The University of Chicago
Models of Status-Correlated Bias in Offspring Sex Ratio
Author(s): Michael Altmann and Jeanne Altmann
Reviewed work(s):
Source: The American Naturalist, Vol. 137, No. 4 (Apr., 1991), pp. 542-555
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Vol. 137, No. 4
The AmericanNaturalist
MODELS OF STATUS-CORRELATED
April 1991
BIAS IN OFFSPRING SEX RATIO
MICHAEL ALTMANN AND JEANNE ALTMANN
Division of Health ComputerSciences, Box 511 UMHC, 420 Delaware StreetSE, Universityof
Minnesota, Minneapolis, Minnesota 55455; Departmentof Ecology and Evolution, Universityof
Chicago, Chicago, Illinois 60637 and Departmentof ConservationBiology, Chicago Zoological
Society, Brookfield,Illinois 60513
SubmittedJuly3, 1989; Accepted January19, 1990
Abstract.-We study a simple model describingthe genealogic structureof a population in
varies witha parent'spositionin a social hierarchy.This model
whichthe sex ratioof offspring
is based on species in which one sex disperses and in whichrank in a stable social hierarchy
of thatsex. We findthatbiased productionof
forthe othersex is "inherited" by the offspring
offspringin such a social systemhas importantimplicationsforgroup-sizeregulation,for the
genetic relationshipsamong individualsat any given time and across generations,and for the
The general
experiencedby a parentand its offspring.
disparitybetweenthe social environment
model predictsthatgroupsize is tightlyregulatedand thatgroupsize is unstablein groups with
mothers.We thenfurtherexamine the particularcase of the
underproductionby high-ranking
parents. In
model in which the nondispersingsex (females) is overproducedby higher-ranking
this case, we findthat females are unusuallyclosely related; all females in a group have a
commonfemaleancestor5-10 generationsback. Moreover,daughtersof mothersin the middle
of the hierarchyexperience the largestintergenerational
disparity.In contrastto daughtersof
high-and low-rankingfemales,theyoccupy ranksmuch lower thantheirmothers'.
Sex biases in offspring
productionmayevolve, althoughoverallparentalinvestof the two sexes will be one to one in diploidorganisms(Fisher
mentin offspring
facultativeadjustmentof offspring
sex ratio
1958; Hamilton 1967). Furthermore,
ratioswithinpopulations. In the
may resultin systematicvariabilityin offspring
sex ratios has fobiological literature,attentionon facultativelybiased offspring
cused on questions of the ultimateevolutionaryresult(e.g., Triversand Willard
1973; Clark 1978; Clutton-Brocket al. 1981; Clutton-Brockand lason 1986) and,
to a lesser extent,on proximatemechanismsthat would produce facultatively
biased primarysex ratios(e.g., James1971, 1983; Myers 1978; Myerset al. 1985).
efHowever, biased offspringproductionhas several strikingintergenerational
fects,and any selective advantage of biased ratios mustpersistover a sufficient
numberof generationsforthe evolutionaryconsequences to be realized or maintained. It is surprising,therefore,thatevolutionarymodels of behaviorand social
structurerarelyfocus on effectsover the timecourse of several generations,and
we know of no such examinationin thecase of offspring
biases. In thisarticle,we
consequences througha simple mathematical
investigatethese intergenerational
model that was initiallymotivatedby the demographyand social structureof
cercopithecineprimates.A mathematicalapproach is particularlyusefulforlongAm. Nat. 1991. Vol. 137, pp. 542-555.
? 1991 by The Universityof Chicago. 0003-0147/91/3704-0006$02.00.
All rightsreserved.
BIAS IN OFFSPRING SEX RATIO
543
lived, slowly maturingspecies such as primatesbecause empiricaldata are available for at most a single generationfor such animals, and even then only for a
few species.
Almostall cercopithecineprimateslive in permanentsemiclosedand matrilocal
social groups: females remain withintheir natal group and reproduce there,
whereas males disperse as adults and, iftheyare successful,reproducein one or
moregroupsin the subsequentyears (reviewin Pusey and Packer 1987). Females
can be orderedin clear dominancehierarchiesthatare usually stable throughout
adulthood, and as a daughtermaturesshe assumes her mother'srelativedominance rank. Even if the relativerankrelationshipamong females stays constant
throughouttheirlives, however,theabsolute rankof a female,by whichwe mean
one more than the numberof females above her in the hierarchy,changes as
females mature and others die. Female rank sometimes,
some higher-ranking
thoughnot invariably,confersreproductiveadvantagethroughearlierage of first
reproduction,shorterinterbirth
intervals(i.e., higherbirthrates), or lower infant
mortality.Althoughmale dominancerankoftencorrelateswithreproductivesuccess in the short term,males, in contrastto females, change dominance rank
oftenduringadulthood,and mother'srankseems littlerelatedto initialor lifetime
differencesin rank or reproductivesuccess (review in Fedigan 1983; Silk 1987;
Waltersand Seyfarth1987; but see Meikle et al. 1984).
THE MODEL
Below, we model mathematicallythe relationshipsbetween maternaldominance rankand offspring
sex ratioand theintergenerational
consequences of such
relationships.The model focuses on the female componentof a group that is
assumed to have the followingbiologicalcharacteristics:(1) a femalestays in her
natal group and reproducesthere; (2) a daughterassumes her mother'srelative
positionin the social hierarchy,or, ifher motheris alive, she occupies a position
directlybelow her mother(the orderingamong sistersis immaterialforour purposes); and (3) the numberof maturedaughtersproduced by a female depends
on the numberof femalesabove herin thehierarchy.This could arise in a variety
of ways; the rank-relateddifferencescould arise fromsecondarysex ratios (i.e.,
in numberof offspring.
throughdifferential
mortality)or fromdifferences
Equivathata femaleproduces could be independentof
lently,the numberof offspring
her rank, but with the sex ratio of the offspringdependenton the numberof
females above her in the hierarchy.It is the terminologyfor this last situation
thatwe use forthe remainderof our explorations.
We reporton the simpleststochasticmodel forthe situationdescribed above:
we break time into discrete generations,and at each generation,a female is
replaced by a randomnumberof daughters,who are all consideredthe same age.
These numbersare independentfordifferent
females and different
generations.
Our investigationthen focuses on threeaspects of the system.Specifically,we
firstask whatare theimplicationsforgroupsize of different
relationshipsbetween
sex ratio; thatis, is group-sizeregulationaffected
dominancerank and offspring
is an increasingor a decreasing
by whetherthe proportionof female offspring
544
THE AMERICAN NATURALIST
functionof dominance rank? Next, using the results of this examination,we
restrictsubsequent investigationsto the case in which the proportionof female
offspringdecreases with decreasingdominance rank, and we ask several questionsabout the numberand identityof theoriginalmatrilinesthatare represented
in the group in subsequent generations.Finally, we examine the relationship
between the absolute rankof a motherand her daughters.
We take a functionf1,k thatgives the probabilitythata femaleof rank i leaves
k daughtersin the nextgeneration.The values of thisfunctioncompletelyspecify
our model. Note that,followingthe conventionin the biological literature,the
individualis assigned rank numberone, rank two is assigned to
highest-ranking
the nexthighest-ranking
individual,and so on. We thendefineRi as the expected
numbersof adult daughtersleftby a femaleof ranki:
Ri
=Ekfi,k
=Z
kR
GROUP-SIZE
REGULATION
We show thatfora species in whichdominancerankis passed frommotherto
mothersoverproduce
daughter,group size is tightlyregulatedifthe high-ranking
daughters,and group size is unstableiftheyoverproducesons.
Suppose thereare initiallys femalesin a group. The ith produces an average
of Ri daughterssuch thatAs, the average change in the group size afterthe next
generation,is
/s
As=
\
Ri
-s,
where here, and elsewhere, the size of the group means the numberof females
in the group. By definitionAs,is positive when the group is expanding and is
negativewhen the groupis shrinking(see fig.1). The sizes at whichthe graphof
A crosses the horizontalaxis (A = 0) are the stationarygroupsizes; thatis, there
is no change in groupsize between successive generations,and a stationarysize
is stable under small perturbationsif the graphcrosses the axis witha negative
slope. If the group declines slightlyfromthe stationarysize, then A is positive
and the group increases in the next generation;if the groupgrows slightly,then
A is negativeand the group shrinksback towardthe stationarypoint (see fig. 1).
situationoccurs ifhigh-ranking
mothersleave relativelyfewer
Quite a different
daughters(and, therefore,relativelymore sons) thando low-rankingmothers.In
thiscase, A is negativeforsmall s, and A would cross the axis frombelow at its
firstcrossing (see the inset to fig. 1). This implies that the nonzero stationary
would tendto move the group size away
groupsize is unstable; any perturbation
fromthe stationarysize, eitherdown to extinctionor up to infinity
(realistically,
otherbiological processes such as groupfissioningwould probablyoccur).
females disproportionConsequently, offspringbiases in which high-ranking
ately produce sons and low-rankingones disproportionately
produce daughters
545
SEX RATIO
BIAS IN OFFSPRING
unstable
U
I
6
change in
group size
4
2
group
*
5
10
group
.
expands
*
I
0-0~~~~~~~~~~~
l
-2
stable groupsize
*
shrinks
*
l
I
15
20
I
25.
30
-4
current numberof females
FIG. 1. A, The changein groupsize (numberoffemales)afterone generation,as a function
of currentgroup size. Calculationsforthe maingraphused the baseline reproductivevalues.
Hor-izontalarrows,thedirectionof changein groupsize. Verticalarrow,the stationarygroup
size, which is stable in this case. The inset shows that the horizontalarrows would be
mothersunderproduceddaughters,makingthe stationarysize unreversed if high-ranking
stable.
lead to a much higherdegree of instabilityof group size and impermanenceof
groupsthanwould unbiased offspring
productionwithstochasticvariability(Cohen 1969). Conversely, if high-ranking
females leave more daughtersthan do
sex-ratiobiases providean indirectmechlow-ranking
females,thenrank-specific
anism for the regulationof group size, withoutany implicationthat the stable
size is itselfadaptive. This resultinvitesa numberof novel analyses and future
research investigations,some of whichwe returnto in the discussion.
On the basis of this findingregardinginstabilityin group size in the case of
male-biasedratiosforhigh-ranking
forthe
females,we develop the model further
females leave fewerdaughters.Indeed, this is the
case in which lower-ranking
situationin a groupof savanna baboons, Papio cynocephalus,in Amboseli Park,
Kenya (Altmannet al. 1988), the site of a longitudinalstudythat has produced
natalityand mortalitydata thatare used forillustrativepurposes below.
We conducted a series of three sets of numericalsimulationsto investigate
intergenerational
changes in matrilinealstructurearisingfromstatus-correlated
offspring
production.In each set of simulations,an exponentialdistributionwas
used to model the distribution
of survivingdaughtersat each rank.In mathematical terms,f1k = ti(I- _t)k, whereoi is a parameter
thatdependson i; this
was
in
different
each
set
of
simulations.
The
choice
of an exponential
dependence
distributionwas motivatedby evidence fromthe Amboseli population that per
annummortalityand fertility
are fairlyconstantformost of a female's reproductive career (Altmannet al. 1988). As our baseline, we chose oi accordingto the
546
THE AMERICAN NATURALIST
survival.This data set provides
Amboseli data on sex ratioat birthand offspring
an example withquite a strongcorrelationbetweenrankand sex ratio. We additionallyconsideredhypotheticalpopulationsin whichthecorrelationwas reduced
to 50% and 33% of the baseline. This reduction,which can be thoughtof as
A by a constant,maintainsthe same stable group size but decreases
multiplying
the correlationbetween rankand numberof survivingdaughters.
CONSOLIDATION
Computer simulationsdemonstratedthat the entiregroup will be descended
froma single ancestral motherwithina small numberof generations.We refer
to this process as consolidation, and we consider here several aspects of the
consolidationprocess.
IfRi is decreasingin i-that is, productionof daughtersis a decreasingfunction
of rank-then eventuallythe entiregroupwill be descended froma singlefemale
because the females at the top of the hierarchyare more than replacingthemselves and thustheirfemaledescendantsoccupy a largerportionof the hierarchy
than they did. We fix some generationas the beginningof time and treateach
female in this generationas the progenitorof a distinctfamilytree. We say that
the descendantsof the femaleof ranki in thisoriginalgenerationare all members
of the ith matriline.The picturethe reader should have is that each generation
of females is subdivided into distinctmatrilinesand that aftera period of time
one of the matrilines,usually one descended froma high-ranking
ancestor, consolidates the entiregroup.
We defineQi as the probabilitythatthe ithmatrilineconsolidatesthe group, Ti
as the average time untilconsolidation,conditionedon the ith matrilinetaking
over, and M,, + 1 as the average numberof distinctmatrilinespresent aftern
generations.(We defineM,,thisway because thereis always at least one matriline
present; M,, measures the numberof excess matrilines.)We present numerical
resultsfor the behavior of Qi, Ti, and Mn and also show thatit is fairlyeasy to
calculate Qi by solving a systemof linear equations. Notice that Qi is the only
quantityconsidered in this articlethattrulydepends on the stochasticnatureof
our model. One could consider Ai,Ti, and Mn even fora deterministic
model and
arriveat comparable results.
The Number of Matrilinesas a Functionof Elapsed Generations
The simplestway to quantifythe consolidationprocess and the consequential
loss in geneticvariabilitythroughthe maternalline is to count the
within-group
numberof distinctmatrilinesthat are present in each generation.Because Ri
decreases in i, thematrilinesat thebottomofthehierarchyhave a highprobability
of becoming "extinct" (no remainingfemale descendants), and thus we would
expect Mnto approach zero quite quickly.Simulationsshowed thatMndecreases
exponentiallyas n increases. For thevalues thatwe used fromthe Amboseli data,
computersimulationsshowed thatM,,decreased by a factorof about 2.4 in each
generation(fig.2). This means thatonly about 40% of the femalesin the present
BIAS IN OFFSPRING
SEX RATIO
547
100
10
* baseline (Amboseli
E * *
1
excess
matrilines
o
0. 1 -
values)
o
*
*
E
*
50% of baseline bias
33% of baseline bias
o
0.01 -
0.001 0
2
Il
4
l
6
l
l Il
8 10 12 1 4 16 1 8 20
generation
FIG. 2. -Number of matrilinesin excess oftheone thatconsolidatesthegroupas a function
of time. Results shown forthreelevels of correlationbetweenrankand sex ratio.
generationcould be expected to leave any female offspring.Afterfive generations, Mn dropped below 0.5, which says that with an initialpopulation of 25
females,only one female in most simulationshad any descendants in the group
afterfivegenerations.Our simulationswithreduced sex-ratiobias showed qualitativelysimilarresults(fig.2). In all threesets, a negativeexponentialfunction
fitMnwithan R2 of at least 0.997. When the bias was reduced to 33% of baseline,
18% of the transientmatrilinesdisappeared in each generation.
Variationin the Time untilConsolidation
The resultsof the precedingsectionindicatethatthe consolidationprocess can
be extremelyrapid, on the order of five generations.Because those analyses
did not consider potentialvariabilityin consolidationtime,dependingon which
matrilinetook over, we now turnto a considerationof Ti,the conditionalaverage
timeuntilconsolidation,conditionedon the ithmatrilinetakingover. Intuitively,
we would not expect Ti to depend stronglyon i. If an initiallylow-rankingmatriline (large i) is going to consolidate, all the highermatrilinesmust die off.The
most likelyway for this to happen is forall the higher-ranking
matrilinesto die
offin the firstone or two generations;otherwisethey will probably grow too
much. Once the higher-ranking
matrilineshave died off,the low-rankingone is
top rankingand thereforebehaves in the same way as a matrilinethatwas initially
high-ranking.
Therefore,we expect that the time untilconsolidation should be
only slightlylongerforconsolidationby low-rankingmatrilines.This expectation
is borne out forall threesets of simulations.The mean timesuntilconsolidation
forthe baseline amountof correlationand for50% and 33% of baseline bias were
6.3, 9.9, and 14 generations,respectively.The coefficientsof variationfor the
threesets of simulationswere 0.091, 0.073, and 0.083.
Again, we see that for the baseline set, the group typicallyhas a common
THE AMERICAN NATURALIST
548
ancestor five generationsback, independentof the rank of that ancestor. This
provides a nice single index of the rate of genetic homogenizationthat can be
groups.
used to compare two otherwise-similar
For the sake of completeness,we note that Ti is also a functionof the size
of the initialgeneration.However, this dependence is extremelyslightbecause
stabilizationof groupsize proceeds even morerapidlythandoes theconsolidation
we always set the initialsize as the
process. Aftersome initialexperimentation,
we
believe
that other initialconditions
and
size
for
our
simulations,
stationary
would have a very small effecton our conclusions.
The Probabilityof Consolidation
Finally,we estimateQi, the probabilityforeach originalfemalethather matriline consolidates the group. Althoughcomputersimulationsagain provide good
estimatesof the quantities,theycan also be estimatedby analyticalmethods,the
details of which are available fromthe firstauthor.
Figure 3 shows that across all ranks the probabilityof consolidationby any
rank is a constantfractionof thatforthe next higherrank; that is, exponential
decay in Qi is a characteristicof all threesets of simulations(all threeR2 values
For rank4 and
exceeded 0.97), althoughthe decay constantsare quite different.
below (higherranknumber),thereis less thana 10% chance of any one matriline
consolidatingthe group,but the sets differmarkedlyin the probabilityof consolidation by the higherrankingmatriline;compare a 64% chance of consolidation
for rank 1 in the baseline with 28% for rank 1 in the set with 33% of baseline
correlationbetweenrankand sex ratio.As thecorrelationincreases,the disparity
between high-and low-rankingfemalesincreases. Withoutgoinginto details, we
note that this exponentialdecay also arises for continuous-timemodels, where
we can show that the decay constantis almost entirelydeterminedby the sexratiobias at the highestranks.
DOMINANCE
RANK OF MOTHERS AND THEIR DAUGHTERS:
THE GENERATION
GAP
A considerable empiricalliteratureassessing the stabilityof dominance rank,
especially in baboons and macaques, has been reviewedby Silk (1987). In addition, Hausfater et al. (1987) have examined the effectsof variabilityin demographicparameterson dominancestructurewithinoverlappinggenerations.Here,
changesin absolutedominancerankbetweensucceswe investigatewithin-family
sive generationswhen transmissionof relativerankis perfect.Even ifgroup size
dominance rank is stable, and relative rank is peris stable, within-generation
fectlytransferablebetweengenerationsfrommotherto daughter,overproduction
femalesof the various ranksresultsin a difference
of daughtersby high-ranking
betweenabsolute ranksof mothersand daughters.An importantconsequence of
the kind of rank-specificsex-ratiobias that we have been investigatingis that
some daughters have an absolute rank that is very close to their mother's,
fromtheirmother's.This differwhereas othershave ranksthatare quitedifferent
ence can be quantifiedas follows.
BIAS IN OFFSPRING
SEX RATIO
549
o
El
0.1 -*.o
consolidation
probability
* baseline (Amboseli
values)
*
0.01 -
U
0.001
0.0001
E 50% of baseline bias
*
El
* 33% of baseline bias
El
-
-
0
_
2 4
6
8 10 12 14 16 18 20
matriline
FIG. 3.-The probabilitythata matrilineconsolidatesthe group. Results shown forthree
levels of correlationbetween rank and sex ratio.
THEOREM: The expected difference
in rank between a motherof rank i and a
randomlyselected daughteris approximatelyAi,withan errortermthatis small
formost choices off]k
PROOF: If the mothersin ranks 1 throughi - 1 produce D daughters,and the
motherof rank i produces d daughters,thenthese d daughtersoccupy ranks D
+ 1, D + 2, .. . , D + d. Therefore,
a randomly
selectedone ofthesedaughters
has rankD + (d + 1)/2,and the differencebetweenthatand the maternalrank
is D + (d + 1)/2 - i. It is not hardto see thatthe means of D and d (conditioned
on d beinggreaterthanzero so thatthereis a daughterto select) are ILz Rr and
Ril(l - jfo).
Some algebrashowsthattheaveragerankdifference
is
E (D
d + I)
2 -1
+
+ Ai
2
~2(1
Ri7ft,
(I
-J>,)V
If it were not forthe factorof 1/(1 - io) in the conditionalexpectationof d, the
right-handside of equation (1) would be exactly equal to interpolatingAXat a
midpointbetween i - 1 and i. The amountby whichthe average rankdifference
deviates from(Ai + Ai-1)/2,the second termon the rightof equation (1), could
in theorybe large,butformostreasonable choices Offk, it stays small. The error
is small because fi0 is usually small forthe values of i forwhichRi is large, and,
when 1 - J0 is small,Ri is also small.
rank differThus, when the errortermis small, the average daughter-mother
ence is approximately/v,and this rank differenceis thereforegreatestnear the
rankwhereA\is greatest.This can be understoodintuitively
because thedaughters
of middle-ranking
females are pushed veryfar down the hierarchyby the large
mothers.High-ranking
overproductionof daughtersfromhigher-ranking
daughters do not experience as much disparitysimplybecause thereare few higher-
550
THE AMERICAN NATURALIST
rankingmothers,and low-rankingdaughtersare not displaced far because the
mothersis almostbalanced by underprooverproductionamong the high-ranking
ductionby mid-to low-rankingmothers.
females overUnder our assumptionthatRi is decreasingin i (higher-ranking
produce daughters),we know that A\has the shape shown in the main part of
figure1. The firstzero of A\is the stable groupsize, and A\peaks near the middle
of the group. We can thereforeconclude thatdaughtersof mid-ranking
mothers
findthe greatestdisparitybetween theirranksand theirmothers'ranks. For the
parametervalues in our examples, the errorneverexceeds halfa rank. The rank
differenceis greatestat i = 11, about halfwaydown the hierarchy.Daughtersof
a rank 11 motherare, on the average, about nine and a half positions lower in
the hierarchythan was theirmother.
DISCUSSION
The present investigationhas departed frommost earlier discussions in the
evolutionaryliteratureby modelingdemographicand behavioralconsequences of
biased offspringsex ratios that vary systematicallywithininternallystructured
groups.The effectswe have examined,moreover,are ones thatare realized after
only a few generations,changingthe social environment,and thus the context
forevolution,withinand among groups. Our models consider species in which
thefemalesof a groupformordered,stable dominancehierarchies,relativedominance is passed frommotherto daughter,offspringsex ratios are a monotonic
functionof maternaldominancerank,and males, but not females,disperse from
theirgroupsof birthby adulthood.
One of the most strikingresultswas the instabilityof group size formodels in
whichhigh-ranking
mothersoverproducesons. In sharpcontrast,overproduction
of daughtersconstitutesan indirectmechanismfor stabilizinggroup size. Differingdirectionsof bias or the absence of systematicbias in offspring
sex ratios
are reflectedin the size distributionof groups withina populationat any given
time and of individual groups over a number of generations. Consequently,
group-sizedistributionscan provide a test of the priorhistoryof offspringsexratiobias in a population.Appropriatetests,however,mustbe based on elaborations of existingmodels of group-sizedistributions(Cohen 1969, 1971, 1972) to
incorporateoffspring
bias. These tests should be designedto use data fromsingle
groupsover timeand fromgroup censuses withinmultigrouppopulations.
Does it matterifgroup size is unstable?The size of a grouphas consequences
forday ranges, social coordination,minimalrequirementsforresources such as
sleepingsites, and so on. If the group size or structurethatan individualexperiences duringmaturationis a poor predictorof the characteristicsof the group
thatthe individualwill experience duringadulthood,especially when raisingits
own offspring,
then the potentialadvantages providedby learningduringa long
stage of immaturity
mightbe dilutedor totallynegatedby thislack of continuity.
The wisdom of individualsand of theirrelativesin the group would be of little
use in a situationof poor predictability,of moderateuse in a neutralsituation,
and of considerableadvantage when variabilityis low.
BIAS IN OFFSPRING SEX RATIO
551
fefemales relativeto lower-ranking
Overproductionof sons by high-ranking
males would probablyproduce additionalreductionin homeostasisof social environmentbeyond that caused by instabilityof group size, because low-ranking
ones. This
maternallineages will expand withina group relativeto high-ranking
would increase the likelihoodthat coalitions of low-rankingrelativeswould be
animalsand
able to challengesuccessfullythedominancestatusof higher-ranking
therebydestabilizethe social structure.The extentof intra-and intergenerational
rank stabilityand the consequences of breakdownsin stabilityshould suggest,
albeit not provide as directa test as would group sizes, the historyof offspring
sex-ratiobiases or absence of persistentbiases in a population.In thefewprimate
populationsforwhich thereare appropriatedata, instabilityof the femaledominance hierarchyseems to be rare (reviews in Samuels et al. 1987; Silk 1987),
females has not been
suggestingthat overproductionof sons by high-ranking
persistentor pervasive.
The remainingpredictionsof the model that we explored incorporatedthe
additional assumption that offspringsex ratios were a decreasing functionof
femalesoverproduceddaughtersrather
dominancerank,thatis, thathigh-ranking
than sons. We found that, althoughdaughtersassume their mothers' relative
fromthat
dominancestatus,theabsolute rankof daughterscan differsignificantly
femalesand the raredaughters
of theirmother.Daughtersof the highest-ranking
of low-ranking
femaleshave dominanceranksclose to those of theirmothers.In
contrast,daughtersof mid-rankingfemales will experience a large generation
gap. If individualsthatare faced with similarconditionsas adults to those they
experiencedas immaturesare at an advantage because theyare betterprepared
females will be at an advantage, in this regard,
(Fairbanks 1989), high-ranking
ones.
relativeto all but the rare lowest-ranking
In additionalanalyses we exploredtherelationshipbetweentherank-biasstructureand the relatednesswithinthegroupsof a populationusingparametervalues
fromdata on baboons in Amboseli. We saw thatin a few generations,the female
componentof a group was composed entirelyof membersof the same matriline
(i.e., all descendantsof a singlefemale)and thatbeforethat"consolidation," the
group experienced approximatelya 60% reductionfromone generationto the
nextin the numberof matrilinesrepresented.The commonfemaleancestor was
female in the startinggeneration;consolidationwas
usually the highest-ranking
rarely(14%) by a matrilineotherthanone of the top two; and the identity(rank
number)of the consolidatingmatrilinehad littleeffecton the time to takeover.
When we examined a hypotheticalexample with a rank-specificsex-ratiobias
that was much weaker than in Amboseli, we found an increased variabilityin
whichmatrilineconsolidatedthe groupand an increased timeto takeover. Even
for the Amboseli model, however-in which consolidationwas usually within
fivegenerationsand was by the highest-ranking
matrilineover 60% of the timein any one generation,the highest-ranking
femaleleftthe most femaleoffspring
less than 20% of the time. Calculations such as these are importantto keep in
mindwhen implicationsare drawnfromempiricalfindings,and theyare particularlyimportantfortaxa such as primatesforwhichdata rarelyspan a fullgeneration and are usually available for only one group or at most one population.
552
THE AMERICAN NATURALIST
Matrilinealaspects of these investigationshave been explored in more detail
usingmodels thattreattimeand, to some extent,rankas continuous(M. Altmann
1988). Genetic data have been gatheredfora few primatepopulations(e.g., Duggleby 1978; Melnick 1987), and other studies are underway.These empiricalinvestigationsare usually undertakenprimarilywitha focus on determining
paternityor differential
reproductionby males, but, ifcombinedwithbehavioraldata,
theyshouldbe able to providethedata necessaryforassessing evidence of persistentpriorrelationshipsbetween rankand offspring
sex ratios.
It is interestingto note thatthe mathematicaltools used in this analysis were
originallydeveloped in the contextof a populationin whichreproductionpotentially varied with social status (Galton and Watson 1874, reprintedin Keyfitz
and Smith 1977). Galton and Watson were concerned withdetermining
whether
upper-class English familynames were disappearingbecause of lower fertility
amongthe upper class or simplybecause of randomfluctuations.Unfortunately,
theyconsideredonlythe nullhypothesisof a homogeneouspopulation,and mathematicalintractability
led the subsequentmathematicaltheoryof branchingprocesses away fromthe originalidea of socially determinedreproduction.Consequently,fewmodels of populationgrowthhave incorporatedboth social structure
and reproductivebehavior.
In themodels analyzed here,we assumed thatfemaleshave theabilityto adjust
and that they do so based on their
facultativelythe sex ratio of theiroffspring
social statusrelativeto the higher-ranking
females. Facultativeadjustmentof sex
ratios is advantageous for species in which thereare conditionaladvantages of
producingoffspring
of each sex and forwhichthese conditionschange duringthe
reproductivespan of a singleindividual,differbetweenparentsand offspring,
or
differamong siblings.Heritabledifferencesin sensitivityto conditionalcues may
be subjectto selection.These cues may arise in several different
milieus;ecological stressmightfavora different
offspring
sex ratiothan social stress. When the
cues varyamong closely related species or perhaps even populationsof a single
species, it is particularlydifficult
to identifythe appropriatelevel foranalysis.
Proximally,sex-ratioadjustmentmay be effectedthroughmechanismssuch as
behavioraltimingof conceptionor physiologicalchanges thatdifferentially
affect
male-and female-bearing
sperm(e.g., Guerroro1970;James1971, 1983). Whether
sex-ratioadjustmentitselfhas been subjectto selectionor whethertheadjustment
is a by-productof mechanismsthathave some otherprimaryconsequence is not
at all clear.
When attentionfirstfocused on the evolutionof facultativeadjustmentof offspringsex ratios,it was arguedthatthe relationshipbetweensex ratioand parental condition or quality "should" be positive (Trivers and Willard 1973), for
example, thathigh-ranking
females should overproducesons ratherthan daughters.The reasoningwas based on the assumptionthatmanyspecies are relatively
polygynous,in which case, males typicallyhave highervariance in reproductive
success thando females.The implicitleap was thenmade to conclude thatmothers in good conditioncould, therefore,influencethe reproductivesuccess of their
sons morethanthatof theirdaughters,and empiricaldemonstrationof a positive
BIAS IN OFFSPRING SEX RATIO
553
relationshipbetween maternalconditionand sex ratio in a populationwas then
equated with demonstrationof adaptive sex ratio (but see Clark 1978; Myers
1978).
Subsequent authors (e.g., J. Altmann 1980; Silk 1983; van Schaik and van
Noordwijk 1983, all focusingon cercopithecineprimates)demonstratedthatthe
modelwas muchless generalthanoriginallysupposed. They pointedout thatsex
differencesin variabilityof reproductivesuccess were overestimatedby dependence on cross-sectional or short-termempirical studies, and that short-term
differencesin male and femalereproductivesuccess were probablyconsiderably
greaterthanlifetimeones preciselybecause of the instabilityof male lifehistories
relativeto those of females (e.g., Hausfater 1975; Samuels et al. 1987).
In addition,even relativelypolygynousspecies oftenexhibitdemographicsystems and formsof social structurein which motherspotentiallyinfluencetheir
daughtersmore than theirsons (see, e.g., Silk 1983). The operationof opposing
factorscould resultin predictedadaptive relationshipsbetween maternalquality
and offspringsex ratio that are positive, negative, or zero, dependingon the
particularbiological systemunderconsideration.This calls into question studies
in which "tests" of adaptive sex ratios were based on the fallacious assumption
that a relationshiphad to be positive to be adaptive and in which a positive
relationshipwas taken as primafacie evidence forthe evolutionof adaptive sex
ratios. If one wanted to follow the general Trivers-Willardapproach and ask
which sex-ratiorelationshipmightbe favored in a given biological system, it
would be necessary to evaluate the Trivers-Willardassumptions by obtaining
estimatesof the relativeeffectsof motherson theirsons' and daughters'reproductive success (see, e.g., Clutton-Brock1985; Hrdy 1987), data that have in
generalbeen unavailable formost species.
In the present series of investigations,we have approached the question of
adaptive sex ratios froma totallydifferentperspective. We have asked about
the consequences of quality-biasrelationshipswithincertaintypes of biological
systems,regardlessof whetherthese relationshipsmighton otherbases, such as
seem to be of selecpolygynyor sex differencesin maternaleffectson offspring,
tive advantage or disadvantage. Despite the apparent ultimateadvantages of a
quality-biasrelationship,the relationshipcan persistonly ifit is compatiblewith
the otherbehavioral, social, and demographicstructuresof the species. Models
of effectson the levels and time scale thatwe have consideredhere are not only
of intrihsicinterestforstudies of demographyand social structure,but theyalso
address the potentialcontextforevolutionof biased offspring
sex ratios.
ACKNOWLEDGMENTS
The authorswould like to thankJ. Cohen and S. Altmannfortheircomments
on earlier draftsand for discussions and supportwhile the manuscriptwas in
preparation.We would also like to thankanonymousreviewersfortheirhelpful
criticism.Supportwas suppliedby the CourantInstituteat New York University,
the SimulationResource at Universityof Minnesota,and grantHD-15007.
554
THE AMERICAN NATURALIST
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