Models of population dynamics generally assume that child survival is independent of maternal survival. However, in humans, the death of a mother compromises her immature children's survival because children require postnatal care. A child's survival therefore depends on her mother's survival in years following her birth. Here, we provide a model incorporating this relationship and providing the number of children surviving until maturity achieved by females at each age. Using estimates of the effect that a mother's death has on her child's survival until maturity, we explore the effect of the model on population dynamics. Compared to a model that includes a uniform child survival probability, our model slightly raises the finite rate of increase λ and modifies generation time and the stable age structure. We also provide estimates of selection on alleles that change the survival of females. Selection is higher at all adult ages in our model and remains significant after menopause (at ages for which the usual models predict neutrality of such alleles). Finally, the effect of secondary caregivers who compensate maternal care after the death of a mother is also emphasized. We show that allocare (as an alternative to maternal care) can have a major effect on population dynamics and is likely to have played an important role during human evolution.
Reproduction requires survival until sexual maturity. Survival until maturity therefore considerably impacts population dynamics and is a major component of fitness. It is a key variable in most models in demography, ecology, and population genetics. Classically, survival until maturity is expressed as a rate or a probability estimated over all individuals of the population. It is an average, independent of the survival of parents during the individuals' juvenile period. However, in many species, offspring survival requires parental care (Clutton-Brock 1991). In humans, females invest energy and time after childbirth into the growth of their offspring (by breast-feeding and foraging) and also into affection, protection, and education. Maternal care is a major determinant of child survival (Brittain 1992; Andersson et al. 1996; Sear et al. 2002; Pavard et al. 2005). Because a mother has to be alive to care for her children, her children's survival depends on the survival of their mother in years following their birth. Thus, each time a female gives birth, she implicitly bets on surviving to rear her child. The child, in turn, bets on the survival of her mother to get maximum care. We therefore refer to this phenomenon as “the bet on mother's survival.”
Several studies have integrated the relationship between a child's survival and that of her mother into an evolutionary perspective. In his seminal paper, Lee (2003) modeled the balance between individuals' production and consumption of resources over all ages. The positive balance of adults frees the resources required by children to survive. The author focused on the effect this production-consumption balance had on population dynamics. The relationship between mother and child survival has also been considered in studies aiming to test an adaptive hypothesis of menopause (Williams 1957). This hypothesis assumes that the cost of reproduction increases with a female's age because late reproduction is associated with a rising risk of death during pregnancy, which in turn compromises the survival of females' already-born but still immature children. A sudden cessation of reproduction might therefore become advantageous because the benefit of continued reproduction is overcome by its cost. Several authors have modeled this effect for late and postreproductive life (Rogers 1993; Peccei 1995; Moss de Oliveira et al. 1999; Shanley and Kirkwood 2001; Sousa 2003). Here we propose a new approach integrating the relationship between a mother's survival and that of her daughters in a demographic framework. This approach provides us with the means to calculate the number of daughters reaching maturity achieved by females at any age, knowing that this number includes the production of new daughters and the care of these daughters throughout their childhood. We explore the implication of integrating this relationship in a demographic, genetic, and evolutionary context.
If we assume that all living mothers care equally for their daughters (i.e., there is no good or bad mother and the care provided by mothers is independent of their age), a given daughter's survival until maturity depends on two components: first, the extent to which maternal care is needed for her to survive until sexual maturity (or, conversely, the extent to which the loss of maternal care following the death of her mother compromises her survival); second, her mother's survival probability during the years following the daughter's birth (or, conversely, her mother's risk of death in these years).
In the first part of this paper, we provide an estimate of the probability for daughters to survive until maturity if their mother dies at any time during their childhood or survives through their whole childhood (i.e., the first component of a daughter's survival). This estimate is based on results from Pavard et al. (2005) for a preindustrial population. Results show death is not inevitable even for motherless children at birth because maternal care is compensated by caregivers other than the mother. The care provided by these secondary caregivers is called allocare. A second function of the effect of mother's death on child's survival until maturity is then modeled for a population with a lower level of allocare to emphasize its specific effect.
In the second part, we present a model that incorporates these estimates at the level of a female's reproductive history. This model allows us to calculate the mean number of daughters surviving at maturity achieved by females at each age. Hereafter this model is called the bet on mother's survival model (BOMS model). Because humans senesce, females' mortality increases with age. The daughters' risk of losing their mother before reaching maturity therefore augments with their mothers' age at their birth: the survival of daughters until maturity is not constant but decreases as a function of mothers' age at childbirth. We explore how this affects population dynamics (i.e., finite rate of increase λ, age-class distribution, and generation time) and discuss the consequences of these impacts in an evolutionary context.
The BOMS model is also extended to the subpopulations of females surviving at (or dying by) a given age (instead of the population average that takes into account females who die and survive at each age). This allows us to calculate the number of daughters surviving to maturity that a female gains by caring for them up to this age. This extended model also allows us to estimate selection exerted on an allele that modifies a female's survival at a given age. We study this effect with particular emphasis on postreproductive females.
The effect of mother's death on child survival is estimated for the population of ancient Quebec (from Pavard et al. 2005) using the population file maintained by the Programme de Recherche en Démographie Historique (PRDH, see http://www.genealogy.umontreal.ca). Present applications use fertility and life tables estimated for this population. The ancient Quebec population comprises the French Canadians who lived in the St. Lawrence Valley from the very beginning of the settlement (1608) to the middle of the 18th century (Légaré 1988). This preindustrial population shows a very high natural fertility rate in the absence of modern contraception and medical care.
The Measure: Effect of Mother's Death on Child Survival
We are primarily interested in estimating the first component of a daughter's survival, which is the extent to which her mother's death impacts her survival.
ANCIENT QUEBEC POPULATION
Pavard et al. (2005) used a large database on the population of ancient Quebec to study the children's relative risk of dying specifically due to the loss of maternal care following the death of the mother. These risks were estimated for fives classes of children grouped according to their age at their mother's death: neonatal, postneonatal, toddler, early childhood, and late childhood. The results provide evidence of higher risk of dying for motherless children starting from their mother's death and continuing long after. These relative risks also decrease with child's age at mother's death. Using these estimates and the hazard function between births to age 15 of children whose mother survived through the period, we calculated the 0–15 children's survival according to their age at mother's death. Let us denote this function wy, with y the child's age at mother's death. Results were fitted from age 0 to maturity to a logarithmic wy=a Ln (y+ 1) +b (with a= 0.0787 and b= 0.4962; see Fig. 1). After maturity (hereafter the age M= 15), daughters' survival is assumed to be independent of their mother's survival status, so wy=wM for y≥M. The function wy continually climbs with y and tends to reach the maximum wM at age M. This maximum wM is the survival until maturity of children whose mothers remain alive during their whole childhood. These children receive therefore the maximum maternal care and wM is called the fully cared for survival probability (FCSP). In the population of ancient Quebec, the FCSP is around 0.71. By contrast, the survival to maturity of a motherless child at birth w0 is the survival of a child receiving no maternal care. The difference between these two values (=wM−w0) is the net benefit in survival of a child due to her mother's survival. Because fathers, relatives, and even the community as a whole can also be care-providers and because nonmaternal breast-feeding and adoption are frequent in human populations, the probability of reaching age M of children whose mother died soon after their birth is not null (by contrast, this does not apply to other species with high parental investment). This phenomenon is mainly the product of human socialization that emphasizes the role of caregivers other than the mother.
POPULATION WITH A LOWER LEVEL OF ALLOCARE
To analyze the role played by nonmaternal care during human evolution, a new function w*y is modeled for a population with a lower level of allocare. This function assumes that motherless children at birth are certain to die because maternal care is not compensated by secondary caregivers. Let us also assume proportionality between w*y and wy. Because it is expected that allocare, as maternal care, decreases with age y, we also assume that a child fully cared for presents the same survival until maturity in the new function w*y. Then w*y is defined as w*0= 0 and w*M=wM. The relationship between wy and w*y is expressed as . Here, we arbitrarily choose a value α that equals 0.5 (see Fig. 1). Note that the net benefit in survival of a child due to a mother's survival (i.e., w*M−w*0) is higher in the case of the new function.
The Model for the Bet on the Mother's Survival
Incorporating the fact that the survival or the death of the mother impacts her daughters' survival, we aim to calculate the mean number of daughters surviving until maturity achieved by females from one age to the next. More precisely, we aim to calculate this age trajectory (1) on average at the population level, for females surviving at each age x; and (2) among these females, for the subpopulations of those who die by, or survive up to, age x+ 1.
Let us first define the notation. For simplicity, we use a discrete framework. For single-year age-classes in humans this does not introduce significant error and simplifies calculations. M is the age at sexual maturity shared by all individuals; px and qx are, respectively, the probability of surviving and dying between age x and x+ 1 of survivors at age x; and sx is the probability of surviving from M to age x of mature females (sx is therefore defined for x∈[M, ∞] and with sM= 1). We denote mx as the age-specific fertility, that is, the average number of daughters born between ages x and x+ 1 to a female surviving at age x.
AT THE POPULATION LEVEL
As only daughters surviving to maturity will reproduce and contribute to the next generation we denote as “effective” a daughter who survives to age M (Austerlitz and Heyer 1998). At the population level, we call effective age-specific fertility, denoted as , the trajectory we are seeking to define: the average number of daughters surviving until maturity achieved by females at each age x. This age-specific effective fertility is the product between the average number of daughters born to a female at each age x (i.e., mx) and these daughters' survival probability until maturity according to the age of their mother when the daughters are born x (hereafter denoted ux), such that . As discussed above, the probability ux depends on two components: (1) the effect of mother's death on daughters' survival until maturity wy (extensively described above), and (2) the mother's risk of death during years following childbirth. Let us calculate ux. To simplify the model, we consider that the mx daughters born between x and x+ 1 to females surviving at age x are all born to females at exactly age x: the mother's survival probability during years following childbirth starts therefore precisely with her survival probability between x and x+ 1 (see Fig. 2). The probability that a daughter loses her mother at age y given her mother's age at the daughter's birth x is therefore the probability that the mother dies at age x + y given that she is alive at age x, (sx+y/sx)qx+y. The probability ux is then equal to the sum over all the daughter's ages y (from 0 to ∞) of the following product: the daughter's probability to survive until maturity according to her age at mother's death wy times the probability (sx+y/sx)qx+y for the mother to die when the daughter reaches this age y. Thus ux can be calculated as follows:
With ux, we can now calculate the age-specific effective fertility as well as the average number of daughters surviving until maturity born to adult females during their lifetime (i.e., the net reproductive rate, ). To compare results from our model to those from models in which daughters' survival is independent of their mothers' survival, we need to calculate the average daughters' survival until maturity . The probability is the proportion of daughters surviving at maturity independently of the survival status of their mothers during their childhood and is given by the ratio of R0 to the average number of daughters born during the lifetime (). Hereafter, we call the reference model all calculations integrating the probability .
To analyze the effect of the BOMS model on population dynamics, we are now interested in calculating the finite rate of increase λ. The λ is defined as the greatest root of the discrete Euler-Lotka equation in the case in which the first age-class is zero (Goodman 1982; see eq. 2a, where lx is female survival probability from birth to age x). Because is the mean females' survival probability from birth to age M and sx is the female survival probability from M to age x, then lx=sx for x > M. From the point of view of population dynamics, it is equivalent to consider all females and their progeny, or only the females who survive at age M and their effective progeny. We can therefore define λREF and λBOMS as the finite rate of increase calculated for the reference model and the BOMS model, respectively (see eqs. 2b and 2c).
If ux varies substantially with x, then the product lxmxux differs from lxmx and so does λBOMS from λREF.
AT THE SUBPOPULATION LEVEL
Consider females surviving at age x. On average, these females achieve effective daughters between x and x+ 1, which is a population average incorporating both females who survive and die between x and x+ 1. However, as a female's death compromises her already born daughters' chances of surviving, a female who dies achieves an average number of daughters between x and x+ 1 lower than the population average . By contrast, a female who survives up to age x+ 1 achieves a higher number of effective daughters than . Therefore a positive (or a negative) deviation in the number of effective daughters achieved by females who survive (or die) between x and x+ 1 is expected compared to the population average . We call effective gain and effective loss the deviations relative to of females who, respectively, survive up to x+ 1 or die by x+ 1. Females who survive to x+ 1 achieve therefore effective daughters and those who die during the interval achieve effective daughters (see Fig. 2). As survivors at age x survive up to x+ 1 with a probability px and die by x+ 1 with a probability qx, then . This implies that equals . Thus, at any age, the ratio of the age-specific effective loss to the age-specific effective gain is equal to the ratio px/qx, which is independent of how maternal care and the daughter's survival are linked. Because the survival or the death of a female at a given age x modifies the survival of all her currently immature daughters, the calculations of the quantities and are more complicated than that of . They are detailed in Appendix 1. These two new functions provide us with the means to calculate the average number of effective daughters achieved during her lifetime for any particular female that survives or dies at a given age. This also allows us to calculate the coefficient of selection associated with an allele leading to the death of a female at a given age (see Appendix 2).
In the population of ancient Quebec, the FCSP value equals 0.71 and the probability of being effective of a motherless daughter at birth is equal to 0.49. Therefore, the net benefit in survival for a daughter owing to her mother's survival is around 0.22. In our model, the daughters' probabilities of being effective ux decreases with the females' age at childbirth x. For example, the probability of being effective for a daughter born to a 15-year-old mother u15 equals 0.7, whereas u30= 0.69 and u45= 0.66.
The mean lifetime number of daughters of a female equals 4.89 in our data set. If all daughters are fully taken care of (i.e., age-specific effective fertility equals mxFCSP), then R0 would be equal to 3.47 (= 4.89 × 0.71) and generation time (i.e., the mean age of the mother of the daughters; Coale 1972) equal to 26.59. In our BOMS model (i.e., age-specific effective fertility ), R0 equals 3.37 and the generation time is equal to 26.53. This result corresponds to a mean daughters' survival until maturity equals 0.69.
Figure 3 shows the effective fertility conditional on survival at each age x, ; the effective gain conditional on survival up to age x+ 1, ; and the effective loss conditional on death by age x+ 1, . As predicted by the properties of the model, and are symmetric around the x-axis. The maximum of occurs at age 23 whereas it occurs at age 32 for , which is nearly 10 years later. This means that the maximum age-specific gain (or loss) due to a female's survival (or death) occurs at age 32. Although decreases to zero along with fertility, decreases to zero later (at around age 55) and thus after the total cessation of reproduction (see Fig. 3). This means that females continue to achieve effective daughters after menopause and that a female who dies after menopause still loses effective daughters. For example, the number of effective daughters of a female who dies at age 50 equals 3.98 () whereas the number of effective daughters of a female who survives at age 65 equals 4.00 (). In other words, females can gain 0.02 effective daughters by surviving 15 years after the total cessation of reproduction, which counts for around 1% of the lifetime reproductive success of such a female.
COMPARISON WITH THE REFERENCE MODEL
To analyze the demographic impact of the difference in the age trajectories of effective fertility between our BOMS model (mxux) and the reference model (), their age-specific differences conditional on survival () are shown in Figure 4. Daughters born to females younger than 28 years old have a higher probability of survival until maturity ux than the population average . The contribution of early effective fertility to R0 is therefore higher in the BOMS model than in the reference model (). The opposite is observed for daughters born to females older than 28 years old. Although all daughters contribute equally to R0 in the reference model, early effective fertility makes a larger contribution to R0 than late fertility in our BOMS model. Females therefore achieve effective daughters earlier in our model than in the reference model.
This impacts the finite rate of increase λ. As λ is the rate of population increase per unit of absolute time, rather than per generation time as for R0, early effective births have a higher weight on λ than late effective births. λ is therefore superior in the BOMS models than in the reference models (equals 1.0494 and 1.0493, respectively). λ may therefore be underestimated in species with a high level of parental investment when the bet on mother's survival is not taken into account. Consequently, generation time and age structure may also be misestimated. However, the difference in λ values calculated for our model and the reference model is very small (less than one part in 10,000). In our example (positive λ, i.e., increasing population size), generation times are similar (26.5 and 26.6, respectively) and, at the stable age-class distribution, the number of immature individuals is raised by only 0.01% to the detriment of the adult age-classes.
POPULATION WITH A LOWER LEVEL OF ALLOCARE
In this model, R0 equals 3.21 and the mean 0 to M-survival probability equals 0.66. The contribution to R0 of early fertility (before age 28) is 1.29% higher in the BOMS model compared to the reference model. The difference in λ between the two models remains small (less than one part in 2000, with λ= 1.0473 and 1.0469 for the BOMS and the reference model, respectively), but the number of immature females is around 1.04% higher in the BOMS model. The generation time is slightly decreased (26.6 in the BOMS model and 26.4 in the reference model).
SELECTION COEFFICIENTS ON A MOTHER DEATH
Selection coefficients σz associated with an allele that leads to the death of a female carrier at a given age z are estimated in the cases of the two models (σBOMSz and σREFz) and for the two functions wy (ancient Quebec) and w*y (lower level of allocare). Selection coefficients sz are calculated using the equations detailed in Appendix 2. Results are shown in Figure 5. For the BOMS model, the resulting selection coefficients are higher than in the reference model for all ages z, and not null the after menopause. Selection coefficients are also higher in the case of the population with a lower level of allocare: the more straightforward the relationship between the survival of the daughter and that of her mother, the more the allele is counterselected.
THE BENEFIT OF MATERNAL CARE
In the ancient Quebec population, the difference in survival from birth to maturity of a motherless daughter at birth and a daughter fully taken care of (FCSP −w0) is equal to 0.22. Thus, the mother's survival is a major determinant of her daughter's survival. When we consider a population without maternal care in which the daughters' probability of being effective is similar to daughters who are motherless at birth (i.e., w0= 0.49), then the corresponding net reproductive rate R0 equals 2.4 (= 4.89 × 0.49). Because R0 calculated in the case of the BOMS model equals 3.37, postnatal maternal care count for around 30% of R0 ([3.37 − 2.4]/3.37 × 100). This effect is far from being negligible but is likely to be less important in humans due to allocare than in other mammal species in which no daughters whose mother dies at birth can survive.
THE COST OF MATERNAL CARE
Because maternal care is conditional on the mother's survival and because all mothers do not survive during the whole rearing time of their daughters, the mean daughter's survival until maturity of a daughter fully cared for (FCSP = 0.71) is not equal to that of an “average” daughter in the population (= 0.69). By comparison with a population in which all daughters would be fully taken care of, the bet on the mother's survival decreases R0 by nearly 3% ([0.71 – 0.69)/0.71 × 100]). Because mothers die, maternal care has a direct cost. The more maternal care is needed for daughters to survive or/and the higher the risk for daughters to not be fully cared for, the higher this cost. In humans this cost is likely to be smaller than in other species due to allocare on the one hand and because most daughters are fully cared for on the other hand. Indeed, the combined effects of the low adult mortality observed in humans and of the menopause (which stops fertility before maternal mortality dramatically increases the mother's risk of death) lead to a low probability for mothers of dying during the rearing time.
By comparison with the reference model in which the daughters' survival is independent of their mothers' survival, the bet on mother's survival increases the contribution to R0 of early effective fertility (before age 28 in our example) to the detriment of later effective fertility. Females therefore achieve effective daughters earlier in our model. Young age-classes therefore contribute more to λ in the BOMS model than in the reference model. In the case of a growing population this leads to a higher λ, a decrease of generation time, and an increase in the frequency of immature individuals. However, this effect is quantitatively small in the case of the population of ancient Quebec. As discussed above, the small magnitude is due to allocare and to the low risk for a mother of dying during her daughters' rearing time in humans. It can also be due to specific characteristics of the population of ancient Quebec. Resource abundance provided mothers with an especially high survival for a preindustrial population and may also have lowered the impact of the loss of a mother on the survival prospects of her daughters. Moreover, families were large and this was likely to lead to a high level of allocare. It would be therefore of interest to study the bet on mother's survival in the case of other human populations to fully address its effect on demography. Finally, in our model we consider only the effect of mother's death on daughters' survival. However, a female's survival may also enhance her daughters' fertility as well as her grandchildren's survival (Hawkes 2003). This might be integrated in our model in future studies.
These results can be important in evolutionary ecology at several levels. First, consider models focusing on the effect of intergenerational transfers on population dynamics. For example, Lee (2003) considers the age-specific production-consumption index of individuals that determines their reproduction and survival in a density-dependent model. It would be interesting to extend Lee's model in cases in which the demographic benefit of a transfer receiver at a given age (i.e., with a negative index) is provided by a transfer donor (i.e., with a positive index) surviving at a specific age. One example would be when parents' survival at a given age is needed for their children to survive until maturity. In optimization theory and perturbation analysis, the bet on mother survival can also be of importance. To put it simply, most optimization models consider the trade-off between benefit of parental care in terms of increasing offspring fitness and its cost addressed in terms of future reproduction and survival (see Clutton-Brock 1991; Gross 2005). However, our model shows that one must take into account parents' survival as a condition of parental care during the whole rearing period to accurately estimate offspring fitness. This can be particularly important in species with high level of parental investment and nonnegligible parental mortality related to the juvenile period length. Finally, perturbation analyses are commonly used to analyze selection pressures on reproduction and survival by estimating elasticities and sensitivities of λ to a relative or absolute change in an age-specific vital rate (extensively studied by Caswell 2001). As our model changes the relative contribution to λ of a given age-class, measures of elasticities are expected to change as well. Further models extend a matrix approach to incorporate the bet on mother's survival (Pavard, unpubl. data).
Here we calculated selection coefficients σBOMSz and σREFz associated with an allele leading to the death of its carrier at age z for the BOMS model and the reference model. Because the mean number of daughters of a female at age x is underestimated in the reference model, σBOMSz is always superior to σREFz. At young ages, this difference between σBOMSz and σREFz is negligible compared to the intensity of selection involved; but at older ages, the difference becomes important. In the reference model, the selection coefficients drop to zero as age-specific fertility tends to 0 (σREF50= 0). For the BOMS model, the selection coefficients σBOMS50 equals 0.0026. Generally, it is considered that the effect of selection outcompetes the genetic drift for a population of an effective size Ne such that 4Ne.s > 1 (Kimura 1983). The minimal effective size Ne satisfying this equation for σBOMS50 is 95. The effective size of the total human population during Pleistocene is estimated at around 10,000 breeding individuals (Harpending et al. 1998). Our results show that if this population was a metapopulation structured into subpopulations of around 100 individuals, then an allele implying the death of its female bearer at age 50 could have been counterselected even in ancestral human populations. Females' death after menopause is therefore likely to be under higher selection than thought and the “selection shadow” (ages at which variations of survival or fertility do not yield a selection effect) may not hold.
POPULATION WITH A LOWER LEVEL OF ALLOCARE
By comparison with the reference model, the impact of the BOMS model on population dynamic parameters increases as the level of allocare decreases. λ is also higher (about one part in 2000) as well as the number of immature females (around 1%). Taking the bet on mother's survival into account may therefore be important in dynamic modeling for populations with a low level of allocare.
To illustrate the evolutionary consequences of allocare, let us consider a 15-year-old woman who immigrated to Quebec in 1800 and let us compare her number of female descendants in year 2000 calculated under two scenarios: (1) allocare estimated for ancient Quebec, and (2) lower level of allocare. At this time, fertility was very high and the population was dramatically increasing. Let us assume that the stable age-class distribution had been reached. The average number of female descendants of this woman 200 years later (that is about 7.5 generations further down) equals λ200. In 2000, her final number of female descendant would equal to 1.0494200= 15,512 in the population of ancient Quebec and is equal to 1.0473200= 10,313 in the modeled population without allocare. The difference in the descendants' numbers equals 5198. Two populations that share the same survival and fertility parameters but differ in the intensity of allocare show dramatically different finite rate of increase. Allocare is therefore an important component of human fitness.
Because it seems unlikely that maternal care is compensated by fathers, siblings, relatives, or the community as a whole when mothers die in other primate species (which corresponds to a value w0 other than null), the effect of a mother's death on fitness is lower in humans than in other primate species. Moreover, if such an allocare effect is ascertained concerning the mother's death, one may assume that every event altering a mother's ability to take care of their children (such as diseases or breast-feeding defects) will also be compensated through allocare and will have a lower effect on maternal fitness in humans.
In our model, allocare is considered as a constant level of care compensating maternal care in case of the mother's death, independent of survival of other caregivers. It is however likely that those caregivers (likely to be kin of the index individual) can also die. Further study should therefore extend our model to a more general “bet on kinship survival” model to integrate the full range of relationships between an individual's survival and all her caregivers' survival.
Taking the bet on the mother's survival into account modifies population dynamics (in terms of finite rate of population increase λ, generation time, and age-class structure). This effect is not important in the case of the population of ancient Quebec because of the high level of allocare in this population, but may be important in other human populations or in other species. More generally, we show that λ is likely to be generally underestimated in species in which maternal investment is needed for offspring to survive (especially if the adult survival probability during the rearing time is high) and further estimates in a wider range of species would be useful to picture the real strength of the bet on mother's survival in evolutionary processes. From a genetic point of view, our results provide evidence that it is especially relevant to take the bet on the mother's survival effect into account in studies that focus on selection pressures on survival at old ages. Finally, we have shown that the level of allocare has a dramatic effect on λ. Thus, variation in the level of allocare between populations will generate significant differences in their rates of increase. Furthermore, because this level of allocare probably varied in ancestral populations, it may have played an important role in human evolution.
Associate Editor: D. Promislow
This study was supported by a grant from the Fondation de France and the Association pour la Recherche sur le Cancer. We thank B. Toupance, R. Mace, J. Oeppen, J. Metcalf, D. Koons, R. Lee, D. Promislow, and two anonymous reviewers for very helpful comments and discussion.
AGE-SPECIFIC EFFECTIVE GAIN DUE TO SURVIVAL,
The quantity is the number of effective daughters achieved between x and x+ 1 by a female surviving up to age x+ 1. It is therefore the difference in the number of effective daughters achieved by females surviving up to exactly age x+ 1 (with x+ 1 excluded from period) minus the number of effective daughters achieved by females surviving up to age x (with x excluded from period). Let us denote Dx as the mean number of effective daughters born to females from age M to x− 1 given that these females are alive at exact age x and Dx+ 1 the similar quantity for females alive at exact age x+ 1. The quantity is therefore equal to the difference between Dx+ 1 and Dx:
To calculate the quantity Dx, we need the survival probability until maturity of all daughters born to a mother surviving at age x, born up to this age. We need therefore the probability that a daughter born to a mother of a years old is effective, given that the mother is alive at a later age x≥a (i.e., the daughter's survival probability until maturity as a function of her mother's age at childbirth a and her mother's current age x). Let us denote ua,x as this probability. It can be calculated as the probability ux described above except that the mother is alive when the daughter reaches her (y=x−a)th year. The daughter's age at mother's death y is defined between x − a and ∞ and ux,a is given by
Dx is therefore the sum over all ages a from age M to age x– 1 of the product between the age-specific fertility m(a) times the probability of being effective ua,x of daughters born to a mother of a years old but alive at age x as follows:
By replacing Dx, Dx+1, and by their expressions in equation (3) we obtain
AGE-SPECIFIC EFFECTIVE LOSS DUE TO DEATH,
Knowing , the quantity is easy to calculate because . However, one can also calculate following a similar method as the one of . The quantity is indeed the difference in the number of effective daughters achieved by females dying between x and x+ 1 (with x+ 1 excluded from period, denoted as D*x+ 1) minus the number of effective daughters achieved by females surviving up to age x (with x excluded from period, i.e., Dx). One should therefore calculate , where D*x+ 1 is the sum over all mother's ages at childbirth a between M and x of the product between the age-specific fertility (ma) and the survival probability wx–a of daughters whose mother dies when they reach the age y=x−a (i.e., ).
COEFFICIENT OF SELECTION ASSOCIATED WITH AN ALLELE THAT IS LETHAL AT AGE Z
Let us consider an allele that is lethal for its carrier between ages z and z+ 1, and σBOMSz and σREFz to be the selection coefficients calculated by using the BOMS model and the reference model, respectively. These selection coefficients can be calculated for any age zM as follows:
where the numerator of the ratio is the expected lifetime numbers of effective daughters of a female carrier and the denominator is the population average, R0. Note that the selection coefficients σREFz are independent of the constant daughters' probability of being effective, .