The Optimal Balance between Size and Number of Offspring
Author(s): Christopher C. Smith and Stephen D. Fretwell
Source: The American Naturalist, Vol. 108, No. 962 (Jul. - Aug., 1974), pp. 499-506
Published by: The University of Chicago Press for The American Society of Naturalists
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Vol. 108, No. 962
The American Naturalist
July-August 1974
THE OPTIMAL BALANCE BETWEEN
SIZE AND NUMBER OF OFFSPRING
CHRISTOPHER C. SMITH AND STEPHEN D. FRETWELL*
Division of Biology, Kansas State University, Manhattan, Kansas 66502
In searchingfor patternsof energyexpenditureby parents on their offspring,two relationshipsseem intuitivelyobvious. (1) As the energyexpended on individual offspringis increased,the numberof offspringthat
parentscan produceis decreased. (2) As the energyexpendedon individual
offspringincreases,the fitnessof individual offspringincreases. The relato visualizeis the one betweenthe energyexpended
tionshipwhichis difficult
and the fitnessof the parent whose genotypeinfluon individual offspring
ences how energyis distributedto the young. Here we presenta graphical
model to representthe relationshipbetweenenergyexpendedon individual
offspring
and the fitnessof parents.The validity of the graphical model is
then demonstratedby standard analytical procedures.
THE MODEL
The model is based on the one assumptionthat at any point in an organism's life historythere is an optimumpercentageof available energy
that should be divertedto reproductionto maximizethe parents' total contributionto futuregenerations.Energy available for reproductionis thus
limitedto a set finiteamount at any given time. Our model demonstrates
sizes and numbersof
how thisfiniteamountcan be distributedinto different
The validityof the basic assumptionis discussedlater.
offspring.
The abscissa of the model in figure1 plots both the size and numberof
individual offspring.Their relationshipon the abscissa is determinedby
the basic assumptionof a set finiteamount of energyavailable for reproStraightlines
duction.The ordinateplots the fitnessof individual offspring.
throughthe origin are fitnessfunctionsor lines of equal fitnessfor the
parent. As we show below, these fitnessfunctionsare similar to those of
Levins (1968), although their orientationis different.Greater slopes of
fitnessfunctionscorrespondto higherparental fitness.No causal relationship betweenthe abscissa and ordinateis implied in the fitnessfunctions.
is linearly proWe do not wish to imply that expectedfitnessof offspring
portionalto theirenergyexpense over a large range of energyexpenses in
any real population. Fitness functionsare simply a consequence of the
values of the ordinateand abscissa.
* Order of authorship was determined by a flip of a coin.
499
THE AMERICAN NATURALIST
500
20SUCCESSFUL
OFFSPRING
0.363
0103~/,1
6"/
0O.2
CAD
C= 0.0
0
5
Iy=EFFORT/OFFSPRING.
OFOFFSPRING
co
Ny=NUMBER
200
10
100
15
67
20
50
25
40
FIG. 1.-Adaptive functions
of parentalstrategies.The verticalaxis describes
parent types with offspringof a certain fitness,as noted. The horizontal
axis describesparent types that put a certainfractionof available energy
into each offspring(or equivalentlyproducea certainnumberof offspring
fromthe 1,000 units of energyavailable). The diagonal lines in the twodimensionalspace are sets of parentaltypesall of whosemembersare equally
to futuregenerations.
Diagonal linesof steeperslope
successfulin contributing
representmore successfulparenttypes.
Real populationshave a curve of expectedfitnessfor individual offspring
(Wy) in relationto their energyexpenseswhich we assume to be similar
in shape to the solid curve in figure2. The curve leaves the abscissa at some
must at least have the energy
point greaterthan zero because an offspring
of a completeset of geneticinformationbeforeit has greaterthan zero fitness. The curve is definedonly for offspringsizes up to that size where all
parental investmentgoes to just one offspring.Beyond that size, expected
fitnessof the young is irrelevant,since the parent could not produce even
convex for the interceptof
one. We consider in detail curves sufficiently
the derivativeto pass throughzero for at least one point on the curve.
Apparentlythis case applies to mostof nature,as mostclutchesare greater
than one, but we cannot prove this. Such convexitymight derive biologically fromtherebeing some limit to the fitnessof individual offspring
that cannotbe reachedno matterhow greatthe parental investment.
Then, when the clutch is small enough (Iy large enough), a unit decrease in the clutch will produce a relativelylarge increase in fy but a
relativelysmall increasein Wy,producingconvexity.If such convexitydoes
not obtain, then, assuming a continuous curve, one can prove from the
shouldbe producedper effort.
graphicalanalysisthatonlya singleoffspring
To determinewhichenergyexpenseper offspring
gives the highestparental fitness,one can draw a series of fitnessfunctionsintersectingthe curve.
The steepestfitnessfunctionthat intersectsthe curve is the fitnessfunction
tangentto the curve (fig. 2). Thereforethe optimumenergyexpense per
offspringis at the point where a fitnessfunctionis tangent to the curve
SIZE AND NUMBER
501
OF OFFSPRING
0.3-
Be
40. 0.2
0.1
0
5
10
15
20
25
5
10
15
20
25
1.0
,,,
0.8
0.6-
an
0.4
?
0.2-
0.00
Iy EFFORT/OFFSPRING
FIG. 2.-Fitness set analysis of parental strategy. The curved line in the
upper part of the figureis the set of possible combinations of parental types,
each type expending a certain amount of effortper offspringand producing offspring of a certain fitness.The shape of this curve is typical of those that will
produce clutches of more than one offspring.The curve can be thought of as
a cause-effect relationship (effort per offspring determines the offspring's
fitness). Two adaptive functions, drawn as in fig. 1, intersect the curve of
possible parental types. The intersection involving the adaptive function of
highest slope determinesthe optimal parental type defined in terms of effort
per offspring.Intersections of other adaptive functions determine the relationship between parental fitness and parental type (effort per offspring),
plotted in the lower graph. Note that all parental types in a single adaptive
function are definedto have equal fitness,so that the two intersectionsof any
lower than optimal adaptive function produce points on the bottom graph
with the same value of parental fitness.
of fitnessfor individual offspring.
The ratio of the slope of a fitnessfunction throughanotherpoint on the curve to the slope of the fitnessfunction
tangentto the curve gives thefitnessto the parent of expendingthe amount
of energyper offspring
representedby that otherpoint (fig.2). The model,
therefore,allows one to visualize in two dimensionsthe fitnessaccruing to
502
THE AMERICAN NATURALIST
both offspringand parent fromvariation in the energyexpended per offspring.
Now, the graphical analysis given above can be presentedanalyticallyto
prove that the point of contactof the sloping line throughthe origin with
size for the parent.
the young's fitnesscurve is the optimumoffspring
Let Wy be the fitnessof the young. We assume that energyinvestment
per young (Iy) increasesthe young's fitness,but that the rate of increase
eitheractually reacheszero or gets close to zero for some high but realistic
value of energyinvestment.This impliesa curve of fitnessof young on cost
of young (fig.2) thatis convexforall Iy greaterthan someparticularvalue.
Let Ny be the numberof youngproducedby the parent,and Ip be the total
energyinvestedby the parent.We assume that
(1)
Ny=Ip/ly.
This is approximate and simplifying;Ny must in reality have integer
values, and yet Ip/ly can take nonintegervalues, unless we restrictly to
certain values (we assume Ip is fixed). Ricklefs (1968) analyzed compromisesthat mightbe made at small clutch sizes, when nonintegervalues
of Ny are optimal.We again proceed,recognizingthat we will obtain noninteger optimal values for Ny, which require furtheranalysis (see Discussion).
Parental fitnessforthe clutchis denotedWp, and thatforthe young,Wy.
We claim that
Wp=Wy*Ny
=Wy-
P
(2)
forvalues on the youngfitnesscurve.
We now assume that productionof young in the presentdoes not influence the futurefitnessof parents.Natural selectionshould then maximize
Wp. We findthe maximumvalue of Wp by maximizingWy/ly in equation
(2) (Ip fixed). But Wy/ly is the tangentof the angle formedby the sloping line throughthe origin and by the ly axis. Since the tangent of an
angle and the angle are positively monotonic,maximizing Wy/ly also
maximizes a. This proves that the line with the maximum slope, which
intersectsthe young's fitnesscurve,also touchesthat curve at the point that
maximizesadult fitness.
These straightlines are analogous to the adaptive or fitnessfunctionsof
fitnesswith size is a fitnessset.
Levins (1968), while the curve of offspring
We now establishthis correspondence.
In an adaptive function,points on a single line or curve representalternative parent phenotypesall of which have the same fitness.However, all
phenotypesmay not be realizable. The statementof an adaptive functionis
this: if we could find a collectionof parental types with certain specified
properties,all would have the same fitness.Here, the propertiesare fitness
Parental fitnessis the product of the fitnessof each
and size of offspring.
timesthe numberof them (eq. [2]).
of its offspring
SIZE AND NUMBER
OF OFFSPRING
If all of a group of parents have the same fitness,then Wp
from equation (2),
WY
503
k, and
k
( N ) -(3)
Ny
From (1),
1
ly
by )
N
Ip
So, substitutingin (3) we obtain
'WY=
k
'p
* 1.
is assumed to be constant; and so
Like k, Ip, the total parental investment,
we have derived an equation for a straightline throughthe origin,with
slope k/lp. This straightline is made up of points each of which defines
a parentwithyoung of a certainsize (ly) and a certainfitness(Wy). All
theseparentshave the same fitness(Wp = k), and so the line is an adaptive
function.Adaptive functionswith higherslopes must have higher Wp (Ip
is constant). By Levins' arguments,we know that the optimumphenotype
is found on the fitnessset (the real possibilities),by taking the point of
intersectionwith the adaptive functionof highestfitness(slope).
DISCUSSION
A numberof life historyphenomenamightappear to contradictour basic
assumptionthat at any stage in an organism's life history an optimum
fractionof its available energyshould be divertedto reproductionto maximize total reproductionduring its lifetime. E. R. Pianka and D. H.
Feener (personal communication)argue that as an individual's reproductive value declines after the onset of reproductivematurity,it maximizes
fitnessby divertinga successivelyhigherfraction of its available energy
to reproductionwith each successive,evenly spaced, reproductiveeffort.
Pianka and Feener's argumentson the effectof changing reproductive
value specifywhat fractionof available energyshould be expended on reproductionat given points througha life history,but not how the fraction
expended should be divided into numbersof offspring.The fact that the
optimumfraction of available energy diverted to reproductionchanges
throughouta life historyin no way negates our model.
For organismslike birds,mammals,and subsocial Hymenoptera(Wilson
as theydevelop,
provisiontheiroffspring
1971), whereparentsprogressively
is considerablygreaterthan that
energyexpended per individual offspring
needed to produce a fertilizedegg. Reproductiveeffortin our model, the
fractionof available energydivertedto reproduction,includes all formsof
parental care. Energy used to provisionprogressivelyor to defend an off-
504
THE AMERICAN
NATURALIST
spring is part of the energyexpended upon that individual offspringand
With progressiveprovisioning,
also part of the parent's reproductiveeffort.
energy expended per offspringis difficultto measure, but in theory our
model still applies.
This model does not hold when the result of one reproductiveeffort
affectsfitnessof individual offspringin successive reproductiveeffortsof
the same parent. Social Hymenoptera (Wilson 1971) and a few birds
(Lack 1968) demonstratesuch an interactionof successive reproductive
effortswith the parental energy expenditureon individual offspring.In
from
fromearly clutcheshelp to provisionoffspring
bothgroups,offspring
later clutches.Feeding of siblingsis thoughtto be an evolutionaryconsequence of kin selection (Hamilton 1964; Wilson 1971), where the unit
under selectionis the extended genetic familyrather than the individual
parent. Our model thereforeapplies to partitioningof energyby the extended geneticfamilyratherthan by an individual parent.
For mammalsexhibitingpostpartumestrus,one littermay decrease the
energy available for another. The same problem may occasionally arise
when young fromsuccessivelittersdepend on a parent at the same time,
as in beaver (Bourliere 1956), chimpanzees (van Lawick-Goodall 1968),
and humans. Overlapping parental care of successivelitters generally reduces the total energyavailable to each litter.It need not affecthow the
in any litter.We know
reducedtotal is divided among individual offspring
that
less
is
shows
of no informationthat
energy expended per individual
offspringin successivelitters.
A parameternot included in our model is the period of time over which
energy is collected for a reproductiveeffort.This parameter is directly
related to a significantproblemof the model. How does selectionrespond
to a situationin whichthe energyavailable for reproduction,when divided
by the optimum energy for each offspring,gives a clutch size of 1.5?
is greatestfor small clutches (Ricklefs
The problemof fractionaloffspring
of
In
clutches
1.5,
1968).
optimal
33% of the reproductiveeffortis unused
by a real clutch of one, while for optimumlittersof 20.5, less than 3%
of the energyis unused by a real clutchof 20. Ricklefs (1968) summarized
extensiveempiricalevidencethat birds with smaller clutches (one or two)
have a greater between-speciesvariance in their developmentrates than
species with larger clutches (three to five). He argues that this difference
in developmentrates could be explained if clutcheswhich would include
fractionalyoungunder maximumdevelopmentrates and maximumfeeding
rates can be increasedto the next largestwholenumberby evolvingreduced
rates of development.Adjusting fractionsin small litterswould lead to a
greaterslowing of developmentrates that would occur for larger litters.
Empirical evidencefrombirds indicatesthat fractionalyoung are adjusted
by changes in the time over which energy is gathered. For our model,
problemsof time of energyaccumulationand fractionaloffspringappear
to cancel each other,the firstbeing adjusted to compensatefor the second.
A flexibilityin the time of energyaccumulationwould give flexibilityto
SIZE AND NUMBER
OF OFFSPRING
505
the numberof youngalong the abscissa of our model and make energycost
per young the only significantvariable on the abscissa (fig. 2).
Althoughour model probably applies to all organisms,it is of greatest
utilityin species with a large clutch size and no parental care. For these
sizes is closelyestimatedby
of different
speciesthe relativecost of offspring
their relative caloric content.If the most common (and presumablyoptisuch as acorns or salmon eggs contained10 units of
mum) size of offspring
energy,it would be safe to assume,on the basis of our model,that an offspringcontainingnine units would have a fitnessless than nine-tenthsthat
of the 10-unitindividual. In mostcases fitnesswill be measuredby relative
survival. However,in species such as some desert annual plants, competition resultsin decreased fecundityratherthan increased mortality(Went
1955). The competitiveadvantage during early growthresultingfrom a
larger parental investment(e.g., larger seed size) may not be expressed
until a seed has grownup to reproduceitself.
Our model integratesassumptionscommonlymade by others. Its main
advantage is in allowing these ideas, which are usually separated, to be
consideredtogether,both graphicallyand analytically,in only two dimensions.Because the model does not specifyhow energyis expended,or what
advantage is gained from it, effectsof competition,predation, and the
physical environmentcan be illustrated simultaneously.The effectsof
these three environmentalfactors can also be isolated and treated independently.For example, a 10% decrease in the energy contentof a seed
mightbe totallyat the expense of endospermand embryoor totally from
seed coats or fromsome combinationof the two. If one could find seeds
within a species that expressed all the above alternatives,one could test
to see whetherthe reductionin endospermand embryoled to reduction
of survivalin competitionwithotherplants and reductionin seed coats led
to increasedmortalityfromseed eaters.This lets one measure the adaptive
tissues. Thus, our model can be
value of the energyexpended in different
used to expressthe effectof competition,predation,or any otheraspect of
the environmenton the fitnessof offspring.
SUMMARY
fitnessof
The relationshipbetweenthe energy expended per offspring,
and parental fitnessis presentedin a two-dimensionalgraphical
offspring,
model. The validity of the model in determiningan optimal parental
strategyis demonstratedanalytically. The model applies under various
conditionsof parental care and sibling care for the offspringbut is most
useful for species that produce numeroussmall offspringwhich are given
no parental care.
ACKNOWLEDGMENTS
We thank Eric Pianka for criticismof the manuscript.Reviewers' commentswere helpful in clarifyingsome arguments.
506
THE AMERICAN NATURALIST
LITERATURE CITED
Bourlibre,F. 1956. The naturalhistoryof mammals.2d rev. ed. Knopf, New York.
364 pp.
Hamilton,W. D. 1964. The geneticevolutionof social behavior.I and II. J. Theoret.
Biol. 7: 1-52.
Lack, D. 1968. Ecological adaptationsfor breedingbirds. Methuen,London. 409 pp.
Princeton UniversityPress,
Levins, R. 1968. Evolution in changingenvironments.
Princeton,N.J. 120 pp.
Ricklefs,R. E. 1968. On the limitationof brood size in passerinebirds by the ability
of adults to nourishtheiryoung.Proc. Nat. Acad. Sci. 61: 847-851.
van Lawick-Goodall,J. 1968. The behaviorof free-livingchimpanzeesin the Gombe
StreamReserve.Anim. Behav. Monogr.1:161-311.
Went,P. 1955. The ecologyof desertplants. Sci. Amer. 192: 68-75.
Wilson,E. 0. 1971. The insectsocieties.Belknap,Cambridge,Mass. 548 pp.