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

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American Journal of Primatology 27245-259 (1992)
Estimating Reproductive Success of Primates and
Other Animals With Long and Incompletely Known
Reproductive Life Spans
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
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
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.
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
c=48 1 1
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:
+ 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
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
1 + (t
+ t/i) - (l/i)c = F
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
+ t/i) - c/il [El = CJPE,
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.
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.
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
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.
O !
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
0 1
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
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
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.
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
+ .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
+ . . . + .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
+ .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.
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
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.
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.
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. To obtain such
a sample even once for baboons will probably take continuous study of an original
troop of animals and their descendants for 25 to 45 years. In the meantime, much
can be learned by measurement techniques not requiring complete reproductive
knowledge for every individual.
This paper was written while the author was a Visiting Fellow a t Clare Hall
College of Cambridge University, and a Fellow of the John Simon Guggenheim
Memorial Foundation. Thanks for clerical assistance are due to H. Rhine, and for
editorial assistance to D. and H. Rhine. Appreciated critiques were received from
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