 3

The dependency of Nchi on a and m is illustrated in Fig. 5. We see that the number of children per woman decreases with a decrease in environmental mortality and it depends very weakly on resource availability.

Fig. 5. The linear approximation of modelled dependency of Nchi, the number of children, on a and m presented as a contour plot of Nchi on a and m. The levels of Nchi are shown by the different hatching and by the numbers on the right-hand side.


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Life expectancy We obtained the following values of life expectancy, Texp, for different combinations of values of a and m:

a m 0.011 0.009 0.007 0.005 0.003 0.001 1.6 52 56 62 68 74 82 1.7 52 57 62 68 75 83 1.8 53 57 62 68 75 83 1.9 53 58 63 69 76 84 2.0 54 58 64 70 77 85 2.1 54 59 64 70 77 85

The corresponding dependency of Texp on a and m is illustrated in Fig. 6. We see that life expectancy increases as environmental mortality decreases, while the relationship with resource availability is weak.

Fig. 6. The linear approximation of modelled dependency of Texp, the life expectancy, on a and m presented as a contour plot of Texp on a and m. The levels of Texp are shown by the different hatching and by the numbers on the right-hand side.


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RESULTS OF DATA ANALYSIS Our results for the relationships between the life-history traits under study two environmental factors (i.e. food and infection), which can be considered surrogates of resource availability and environmental mortality, are as follows of countries and areas are indicated on the figures by their Internet codes as Appendix 2). Female age at menarche The age at menarche Tmnr is one life-history trait that characterizes the age at maturity. The regression of Tmnr on `food' and `infection' is as follows: Tmnr = 16.304 - 0.981 â food - 0.0114 â infection (n = 65, R = 0.26, Pfood < 0.002, Pinfection > 0.83) This dependency is illustrated in Fig. 7. We see that female age at menarche is a decreasing function of food supply, but does not depend significantly on infection. and the as good (names given in

Fig. 7. A infection shown by Appendix

linear approximation of observed dependency of age at menarche, Tmnr, on food and presented as a contour plot of Tmnr on food and infection. The levels of Tmnr are the different hatching and by the numbers at the top. The country codes are given in 2.

Female age at marriage The female age at marriage, Tmrg, is another life-history trait that characterizes the female age at maturity because it is closely associated with the beginning of reproduction. The regression of Tmrg on `food' and `infection' is as follows:


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T

mrg

= 26.598 - 0.162 â food - 0.555 â infection
infection

(n = 90, R = 0.48, Pfood > 0.83, P

< 0.0001)

This dependency is illustrated in Fig. 8. We see that female age at marriage across countries increases as infection decreases, but does not depend significantly on food availability.

Fig. 8. A linear approximation of observed dependency of age at marriage, Tmrg, on food and infection presented as a contour plot of Tmrg on food and infection. The levels of Tmrg are shown by the different hatching and by the numbers on the right-hand side. The country codes are given in Appendix 2.

Female adult stature We use female adult stature, Xfem, to characterize body size at maturity because stature is highly correlated with body weight. The linear regression of this variable on `food' and `infection' is as follows: Xfem = 135.304 + 7.440 â food + 0.366 â infection (n = 44, R = 0.64, Pfood < 0.00005, Pinfection > 0.18) This dependency is illustrated in Fig. 9. We see that female adult stature increases with an increase in food availability, but does not depend significantly on infection.


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Fig. 9. A linear approximation of observed dependency of adult female stature, Xfem, on food and infection presented as a contour plot of Xfem on food and infection. The levels of Xfem are shown by the different hatching and by the numbers at the top. The country codes are given in Appendix 2.

Weight at birth The regression of weight at birth, Xbth, on food and infection is as follows: Xbth = 2.631 + 0.219 â food - 0.0116 â infection (n = 108, R = 0.73, Pfood < 0.00001, Pinfection < 0.04) This dependency is illustrated in Fig. 10. We see that weight at birth increases significantly with a decrease in infection and with an increase in food. Fertility The regression of female fertility, Nfrt, measured as the mean lifetime number of children for a woman, on food and infection is as follows: Nfrt = 5.982 - 1.378 â food + 0.298 â infection (n = 130, R = 0.76, Pfood < 0.00001, Pinfection < 0.000001) This dependency is illustrated in Fig. 11. We see that human fertility across countries is a decreasing function of both parasitic pressure and food availability in the environment.


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Fig. 10. A linear approximation of observed dependency of weight at birth, Xbth, on food and infection presented as a contour plot of Xbth on food and infection. The levels of Xbth are shown by the different hatching and by the numbers at the top and on the right-hand side. The country codes are given in Appendix 2.

Fig. 11. A linear approximation of observed dependency of female fertility, Nfrt, on food and infection presented as a contour plot of Nfrt on food and infection. The levels of Nfrt are shown by the different hatching and by the numbers at the top and on the right-hand side. The country codes are given in Appendix 2.


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Female life expectancy The regression of female life expectancy at 1 year old, Tfem, on food and infection is as follows: Tfem = 54.721 + 8.521 â food - 1.361 â infection (n = 127, R = 0.80, Pfood < 0.000001, Pinfection < 0.000001) This dependency is illustrated in Fig. 12. We see that female life expectancy increases as parasitic pressure decreases and as food supply increases.

Fig. 12. A linear approximation of observed dependency of female life expectancy, Tfem, on food and infection presented as a contour plot of Tfem on food and infection. The levels of Tfem are shown by the different hatching and by the numbers at the top and on the right-hand side. The country codes are given in Appendix 2.

DISCUSSION In this study, we have built a dynamic model of human female body growth and reproduction based on evolutionary optimization of the strategy of partitioning an individual's resources between the needs for growth, reproduction and survival. We have tried to make this model as simple as possible because analysing results obtained with complex models is usually cumbersome. We disregarded some aspects of the evolutionary optimization of human life histories considered elsewhere, as some previous analyses have suggested that their role would be unimportant in the present analysis. In particular, we did not take into account the investment of resources into repair. The effect of such investment was considered in detail in Teriokhin (1998). In addition, the evolutionary optimization of human sexual dimorphism in body size and in life expectancy were investigated by GuÈgan et al. (2000) and Teriokhin et al. (2000), respectively. Here


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we restricted our analysis to the consideration of only female life histories. We did not optimize the age at menopause in females as this was done by Teriokhin and Budilova (2000). We simply took its average value of 50 years. In accordance with our data, the age at menopause does not depend significantly on food and infection, and in any case its worldwide variation of the order of 1 or 2 years is much less than that predicted by optimization models (Teriokhin and Budilova, 2000; Thomas et al., 2001). We were obliged, however, to complicate the model by allowing the accumulation of reproductive energy to make possible the modelling of the dependency of female fertility on environmental factors. In this study, we used two model variables associated with age at maturity. These were age at maturity, Tmat (i.e. the age at the beginning of reproductive accumulation), and age at release, Trel (i.e. the duration of parental care). Both of these variables depend strongly on environmental mortality (e.g. parasitic pressure), m, and very weakly on resource availability, a. In addition, both increase when environmental mortality decreases. There are also two empirical characteristics of age at maturity. These are the age at menarche, Tmnr, and the age at marriage, Tmrg. The first characteristic increases with decreasing food availability, and the second increases with decreasing infection ­ that is, it is the age at marriage which matches the optimal dependency predicted by the model. We may thus suppose that this rather cultural rather than biological characteristic better reflects the optimal age of maturity. The importance of cultural aspects in human evolution has been discussed in particular by Laland et al. (2001). The two model variables associated with female body size at maturity ­ that is, body size at maturity, Xmat (i.e. the body size at the beginning of reproductive accumulation), and body size at release, Xrel ­ both increase with an increase in resources and a decrease in environmental mortality. The two empirical characteristics of age at maturity, female stature and weight at birth, decrease with decreasing food supply too, but only weight at birth decreases significantly with an increase in infection. Female stature was observed not to decrease with increasing infection (there was even some increase in stature with increasing infection in our analysis, although not significantly so), though it is predicted by our simple optimization model. This phenomenon was considered in detail by GuÈgan et al. (2000), who conjectured that some increase in female stature with increasing fertility (which increases with increasing environmental mortality due to parasitic pressure) might be an adaptation for mitigating high risks of maternal mortality caused by an insufficient female body size. The number of children, Nchi, in accordance with the optimal model should increase with increasing environmental mortality, and we also observed a similar trend for empirical fertility, Nfrt, with increasing infection. Fertility varies significantly throughout the world with a tendency to decrease with increasing wealth. This phenomenon, known as demographic transition, has been investigated using different approaches (see Mason, 1997; Borgerhoff Mulder, 1998; Mace, 1998, 2000). In our model, the effect of a decrease in the number of children with an amelioration of environmental conditions (see Fig. 5) emerges as a consequence of the maximization of lifetime reproductive output under the assumption of non-linear dependence of reproductive output on reproductive investment. Observed fertility, Nfrt, increases significantly not only with increasing environmental mortality but also with decreasing food supply, whereas this is not the case for Nchi, the model characteristic of fertility. One possible explanation is that the variable `food', which is highly correlated with the general level of development of a country, contains some


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information on mortality additional to that contained in the variable `infection' (e.g. the level of medical care); Comparing the tendency for increasing Nchi with the tendency for decreasing Trel and Xrel in the model, we conclude that the increase in the number of children is accompanied by a decrease in the duration of parental care and body size of offspring at release. The modelled life expectancy, Texp, increases with increasing environmental mortality but it depends very weakly on resource availability. Similarly, the empirical life expectancy, Tfem, increases significantly with decreasing infection, but it also increases with increasing food. One explanation for this disparity in results obtained from modelling and statistical analysis is the same as for fertility: the variable `food' may also contain some information about mortality. A more detailed statistical analysis of the influence of infection on human life expectancy has been conducted by GuÈgan and Teriokhin (2000). CONCLUSION We conclude that the empirical dependencies of some human life-history traits on environmental conditions roughly coincide with the predictions obtained from optimization models, in the sense that there is no empirical dependency that is opposite to that predicted by modelling. It would appear that the observed dependencies of human life-history traits on environmental conditions are generated, at least partly, by the process of evolutionary adaptation. In addition, this optimization can be ensured by means that are cultural (e.g. by changing the age at marriage) rather than biological (e.g. by changing the body size or the birth weight). The general feature of the discrepancies between the data analysis and modelling is that the empirically observed dependencies of life-history traits on the availability of food supply predicted by the model are worse than the dependencies on environmental mortality. So an additional study is necessary to clear up the source of these discrepancies: whether they are due to inadequacies of the model or to disparities in the matching of modelled and observed variables. ACKNOWLEDGEMENTS
The authors are grateful to P. Taylor for helpful comments. The work was supported by a CNRS senior research fellowship for A.T. and by grants from EGIDE (2001) to E.B. to visit CEPM and RFBR (01-04-48384) to E.B. and A.T.

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Borgerhoff Mulder, M. 1998. The demographic transition: are we any closer to an evolutionary explanation. TREE, 13: 266­270. Eveleth, P.B. and Tanner, J.M. 1976. Worldwide Variation in Human Growth. Cambridge: Cambridge University Press. GuÈgan, J.F. and Teriokhin, A.T. 2000. Human life-history traits on a parasitic landscape. In Evolutionary Biology of Host­Parasite Relationships: Theory Meets Reality (R. Poulin, S. Morand and A. Skorping, eds), pp. 143­161. Amsterdam: Elsevier. GuÈgan, J.F., Teriokhin, A.T. and Thomas, F. 2000. Human fecundity variation, size-related obstetrical performance and the evolution of sexual stature dimorphism. Proc. R. Soc. Lond. B, 267: 2529­2536.


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GuÈgan, J.F., Thomas, F., Hochberg, M.E., de MeeØs, T. and Renaud, F. 2001. Disease diversity and human fertility. Evolution, 55: 1308­1314. Hamilton, W.D. 1966. The moulding of senescence by natural selection. J. Theor. Biol., 12: 12­45. Kozlowski, J. and Teriokhin, A.T. 1999. Energy allocation between growth and reproduction: Pontryagin maximum principle solution for the case of age- and season-dependent mortality. Evol. Ecol. Res., 1: 423­441. Laland, K.N., Odling-Smee, J. and Feldman M.W. 2001. Cultural niche construction and human evolution. J. Evol. Biol., 14: 22­33. Mace, R. 1998. The coevolution of human fertility and wealth inheritance strategies. Phil. Trans. R. Soc. Lond. B, 353: 389­397. Mace, R. 2000. Evolutionary ecology of human life history. Anim. Behav., 59: 1­10. Mangel, M. and Clark, C. 1988. Dynamical Modelling in Behavioural Ecology. Princeton, NJ: Princeton University Press. Mason, K.O. 1997. Explaining fertility transitions. Demography, 34: 443­454. McNamara, J.M. and Houston, A.I. 1996. State-dependent life histories. Nature, 380: 215­221. Mylius, S.D. and Diekmann, O. 1995. On evolutionary stable life histories, optimisation and the need to be specific about density dependence. Oikos, 74: 218­224. Perrin, N. and Sibly, R.M. 1993. Dynamic models of energy allocation and investment. Annu. Rev. Ecol. Syst., 24: 379­410. Sokal, R.R. and Rohlf, F.J. 1994. Biometry: The Principles and Practice of Statistics in Biological Research, 3rd edn. New York: Freeman. Stearns, S.C. 1992. The Evolution of Life Histories. Oxford: Oxford University Press. Teriokhin, A.T. 1998. Evolutionarily optimal age schedule of repair: computer modeling of energy allocation between current and future survival and reproduction. Evol. Ecol., 12: 291­307. Teriokhin, A.T. 2002. Models of Competition: Population Dynamics and Phenotype Evolution (in Russian). Moscow: MaxiPress. Teriokhin, A.T. and Budilova, E.V. 2000. Evolutionarily optimal networks for controlling energy allocation to growth, reproduction and repair in men and women. In Artificial Neural Networks: Application to Ecology and Evolution (S. Lek and J.F. GuÈgan, eds), pp. 225­237. Berlin: Springer-Verlag. Teriokhin, A.T. and Budilova, E.V. 2001. Evolution of life history: models based on optimization of energy allocation (in Russian). J. Gen. Biol., 62: 286­295. Teriokhin, A.T., Budilova, E.V., GuÈgan, J.F. and Thomas, F. 2000. Human sexual lifespan dimorphism: an evolutionary optimization view. In 2nd European Congress on Biogerontology: From Molecules to Humans, St Petersburg, 2000. Adv. Gerontol., 5: 84­85. Thomas, F., Teriokhin, A.T., Renaud, F., de MeeØs, T. and GuÈgan, J.-F. 2000. Human longevity at the cost of reproductive success: evidence from global data. J. Evol. Biol., 13: 409­414. Thomas, F., Renaud, F., BÈnÈfice, E., de MeeØs, T. and GuÈgan, J.-F. 2001. International variability of ages at menarche and menopause: patterns and main determinants. Hum. Biol., 73: 271­290. West, G.B., Brown, J.H. and Enquist, B.J. 1999. The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science, 284: 1677­1679.


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APPENDIX 1
An example of optimal life-history traits values (for a = 1.7 and m = 0.9)
t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 u v wt 67 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z Et 2.3 2.5 2.7 2.8 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.8 3.9 4.0 4.0 4.1 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 Lt 0.99 0.98 0.96 0.95 0.94 0.93 0.92 0.91 0.91 0.90 0.89 0.88 0.87 0.86 0.86 0.85 0.84 0.83 0.82 0.82 0.81 0.80 0.79 0.78 0.78 0.77 0.76 0.75 0.74 0.74 0.73 0.72 0.71 0.71 0.70 0.69 0.68 0.68 0.67 0.66 0.66 0.65 0.64 0.64 0.63 0.62 0.62 0.61 0.60 X Y

t

t

t

t+1

t+1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80

33 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

no no no no no no no no no no no no no no no no no no no no no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no

4.5 6.4 8.3 10.4 12.6 14.9 17.3 19.7 22.2 24.8 27.4 30.1 32.9 35.7 38.5 41.5 44.4 47.4 50.5 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4


Evolutionary optimization model of human life history
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 93 93 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 7 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no no no yes no yes no yes 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 0.60 0.59 0.58 0.58 0.57 0.57 0.56 0.55 0.55 0.54 0.54 0.53 0.53 0.52 0.52 0.51 0.51 0.50 0.49 0.49 0.48 0.48 0.47 0.47 0.46 0.46 0.45 0.45 0.45 0.44 0.44 0.43 0.42 0.42 0.41 0.41 0.40 0.39 0.39 0.38 0.38 0.37 0.37 0.36 0.35 0.35 0.34 0.34 0.33 0.32 0.31 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6 53.6

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6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.4 6.8 10.2 13.6 3.7 7.4 11.0 14.7 3.7 7.4 11.0 14.7 3.7 7.4 11.0 14.7 3.7 7.4 11.0 14.7 3.7 7.4 4.0 7.9


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APPENDIX 2
Two-letter codes for countries and autonomous areas (Internet codes are used)

Code ae ag an ar at au bb bd bf bg bi bj bn bo br bw bz ca cf cg ch ci cl cm cn co cr cu cz dm do dz ec eg es fi fj fr gd gf gh gm gn gp gr gt gy

Country United Arab Emirates Antigua & Barbuda Antilles (NL) Argentina Austria Australia Barbados Bangladesh Burkina Faso Bulgaria Burundi Benin Brunei Bolivia Brazil Botswana Belize Canada Central African Rep Congo Brazz Switzerland Cote d'Ivoire Chile Cameroon China Colombia Costa Rica Cuba Czech Rep Dominica Dominican Rep Algeria Ecuador Egypt Spain Finland Fiji France Grenada Guyana (FR) Ghana Gambia Guinea Guadeloupe (FR) Greece Guatemala Guyana

Code hn ht id ie il in is it jm jp ke ki kn kp kr kw lc lk lr ls ly ma mg ml mm mn mq mr mw mx my mz nc ne ng nl no np nz pa pe pf pg ph pk pl pt

Country Honduras Haiti Indonesia Ireland Israel India Iceland Italy Jamaica Japan Kenya Kiribati St Kitts & Nevis North Korea South Korea Kuwait St Lucia Sri Lanka Liberia Lesotho Libya Morocco Madagascar Mali Myanmar Mongolia Martinique (FR) Mauritania Malawi Mexico Malaysia Mozambique New Caledonia (FR) Niger Nigeria Netherlands Norway Nepal New Zealand Panama Peru Polynesia (FR) Papua New Guinea Philippines Pakistan Poland Portugal


Evolutionary optimization model of human life history
py ro ru rw sa sb sd se sg sl sn so sr st sy sz tg th tn Paraguay Romania Russia Rwanda Saudi Arabia Solomon Is Sudan Sweden Singapore Sierra Leone Senegal Somalia Suriname Sao Tome & Principe Syria Swaziland Togo Thailand Tunisia to tr tt tz ug uk us uy vc ve vu ws ye yu za zm zr zw Tonga Turkey Trinidad & Tobago Tanzania Uganda United Kingdom United States Uruguay St Vincent & Grenadines Venezuela Vanuatu Western Samoa Yemen Yugoslavia South Africa Zambia Congo Dem Rep Zimbabwe

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