# Estimating reproductive success of primates and other animals with long and incompletely known reproductive life spans.

код для вставкиСкачатьAmerican Journal of Primatology 27245-259 (1992) Estimating Reproductive Success of Primates and Other Animals With Long and Incompletely Known Reproductive Life Spans RAMON J. RHINE Department of Psychology, University of California, Riverside Lifetime reproductive success is difficult to obtain for adequate numbers of wild, long-lived animals because of incomplete knowledge of reproductive life histories and fates of offspring. A procedure is described that provides a standard estimate of lifetime reproductive success from either complete or incomplete reproductive life spans. For this procedure, reproductive success is defined as the following ratio: the number of offspring living to a criterion age in a given time period divided by the number expected for that age and period. The number expected is based upon the mean interbirth interval of the population, the length of the time period, the age criterion of reproductive success chosen by the researcher (e.g., the average age of menarche), and the probability that an offspring will live t o the criterion age. The effect of error upon the estimation of parameters is analyzed using the yellow baboons of Mikumi National Park as an example. The method can be used for both sexes, any litter (clutch) size, any mating system, any mean length of interbirth interval, any age criterion of reproductive success, and any study length that allows a reasonably sized sample of individuals for each of whom a substantial proportion of their reproductive life span is known. o 1992 Wiley-Liss, Inc. Key words: measurement; baboon females; long-lived animals INTRODUCTION Reproductive success, which links evolutionary theory to research, is difficult to assess adequately for individual long-lived animals. In principle, the best way to determine reproductive success is through observations of natural populations of individuals and their offspring over their lifetimes [Arnold & Wade, 1984; CluttonBrock, 1988b1. In practice, studies of free-ranging, long-lived animals, such as the great apes and cercopithecine monkeys, are plagued by incomplete knowledge of life histories of parents and of fates of offspring to at least reproductive age [e.g., Fedigan, 1983; Hrdy, 1986; Bercovitch, 1987; Altmann et al., 1988; Cheney et al., 1988; Packer et al., 1988; Thomas & Coulson, 19881. Resulting small samples of complete reproductive data hinder convincing comparisons of individuals differing Received for publication January 31, 1991; accepted September 24, 1991. Address reprint requests to Ramon J. Rhine, Dept. of Psychology, Univ. of California, Riverside, CA 92521. 0 1992 Wiley-Liss, Inc. 246 / Rhine in significant characteristics, such as dominance, thought to be related to reproductive success [Fedigan, 19833. This situation can be improved by finding a meaningful way to use estimates of reproductive success based upon different fractions of reproductive life span. To that end, this paper focuses on a method that assigns to an individual one number that represents the relative degree of lifetime reproductive success. Troublesome variations encountered in attempting to arrive at a measure of reproductive success applicable to different fractions of reproductive life span will be illustrated by a 10-year study of free-ranging, adult-female baboons of Mikumi National Park, Tanzania [Norton et al., 1987; Rhine & Westlund, 1981; Rhine et al., 19881. Baboons can live up to 25-30 years [Napier & Napier, 19671. Some Mikumi females were young, middle-aged or old adults when observations began, at which time they were either lactating, pregnant, or menstrual cycling. Others were reproductively immature at the onset of observations, and still others were unborn. In some cases, a female’s reproductive history could be recorded from menarche to death; for others, because one or both of the menarche or death dates were unknown, the female was observed for an unknown fraction of her reproductive life span. The question arises as to what can be used as a single, standard measure of reproductive success that takes such variations into account. Neither the observed number nor the observed rate of a female’s offspring reaching a criterion is a satisfactory measure of reproductive success. In regard to number, suppose one female of unknown age was reproductive a t the start of a 10-year study and lived t o the end, a second was reproductive a t the start and died after 4 years, and a third reached menarche after 4 study years and was still alive after the remaining 6. If all were observed to have two offspring who lived to criterion, it is not appropriate t o conclude that the three females had equal reproductive success: The one studied the longest is expected to have the most observed offspring reach criterion and the one studied the shortest is expected to have the least. In regard to rate, suppose two females are observed from the time of menarche. If one has three offspring reach criterion during 10 years of observations, she has a rate of reproductive success of .3 per year. If the other dies 2 years after menarche, and has one offspring who reaches criterion, she has a rate of .5 per yer. But the second female is less rather than more reproductively successful than the first who has three times as many offspring reaching criterion and who may have more after observations cease. To address such problems, a measurement model is developed using a standard against which reproductive success over varying fractions of complete or incomplete reproductive life spans can be compared. The approach taken is analogous to a statistical procedure in which an empirical finding is evaluated by comparing it to a relevant theoretical distribution of what is expected. The model is first developed for sexually reproducing females having a specifiable interbirth interval and a litter (clutch) size of one. Then it is generalized to any litter (clutch) size and to both sexes. THE MODEL Lifetime reproductive success is usually defined as the number of an individual’s offspring living to a logically chosen criterion, such as the average age of reproduction [e.g., Grafen, 1988; Thomas & Coulson, 19881. In theory, inclusive reproductive success as the agent of selection is the individual’s contribution to the gene pool via viable descendants and via the individual’s effect on the reproductive success of relatives. In practice, adequate estimates of numbers of surviving grandoffspring, remote descendants or relatives, from field studies of long-lived species, Estimation of Reproductive Success / 247 ki =21+4 >I t=114 t 1 2 3 ‘ I I 4 5 6 I 1 I BIRTHS 1 c=48 1 1 3 2 4 CRITERION REACHED I 0 I I I 20 I / I I I I I 40 I / I I I I 80 60 I I l 100 / I I l 120 STUDY TIME IN MONTHS Fig. 1. Timing of six births during a 10-year study of a hypothetical average female baboon who had four infants reach a criterion of reproductive success. The first infant was born 6 months after the study began; t is the time from the female’s first observed birth to the study’s end; i is the mean interbirth interval of the population; and c is the criterion of reproductive success (offspring lives at least 48 months). are made unlikely a t present by amplification of complexities in obtaining complete and unbiased information even on offspring [e.g., Bercovitch, 1987; Altmann et al., 1988; Cheney et al., 1988; Cook & Rockwell, 1988; Packer et al., 1988; Le Boeuf & Reiter, 19883. Consequently, an individual’s reproductive success is usually assumed to be adequately approximated from information about offspring. For example, Clutton-Brock’s [1988al edited book on reproductive success contains 25 representative, empirical studies of species ranging from insects to humans, and all concentrate upon offspring, as does the measurement model developed below. Figure 1 represents events in a 10-year study of a female’s reproductive life, starting with her first observed birth. It illustrates the quantities used to define expected (not actual) fecundity, F, and potential (not actual) reproductive success, P, and it continues the baboon example for which the mean interbirth interval, i, is 21 months. In this example, the total observation time, t, which goes from the day of the first observed birth until the end of observations, is 114 months, and the criterion of reproductive success, c, is offspring survival to 48 months. Any other duration could be used as the criterion if it better fits the research situation or species. Similarly, it is not necessary to use the same criterion for both sexes. For convenient reference, definitions of the above and other quantities used in this paper are presented in Table I. An individual’s actual (observed) fecundity is offspring production per unit time, such as production per year [e.g., Brown, 19881, whereas a population’s expected fecundity during a given length of time (t)is defined here as the number of births expected of the average reproductive female during that time. In this definition, expected fecundity is an inverse function of mean interbirth interval [Strum & Western, 19821, as follows: F = 1 + tii. As Figure 1 illustrates, in 114 months, starting with the first birth, the average reproductive female a t Mikumi is expected to have six births, plus .43 of a seventh, 248 / Rhine TABLE I. Definitions of Svmbols ~ ~~ Symbol C c, E F 1 P PE R t Definitions Criterion in months of reproductive success (herein 48 months of life) No. of offspring reaching the criterion (not necessarily a whole number due to prorating of live offspring at study’s end) Proportion of offspring in the population expected to reach the criterion (herein .5) Expected (not actual) fecundity, the no. of births expected of the average reproductive female during the study time, t Mean interbirth interval of the population (herein 21 months) Potential reproductive success, the number of the average individual’s offspring that could reach criterion during the study time, t The product of P and E is called expected reproductive success Relative reproductive success, indicating a given individual’s reproductive success relative to expectation; a n estimate of lifetime reproductive success The time from a female’s first observed birth to t,he end of observations, or, if a female of known menarche and below average reproductive life span dies during the study, the average reproductive life span of the population, if known, or otherwise of the species (herein 144 months) i.e., F = 1 + 114121 = 6.43.The mean number of births expected is influenced by factors such as absorbed embryos, miscarriages, and neonate deaths. The effect upon reproductive timing of these factors is taken into account via their effect upon the mean interbirth interval. Potential reproductive success, P, which is a function of expected fecundity and the size of the chosen criterion of reproductive success, is the maximum number of the average female’s offspring that could reach the criterion during the study period. As will be seen from Figure 1, potential reproductive success is P = 1 + (t - c)/i = (1 + t/i) - (l/i)c = F - (l/i)c. In the example of Figure 1, if all offspring of a female with average fecundity lived to the criterion age, then in 114 months four would be expected to reach criterion, ‘plus .14 of a fifth, i.e., P = 6.43 - 48/21 = 4.14. Notice for future reference that P is a linear function of c with an intercept constant of F and a slope constant of -Ui. The measure of reproductive success for a given population is relative reproductive success, R, which is a function of Cn, the number of offspring actually reaching criterion, and PE, the expected reproductive success. To take into account mortality before the criterion is reached, potential reproductive success is multiplied by E, the proportion of offspring born into the population who are expected to reach the criterion. Relative reproductive success is R = CJ(1 + t/i) - c/il [El = CJPE, t 2 c. The restriction t 2 c deletes from the sample females who were not observed long enough €or even one offspring to reach criterion. Circumstances may arise, such as a quite large N, where a researcher judges a stronger restriction t o be appropriate, such as t z 2c. Suppose in the baboon example that E is .5. Then, for a given female baboon, if two offspring reach criterion and P is 5, R = 245 . 50) = .80. This measure indicates the degree to which a female’s reproductive success is greater or smaller than what is expected in the population being studied. It is zero if no offspring reach the criterion. In theory, it is less than one, as in the example, if the number Estimation of Reproductive Success I 249 reaching criterion is fewer than expected in the study time; it is greater than one if C, exceeds expectation; and it is one if C, equals the number expected. Also in theory, if R is two, the number meeting the criterion is twice the expectation; if R is three, the number is thrice; and so on. If the criterion of reproductive success is reduced to c = 0 (birth), then R becomes the ratio of the number born in time t to the expected number during t, i.e., the ratio of actual fecundity to expected fecundity. In practice, conditions may arise that prevent these theoretical expectations from occurring. In such cases R can be interpreted as a relative measure for which the larger of two values still signifies greater reproductive success, but for which an R of one may no longer mean that a female’s reproductive success is equal to the expected value of the population. APPLICATIONS Cases l a and l b In case la, R is obtained for a female whose complete reproductive life span is known from the date of her first offspring’s birth to the female’s death, and the study is long enough to determine if each of her offspring reach criterion. Reproductive life span is likely to be an important covariate of reproductive success in most long-lived animals [e.g., Fedigan et al., 1986; Altmann et al., 1988; Brown, 1988; Cheney et al., 1988; Clutton-Brock, 1988c; Ollason & Dunnet, 1988; Scott, 19883; therefore, if a female dies after a short reproductive life, R will be inflated using a t of her total observation time. Instead, since her entire reproductive performance is known, it is compared with that expected during an average reproductive life span. If a dead female’s reproductive life span was shorter than average, the t used in the calculation of her R is the average female reproductive life expectancy of the study population, if known, or otherwise of the species. An estimate from the study population usually is preferred because reproductive life expectancy may differ among populations of a single species living in habitats of varying textures and qualities. Altmann et al. [19881estimated a reproductive life expectancy of 10-12 years for yellow baboons living in a degraded habitat [Western & van Praet, 19731. Using the upper end of that range to calculate t for the resource-rich habitat of Mikumi [Norton et al., 19871, and an E of .50, the Rs of females having 0 , 1, 2, 3, or 4 offspring reach criterion, respectively, are .OO, .36, .72,1.08, and 1.43. Thus, for the parameters of the present example, a female with an average reproductive lifetime needs to have three offspring reach criterion to exceed the norm of reproductive success. In case lb, the female’s history of births is again fully known, but unlike case l a one or more of her infants have not reached criterion when observations end. Small baboon infants who are orphaned are unlikely to live, but offspring aged 1 year or more are capable of surviving without their mother [Altmann, 1980; Rhine et al., 1980; Hamilton et al., 19823. Suppose at a female’s death she had two offspring who reached the 48-month criterion, and a third who was 24 months of age when observations ended. To account for the third offspring, the number reaching criterion is prorated as 2 + 24/48 = 2.5. The resulting R of .90 is the mid-point between .72 and 1.08, the Rs above for two and three offspring reaching criterion in case la. If the data are available, a more refined prorating can be achieved by using the probability that a 2 year old will live to 48 months. For Mikumi baboons, the probability is .77 of an offspring being alive at the end of 48 months if it was alive a t the end of 24. Using this information, C, in the above example is 2.77 and R is .99, which is .77 of the distance between .72 and 1.08. Similar prorating is used in case 2 if offspring are alive a t the end of observations. 250 I Rhine Case 2 In case 2, the female’s history of births is only partly known for one of three reasons: 1) she died before observations ended, but she was reproductive before they began, 2) she became reproductive during the study, but she was alive when observations ended, and 3) she was both reproductive when observations began and alive when they ended. For none of these alternatives is it appropriate to use expected reproductive life span in the calculation of P since a female may have additional offspring during the unknown periods of reproductive life. Using average reproductive life span will then tend to underestimate R because C, relative to the average will be too small. In case 2, P is calculated using observation time, from the occurrence of a female’s first known birth until the end of observations. If a female baboon already has with her a small infant when observations begin, her start date can be pushed back using developmental benchmarks decribed by Altmann et al. [19811, Rasmussen [19791, and Rhine et al. [1984] to estimate the infant’s birth date. Suppose two females who were alive at the end of observations (case 2) each had two offspring reach criterion, and suppose these females were observed, respectively, for 6 and 10 years of reproductive life. Since they both had the same number of offspring reach criterion, R should be most for the one observed the shortest time and least for the one observed the longest. Using the same parameters as before, the respective Rs are 1.87 and .90. Now suppose a 10-year study in which one female’sreproductive life history is known from menarche to death (case l),and a second female had her first offspring 4 years after the start of observations and was alive at the end (case 2). Each had two offspring reach criterion. Since the first female had two offspring reach criterion in a reproductive lifetime and the second had two in a fraction of a reproductive lifetime, R should be greater for the second. R is .72 for the first female and 1.87 for the second. When P is calculated using observation time, it is assumed that known reproductive performance predicts unknown, in other words, that the rate of C, occurring during the observation period approximates that occurring before or after. Results obtained under this assumption should be more accurate the longer the female is observed. Therefore it may be necessary to establish a rule requiring the study of an individual for a substantial portion of the average reproductive life span, such as half or more, before including her in the study. The assumption should have little effect upon the calculation of average reproductive success from representative, reasonably sized groups of females, but some individual estimates may suffer, which will tend to increase error variance. Statistically significant differences in reproductive success between contrasted groups, such as dominant versus subordinate females, will then be harder to achieve. In regard to female baboons, the assumption of known outcomes being representative of unknown is least tenable when the female is observed for a relatively small proportion of her reproductive life, and the proportion observed is her earliest reproductive period or the period of very old age. If fecundity and C, deviate noticeably from her lifetime average, it is most likely to occur during these periods [Strum & Western, 19821, though such deviations could not be demonstrated in an unusually complete analysis of reproductive success in Japanese macaques [Fedigan et al., 19861. A procedure for ameliorating possible age effects is suggested below. ACCURACY OF ESTIMATES The accuracy of R depends upon three parameters that are subject to sampling error: i, E, and t (as the average reproductive life span of a population). It also Estimation of Reproductive Success / 251 depends upon the selection by the researcher of a meaningful c and upon research techniques which produce accurate values of C,. If it is not possible to obtain an accurate count of the number of a long-lived female’s offspring reaching criterion during the study period, then an attempt to specify reproductive success is inadvisable. The criterion of reproductive success is chosen by the researcher, presumably on the basis of theory, research logic and the realities of methodology and data. A meaningful standard criterion, which can be used across species, is offspring life to the age of reproduction or to a nearly equivalent earlier age [e.g., Fairbanks & McGuire, 1984; Clutton-Brock et al., 19883. For Mikumi, 48 months is nearly equivalent to the age of reproduction since 94% of all offspring who had a known birth date and lived to 48 months also lived to the age of menarche (females) or transfer (males). Using 48 months instead of the age of menarche or transfer increases the number of offspring who can be observed to criterion. Sometimes sample size or other study conditions dictate a choice of criterion considerably briefer than the offspring’s age of reproduction [e.g. Cheney et al., 1988; Gouzoules et al., 1982; Paul & Thommen, 1984; Bulgar & Hamilton, 1988; Le Boeuf & Reiter, 1988; Thomas & Coulson, 19881, such as birth (c = 0)or the age of weaning, which may sometimes be chancy indexes of reproductive success. Correlations were calculated for Mikumi females between R based upon a c of zero and Rs based upon cs of 12, 24, 36, 48, 60, and 72 months. These six correlations range from .63 to .76, suggesting that, although birth is a chancey criterion, it could be useful in a pinch or in preliminary studies. Although c is not empirically determined, one can ask if the estimation accuracy of other empirical quantities varies with c, and if so, how and to what degree. Effect of Error on Expected Reproductive Success (PE) Relation of P to c and reproductive life span. As illustrated in Figure 2, P is a linear function of c and t. The lines in the figure correspond to five values of t. For a given i, the slopes are constant, and the intercepts, equal to F, vary. The difference between pairs of parallel lines is proportional to the difference between t’s; for any c, the larger the t, the larger the P. Where t = c, P = 1,as illustrated in the figure for t’s of 0, 60, and 120. A c of zero means that P equals expected fecundity. A P of less than 1 is meaningless. It occurs in the figure where the criterion of reproductive success is greater than the length of the study (hence, the restriction to R o f t 2 c). Effect upon PE of error in t (as average reproductive life span). The effect upon PE of an inaccurate estimate of mean reproductive life span is illustrated in Figure 3. Since P is linear, so is P multiplied by the constant, E. Suppose the population value of mean reproductive life span is 144 months. Then if 132 months is erroneously used for life span instead of 144, PE will be in error by .29, which is the distance in Figure 3 between the middle line (t = 144) and the ones immediately above or below it (t = 156 and 132). Every additional year of error adds another increment of .29. Thus, in Figure 3, if 180 or 108 are used as the estimate of reproductive life span, the error in PE is .87. With a criterion of 48 months, the average female is expected to have a PE of 2.79 in 12 years. An error of 1year is 10.39%of 2.79, and of 1month is 37%. The parallel lines indicate that the amount of error will be the same for all c’s. As the criterion of reproductive success becomes tougher (increases), PE decreases, as indicated by the negative slope of the lines; therefore, the percent of error increases as c increases. The choice of a criterion is a trade off between the minimal time the offspring is required to live and the percent of error tolerable. 252 / Rhine 10 P - 5- 0- - -5 20 0 40 60 80 100 120 c in months Fig. 2. Potential reproductive success (P) as a function of the criterion of reproductive success (c), for an interbirth interval of 21 months and observation times (t) from 0 to 240 months. 51 PE O ! 0 20 40 60 80 100 120 c in months Fig. 3. Expected reproductive success (PE, where E = .5) as a function of the criterion of reproductive success (c), for an interbirth interval of 21 months and observation times (t) from 108 to 180 months. Estimation of Reproductive Success I 253 PE 0 1 0 20 40 60 80 100 120 c in months Fig. 4. Expected reproductive success (PE, where E = .5) as a function of the criterion of reproductive success (c), for an observation time of 120 months and interbirth intervals (i),from the top to bottom lines, of 19,20,21, 22, and 23 months. Effect upon PE of error in i. Figure 4 contains a set of curves for which t = 120 and i varies from 19 to 23. In these curves both the intercept and slope constants depend upon the value of i. The difference between pairs of curves for a given c is greatest when the criterion is birth (c = 0) and decreases steadily as the value of c increases. Differences between curves decrease as i increases. All curves converge a t the point where P = 1 and c = t. These patterns occur for any combination of i’s, c’s, and t. Error in PE due to error in the estimation of i will be illustrated by taking the i of the middle curve of Figure 4 as the population value (21). An under-estimate of i yields a greater error in PE than will an equal over-estimate. For example, an under-estimate of 2 months for a c of zero produces an error in PE of .30,and an over-estimate of 2 months an error of .25. As the converging curves illustrate, the larger the c that can be tolerated without unduly restricting sample size, the less the impact on PE of either an over- or an under-estimate of i. In this regard, birth is the worst choice of c. Criteria of zero versus 48 months have, respectively, errors in PE of .30 and .18 for a 2-month under-estimate of i and of .25 and .15 for a 2-month over-estimate. Effect upon PE of error in E. Curves illustrating the effect upon PE of error in E are shown in Figure 5. The middle line in Figure 5 is for the E of .50 used in the baboon example, and the remaining lines are for increases or decreases by increments of .05. The five lines are negatively sloped and not parallel, but they never meet. Each goes through the point where c = t and P = 1 (i.e., PE = E). The distance between pairs of curves for a given c is proportional to the difference between Es. In Figure 5, the difference between adjacent curves is equal for a given c because the Es of these curves all differ by the same amount (.05). Similarly, the 254 I Rhine PE " I 0 20 40 60 80 100 120 c in months Fig. 5. Expected reproductive success (PE) as a function of the criterion of reproductive success (c), for an observation time of 120 months, an interbirth interval of 21 months, and the proportion of offspring born into the population who are expected to reach criterion (El of .40, .45, 50,.55, and .60. error in PE from Es of .40 to .50 is twice the error from .45 to 5 0 . If the middle curve of Figure 5 is taken as the population value (.50),then for a c of 48, a t of 144, and an i of 21, the error in PE due to a 1%error in E is .028. The effect upon PE of an error in E lessens as c increases. Effect of errors upon relative reproductive success (R).An error in PE due to mean reproductive life span, or i, or E carries over to R as a constant proportion of C,. This is illustrated for i in Figure 6. The lines in that figure, and also in similar figures for varying values of mean reproductive life span and E, all have the same form, R = (l/PE) (C,), which is a linear equation having a slope constant of 1PE and an intercept constant of zero (R is always zero if C, is zero). Since the radiating lines all pivot around zero, error in R for different values of C, varies as a function of the line's slope, that is, as a function of the tangent of the angle between the C,-axis and the line. Consequently, the total amount of error in R will increase as C, increases, but error expressed as a proportion of Cn will remain the same. For example, if 21 in Figure 6 is taken as the population value, then an under-estimate of 2 months (i = 19) leads to errors in R for C,s of 1 to 4, respectively, of .034, .068, .102, and .136, which are all equal to .034C,. Families of R-lines for varying values of mean reproductive life span or E take the form shown in Figure 6, but they differ in the tangents of their angles and therefore in the separation between lines. In Figure 6, the differences between pairs of lines are very small, but they do occur, with the difference between lower pairs of lines being slightly greater than between upper pairs. Equivalent graphs for mean reproductive life span and E have larger differences. In all three cases, errors in R are greater for under- than for over-estimates. Data are available from 50 Mikumi baboons to determine confidence limits for i, and this allows a further example of error analysis. The mean of i for the 50 Estimation of Reproductive Success I 255 0.00 r 0 2 1 3 4 Cn Fig. 6. Relative reproductive success (R)as a function of the number of offspring (C,) reaching a criterion of 48 months, for an observation a time of 120 months and interbirth intervals (i),from the bottom to top lines, of 19, 20, 21, 22, and 23 months. baboons is 21.29 for which the 95%confidence limits are 20.05 and 22.53. Rs were calculated using i’s of 20.05,21.29, and 22.53, a C, of 4, and the values oft, E, and c upon which the lines of Figure 6 are based (t = 120, E = .50, and c = 48). These Rs are 1.743 for the lower confidence limit of i, 1.826 for the sample mean, and 1.907 for the upper limit. The differences between 1.826 and the other two values of R are .081 and .083. Therefore, the error in R due to sampling error in the estimation of i is unlikely to be greater than .083 for a C, of 4 or less. MALES A male’s potential reproductive success may or may not differ from his mate’s(s’). His E is the same as for females of his population, and his C, is the number of offspring he sired that reached criterion. The potential reproductive success of the average male is the sum of the Ps, or fractions thereof, of females he might fertilize. If a pair is strictly monogamous, the male’s potential (and actual) reproductive success is the same as the female’s. If a male has exclusive mating access to two or more females, and to none other, his potential (and actual) reproductive success is the sum of theirs. If he has access to one female for .5 of her observation time, to a second for .7, and to none other, his potential reproductive success (but not necessarily his actual) is the sum of the fractions of the females’ potential successes for which he might share responsibility, i.e., his potential reproductive success is P = .5P, + .7P,, where the subscripts refer to the two females. 256 / Rhine The number of the average male’s offspring that could reach criterion during the study (potential reproductive success) depends upon the number of breeding females and the number of male competitors. As for females, a male’s potential reproductive success usually will be different from his actual success. Among savannah baboons, for whom a given male can mate with several females and a given female with several males, the average male’s potential reproductive success is the sum for his troop of all female Ps multiplied by the equal probability that each adult male in his troop will sire each offspring. For example, if there are ten reproductive females and two adult males in the troop, then each male’s potential (not necessarily actual) reproductive success is .5P, + . . . + .5P10 = .52Pk, + where the k subscripts refer to the ten females. If he transfers t o another troop where there are 12 females and four males, including him, his potential reproductive success for both troops combined is P = .52Pk1 + .25CPk,, where the numerical subscripts refer to troops 1and 2. In harem-breeding species, such as elephant seals, the male’s potential reproductive success is the sum of the female Ps of an average sized harem, and his R is the sum of all offspring he sired during the study divided by E times the sum of the Ps for that period. In species with a discrete breeding season, a male’s P is based upon his several seasons and is calculated in the same manner as a P based upon two baboon troops. If more than one male shares a harem, as for lions [Packer et al., 19881,or if several males share a polyandrous female, then a given male’s P requires appropriate proportions, as illustrated above for baboons. Although in principle the model can be extended to males, the fact that males do not bear the offspring complicates the assessment of their reproductive success. Paternity over the years often is difficult to determine with the same degree of certainty as maternity [Fedigan, 1983; Bercovitch, 19871. It is sometimes assigned using consort information or genetic techniques. Long-term application of genetic paternity analysis is promising, but it has not yet been successfully accomplished with any long-lived, free-ranging species. Estimates based upon consort information are likely to be less reliable and valid for some species, such as common baboons [Howard, 1979; Bercovitch, 19871, than for others, such as animals with clearly demarcated mating seasons and a harem mating structure [e.g., CluttonBrock et al., 1988; Le Boeuf & Reiter, 19881. To date, no one has determined from consort data the probable C,s of a reasonable sample of free-ranging male baboons over substantial proportions of their reproductive life spans. Even if a male baboon spent his entire reproductive life in the same troop, which he usually does not [Packer, 1979; Rasmussen, 1981; Altmann et al., 19881, it would still be necessary to keep track of all consorts over a substantial proportion of his reproductive life. In the end, this staggering undertaking often would not allow a confident choice between multiple male consorts. LITTER (CLUTCH) SIZE For purposes of this paper, a litter (or clutch) is one or more offspring born to a female close in time, followed by a much longer time before she gives birth to other offspring. In Figure 1,six litters are shown on the birth line separated by an inter-litter interval of 21 months. So far the model has been developed for a litter of one. Larger litters can be taken into account by incorporating into R an estimate Estimation of Reproductive Success f 257 of mean litter size, L, of a population, if available, or otherwise of the species. R for any litter size is R = CJLPE. This expression is general, applying to both sexes, all mating systems and any litter size, interbirth interval, criterion of reproductive success, and study length that includes substantial proportions of reproductive life spans. AGE In some situations (case 2), an R based upon an incomplete reproductive life span is assumed to be representative of R for the entire span. This assumption is sensitive to variations associated with age. If violation of the assumption is judged likely to create serious problems in data interpretation, either the study should be long enough so that complete reproductive life histories are obtained for most or all animals, or age should be taken into account statistically or by stratifying comparison groups. For example, to determine if reproductive success is different in large versus medium-sized groups, stratified samples could be chosen from groups of both sizes with approximately the same proportion of young, old, and in-between ages in each. Alternatively, statistical control can be achieved with techniques that either partial age out or include age as a factor in the design, for example, a two-way analysis of variance using group size as one factor and age as the second. This allows a statement of variance associated with group size after any agerelated variance is removed. In order to determine the relationship between reproductive life span and reproductive success, it is necessary to know the dates when parents reach reproductive age and when they die, and the number of their offspring that reach criterion. For case 2, not all of this information is available. Yet, reproductive life span may be an important component of reproductive success [Brown, 19883, especially among long-lived animals who continue to reproduce into old age. For some species, baboons included, it takes a very long study to acquire an adequate sample of animals who were born during the study and who died in late maturity or old age. Although old Mikumi females can be identified, information is lacking on the number of their offspring reaching criterion before the study started. These old females can be used to determine relationships with age, such as the effect of age upon infant survival or the length of the interbirth interval, but not to determine the relationship between reproductive life span and reproductive success. CONCLUSIONS 1. A measurement model for reproductive success has been proposed to make comparable use of complete and incomplete reproductive histories of long-lived animals. The model compares an individual's successful reproductive output with what can be expected in the population during the length of time the individual was studied. It is applicable to both sexes, any litter size, and any mating system. 2. The effect of errors in parameters of the measurement model upon measures of expected reproductive success was analyzed and illustrated by application to a 10-year study of female baboons. Measurement of reproductive success in field studies of long-lived animals is a difficult and imperfect endeavor. As in all such measurement, sampling and reliability errors occur, although they are not always recognized or well understood. Small samples and unrecognized error contribute to inconsistent conclusions among studies. 3. Currently, it is impractical to depend entirely upon data from complete life 258 / Rhine histories of individual, long-lived animals. As longitudinal field studies of longlived animals remain operative for longer and longer periods of time, the chances increase of acquiring a moderately large sample of the entire reproductive lives of individuals whose offsprings’ fates are known to a strong criterion. 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