The Auk 118(4):973-982, 2001 HOW MANY BASKETS? CLUTCH SIZES THAT MAXIMIZE ANNUAL FECUNDITY OF MULTIPLE-BROODED BIRDS GEORGE L. FARNSWORTH • AND THEODORE R. SIMONS CooperativeFish and Wildlife Research Unit, Departmentof Zoology,North CarolinaState University, Raleigh,North Carolina27695, USA ABSTRACT.--Wedeveloped deterministic models on the basisof nest survival rates and renestingbehavior capable of predicting annual fecundity in birds. The modelscalculate probabilitiesof fiedgingfrom oneto four nestswithin a discretebreeding season. We used thosemodels to addresstheoretical issues related to clutch size. In general,birds require at least one day to lay an egg, and many speciesdelay incubationuntil their entire clutch is laid. Becauseit takeslonger to completea larger clutch,and fewer suchclutches can fit into a limited breedingseason, there exists a clutchsize for whichannual fecundity is maximized. We asked,for a givenamount of reproductiveeffort (i.e. a setnumber of eggs),does the age- old maxim "don't put all your eggsin one basket" apply? If so, in how many "baskets" shoulda nestingbird placeits eggs?The answerdepends on both likelihoodof nestpre- dation and length of the breedingseason. Those results are consistentwith the observed increasein clutchsize with latitude (shorterbreeding season length) and largerclutch sizes characteristicof cavity-nestingspecies (with highernest survival rates). The modelsalso pre- dict that the size of replacementclutches should decrease as the breedingseason progresses, and that intraseasonaldecline in clutchsize shouldbe more pronouncedwhen the breeding seasonis short.Received 24 January2000, accepted24 April 2001. ECOLOGISTSHAVE BEEN fascinated by factors vival rate of nests and the trade-off between constrainingavian clutchsize for more than 50 clutchsize and predationrisk. He estimatedfe- years. Lack (1947, 1954) proposedthat clutch cundity (?) as, size should reflect maximum number of young F = ns'•+L (1) that adults can raise. Most discussionsof opti- mal clutch size in birds have centered on pro- where n representsthe clutchsize, L is the num- ductivity of a single clutch of eggs,but some ber of days of incubationneeded to hatch the have looked at the influence of clutch size on eggs,and s is daily nest survival rate. We have lifetime reproductive success(for a recent re- changedPerrins' notation slightly to matchno- view, seeMonaghan and Nager 1997).Another tation used in our models.By taking the partial important measure of productivity is annual derivativewith respectto n, Perrinsrelated op- fecundity becausemany speciesattempt to nest timal clutchsize to daily survival rate of nests: more than once within a limited breeding sea- -1 son. In those species,length of the breeding (2) seasonmay be an important constraint on an- F/opt= ln(s) nual productivityleading to selectionon clutch For example,with a daily nest survival rate of sizes that maximize annual fecundity. 0.95, the predicted optimal clutch size is 19.5 Many birds lay one egg a day until their eggs. clutchesare complete and then begin incubat- The optimum predicted by Perrins' model ing. That means that larger clutchesare ex- would be reduced if daily nest survival rate posed to predatorsfor a longer period of time. were modeledto decreasewith larger clutches. For many species,predation usually resultsin Skutch (1949) hypothesized that larger broods loss of the entire clutch or brood (e.g. Wood have higher predation rates due to increased Thrush, Hylocichlamustelina; Farnsworth 1998). activity at the nest.He reasonedthat in altricial Perrins (1977) related optimal clutch size di- birds, more begging chicksand more frequent rectly to risk of predation.He proposeda math- feeding trips by adults could attract attention ematicalmodel based on a constantdaily sur- of predators. There has been some empirical supportfor the mechanismsof that hypothesis l E-mail: [email protected] (Mullin and Cooper 1998, Martin et al. 2000), 973 974 FARNSWORTHAND SIMONS [Auk, Vol. 118 but others have failed to find that (Roper and we investigatehow intraseasonalvariation in Goldstein 1997). Ricklefs (1977) developedan clutchsize may affect annual fecundity. optimalclutch-size model for singlenesting at- Other models have been developedto esti- tempts that incorporatedlower nest survival mateannual fecundity in multiple-broodedbird for larger clutchsizes. He concludedthat even species(Ricklefs 1970, Peaseand Grzybowski with higher predation rates on larger clutches, 1995),but thosehave not been appliedto ques- that alone was not likely to be a major factor tionsof optimal clutchsize. Ricklefs(1970) de- determining evolutionof clutchsize. velopeda modelthat convertedestimates of nest Departing from traditional inquiries about survival,season length, and renestingbehavior clutchsize at which fecundity of a singleclutch into estimatesof annualfecundity. It assumeda is a maximum, someinvestigators have exam- constantrate of nest initiation throughoutthe ined how clutchsize may affect lifetime repro- breedingseason. That rate was setequal to rate ductivesuccess. Lima (1987)developed a mod- of nest termination,keeping the number of ac- el in which an increase in clutch size was tive nestsat equilibrium.In this model,survival associatedwith an increasein predationrisk to rate of nestsdetermined the proportionof suc- nesting femalesas well as to their dependent cessfulnests and the averageduration of a nest- chicks,thus reducingher future reproductive ing attemptbecause higher predation rates in- potential.That modelshowed that under high creaseproportion of neststhat fail early in the nest-predationrates, smaller clutchsizes were nestingcycle. The resultingmodel calculated a more productive than larger clutchsizes. Mur- constantnumber of fledglingsproduced per day ray (1979)used an interestingapproach to evo- for a large population(see also Ricklefsand lution of clutch size by assumingthat females Bloom1977). Pease and Grzybowski(1995) de- are selected to produce the minimum clutch velopedan approachthat incorporated pulses of size necessaryto replacethemselves based on nestingactivity by allowing the numberof ac- the Euler-Lotkaequation. That model has pre- tive neststo fluctuatethroughout the breeding dicted clutch sizes close to those observed for season.Their modelapplied instantaneous rates at least two species(Prairie Warbler[Dendroica of nestpredation and brood parasitism in a con- discolor];Murray and Nolan 1989,and Florida tinuous-timemodel adaptedfrom VonFoerster Scrub-Jay[Aphelocoma coerulescens]; Murray et equations.The model does not assumea con- al. 1989), but the model also requires the as- stant rate of nest initiation throughout the sumption that larger clutchesincur a larger breedingseason. Our approachis basedon a survival cost to the female,reducing her pros- discrete-timeprobability model, roughlyanal- pectsfor future breedingsuccess. ogousto a specialcase of the modelof Peaseand All of those studieshave investigatedinflu- Grzybowski (1995) with constantparameters enceof clutchsize on either fecundity of a sin- and no parasitism.However, our model is ca- gle nestingattempt or lifetimefecundity of a fe- pable of limiting number of eggs a female can male. In this paper, our goal is to investigate lay per season;previous models have not in- the relationshipbetween clutchsize and a fe- cluded that potentially important constraint. male'sannual fecundity.We soughtto begin by Birds do not have unlimited resources(includ- using a minimum number of assumptions.We ing a limited time remaining in the breeding describe here two mathematical models that season),and that limitation may have implica- are capableof incorporatingthe renestingbe- tions for evolution of clutch size. havior of multiple-brooded species. These models do not invoke mechanisms that have di- MODEL DESCRIPTIONS rect negative effectson larger clutchessuch as thoseemployed by Ricklefs(1977), Lima (1987), Our approach is based on the assumption and Murray (1979). This approachprovides a that a nestingfemale will renestafter her nest new framework for testing ideas about clutch fails or she fledgesa brood, provided there is size. We use the models to find clutch size at enoughtime remainingin the breedingseason which annual fecundityis a maximum.We ex- to fledge anotherbrood (for a list of parame- amine how that optimal clutch size changes ters,see Table 1). The logicalformulation of the with changesin both the length of breeding model is as follows.A breeding femalebegins seasonand survival rate of nests.Additionally, the seasonwith an initial nesting attempt.She October2001] ClutchSize in RenestingBirds 975 TABLE1. Parametersused to calculateannual fecundity in multiple-broodedbird species. Param- eter Value(s) Explanation L 25 days Lengthof nestingcycle from the time the last egg is laid to the time all chicks leave the nest d 8 days Number of daysbetween failure of one nest and first egg of subsequentnest D 14 days Numberof daysfrom a successfulnest to the first eggof subsequentnest t 60, 90, 120 days Length of breedingseason s 0.93, 0.95, 0.97 Daily nest survival rate n 1 to 30 eggs Clutch size rn 2,3,4 Maximum number of nesting attemptsper season lays one egg each day until her clutch is com- all the independentways to attempt the ith plete. The nest is subjectto a constantproba- brood with a maximum of m nestingattempts. bility of predationevery day from the day the The annual fecundity is simply the product of first egg is laid. If the first nest fails,