The Optimal Clutch Size of Insects When Many Females Oviposit Per Patch Author(S): Anthony R

The Optimal Clutch Size of Insects When Many Females Oviposit Per Patch Author(s): Anthony R. Ives Reviewed work(s): Source: The American Naturalist, Vol. 133, No. 5 (May, 1989), pp. 671-687 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2462074 . Accessed: 15/02/2012 07:21

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http://www.jstor.org Vol. 133, No. 5 The American Naturalist May 1989

THE OPTIMAL CLUTCH SIZE OF INSECTS WHEN MANY FEMALES OVIPOSIT PER PATCH

ANTHONY R. IVES* Departmentof Biology, PrincetonUniversity, Princeton, New Jersey08544

SubmittedMay 4, 1987; Revised July25, 1988; Accepted August 2, 1988

Manyinsects lay clutcheson discretepatches of larval food. Examples include parasitoidwasps ovipositing on insecthosts, flies on fallenfruit, and butterflies on theleaves of isolatedhost plants. For theseinsects, determining the number of eggsthat a femaleshould lay per patchis akinto an optimal-foragingproblem: how shoulda femaledistribute her eggs among patches to optimizeher lifetime fitness?When more than one femalelays eggson thesame patch, the best clutch size dependson theclutch sizes adoptedby all of theother females in a population.One way of examiningthis type of group situation is to applythe idea of an evolutionarilystable strategy (ESS). Whenadopted by all individualsin a population,an ESS cannotbe betteredby an individualemploying a differentstrategy (MaynardSmith and Price 1973; Oster and Wilson1978). Parker and Begon (1986) presenteda modelto analyzethe effectsof differentenvironmental and pheno- typicparameters on theoptimal clutch size and eggsize forinsects whose larvae competeon discreteresource patches. One oftheir results repeats a conclusionof Parkerand Courtney(1984) and Skinner(1985): as theaverage number of females ovipositingper patch increases, the evolutionarily stable clutch size decreases.In derivingthis result, these authors considered only a fewof the possible functions describingcompetition among larvae in a patch.For different,but also realistic, curves describinglarval competition,the evolutionarilystable clutchsize in- creasesas theaverage number of femalesovipositing per patch increases. Here I discusshow theevolutionarily stable clutch size of insectsdepends on the type of competitionamong larvae withindiscrete patches. This requires examiningdifferent functions that describe larval competition. I first show which typesof larvalcompetition promote increases or decreasesin theevolutionarily stableclutch size withincreasing numbers of females ovipositing per patch. This discussionuses a simplemodel governing female oviposition behavior. Second, I extendthe simple model to ask whetherthe conclusions it generatesare qualita- tivelyaltered by complexitiesthat affect oviposition in nature.Third, I construct larval-competitionfunctions based on assumptionsabout the feeding behavior of

* Present address: Departmentof Zoology, NJ-15,University of Washington,Seattle, Washington 98195.

Am. Nat. 1989. Vol. 133, pp. 671-687. ? 1989 by The Universityof Chicago. 0003-0147/89/3305-0008$02.00.All rightsreserved. 672 THE AMERICAN NATURALIST

Z <,, 25- z 20- - U. '2 15]

.0E L) o 5 1 1'5 20 25 Numberof Eggs, N

FIG. 1.-The combinedfitness of all larvae, Ns(N), as a functionof the numberof eggs laid on a patch, N. Functionsused fors(N) are givenin equation (1): dottedline, s1(N) withP,u =

4.0 andet = 0.05; dashedline, s2(N) with[L2 = 5.44and r = 0. 1; solidline, s3(N) withp3 = 6.32, a = 1/3o,and b = 4. larvae withinpatches. This analysis shows that small, quantitativedifferences in larval feedingbehavior can produce competitionfunctions that yield opposite predictionsabout how the evolutionarilystable clutch size varies withincreasing numbersof femalesovipositing per patch. This warnsagainst inferring the behavior of the evolutionarilystable clutch size fromonly qualitative knowledge of larval feedingbehavior. Fourth, I give examples fromthe literatureof insects showing diverse patternsof larval competitionwithin patches. These examples include larval-competitioncurves that promoteeither increases or decreases in the evolutionarilystable clutch size as more females oviposit per patch.

A SIMPLE MODEL FOR THE EVOLUTIONARILY STABLE CLUTCH SIZE Parkerand Begon (1986) analyzed manyof the factorsthat influence clutch size in insects. To examinejust one of these factors,I presenta simple,special case of the model given by Parkerand Begon (see also the similarmodels in Parkerand Courtney1984; Skinner1985). Assume thatN eggs are ovipositedby one or more femaleson a discretepatch of larvalfood. Also, assume competitionamong larvae such that the per capita fitness(survival times the adult fecundityof females or timesthe matingsuccess of males) of the larvae, s(N), is a decreasingfunction of N. The formsof s(N) used here are

IRI(I - tN) N < 1I/o s1(N) = a>0, RI >0 0O N2 I/otl

s2(N) =p2e r > 0, R2 > (

s3(N) = ,u3(1 + aN)-b a > 0, b > 1, R3 > 0. For each of these larval-competitioncurves, figure1 gives a graph of the combined fitnessof all larvae survivingper patch, Ns(N), as a functionof the number OPTIMAL CLUTCH SIZE OF INSECTS 673 of eggs laid. The curve sl(N) is thatused by Parkerand Begon (1986) and implies severe competitionamong larvae. Curves s2(N) and s3(N) are taken fromthe ecological literatureas popular descriptionsof competition(see May 1981). In s3(N), larval fitnessdrops more rapidlywith increasing N whenthe parameterb is large,implying severe competitionat highdensities. As b approaches infinityand a approaches zero, curve s3(N) converges to s2(N) as a limitingcase. For all curves, ,u scales the overall larval fitnessand is set to give each curve the same maximumvalue of Ns(N). To calculate the evolutionarilystable clutchsize forthe simplestpossible case, assume thatthe numberof eggs a female lays on one patch has no effecton the numbershe can lay on otherpatches. Therefore,the evolutionarilystable clutch size fora given female depends only on the costs and benefitsassociated witha single oviposition episode. This is the same assumption used to calculate the "Lack optimalclutch size" when only one femaleoviposits per patch (Charnov and Skinner1984; Godfray1986). Furtherassume thatF femalesoviposit on each patch. If the fates of the eggs and resultinglarvae fromall femalesare identical, thenthe fitnessof a female layingn eggs per clutchwhen all otherfemales lay n' eggs per clutch, w(nIn,),is given by

w(n|n,) = ns[n + (F - 1)n]. (2) The evolutionarilystable clutchsize, n*, is the solutionof the equation (Oster and Wilson 1978; Parker and Begon 1986)

awl/nln=h = 0 , (3a) providedthat

a2W 0n.2n n* ? (3b) For each larval-competitioncurve, the evolutionarilystable clutch sizes are

n*= 1/t(F + 1),

n*= 1/r, (4) n*= 1/a(b-F) forF

U) U) 4

0. 4

0 5 10 15 20 25 Clutch Size, n

FIG. 2.-The combined fitnessof the larvae froma single female when she oviposits a clutch size n on a patch already containing20 eggs laid by two otherfemales. Dashed line, s2(N); solid line, s3(N); parametervalues as in figure1. Survivalfor the thirdfemale is zero forsl(N) and is thereforenot shown. some value n. If the optimalclutch size forthe single,deviating female is the same as the clutchsize assumed forthe otherfemales, then n' is theevolutionarily stable clutch size; any female that deviates fromthe ESS suffersa suboptimalclutch size. To see the effectof the threedifferent larval-competition functions given in figure1, consider the case in which three females oviposit on the same patch. Assume thattwo of thefemales each oviposit 10 eggs; 10 eggs is the optimalclutch size when females oviposit singlyper patch, because n = 10 gives the maximum value of ns(n) for all three larval-competitioncurves. The fitnessof the third female,depending on her clutch size, is given in figure2. For s3(N), the optimal clutch size for the thirdfemale is 17. This implies thatthe evolutionarilystable clutch size is greaterthan 10, because if all threefemales had clutch size 10, any one of themthat increased her clutchsize would obtaingreater fitness. For s1 (N), the survivalrate of the eggs fromall of the females is zero, because s1(20) = 0. Therefore,the evolutionarilystable clutch size mustbe less than 10. The actual evolutionarilystable clutch sizes fors1(N) and s3(N) are 5 and 30, respectively. For s2(N), the optimalclutch size forthe thirdfemale is 10; consequently,10 is the evolutionarilystable clutchsize. This confirmsthe resultsin equations (4) thatthe evolutionarilystable clutchsize is independentof the numberof femalesovipositing when larval competitionhas the forms2(N). The conclusion thatthe evolutionarilystable clutchsize increases withincreasing numbersof females ovipositingper patch resultsfor a wide class of larval- competitioncurves, notjust for the specificcurve s3(N). Appendix A derives a generalcriterion for competition curves thatshow thisbehavior. This criterionis based on the functionh(N), definedas h(N) = s(N)ls(N). (5) Here, s(N) is the derivativeof s(N) withrespect to N. If the derivativeof h(N), h(N), is negative,as it is for s1(N) (Appendix A), then the evolutionarilystable clutch size decreases withincreasing F (i.e., as morefemales oviposit per patch). OPTIMAL CLUTCH SIZE OF INSECTS 675

If h(N) is positive, then the evolutionarilystable clutch size increases with increasing F, as for s3(N). For the functions2(N), h(N) is zero. Generally speaking,the evolutionarilystable clutch size increases withincreasing numbers of ovipositingfemales wheneverthe curve given by Ns(N) increases rapidlyfor small N but decreases slowly for large N (see fig. 1). Althoughh(N) retainsthe same sign across all N for the threeexamples of s(N) in equations (1), this need not be true, and the evolutionarilystable clutch size can increase and decrease withF over differentparts of its range. (An example of thisis givenin the section "Inferringfitness curves fromlarval feedingbehavior.")

ADDING COMPLEXITIES TO THE SIMPLE MODEL In the simple model described above, the numberof females ovipositingon every patch was assumed to be the same, and femaleswere assumed to optimize theirclutch size on a per-patchbasis. This section relaxes these assumptionsby consideringthe cases when (1) a variable numberof femalesoviposits per patch; (2) the clutch size of a female on a single patch cannot be treatedin isolation, because the clutch size on one patch will affecteither the clutch sizes on other patches or the numberof patches a femalecan findin herlifetime; and (3) females can assess whethereggs were previouslylaid on a patch and adjust theirclutch sizes accordingly. The resultusing s3(N) in the simplemodel, thatthe evolutionarilystable clutch size increases as more femalesoviposit per patch, is counterintuitive.Therefore, throughoutthis section, emphasis is placed on larval-competitioncurve s3(N). Cases 1-3 above are analyzed to determinewhether the conclusion fromthe simple model can be reversed: can the evolutionarilystable clutch size decrease as more females oviposit per patch, even thoughlarval competitionhas the form S3 (N)? Variable Numbers of Females Ovipositingper Patch In nature,not all patches are visitedby the same numberof females. Let p(i|F) be the probabilitythat i females oviposit on a patch given that F is the mean numberof females ovipositingper patch. The distributionsused here forp(i|F) are the Poisson distributionand the negative-binomialdistribution, which has a greatervariance than the Poisson distribution.Further assume thatfemales produce the same clutchsize on everypatch theyfind; this assumption is realisticfor insects with fixed clutch size (e.g., Calliphoridae, Diptera [Ullyett 1950]) or insects that cannot assess the presence of eggs already on patches (e.g., Hy- menoptera,Gregopimpla himalayensis [Shiga and Nakanishi 1968]; Lepidoptera, Euptychia arnaea and E. libye [Singer and Mandracchia 1982]). Under this assumption,the expected fitnessof a female with clutch size n when all other females have clutch size ntis

E[w(n|nr)] = [nsn + (i - 1)n] ip(iIF)/F. (6) i=O Here, larval competitionis assumed to take the formof s3(N). 676 THE AMERICAN NATURALIST

30 -

N 20 -

0 0 1 2 3 4 Average Numberof OvipositingFemales, F

FIG. 3.-The evolutionarilystable clutch size, n*, as a functionof the average numberof females ovipositingper patch, F. Larval competitionhas the formof s3(N) withparameters as in figure1. Solid circles, Females distributedsuch thatexactly F ovipositper patch(at F = 4, the evolutionarilystable clutchsize is infinite);solid line,females distributed according to a Poisson distribution;dotted line, femalesdistributed according to a negative-binomialdistribution (with the parameterk = 1) (eq. 6).

The evolutionarilystable clutch size, n*, can be calculated forequation (6) in the mannerof equations (3), and figure3 shows n* graphedas a functionof F, the mean numberof females ovipositingper patch. Withvariability in the numberof females ovipositing per patch, the evolutionarilystable clutch size does not approach infinityfor some values of F. However, the evolutionarilystable clutch size still increases as more females oviposit per patch. Also, comparingthe Poisson distribution(solid line) with the negative-binomialdistribution (dotted line) shows thatas the degree of variabilityin the numberof femalesovipositing per patch increases, the predictedevolutionarily stable clutchsize increases more slowly withincreasing F. LifetimeTrade-offs For many insects, the numberof eggs a female lays on one patch affectsthe numberof eggs she should ovipositon otherpatches. In thiscase, determiningthe evolutionarilystable clutch size requires analyzing the trade-offsin a female's lifetimefitness. As an example, suppose a femaleis limitedby the amountof time that she can spend searchingfor patches. The female should optimizeher fitness per unit of time, analogous to the situation depicted by the marginal-value theorem(Charnov 1976; Parkerand Courtney1984; Skinner1985). Let Ts be the time requiredfor a female to finda patch, and let TH be the time it takes for a female to produce a single egg; TH includes feedingtime iffemales requirefood beforematuring eggs and also the timeit takes to oviposita singleegg once a patch is found. It is assumed that the handlingtimes for eggs are additive: the time it takes to produce n eggs is nTH. A female's fitnessper unit of time when she oviposits n eggs and all otherfemales oviposit nieggs, w(nIn), is given by (Parker and Courtney 1984; Godfray1986) w(nln,)= ns3[n + (F - l)ni] r(n), OPTIMAL CLUTCH SIZE OF INSECTS 677

40-

0 30 - N 0

20- 0

A) 1010 - 1 K I

0 1 2 3 4 Number of Ovipositing Females, F

FIG. 4.-The evolutionarilystable clutchsize, n*, as a functionof the numberof ovipositingfemales per patch whenfemales are limitedby the timethey can searchfor patches (eq. 7). Larval competitionhas the formof s3(N) withparameters as in figure1. Squares, Ts = 100, TH = 1.0; triangles,Ts = 10,TH = 1.0; solidcircles, evolutionarily stable clutchsize when thereis no lifetimetrade-off, that is, TH = 0 (at F = 4, the evolutionarilystable clutchsize is infinite). where

r(n) = 1I(TS + nTH). (7) Here, it is assumed thatexactly F femalesoviposit per patch. Figure4 shows the resultingevolutionarily stable clutch size as a functionof F fordifferent values of Ts, the searchingtime required to findpatches. As Ts decreases, the evolutionarily stable clutchsize, n*, increases withF more slowly. However, even small values of Ts do not cause the evolutionarilystable clutchsize to decrease withincreasing F, provided thatlarval competitionhas the formof s3(N).

Knowledgeable Females The complexitiesadded to the simplemodel in the last two subsectionscan be combined by assuming that variable numbers of females oviposit on different patches and thatfemales choose theirclutch size to optimizetheir lifetime fitness. A thirdassumption can be added: females can accuratelyassess whethereggs have been previouslylaid on a patch and can adjust theirclutch sizes accordingly. The abilityto assess whetherother females have previouslyoviposited on a patch has been shown for several insects (Hymenoptera,Nasonia vitripennis[Holmes 1972],Apanteles glomeratus[Ikawa and Suzuki 1982]; Lepidoptera,Pieris brassicae [Rothschild and Schoonhoven 1977]; Diptera, Rhagoletis pomenella [Roitberg and Prokopy 1981]). This subsection explores whether the evolutionarilystable clutch size for knowledgeable females may decrease as more females oviposit per patch, even thoughlarval competitionis given by s3(N). When females can assess the presence of eggs on patches, calculating the evolutionarilystable clutch size formany females ovipositing per patch becomes mathematicallycomplex. Therefore,for the sake of simplicity,I consideronly the case when at most two females oviposit on the same patch. This case is also 678 THE AMERICAN NATURALIST consideredby Parkerand Courtney(1984) and by Parkerand Begon (1986) (fora general discussion, see also Godfray1987, pp. 142-145). Let p(O),p(1), andp(2) denotethe probabilities that 0, 1,and 2 femalesoviposit on the same patch,given that the mean numberof femalesovipositing per patchis F (O ' F ' 2). If ovipositingfemales are distributedrandomly, then the values of p(i) will be binomiallydistributed. Let qo, ql, and q2 denotethe probabilitiesthat a given female oviposits alone, first,or second in a patch, respectively:

qo = p(l)I[p(l) + 2p(2)], (8) q, = q2 = p(2)1[p(l) + 2p(2)]. For a given female, let n1 be her clutch size if she oviposits on a patch before any otherfemale, and let n2 be her clutch size if she oviposits on a patch already containingeggs. Let nt and n2 be defined similarlyfor all other females in a population. Because the larvae fromthe firstfemale may get a head start in feedingon the patch, they mighthave a competitiveadvantage over the larvae fromthe second female. To allow for this type of first-clutchadvantage, let K denote the reductionin fitnessof the larvae fromthe second femalerelative to the first;K rangesfrom zero, when all of the larvae fromthe second femaledie, to 1, when there is no first-clutchadvantage. Under these assumptions,the expected lifetimefitness of a female per unitof time is

E[w(n1, n2lni, n2)] - nIS3(nl)qo + + n2)ql + Kn2s3(,l + n2)q2 Ts +nIs3(n, TH(nlqO + nlql + n2q2) Solving equation (9) for the evolutionarilystable clutch sizes gives values for the firstand second clutch laid on a patch, nl and n*. Figure 5 shows nl and n* graphedas functionsof the mean numberof ovipositingfemales, F. In figuresSA and SB, thereis no first-clutchadvantage (K = 1); in figuresSC and SD, K = 0.7. Moreover, in figuresSA and SC, the searchingtime (Ts = 100) is longerthan in figuresSB and SD (Ts = 10). As foundfor the models in the previoussections, the evolutionarilystable clutch sizes forboth the firstand the second femaleincrease withan increase in F, the mean numberof females ovipositingper patch. Althoughthe evolutionarilystable clutchsizes increaseas F increases,whether the second female should lay more or fewereggs than the firstfemale depends on the magnitudesof the searchingtime, Ts, and the first-clutchadvantage, K. When the searchingtime, Ts, is shortrelative to the handlingtime, TH, patches are easy to find.Therefore, a second femaleis more likelyto profitfrom withholding some of her eggs, because she can more easily findan emptypatch on which to deposit them. Consequently, when Ts is short, the clutch size of the second femaletends to decrease relativeto thatof thefirst. Similarly, when there is a first- clutch advantage (K < 1), the second female ovipositingon a patch incurs a greatercost, and withholdingeggs for an emptypatch becomes more profitable. This also promotessmaller clutch sizes forthe second female. In the examples above, it is always optimalfor the second femaleto ovipositat least one egg. This can be explained as follows. The gain in fitnessthat the first femaleobtains by ovipositingher nthegg is the fitnessof thategg, s(n)r(n), minus OPTIMAL CLUTCH SIZE OF INSECTS 679

12 A 12 C

8 8

4 4

= 12 -B12 D

8-4 8 8

4 4

0 0 0 1 2 0 1 2 Average Number of Ovipositing Females, F

FIG. 5.-The evolutionarilystable clutch sizes for females ovipositingon a patch as a functionof the average numberof females ovipositingper patch, F (eq. 9): solid line, first female;dotted line, second female. Larval competitionhas theform of sO(N) withparameters as in figure1. A, B, No first-clutchadvantage (K = I); C, D, K = 0.7. Searchingtimes, Ts: A, C, 100; B, D, 10. Handlingtime, TH, is 1.0 in all fourviews. the decrease in fitnessof all of her previouslylaid eggs, (n - 1)[s(n - l)r(n - 1) - s(n)r(n)]. The firstfemale will continueto lay eggs untilthe rate at which she gains fitnessby ovipositingeggs drops below the expected rate she would obtain by searchingfor another patch. Therefore,if the evolutionarilystable clutchsize forthe firstfemale is nI, the expected rateof fitnessgain for a femalesearching for a new patch is s(n'l)r(n'*) - (n'l - 1)[s(n'l - l)r(n'l - 1) - s(n'l)r(n'*)]. For the second femaleovipositing her firstegg, the fitnessshe gains is Ks(n'* + I)r(n'* + 1), but there is no loss in fitnessthrough previously laid eggs. Therefore,the second female should lay at least one egg, provided that the fitnessshe gains throughlaying that egg is greaterthan the expected rate of fitnessshe would gain by searchingfor another patch, thatis, if Ks(n'l + l)r(n'l + 1)

> s(n'l)r(n l) - (n - 1s - 1)r(n'l - 1) - s(n'l)r(n'l)] (10) > n*s(n*)r(n*) - (nW- I)s(nW - I)r(nW- 1). This inequalitywill oftenbe satisfied.In particular,for large n, inequality(10) is approximately Ks(n)r(n) > s(n)r(n) + ns(n)r(n) + ns(n)r(n). (11) 5~~~~~~

2 3 4 Food Intake

25- B zCh 20 - i 0 - C 15 LL 10 II

C 5

E 0 I 0 10 20 30 40 50 Numberof Eggs, N

05 E 3

o 0-

0 1 2 3 4 5 Numberof Females,F

FIG. 6.-The evolutionarilystable clutch size as a functionof the numberof females ovipositing per patch for two larval-competitioncurves constructedfrom larval feeding characteristics.A, Two examples (i and ii) of the relationshipbetween the fitnessof a larva and the amountof food it eats. The functionfor larval fitnesshas the form(cl + c2x)I(c3 + C4X), wherex is the amountof food intakeand the ci's are constants.B, The combinedfitness of larvae, Ns(N), as a functionof the numberof eggs laid per patch,N, determinedfrom the curves in A. The ci's in A were chosen to give the curves in B the same maximaas the curves in figure1. C, The resultingevolutionarily stable clutch sizes as functionsof the numberof females ovipositingper patch, F. At F = 1, the solid circle and the open square coincide. OPTIMAL CLUTCH SIZE OF INSECTS 681

Because boths(n) and r(n) are negative,this inequality always holds whenthere is no first-clutchadvantage (K = 1). However, a large first-clutchadvantage (K small) can overridethis and cause the second female to withholdall of her eggs.

INFERRING FITNESS CURVES FROM LARVAL FEEDING BEHAVIOR The distinguishingfeature of larval-competitioncurve sl(N) is that,when the numberof eggs on a patch is large, no larva on the patch survives. This type of catastrophicmortality can resultfrom scramble competition; if there are too many larvae, none can sequester enoughfood to survive.Appendix B shows thatwhen catastrophicmortality occurs at high larval densities, the evolutionarilystable clutch size necessarily decreases when the numberof females ovipositingper patch is very high. However, this need not be true when the numberof females ovipositingper patch is low. This section derives an example of a scramble- competitionfunction with catastrophic mortalityfor which the evolutionarily stable clutch size initiallyincreases as the firstfew females oviposit on a patch, and then decreases as more females oviposit. A functiondescribing larval competitioncan be derived by knowinghow the fitnessof a larva varies with the amount of food it eats. Figure 6 gives two hypotheticalexamples forlarvae thatcompete for food but do not interactaggres- sively towardone another.Figure 6A shows the fitnessof larvae as a functionof the amount of food they eat, and figure6B translatesthis into a curve for the numberand fitnessof larvae produced per patch as a functionof the numberof eggs laid. The curves in figure6B are equivalentto the curves ofNs(N) in figure1; and in fact, curve i in figure6B is identical to Nsl(N). Figure 6C gives the evolutionarilystable clutch sizes as functionsof the numberof ovipositingfemales, F, assumingthat F is the same forall patches. The importantmessage from figure6 is thatunder certain circumstances (i.e., case ii), the evolutionarilystable clutch size may firstincrease and then decrease withF. This example clearlyshows thatwhen no larvae surviveto adulthood,this zero larval survivorshipafter high initiallarval densitiesdoes not necessarilylead to evolutionarilystable clutch sizes that monotonicallydecrease as more females oviposit per patch. It also shows that qualitative knowledge about the feeding behavior of larvae is not sufficientfor drawinga conclusion about the evolutionarilystable clutch size. In both of the examples above, competitionamong larvae was assumed to be a scramblefor limited resources; the examples differed only in the quantitative details of the relationshipbetween fitnessand food consumption.To predictthe evolutionarilystable clutch size, it is necessary to analyze larval fitnesscurves generated by detailed competitionstudies across manylarval densities. This is done in the next section.

EXAMPLES OF LARVAL-COMPETITION CURVES I searched the literatureto find examples of larval-competitioncurves for hymenopteranparasitoids in which females lay more than one egg per clutch. I chose to concentrateon hymenopteranparasitoids because hymenopteransoften 682 THE AMERICAN NATURALIST

TABLE 1

LARVAL-COMPETITION CURVES

Parameter Species Resource Estimates* Measuret Source

CATEGORYi: s1(N) = .,(jl - otN) Hymenoptera Bracon greeni Eublemma amabilis 0.17 fitness Pramanik& larvae Choudhury1963 Dahlbominus Neodipron leconti 0.00095 survivorship Wilkes 1963 fuliginosust larvae Microbracon gram flourmoth 0.023 survivorship Narayanan & Sub- gelechiae larvae ba Rao 1955 Coleoptera Callosobruchus cowpeas 0.028 survivorship D. P. Giga (in maculatus Smith& Les- sells 1985)

CATEGORY 2: S2(N) = pU2e`N Hymenoptera Apanteles tobacco hornworm 0.0024 survivorship Beckage & Rid- congregatus larvae diford1979 Pteromalus Papilio xuthus 0.0024 fecundity Takagi 1985 puparum pupae CATEGORY 3: S3(N) = ii3(l + aN) b Hymenoptera Bracon almond mothlarvae 2.3 0.58 fitness Benson 1973 hebetort? Mediterranean 0.050 0.57 fitness Taylor 1984, 1988 flour-mothlarvae Trichogramma Mamestra bras- 0.36 6.0 survivorship N. Pallewatta (in evanescens sicae eggs Waage & God- fray1985) Trichogramma Mediterranean 2.4 1.2 fitness Klomp & Teerink embryophagumt flour-motheggs 1967 Diptera Phaenicia mammalcarcasses 0.062 0.87 fitness Ives 1988 coeruleiviridis

* Category 1, co;category 2, r; category3: left-handcolumn, a; right-handcolumn, b. t The measure of competitionused in the study: fitness,survivorship, or fecundity. t Competitioncurve differedsignificantly from the formof s2(N). ? Competitioncurve differedsignificantly from the formof s2(N) only forthe data in Benson 1973. show complex ovipositingbehavior and their clutch size is consequentlywell studied. Nine laboratorystudies of eightdifferent species gave sufficientdata to determinelarval-competition curves; these are listed in table 1. The table also includes two representativestudies on larval competitionin non-hymenopteran insects. The insects are divided into categories 1, 2, and 3, correspondingto larval- competitioncurves si (N), s2(N), and s3(N), respectively.Each of thethree larval- competitioncurves was fittedto the data sets using nonlinearregression, and category1, 2, or 3 was assigned accordingto whichcurve gave the lowest sum of squared residuals. For manyof the data sets, onlythe mean survivorshipor fitness OPTIMAL CLUTCH SIZE OF INSECTS 683 was reportedfor replicates at each larval density.Therefore, it is not possible to calculate the percentage of variationexplained by the fittedcurves. However, AppendixC derives a testfor statistically distinguishing between sI (N) and S3(N), with s2(N) servingas the null hypothesis.Insects markedwith a double dagger have larval-competitioncurves statisticallymore like sI (N) (category1) or s3(N) (category3) than s2(N) (category2) at the P < 0.05 level. Several commentsare necessary about the data sets I analyzed for the table. The effectof competitionon larvae was measured differentlyin differentstudies. In some cases, only survivorshipwas determined;in others, the fecundityof female offspringwas also measured, allowing estimates of female fitness.For simplicity,I have assumed thatmale fitnessvaries withlarval densityin the same way as femalefitness. In one study(Takagi 1985) densitywas foundnot to affect survivorship,although female fecunditydecreased with larval density.Many of the studies withhymenopterans showed changes in the sex ratio withincreasing numbersof larvae per host; some of the studies showed thatthis was caused by femalesvarying the sex ratioof theireggs. I ignoredthese changes in sex ratioand summedthe numbersof female and male offspringto calculate survivorship. A problem in interpretingthe data in the table is the possibilitythat zero survivorshipwould occur for insects in categories2 and 3 if the researchershad explored sufficientlyhigh larval densities.In fact,zero survivorshipwas obtained by N. Pallewatta (in Waage and Godfray1985) forTrichogramma evanescens. As shown in Appendix B, the evolutionarilystable clutch size must eventually decrease when zero survivorshipis approached. Therefore,it is possible that some of the examples in category3 show the patterngiven in figure6C forcase ii, wherean increase in the evolutionarilystable clutchsize is followedby a decrease as more femalesoviposit per patch. However, forother species, competitionmay never be sufficientto produce zero larval survivorship,as suggestedby Klomp and Teerink (1967) for Trichogrammaembryophagum. The table is meantto give representativelarval-competition curves, not to make predictionsabout the evolutionarilystable clutchsize forthese particularinsects. There are of course manyfactors that influence clutch size beyond those considered in thispaper (see, e.g., Parkerand Begon 1986; Godfray1987; Mangel 1987). However, the table shows that all three curves sl(N), s2(N), and s3(N) are biologicallyrealistic. Therefore,the potentialexists for both decreases and increases in the optimalclutch size of insects as the numberof femalesovipositing per patch increases.

DISCUSSION Here I have addressed how, undera varietyof differentcircumstances, a female insect's optimal clutch size changes as more females oviposit per patch. The optimalclutch size mightbe obtainedbehaviorally if individual females can assess average population densities, or the optimal clutch size mightrespond on an evolutionarytime scale to changes in average populationdensities. One of the more strikingpredictions to emergefrom the analysis is the behavior of a second female arrivingon a patch thatshe knows already containseggs. The 684 THE AMERICAN NATURALIST evolutionarilystable clutch size of this second female is predictedto be greater than that of the firstonly if (1) the larval-competitioncurve is like s3(N), (2) the timerequired to finda patch is long relativeto the timerequired to produce eggs (Ts > TH), and (3) any first-clutchadvantage is not large. Since most parasitoids do not satisfyall three of these criteria,in general, second females mightbe expected to oviposit fewer eggs than the first.Studies of the hymenopteran parasitoidsNasonia vitripennis(Holmes 1972) and Apanteles glomeratus(Ikawa and Suzuki 1982) reportedthat the second femaleovipositing on a host lays fewer eggs thandoes the firstfemale. However, Werren(1980, reportedin Skinner1985) showed examples fromNasonia vitripennisin whichthe second femaleslaid both fewerand moreeggs thanthe first.This variabilityin the second females'behavior has been explained by assuming thatthe second females differin the amountof timesince theylast laid eggs; a femalethat has searchedfor a long timeshould be more likelyto oviposit (Iwasa et al. 1984; Mangel 1987). Data fromWylie (1965) suggestthat larval competitionfor N. vitripennishas roughlythe formof s1(N), althoughthe data cannot supporta statisticalanalysis. Therefore,it is unlikely thatthe formof larval competitionplays a role in explainingwhy the second N. vitripennisfemale sometimes lays more eggs thanthe first.Nonetheless, it may be possible to findother hymenopteran parasitoids that meet the threecriteria for the optimal clutch size of a second female to be greater than that of the first; hymenopteranparasitoids certainly exist forwhich the larval-competitioncurves are like s3(N). This analysis may also help to explain why some insects do not behaviorally alter theirclutch size. For example, the blow flyPhaenicia coeruleiviridisis a batch-layingspecies, withfemales always ovipositingtheir entire complement of eggs (Ives 1988). If the conclusions of previous authors (Parker and Courtney 1984; Skinner1985; Parkerand Begon 1986) were applied to P. coeruleiviridis,it would seem thatfemales should have evolved a means of assessing the numberof eggs on carcasses and decreasingtheir clutch sizes on heavilyinfested carcasses. However, competitionamong P. coeruleiviridislarvae takes the formof s3(N) (table), and fromthe analysis in this paper, females may in fact lose little or nothingby not decreasing theirclutch size. The counterintuitiveresults presented here are explained by the structureof ESS models. AlthoughESS models explicitlyconsider the behaviorof groups of interactingindividuals, the evolutionarilystable solutionis foundby considering the fitnessof an individualthat deviates fromthe behavioralnorm. In the models examined here, a female insect optimizesher clutch size againsta backgroundof eggs laid by otherfemales. Depending on the formof competitionamong larvae, this background can either increase or decrease the marginal-fitnessgain of a femalelaying additional eggs on the same patch. Therefore,although the eggs laid by otherfemales will always decrease the overall fitnessof her eggs, a female's optimalclutch size will nonethelessincrease for some formsof larvalcompetition.

SUMMARY When more than one female insect oviposits on the same patch of larval food, the optimal clutch size depends not only on the numberof other females that OPTIMAL CLUTCH SIZE OF INSECTS 685 ovipositbut also on theform of larval competitionwithin patches. I use models to show that,depending on the type of larval competition,the evolutionarilystable clutch size can eitherdecrease or increase as more females oviposit per patch. The models predictdecreases in the evolutionarilystable clutchsize withincreasing numbers of females ovipositingper patch when larval competitioncauses rapid mortalityat highdensities, and theypredict increases in the evolutionarily stable clutchsize when larval competitionis more benign.I also give examples of both extremesof larval competitionby analyzingdata fromthe literaturefor 10 insect species.

ACKNOWLEDGMENTS I thankH. C. J. Godfrayfor many stimulatingdiscussions, and A. Taylor for helpingme findthe parasitoid examples in the table and providingme with his data. J. Bergelson,H. C. J. Godfray,G. Head, J. Herron,P. Kareiva, C. Lord, G. Parker,A. Taylor,J. Werren,and threeanonymous reviewers made manyhelpful suggestionson earlierversions of the manuscript.I am now being supportedas a Glaxo Fellow of the Life Sciences Research Foundation at the Universityof Washington,Seattle.

APPENDIX A Here I derivethe general criterion for the response of theevolutionarily stable clutch size to thenumber of femalesovipositing per patch. Using the definitionof h(N) (eq. 5), equation (3a) can be rewrittenas h(n*F) = - lIn*. Applyingthe implicit-functiontheorem and simplifyinggives An* -n3h(N) (Al) aF Fn2h(N) - 1 n=n* It can be shown thatthe denominatorin equation (Al) is always negativewhen n* exists; consequently,the sign of an*IaF depends only on thesign of h(N). Whenh(N) is negative, the evolutionarilystable clutch size decreases withF, the numberof females layingeggs per patch; when h(N) is positive,the opposite is true.The values of h(N) fors1 (N), s2(N), and s3(N) are -o2(l - an*) -2, 0, and a2b(I + an*3)-2,respectively. Comparing these to the explicit expressions for the evolutionarilystable clutch sizes (eqs. 4) supports the predictionsmade in equation (Al). Note thatthe shape of h(N) depends not on the number of femalesovipositing per patch,but onlyon the totalnumber of eggs.Therefore, the responseof the evolutionarily stable clutch size to increasingnumbers of females ovipositingper patch is determinedonly by the type of interactions among larvae and the resulting formof theircompetition curve.

APPENDIX B Here I showthat for larval-competition curves with catastrophic mortality, the evolutionarilystable clutch size decreaseswhen large numbers of femalesoviposit per patch. To argueby contradiction, assume that the evolutionarily stable clutch size remainsfixed at thevalue fi* as morefemales oviposit per patch. Let N denotethe number of eggs on a patchat whichsurvivorship drops to zero,and letF be themaximum number of females ovipositingbefore zero survivorshipis reached;fi*F < N ? n*(F + 1). IfF + 1 females ovipositon a patchand one is allowedto alterher clutchsize (followingthe logic of evolutionarily-stable-strategymodels), then she shouldlower her clutch size to allowsome 686 THE AMERICAN NATURALIST of her larvae to survive. This impliesthat the evolutionarilystable clutchsize mustbe less than n*. As an alternativeproof, it can be shown thath(N) is negativeas N approaches N forany continuouslydifferentiable function s(N) (see Appendix A).

APPENDIX C Categorizinglarval-competition curves by which function,sl(N), s2(N), or s3(N), best fitsthe data can be misleading.For example, s2(N) or s3(N) will fitan observed larval- competitioncurve thatis curvilinearbetter than sl(N) will,although curvilinearity does not guaranteethat the evolutionarilystable clutch size will stay the same or increase as more females oviposit per patch; a functionof the forms(N) = ,L(l - cN)2 is curvilinearbut causes evolutionarilystable clutch sizes to decrease as more females oviposit per patch (Skinner1985). To separate statisticallycategories 1, 2, and 3, a testcan be used based on the sign of h(N) (Appendix A). If s2(N) is taken as the null hypothesis,departures from s2(N) can be testedby addinga quadratic termto the exponent of s2(N) to give s(N) = VLexp(-rN + yN2). (C1) For s(N), h(N) = 2,y.Therefore, statistically significant departures of y fromzero indicate that the evolutionarilystable clutch size tends to decrease ('y negative) or increase ('y positive) as more females oviposit per patch.

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