Ecology and Evolution. 2018;1–15.  | 1www.ecolevol.org

Received:15May2017  |  Revised:11October2017  |  Accepted:15October2017DOI:10.1002/ece3.3730

O R I G I N A L R E S E A R C H

Twelve fundamental life histories evolving through allocation- dependent fecundity and survival

Jacob Johansson1,2  | Åke Brännström1,3 | Johan A. J. Metz1,4,5 | Ulf Dieckmann1

ThisisanopenaccessarticleunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsuse,distributionandreproductioninanymedium,providedtheoriginalworkisproperlycited.©2018TheAuthors.Ecology and EvolutionpublishedbyJohnWiley&SonsLtd.

1EvolutionandEcologyProgram,InternationalInstituteforAppliedSystemsAnalysis,Laxenburg,Austria2DepartmentofBiology,TheoreticalPopulationEcologyandEvolutionGroup,LundUniversity,Lund,Sweden3DepartmentofMathematicsandMathematicalStatistics,UmeåUniversity,Umeå,Sweden4SectionofTheoreticalBiology,InstituteofBiologyandMathematicalInstitute,LeidenUniversity,Leiden,TheNetherlands5NaturalisBiodiversityCenter,Leiden, TheNetherlands

CorrespondenceJacobJohansson,DepartmentofBiology,TheoreticalPopulationEcologyandEvolutionGroup,LundUniversity,Lund,Sweden.Email:jacob.johansson@biol.lu.se

Funding informationSvenskaForskningsrådetFormas,Grant/AwardNumber:2015-839;Vetenskapsrådet,Grant/AwardNumber:2015-00302

AbstractAnorganism’slifehistoryiscloselyinterlinkedwithitsallocationofenergybetweengrowthandreproductionatdifferentlifestages.Theoreticalmodelshaveestablishedthatdiminishingreturnsfromreproductiveinvestmentpromotestrategieswithsimul-taneousinvestmentintogrowthandreproduction(indeterminategrowth)overstrate-gieswithdistinctphasesofgrowthandreproduction(determinategrowth).Weextendthistraditional,binaryclassificationbyshowingthatallocation-dependentfecundityandmortalityratesallowforalargediversityofoptimalallocationschedules.Byana-lyzingamodeloforganismsthatallocateenergybetweengrowthandreproduction,we find twelve typesofoptimal allocation schedules, differingqualitatively inhowreproductive allocation increaseswith bodymass. These twelve optimal allocationschedulesincludetypeswithdifferentcombinationsofcontinuousanddiscontinuousincrease in reproduction allocation, inwhichphasesof continuous increase canbedeceleratingoraccelerating.Wefurthermoreinvestigatehowthisvariationinfluencesgrowthcurvesand theexpectedmaximum lifespanandbodysize.Ourstudy thusrevealsnewlinksbetweeneco-physiologicalconstraintsandlife-historyevolutionandunderscores how allocation-dependent fitness componentsmay underlie biologicaldiversity.

K E Y W O R D S

determinategrowth,dynamicprogramming,indeterminategrowth,marginalvaluetheorem,reproductiveallocation

1  | INTRODUCTION

Simplelife-historymodelsoftenpredictthatitisoptimaltoallocateallsurplusenergytogrowthearlyinlife,beforeswitchingtoallocateallenergytoreproduction.Thisallocationpatternisoftenreferredtoasa“bang-bangcontrol”andleadstodeterminategrowth.Yet,simultane-ousinvestmentintogrowthandreproduction,leadingtoindeterminategrowth,iscommoninnature.Muchtheoreticalresearchonreproduc-tiveallocationhasthereforeinvestigatedmechanismsandconditions,whichcanpromoteevolutionof indeterminategrowth, forexample,stochastic environments (King & Roughgarden, 1982), diminishing

returns of reproductive investments (Sibly,Calow,&Nichols, 1985;Taylor,Gourley,Lawrence,&Kaplan,1974),orstructuralconstraints(Kozłowski&Ziólko,1988).Thisresearch(reviewedinHeino&Kaitala,1999;Kozłowski,1991;Perrin&Sibly,1993)hasestablishedthatsi-multaneousinvestmentintogrowthandreproductioncanbeoptimal,at least during someperiodof anorganism’s life. Less researchhasfocusedoninvestigatingtheshapeandnatureoftheresultingmixedallocationpatterns.Whereasbang-bangcontrolstrategiescansimplybecharacterizedbytheagesorsizesatwhichtheswitchfromgrowthto reproduction occurs, it is less clear how to characterize and un-derstandallocationschedulesthatcausereproductiveinvestmentto

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2  |     JOHANSSON et Al.

changegraduallyoveralifetime.Whichshapescanweexpect?Whatconditionsfavordifferentfeaturesintheseshapes?Howdoparticularoptimal allocation schedules affect key life-history features, suchasgrowthcurves,averagelifespans,orasymptoticbodysizes?

Here,we pursue these questions for life-history strategies thatevolve under allocation-dependent fecundity andmortality.Wewillspecifically consider cases where fecundity and mortality rates in-creasewith reproductive allocation, but at rates thatmaybeeitherfaster (henceforth referred toasaccelerating)or slower (henceforthreferredtoasdecelerating) thanproportional. It is theoreticallywellestablishedthatsimultaneousinvestmentintogrowthandreproduc-tionisfavoredwhentherearediminishingreturnsofreproductivein-vestments.Thisoccurswhenfecundityincreasesatadeceleratingratewiththefractionofsurplusenergyinvestedintoreproductionorwhenmortality increases at an accelerating rate with the reproductive-investmentfraction(León,1976;Siblyetal.,1985;Tayloretal.,1974).

There are many biological reasons why fecundity may dependnonlinearlyonreproductiveallocation.Competitionbetweentheoff-springorbetweenthegametesproducedbyanindividualmaycausediminishingreturnsfromenergyinvestedintoreproduction.Eggsfromasinglemothermaycompete for resources,andspermcompetitionis common for both animals and plants (Andersson& Iwasa, 1996;Scharer,2009).Structuralconstraintswithinanorganism,forexample,limitedsizeofbroodchambersincladocerans(Perrin,Ruedi,&Saiah,1987),canalsoleadtodiminishingreturnsbyimpedingefficientuseofsurplusenergy(cf.Kozłowski&Ziólko,1988).

Acceleratingreturnsfrominvestmentintoreproductionmayoccuramongplantsinwhichincreasinginvestmentsattractmoreseed-dis-persingorpollinatinganimals.Forexample,Sallabanks (1992)foundthattheproportionofseedsdispersedperplantincreasedwithfruitabundance in hawthorn, Crataegus monogyna. By a similar token,SchafferandSchaffer(1979)foundacceleratingreturnsproducedbypollinatorspreferringthelargerflowersinAgavaceae.Significantpartsofthetotalenergyinvestedintoreproductionmaynotbechanneleddirectly to offspringbodymass but rather intoorgansor capacitieswhich facilitate reproduction, for example, ovaries, inflorescences,or shells.Accelerating returnsmay thenoccurdue toeconomiesofscale,as theefficiencyofsuchsupportivefeatures, in termsofcostperoffspring,mayincreasewiththesizeoftheoperation,forexam-ple,viareducedvolume-surfaceratiosorbecausethesamefacilitiescanbeusedmanytimes.ThefindingbyGreeneandJohnson(1994)thattreeswithlargerseedsinvestasmallerproportionofenergyintostructuresforprotectionanddispersalsupportsthisidea(butseeLord&Westoby,2012).Anotherexampleislearninginseabirds,wherebytheprobabilitythatabreedingattemptissuccessfulincreaseswiththenumberofattempts,thatis,withexperience(Goodman,1974).

Mortality is generallyexpected to increasewith investment intoreproduction (Calow, 1979; Calow &Woollhead, 1977; Sletvold &Ågren,2015).Animalsmay,forexample,bemorevulnerabletopre-dationwhentheyarebreedingorsearchingforpartners,orbemoresusceptibletodiseaseduringthereproductivephase(Orton,1929).Aswithfecundity,thereareseveralreasonswhyalsomortalitymaybeanonlinearfunctionofreproductiveallocation.

Mortality will increase at an accelerating rate with reproduc-tive allocation if survival costs increase sharplywhen reproductive-investment levels pass a threshold. For example, many systems ofdefencetodiseasesinplantsrequireaminimalproductionofsecond-arytissue(Feeny,1976;Fraenkel,1959).Ifinvestmentintogrowthisreducedasaresultofsurplusenergybeingchanneledintoreproduc-tion,productionofthosetissueswilldecreaseand leadtoasharplyincreasedmortalityrate.Further,asdiscussedbyBell(1980),growthofgonads in fishmayhave littleeffectonmortalityas longas theyarerelativelysmall,butexertastrongnegativeeffectiftheyexceedacriticalproportionofthetotalbodysizeandstartcompromisingthefunctionalityoforgansnecessary for survival.Bycontrast,mortalitymayincreaseatadeceleratingratewithreproductiveallocationwheninitialinvestmentstoreproductionareriskierforadultsthanadditionalinvestments.Asanexample,themortalityrateofzooplanktonoftenincreasesduringreproduction,ascarryingeggsincreasesthechancetobedetectedbyvisualpredators(e.g.,Svensson,1997).Becausetheriskofbeingdetected is related to surfacearea rather thanvolume(e.g.,Aksnes&Giske,1993),onemayexpectthatpredationriskwillincreaseatalessthanproportionalratewiththetotalnumberofeggscarriedbyafemale.Otherorganismsthatinitiallysufferhighmortal-ityrisksarethosethatundertakelongmigratoryjourneysbeforere-producing,suchasanadromousfish(cf.Bell,1980;Gadgil&Bossert,1970).Forthem,producingthefirstfeweggsconfersahighmortalityrisk,whereascontinuedeggproductionlikelyconfersonlylittleaddedrisk.

Important qualitative insights into how the shape of optimalreproductive-allocationschedulesdependsonfecundityandmortalityrateshavebeenobtainedthroughmathematicalinvestigations(León,1976;Siblyetal.,1985;Tayloretal.,1974).However,althoughthesestudiesareofageneralnature,theydonotgiveanoverviewofex-pected shapes of nonbang–bang reproductive-allocation schedules.Numerical investigations in early pioneering studies (e.g., Gadgil &Bossert,1970)areinstructiveinsuggestingdifferentpossibleshapes,buttheseinvestigationsarelimitedtoscenarioswithasmallnumberofageclasses.Here,weattemptasystematicoverviewofthediversityof life-historytypesthatcanarisefromvariation intheshapeoffe-cundityandmortalityfunctions,andoftheconsequencesthatdiffer-entoptimalreproductivescheduleshaveforthegrowth,expectedlifespan,andultimatesizeoforganisms.Weshallfocusongraduallyin-creasing,accelerating,anddeceleratingfecundityandmortalityfunc-tions,bothbecausewebelievethesecasesarebiologicallyrelevant,asdescribedabove,andforcontinuitywithprevioustheory(e.g.,Bell,1980;Gadgil&Bossert,1970;Siblyetal.,1985).

The remainder of this article is structured as follows. First, weintroduceanddescribeageneral life-historymodelofontogenyandprocreation. Using dynamic programming, we determine optimalreproductive-allocationschedulesforeachcombinationofgenericfe-cundityandmortalityregime.Wesystematicallyclassifytheemergingallocationschedulesintotwelvedifferentclassesdependingontheircharacteristicshapes.Wethenstudygrowthpatternsand lifespansassociatedwiththetwelvedifferenttypesofoptimalallocationsched-ules.Weproceedbyshowinghowourresultscanbeunderstoodat

     |  3JOHANSSON et Al.

leastinpartbystudyingthemarginalvalueofreproductiveinvestmentatdifferentlifestages.Finally,wesynthesizeourfindingsintoagen-eral,conceptualframeworkanddiscusshowtheyconnecttoempiricalpatterns.

2  | MODEL

Ourmodelbuildsonanestablishedtraditioninearlierstudiesofop-timal allocation to reproduction (Cohen, 1971;Kozłowski&Ziólko,1988;Perrin,Sibly,&Nichols,1993;Siblyetal.,1985).Anindividualisassumedtoproduceenergyatamass-dependentrateE.Afractionuofthisenergyisallocatedtoreproductionandtheremainingfrac-tion,1−u,isallocatedtosomaticgrowth.Themassmofanindividualincreasesas

TheprobabilityPofanindividualsurvivinguntilagetdecreasesas

Intheaboveequation,q(u)denotestheinstantaneousmortalityrate,whichweassume tobe an increasing functionof the fractionu ofavailableenergyallocatedtoreproduction.ThisassumptionisinlinewithCalow(1979)whoarguedthatamongseveraltraditionalalterna-tives,energyinvestedintoreproductionasaproportionofthetotalenergy income is themostsuitablepredictorof reproductivecosts,such asmortality, and best reflects that these costs typically arisebecause reproduction competes for energy with other importantphysiologicalprocessesoractivities.Italsoformsthebasisforprevi-oustheoryby,forexample,MyersandDoyle(1983)andSiblyetal.(1985). The energy devoted to reproduction, uE, is converted intooffspringbiomassatarateb(uE).Unlikethemortalityrateq,whichdepends on the fraction invested into reproduction, the fecundity

ratebisassumedtodependonthetotalamountofenergyinvestedintoreproduction.

Inadherencetotheexistingtradition,wedescribethedependenceof vital rates on reproductive allocation using power functions(Figure1).Specifically,weassumethatthemortalityrateisgivenby

andthepotentialfecundityrateby

Asdescribedbelow,wedeterminefromthepotentialfecundityratea realized fecundity rate b that determines the expected numberof offspring of an individual. The two exponents kq and kb controlwhetherthemortalityrateandfecundityrateincreaseatanacceler-ating(ki>1),proportional (ki=1),ordecelerating(ki <1)ratewithu (seeFigure1).

Thepotentialfecundityrate,Equation(4),hastheunrealisticandun-desirable feature that the slopebecomes infiniteat zero reproductiveinvestmentwhenkb<1.Thismeansthatasmallincreaseinenergyal-located to reproductioncanconveyanunrealistically large increase intherateatwhichoffspringareproduced,potentiallycausingimprobableevolutionarypredictions.Weavoidthisproblembyassumingaphysio-logical limitsuchthatwheneverkb<1therealizedfecundity isalwayslessthanafactorptimestheenergyallocated(Figure1c).Next,wecon-structacontinuousanddifferentiablerealizedfecundityrateasasmoothminimumofthepotentialfecundityandthephysiologicallimitbysettingb(uE)= (b̂

(

uE)s+ (puE)s)1∕sforsomevalueofs<0,withincreasinglyneg-

ativevaluesofsimplyingacloserapproximationoftheminimum.Followingthecommonassumptionthatbiomassintakescaleswith

massaccordingtoapowerlaw(e.g.,Kozłowski&Wiegert,1986;Reiss,1989;Roff,1992),theorganismisassumedtoacquireenergyataratethatdependsonitscurrentmassas

(1)dm∕dt=(

1−u(m))

E(m), m(0)=mbirth.

(2)dP∕dt=−Pq(u), P(0)=1.

(3)q(u)= c1+c2ukq ,

(4)b̂(uE)= c3(uE)kb .

(5)E(m)= c4mke .

F IGURE  1 Fecundityandmortalityratesasfunctionsofreproductiveinvestment.Thepanelsillustratehowdifferentparametersaffecttheshapesofthesefunctions.Thecurvaturesofthemortalityfunctionsin(a)dependonkqandthecurvaturesofthefecundityfunctionsin(b)dependonkb.(c)showshowtherealizedfecundityfunction(b,blackline)isconstructedasasmoothminimumofthepotentialfecundity(b̂,grayline)andthephysiologicallimit(dashedline)

(a) (b) (c)

4  |     JOHANSSON et Al.

Theexpectedlifetimereproductionoftheindividualisthengivenby

Weaimtodetermineoptimalmass-dependentallocationsched-ules that maximize expected lifetime reproduction as given byEquation(6).Thiscontrolproblemissolvedbyapplyingthemethodofdynamicprogrammingtoatime-discreteversionoftheordinarydifferential equations (Equations[1 and 2]; see, e.g., Bertsekas,1987orHouston&McNamara,1999foranoutlineofthestandardproceduresused).Primarily,wevary theparameterskq,kbandthephysiological limitp,because they influence thecurvaturesof thefecundityandmortalityfunctions.Inlinewithpreviousstudies(e.g.,Charnov,1993;Kozłowski&Uchmanski,1987),wesettheproduc-tion exponent ke to 3/4.Without lossof generalitywe reduce themodeldimensionalitybyadjustingthetimescalesothatthebaselinemortalityrate,c1,equals1.The remainingparametersareassignedvaluesmotivatedby simplicityor chosen for illustrativepurposes.Weconductarobustnesscheck(seeAppendixB)toclarifyhowvari-ationofparametersettingsandselectedmodelassumptions influ-enceoptimalallocationschedules.

Theoptimalmass-dependentallocationschedulesarerepresentedby the proportionsu*(m) (where the asterisk denotes optimality) ofenergyinvestedintoreproductionforanyindividualbodymassm.Wedivideallocationschedulesintodifferentcategoriesbyconsideringthecurvatureofthefunctionu*(m)forearlyandlatelifestages.Tocovermostofthebiologicallyrelevant lifespan,wecalculatethetrajecto-riesofu*andm fromt=0toa time-pointof lowsurvivalprobabil-ity,P = 10−6. Ifuhasnot reached1withinthis timespan,growth isconsideredtobe indeterminate.Wealsoestimatethemaximumlifespanandmaximumbodysizecorrespondingtotheoptimalschedules.Followingacommonpracticeinanimalstudies(e.g.,Benedettietal.,2008;Satohetal.,2016),wedefinethemaximumlifespanastheav-erageageatdeathofthe10%mostlong-livedindividualsinacohort.Wethendefinemaximumbodysizeastheaveragefinalsizeoftheseindividuals.

3  | RESULTS

3.1 | Types of optimal allocation schedules

By exploring the salient parameter space, we identify twelvequalitativelydifferent typesofoptimalallocation schedulesu*(m).Figure2ashowsexamplesofeachofthetwelvetypeswithparame-tercombinationsselectedforvisualclarity.Reproductiveallocationincreaseswithmasseitherstepwiseorgradually.Asaconsequence,theoptimalallocationschedulesconsistof intervalswithcontinu-ous increase andpointswith discontinuous increase (denotedD).Intervals of continuous increase can furthermore be divided intoaccelerating (denoted a) or decelerating (denoted d) functions ofbodymass.Wealsonotethattheoptimalallocationschedulescanhaveadifferentshapeduringtheonsetofreproduction(whenu*(m)

increasesfromzero)andduringthecompletionphase(whenu*(m)approaches1oranasymptoticvalue).

Four of the twelve types of allocation schedules (top row inFigure2a) exhibit a discontinuous onset of reproduction, whereastheothereightarecontinuousinthebeginning,increasinggraduallyfromzero.Inthecompletionphase,fullreproductiveallocation(u=1)isapproacheddiscontinuouslyinthreeofthetwelvetypes(leftmostcolumninFigure2a).Intheotherninetypes,theallocationcurvein-tersectswithu=1asa continuouscurve that is either acceleratingordecelerating(withthecompletionphasedenotedaord).Inthreetypes (rightmost column inFigure2a), the allocation curve isdecel-eratingwithouteverreachingfull reproduction (u=1),andthisspe-cialcaseiscategorizedasindeterminategrowth(withthecompletionphasedenotedi).

Combinationsofthethreecategoriesofshapesintheonsetphase(D,a,ord)andthefourcategoriesofshapesinthecompletionphase(D,a,d,ori)constitutethetwelvequalitativelydifferentreproductive-allocationschedulesshowninFigure2. Inthefollowing,wewillusetheseonset-andcompletion-phaseshapecategoriestodescribethedifferent optimal mass-dependent reproductive-allocation sched-ules in abbreviated form.As an example, the string Dd refers to areproductive-allocation schedule with a discontinuous onset phaseandacontinuous,deceleratingcompletionphase.

3.2 | Growth patterns of the optimal types

DeterminedaccordingtousingEquation(1),thegrowthcurvesofthetwelvenumericallyobtainedoptimaltypesdescribedaboveareshownin(Figure2b).Thegrowthcurvesallhaveanacceleratingphaseinthebeginning,sincetheninvestmentintoreproductionislowandalmostallenergy isusedforgrowth.Thegrowthcurvesdiffermore inthelaterstages.Thetypeswithadiscontinuouscompletionphasestopgrowingsuddenly(growthtypeI,firstcolumninFigure2b).Thetypeswithacontinuouscompletionphasethatreachmaximalinvestmentinto reproduction,u=1, before the survival probability falls belowthestipulatedthresholdofP = 10−6(growthtypeII,secondcolumninFigure2b),growasymptoticallytowardafinalmass.Growthcurvesforthetypesthatdonotreachu=1withinthistimeframeareeitherdecelerating (growth type III, thirdcolumn inFigure2b)oracceler-ating(growthtypeIV,fourthcolumninFigure2b). Insummary,thetwelvequalitativelydifferenttypesofreproductive-allocationsched-ulescorrespondtofourqualitativelydifferentmodesofgrowthlateinlife.

3.3 | How optimal allocation schedules depend on fecundity and mortality curvature parameters

TofindalltwelvetypesofoptimalallocationschedulesinFigure2a,itisnecessarytovarymorethantwoparametersatatime.Itisthere-forenotstraightforwardtogiveafulloverviewofwhereintheparam-eterspacedifferenttypesareoptimal.However,severaltypescanbefoundbyvaryingkbandkqwhilekeepingtheotherparametersfixed

(6)R0=∫∞

0

P(t) b(u[m(t)] E[m(t)]) dt.

     |  5JOHANSSON et Al.

(Figure3a).Weusethisasastartingpointforelucidatingwhichcondi-tionsfavordifferenttypes.

First,thetwocurvatureparameterskbandkqdeterminewhetherthe optimal allocation schedule is continuous, discontinuous, or acombinationofthetwo.ThefourquadrantsinFigure3acorrespondto combinations of accelerating (convex) anddecelerating (concave)fecundityandmortalityrates.Inthetop-leftquadrant,mortalityisac-celeratingandfecundityisdecelerating.Theseconditionsareknownto favor gradually increasing allocation to reproduction (see Siblyetal.,1985).Therefore,exclusivelytypeswithcontinuouslyincreasingreproductive investment (aa,ad,ai,anddi)arepresent there. In theopposite,lower-rightquadrant,noneoftheseconditionsarefulfilled,and thus,only thebang-bangcontrol strategy (DD)occurs there. Inthetop-rightandbottom-leftquadrants,oneoftheconditionsforin-termediateallocationisfulfilled,butnottheother.Itisonlyinthesetwoquadrantsthatwefindtypeswithmixeddiscontinuousandcon-tinuousphases (Dd,Di, andDa).Heuristically, the combinationof acurvatureparameterthatfavorsreproductiveallocationofbang–bangtypeandacurvatureparameterthatfavorsgradedallocationmayyieldtypesthatareamixtureofthetwo.Amoretechnicalunderstandingcanbegainedbyconsideringthemarginalvaluesofreproductivein-vestmentatdifferentlifestages,aswewilldetailinSection3.5.

Second, the curvature parameters kb and kq determinewhetheroptimalallocationscheduleswithgraduallyincreasingreproductiveal-locationhaveacceleratingordeceleratingphases.Thisvariationcan

beexplainedbythemagnitudeofthecurvatures.Note,forexample,thatdecreasingthecurvatureparameterkbofthefecundityfunctionandmovingfrom(kb,kq)=(1,1)to(kb,kq)=(0.5,1)inFigure3inducesatransitionfromanacceleratingtoadeceleratingcompletionphase,andfinallytoindeterminategrowth,whichisastronglydeceleratingcompletionphase.Similarly,increasingkqinthethreeleftmosttypesinthemiddlerowofFigure2leadsfromacceleratingtoincreasinglydeceleratingcompletionphases.

Inmanycases,theshapesintheonsetandcompletionphasesdonotcoincide(alltypesexceptDD,aa,anddd).Consider,forexample,theDd and the dD types. Both exhibit continuously anddiscontin-uously increasingallocation,but theDd typehas thediscontinuousphase first and the dD type has the discontinuous phase last. Thevariation in the order by which different shape categories appearcanbeunderstoodbyobservinghowthecurvaturesofthefecundityandmortalityfunctionsvarywiththelevelofinvestment.Forexam-ple, thephysiological-limit parameterp affects the curvatureof thefecundityfunctionatlowlevelsofinvestment(Figure1c).Therefore,variationinpmainlyaffectstheshapeofthereproductive-allocationscheduleinitsearlyphases.Thisexplainswhytheearlyphasesofthereproductive-allocationschedulesinthesecondandthethirdrowinFigure2a,whichdifferonlyintheparameterp,havedifferentshapes(acceleratinganddecelerating,respectively).Next,considerthetypesaa,ad,da,andddinFigure2.Thesedifferinthephysiological-limitpa-rameterpandinthekqparameteronly.Variationinpcausesachange

F IGURE  2 Differenttypesofoptimalreproductive-allocationschedules(a)andcorrespondinggrowthcurves(b).Lettersindicatetheshapesoftheallocationschedulesintheonsetphase(rows)andinthecompletionphase(columns)ofreproductiveinvestment.Foreachallocationscheduleshownin(a),thecorrespondinggrowthcurveisshownin(b).ThetypeofgrowthcurveisindicatedbytheromannumbersI–IVin(b).Thecirclesrepresentthemeanageormeansizeofthe10%oldestindividuals.Withsomeexceptionsdetailedbelow,weassumethefollowingparametersettings:kq =0.95,1.25,3.5,and10inthecolumnsfromlefttoright,kb =1.02,0.92,and0.92intherowsfromtoptobottom,p =15,15,and1.2intherowsfromtoptobottom,c2=0.04,c1 = c3 = c4=1,mbirth=0.01,andke =3/4.TheexceptionsarethatforDawesetc2 = 1.15 andkb =1.1;forDi,wesetc2=0.8;foraD,wesetc1 = c3=2andkb = kq =0.95;andfordD,wesetc1 = c3=2,kb = kq =0.95,andp = 2.5

(b) (a)

6  |     JOHANSSON et Al.

between accelerating and decelerating curves in the early phasesof investment,whilevariation inkq causes the changebetween ac-celeratinganddeceleratingcurves in the laterphasesof investment(Figure1a).Hence, by changing theseparameters independently, allcombinationsaa,ad,da,andddcanbeobtained.

InAppendixBwedescribehowvariationintheothermodelpa-rameters (c1,…,c4,ke,P, and s) qualitatively influence the shapes oftheoptimalschedules.Wegenerallyfindthatthediversityofoptimaltypesisrobusttomoderatevariationintheseparametervalues,butcanbereducediftheytakeonverylargeorverysmallvalues.Forex-ample,severaltypesinourschemeappearbecauseofaninteractionbetweenthemortalityandthefecundityfunctions.Wemightthere-foregetalowerdiversitywheneffectsfromoneofthesedominateforexample,ifc2issmallcomparedtoc3suchthatvariationinuwillaffectfecunditymuchmorethanmortality.

Wealsostudyhowtheoptimalallocationschedulesareaffectedby fixed costs of reproduction and size-dependent mortality (seeAppendixB).Wefindthatfixedcostsofreproduction,suchthatfe-cundityiszerountilaminimumamountofenergyisallocatedtorepro-duction,haveverysimilareffectsontheshapesofoptimalschedulesas assuming accelerating fecundity.With regard to size-dependentmortality,wefindthatifmortalityisinitiallyveryhighanddecaysonlyslowlywithsizewecangetextremeandunrealisticoutcomeswhereit isoptimalto investallenergytoreproductionalreadyfrombirth.However,ifmortalitydropsrelativelyfastwithsizeorisnottoolargeoverall,theeffectsontheshapesoftheoptimalallocationschedules,andthusalsoonoveralldiversity,shouldberelativelymild.

3.4 | How growth- curve types and life- history attributes depend on fecundity and mortality curvature parameters

The allocation schedule determines the growth curve and, conse-quently, also themaximum life span andmaximumbody sizeof anorganism.Forexample, themaximumbodysize increasesas theal-location schedule becomes increasingly decelerating (e.g., secondrow inFigure2b). Inorder togiveamore systematicoverview,weinvestigatehowsalientlife-historycharacteristicsareaffectedbytheshapesofoptimalreproductiveallocationundervariationofthecur-vatureparameterskbandkq(Figure3b-d).Whenmovingtowardtheupperleftcorner(lowkbandhighkq)intheparameterspace,thetypesofgrowth (Figure3b)changefromendingabruptly (I)orasymptoti-cally (II) tocontinued, indeterminategrowth (III, IV).Whencompar-ingFigure3a,b, it isalsoapparentthatthereisnocompleteoverlapbetweentheoptimaltypesandthegrowthcurves intheparameterspace.Only thegrowth type that stopsabruptly (I) exclusively cor-respondstotheDDtypeinFigure3.However,withotherparametersettings,thistypeofgrowthcurvemayalsoarisewithanaDoradDtype(Figure2b).

Variationingrowthcurvesisassociatedwithvariationinmaximumlifespanandmaximumbodysize.Asshown inFigure3d, thebang-bangcontrolstrategies(DD)andnonbang-bangcontrolstrategiesareaffecteddifferentlybyvariationinthecurvatureparameters.Forbang-bangcontrolstrategies,theswitchfromnonetofullreproductivein-vestmentoccursatincreasinglylargersizeswhenkbisincreased.The

F IGURE  3 Overviewofhowthecurvatureparameters(kb,kq)ofthefecundityandmortalityfunctionsaffecttheshapeoftheoptimalreproductive-allocationscheduleandtheirproperties.In(a)and(b),thelinesindicatebordersbetweendifferenttypes.ThetypesareclassifiedasinFigure2.(c)and(d)arecontourplots.In(b–d)thedashedlineindicatestheborderbetweenbang-bangcontrolstrategies(DD)andothers.Thegraylinesin(a)and(b)correspondtokb = 1 andkq =1,respectively.Theinsetinpanel(d)showstheoptimalallocationschedulesu*(m)attheparametervaluesindicatedbyXandYinthecontourplot.Parameters:c1 = c2 = c3 = c4=1,mbirth=0.01,ke=3/4,andp = 15

(a) (b)

(c) (d)

     |  7JOHANSSON et Al.

reasonisthatthestrongeracceleratingfecundityreturnsmakeitop-timaltowaitlongertoreproduce.Hence,bothmaximumlifespanandmaximumbodysizeincreasewithkb.Thebang-bangcontrolstrategiesare,however,notaffectedbykq,astheyonlytakethevaluesu* = 0 andu*=1,andarethereforeindependentofkqaccordingtoEquation(3).Fornonbang-bangcontrolstrategies,theeffectofincreasingkbisthatu*decreasesatlowersizesandincreasesathighersizes(movingfrompointsXtoYinFigure3d,inset),andasaconsequence,individ-ualswiththeoptimalallocationscheduleexperiencelowermortalityand grow faster when young, and experience higher mortality andgrowslowerwhenold.Theneteffectonmaximumlifespan(maximumbodysize)dependsonwhetherornottheincreasedsurvival(growth)intheearlierstagesiscompensatedbythereducedsurvival(growth)inlaterstages.

3.5 | The marginal values of reproductive investment

Qualitativeinsightsintohowfecundityandmortalityfunctionsaffecttheshapeoftheoptimalallocationschedulecanbegainedbystudyingthereturnsofashortchangeinreproductiveinvestmentforthelife-timereproductivesuccessofanindividual.FollowingMetz,Staňková,andJohansson(2016)andinthespiritofthemarginal-valuetheorem(Charnov,1976;seealsoWilliams,1966),wederiveanexpressionforsuch fitness returns,denoted r for short (AppendixA).For thepur-posesofourarguments,rcanbewrittenas

Here, the first term gives the instantaneous return from reproduc-tion ifthereproductiveallocation is increasedfromu(t) toũattimetandthesecondandthirdtermcorrespondtothefuturereduction

inreproductionfromtheassociateddecreaseinfuturesizesandde-crease insurvivalprobability,respectively.Theexpressionallowsustocomparethefitnessreturnfromchangingthelevelofreproductiveinvestmentfromutoũatagivenmassmofanindividual.Here,ũcantakeonanyvaluebetween0and1,incontrasttotheexpressionde-rivedbyMetz,Staňková,andJohansson(2016)thatgivesthemarginalfitnessreturns,i.e. is,effectsofsmalldeviationsfromu.Thefactorsy1(t,u)andy2(t,u)dependonthestrategyuasawhole;explicitex-pressionsaregiveninAppendixA.Areproductive-allocationscheduleu*isoptimalifandonlyifthefitnessreturnismaximizedatũ=u∗(m) foranym≥mbirth.Anydeviationfromtheoptimalstrategywill thuseitherdecreaselifetimereproductionorleaveitunchanged,andthefitnessreturnfortheoptimalstrategyisalwayszero.

Byanalyzingtheexpressionforfitnessreturn,weseethatthecur-vatureofthefitnessreturnrdependsonthecurvaturesofthefecun-dity functionb and themortality functionq. In fact, rhasone termthatisequaltob(ũE)andonetermthatisproportionalto−q(ũ). The futurefitnessreturnscanbeseenasaweightedmeanofthesetwocomponents.Forexample,whenbothband−qareaccelerating(i.e.,haveincreasingslopes),theirweightedmean,andthusr,willalsobeaccelerating(Figure4a).Likewise,whenbothband−qaredecelerating(i.e.,havedecreasingslopes), rwillbedecelerating too (Figure4b,c).Finally,whenband−qhavedifferentcurvatures,rmaytakeonasig-moidalshape(Figure4d).

Basedontheseobservations,wecanmakequalitativepredictionsabouttheoptimalallocationscheduleswhenband−qhavetheformsinFigure4a–d.Withanacceleratingr(Figure4a),therecannotbeaninteriormaximum, sowhen its slope changes, changes in allocationmustoccur in the formofa sudden,discontinuous shift (Figure4e).Withadeceleratingr(Figure5b,c),therecanbeaninteriormaximum.Astheslopeofrchangesinthesecases,themaximumwill increase

(7)

r(ũ,t;u)=[

b(ũE(m))−b(uE(m))]

(t)−[

(ũ−u)E(m)]

(t)y1(t,u)

−[

q(ũ)−q(u)]

(t)y2(t,u).

F IGURE  4  Illustrationofhowshapesoftheoptimalreproductive-allocationschedulescanbededucedusingthefitness-returnapproach.Foranindividualinagivenstate,thefitnessreturn(rinEquation(7)ofchangingtheallocationlevelfromutoũcorrespondstoaweightedmean(greylines)ofthefunctionsforfecundity(b(ũE),continuouslines)andnegativemortality(−q(ũ),dashedlines).Panels(a–d)showhowtheweightedmeanqualitativelydependsontheshapethefecundityandmortalityfunctions.Panels(e–h)illustratehowtheshapeoftheoptimalallocationschedule(u*,thicksolidlines)dependsontheshapeofthefitness-returnfunction(r(ũ,t;u∗),greylines)atdifferentagest.Thefitness-returnfunctionsineachpanel(e–h)correspondtotheweightedmeaninthepanels(a-d)above

(a) (b) (c) (d)

(e) (f) (g) (h)

8  |     JOHANSSON et Al.

gradually from 0 to 1, rendering a continuous allocation schedule(Figure4f,g).Finally,withasigmoidalr(Figure4d),wemaygetamix-tureofdiscontinuousandcontinuousallocationschedules(Figure4h).Noticethatinthisexample,theweightedmeanisacceleratingforlowvaluesofũanddeceleratingforhighvaluesofũ(Figure4d),whichex-plainswhytheoptimalallocationschedulefirstexhibitsadiscontinu-ousincreasetothenincreasingcontinuously.

Whether a gradually increasing optimal allocation schedule willbe accelerating or decelerating as a function of mass (e.g., typesaaordd)dependsonmore subtlepropertiesof the fitness compo-nents. Consider theweightedmean of the fecundity andmortalityinFigure4b.Here,thebirthrateb isnearlylinearinũandq ismorecurvedforhighervaluesofũ.Theweightedmeanwillthereforehavemostcurvatureathighervaluesofũ.Astheslopechangeswithageinthiscase,themaximumwillincreaserelativelymuchatyoungagesand relatively little at older ages, thus giving rise to a deceleratingshape.InFigure4c,bycontrast,bhashighercurvaturethanq,andtheweightedmeanhashighestcurvaturefor lowvaluesofũ.Therefore,themaximumwillincreaseslowlyatyoungagesandfastatoldages,resultinginanacceleratingshapeoftheoptimalallocationschedule.Note, however, thatwhether the accelerating function of time alsowillbeanacceleratingfunctionofmass, inadditiondependsonthegrowthcurves(whichdescribemassasafunctionoftime).Thus,therelationshipbetweenearlyandlatecurvatureoftheweightedmeanandacceleratinganddeceleratingshapeofu*(m)shouldbeseenmoreasatendencythanasastrictrule.

3.6 | Summary and synthesis

Three key observations from numerical investigations (Figures2and 3) and from studying the fitness returns of reproductive

investments (Figure4) can be used to summarize our results. First,an intervalof a fitness-returncurve that is accelerating leads toanoptimal reproductive-allocation schedule with discontinuous in-crease(cf.Figure4a,e),whiledeceleratingintervalsleadtoanoptimalreproductive-allocationschedulewithgraduallyincreasingallocation(cf. Figure4b,f). Second, the slope of a decelerating fitness-returncurveaffectstheshapeofgraduallyincreasingoptimalreproductive-allocationschedules,suchthatmoredecelerationcausesatransitionfromanacceleratingtoadeceleratinggradualincrease(cf.Figure4b,fwithc,g).Third,theshapeofthefitness-returncurveforlow(high)lev-elsofreproductiveinvestmentaffectstheshapeofoptimalreproduc-tive-allocationschedulesatlow(high)agesormasses(cf.Figure4d,f).

We synthesize these relationships into the general classifica-tion scheme of optimal allocation schedules shown in Figure5(corresponding to the numerically obtained allocation schedulesin Figure2). Variation in fitness-return curvatures in the earlystages (small investment levels) leads to three different catego-riesof shape in theonsetphase (rows inFigure5).Theonsetofreproductionmay be discontinuous (D) or continuous dependingonwhether the fitness-return curve at early stages is accelerat-ing or decelerating. Depending on how strong the decelerationis, a continuousonset canbedecelerating (d) or accelerating (a).Variation in fitness-returncurvatures in the later stages (high in-vestmentlevels)leadstofourdifferentcategoriesofshapeinthecompletion phase (columns in Figure5). Depending on whetherthefitness-returncurveattheseinvestmentlevelsisacceleratingor decelerating, full reproductive allocation may be approacheddiscontinuously (D)orgraduallybyacontinuouscurve(a,d,or i).Dependingonhowstrongthedecelerationis,thecontinuousallo-cationcurvemaybeaccelerating(a),decelerating(d),ordecelerat-ingwithouteverreachingu=1(i).

FIGURE  5 Conceptualmapfromallocation-dependentfitnessreturnstooptimalallocationschedules.Thecurveswithintheboxesframedingreyrepresenttheshapeofthefitnessreturn(verticalaxes)asafunctionofreproductiveallocationu(horizontalaxes)forearlyages(leftcolumn)andlateages(toprow),respectively.Theseshapesinturnmaptoshapesoftheoptimalreproductive-allocationschedule,attheonsetandcompletionofreproductiveinvestment,respectively,asindicatedbythegreyarrows.Thetwelvetypesofoptimalallocationschedules(boxesfilledwithgrey)arethenobtainedascombinationsofthethreetypesofshapesattheonsetphaseandthefourtypesofshapesatthecompletionphase.AsinFigure2a,optimalallocationschedulesareshownasfunctionsdescribinghowtheallocationlevelu*(verticalaxes)dependsontheofsizem (horizontalaxes)

     |  9JOHANSSON et Al.

Compared to the traditional classification of life histories intotypeswithdeterminategrowthandtypeswithindeterminategrowth,threetypes(DD,dD,andaD)clearlyexhibitdeterminategrowthandthreetypes (Di,di,andai)clearlyexhibit indeterminategrowth.Fortheremainingsixtypes,however,itisasomewhatarbitraryjudgmentwhethertheycouldbesaidtoexhibitdeterminateor indeterminategrowth(cf.Figure3a,b).

4  | DISCUSSION

Ourresultsshowthatvariationintheshapeofallocation-dependentfecundityandmortalityratescangiverisetoasurprisingdiversityinoptimalallocationschedules,hithertonotappreciatedintheliterature.Belowwediscusshowourfindingsarelinkedtoprevioustheory.Wealsodiscusshowtheresultscanbeusedto interpretempiricalpat-ternsinallocationandgrowthstrategiesandtounderstandhoweco-logicalandphysiologicalconstraintsinfluencelife-historyevolution.

4.1 | Relation to other theoretical approaches

Our results extend the more qualitative results reported in earlieranalytic studies (e.g., Sibly etal., 1985; Taylor etal., 1974). For ex-ample,Siblyetal.(1985)discussedwhichcombinationsofcurvaturesin fecundity andmortality rates render graded allocation schedulesoptimal,withoutgoing intofurtherdetailabouttheshapesoftheseschedules.By contrast, our results give concrete insights intowhatkindofdiversitywecanexpecttoarisefromthesemechanisms.Wealsoshowhowthisdiversitycanbeunderstoodbystudyingthefit-ness returns of reproductive investment (Equation[7], Figure5,AppendixA)andthusshedlightonhowspecificfeaturesoffecundityandmortalityratesinfluenceoptimalallocationschedules.

Thefitnessreturn,whichcanbedeterminedforanyu,isinmanywayssimilar to the fitnessgradient inadaptive-dynamics theory fortheevolutionoffunction-valuedtraits(Dieckmann,Heino,&Parvinen,2006;seealsoMetzetal.,2016).Thisconnectionprovidesaninroadto study allocation problems under the influence of environmentalfeedback (seealsoParvinen,Dieckmann,&Heino,2013).This isaninteresting possibility for extensions of our presentmodel, as opti-mizationapproaches inherentlyaredirectlyapplicabletoarelativelynarrowrangeofcompetitivescenarios, suchasnurserycompetition(Metz,Mylius,&Diekmann,2008).Withanenvironmentalfeedbackloop in place, we have to distinguish between primary parametersand parameters modified by the environmental feedback, like thefullrangeofpossiblefertilitiesandthatoffertilitiesinenvironmentsdepleted by the corresponding equilibrium populations. Ifwe allowmaximalfreedomintheprimaryparameters,uptonaturalphysiologi-calrestrictions,themodifiedparameterscanonlybemorerestricted.Hence, incorporating an environmental feedback can only decreasethepossiblevariation inoutcomesof theoptimizationproblem.Wealsonotethattheuseoffitnessreturnstogetherwithanassociatedmarginal-valuetheoremtosolveallocationproblemsmethodologicallyconnectsourworktoalargerangeofotherproblemsinevolutionary

theory,includingmatingbehavior(Parker,1974)andoptimalforaging(Charnov,1976).

Whilewehavestudiedtheinfluenceofallocation-dependentfe-cundity andmortality rates on optimal allocation schedules, similartypesandsimilardiversitymayresultfromothermechanismsaswell.KingandRoughgarden(1982)found,forexample,thatadiscontinu-ousonsetofreproductionfollowedbyacontinuouscompletionphasecanbeoptimalforannualspecieswhenseasonlengthsfluctuatesto-chastically.Asanotherexample,Janczur(2009)foundthataqualita-tivelysimilarallocationschedulecanbeoptimalforplantsaffectedbyherbivory,andfurthermore,thattheexactallocationlevelduringthecompletionphasedependedonthecostsandefficienciesofchemicaldefensesubstances.

Theoretical studies have also predicted optimal reproductive-allocationscheduleswithqualitativelydifferentshapescomparedtothoseweidentifyhere.OneexampleisgivenbyMcNamara,Houston,Barta,Scheuerlein,andFromhage(2009),whoinvestigatedamodelinwhichdamageaccumulatedoveranorganism’s lifetimemakesimul-taneousallocationtogrowthandreproductionoptimal.Incontrasttoourmodel,inwhichreproductiveallocationalwaysincreaseswithageorremainsconstant,theyshowedthatreproductiveallocationcoulddecreaseatlaterstagesofanorganism’slifewhenphysiologicalcon-ditionsdeteriorate.Asanotherexample,models taking intoaccountseasonalvariationintheenvironmentpredictthat itmaybeoptimalforperennial lifehistories to switchbetweenphasesofgrowthandreproductioneveryyear(e.g.,Kozłowski&Uchmanski,1987)insteadofsimultaneouslyallocatingtogrowthandreproduction,aswehavestudiedhere.Seasonallyvaryingsurvivalprospects for theoffspringcanfurtherinfluencetheoptimalshapeofsuchindeterminategrowthpatternsandaffectwhether it isoptimalto invest intoreproductionbeforeoraftergrowthwithinasingleseason(Ejsmond,Czarnołeski,Kapustka,&Kozłowski,2010).

Becausedifferentalternativemechanismsmayfavorindetermi-nategrowth, itwouldbeof interesttoknowhowourpredictionsregardingshapesofoptimalallocationschedulesandtheirdiversitymightbeaffectedbyalternativemechanisms,whichalsomayex-plainindeterminategrowth.Someinsightsintothisquestionwereprovided in Klinkhamer, Kubo, and Iwasa (1997)who consideredseasonal variation aswell as allocation-dependent fecundity andmortality functions (corresponding tokbandkqbeing nonzero) inamodelofperennialplants.Notethatotherstudies(e.g.,Ejsmondetal., 2010;Kozłowski&Uchmanski,1987)ofoptimal reproduc-tiveallocation in seasonalenvironmentsassume that fecundity isproportional to reproductive allocation (kb=1). In line with ourconclusions,Klinkhameretal.(1997)foundthattheoptimalyearlyallocationtoreproductionincreasedgraduallywithagewhentherewerediminishingreturnsfrominvestmentintofecundity.However,whenmortalityincreasedatadeceleratingratewithreproductiveallocation, inwhichcaseourmodelpredictsbang-bangcontrol isoptimal,theyinsteadfoundthat itwasoptimaltoreproduceonlyin certain years, with nonreproductive years in between, a pat-terncomparabletomasting.AsnotedbyKlinkhameretal.(1997),suchpatternsof intermittentreproduction inpracticecorrespond

10  |     JOHANSSON et Al.

tobang-bangcontrolifsurvivalafterthefirstreproductiveyearisverylow.Thesecomparisonsshowthatsomeofourresultscanbecarriedovertomorecomplexscenarios,butthatadditionalmech-anisms, e.g. associatedwith the presence of storage organs, canyieldqualitativelydifferentpredictions.

Previous theoretical studies have mostly focused on studyingwhetherornot,orunderwhichconditions,acertainmechanismcangiverisetoindeterminategrowth.Bycontrast,exploringtheeffectsofvariationinparametersthatarestructurallyimportantfortheshapeof reproductive schedules, as we have done here, clarifies whichpotentialagivenmechanismhas togeneratediversity in lifehisto-ries.An interestingavenuefor futureresearch is tocomparewhichpatternsof life-historyvariationaregeneratedbywhichalternativemechanisms.Itwouldalsobeinterestingtostudymoresystematicallyinteractions among different mechanisms that independently mayfavorindeterminategrowth(cf.Klinkhameretal.,1997).

4.2 | Predictions about empirical patterns

Ouranalysisreveals linksbetweenecologicalandphysiologicalcon-straints on life-history evolution, on the one hand, and shapes andcharacteristics of the expected reproductive-allocation and growthstrategies,ontheotherhand.Inparticular,ourmodelbridgesbetweentheallocationpatternsexpectedfromevolutionandthecurvaturesofvitalratesthataffectfitnessreturns.Ifthesecurvaturesareknown,thetypeofoptimalallocationschedulecanbepredicted.Similarly,theshapeofanoptimaltypecanbeusedtoinferaspectsoftheshapeoftheunderlyingfitness-returnfunction.

Theusefulnessofestablishingtheselinksdependsonhowwellthecurvaturesofthefitness-returnfunctionsandthefeaturesoftheal-locationpatternscanbeobservedempirically.Empiricalstudieshaveidentified reproductive-allocation schedules, that can be related tothosefoundinourstudy.Forexample,WenkandFalster(2015)eval-uated the existing empirical evidence for diversity of reproductive-allocation schedules among perennial plants.Among the 32 speciesincluded in their review forwhich reproductive-allocation scheduleshadbeenquantifiedor couldbe inferred, they identified sixdistincttypes.Specifically,theirtypescorrespondedtoDD,Dd,dd,twover-sionsofdi(dependingonwhetherreproductiveallocationapproachedanasymptoteorcontinuedtoincrease),andonetype,notpresentinourmodel,characterizedbyacompletionphasewithdecliningrepro-ductive allocation.As anotherexample,Ware (1980)estimatedhowmuchsurplusenergywasallocatedtoeithergrowthorreproductioninAtlanticpopulationsofdifferentfishspeciesusingallometricfunc-tionsofsize.Thestudyreportedthatthefunctionsdescribingenergydevoted to reproduction (corresponding touE in ourmodel) for thedifferentspecieshad largerexponents thanthefunctionsdescribingsurplusenergy(correspondingtoEinourstudy).Ifweassumethatre-productiveeffortuinourstudyforeachpopulationcorrespondstotheratiobetweenthetwofunctions,andbecausethedifferencebetweentheestimatedexponentsinallcaseswereabove0andbelow1(Table3inWare,1980),wecandeducethatushouldbeadeceleratingfunctionofmass(correspondingtotheddorditypesinourframework).

Forsomespecies,researchershavealsoputforwardempiricalev-idenceforwhyfecundityisanonlinearfunctionofenergydevotedtoreproductionandhaveusedthistoexplainwhyacertainreproductivestrategymightbeoptimal.Forexample,SchafferandSchaffer(1979)relatedpollination-drivenacceleratingreturnsofreproductiveinvest-ments to the bang–bang strategy (DD) of Yucca wipplei andMiller,Tenhumberg&Louda(2008)relateddiminishingreturnsofreproduc-tionowingto insectherbivorytoagradedallocationpattern (corre-spondingtotheddorditypesinourframework)inaspeciesofcactus(Opuntia imbricata).

Estimatingenergyallocatedtogrowthandreproductionduringthe lifetime of an organism can be complicated and expensive,which may explain why such observations are rare (cf. Myers &Doyle,1983;Wenk&Falster,2015).Estimatingshapesoffecundityandmortalityfunctionsisperhapsevenmorechallengingowingtothetemporaldecouplingbetweentheallocationdecisionandfinaleffectonlifetimereproduction.However,evenifitmaybehardtogetafirmgriponthesekeyfeaturesfromdata,theycorrelatewithother patterns, that may be easier to observe. To startwith, ourmodel predicts connections between different types of allocationschedulesandqualitativelydifferentgrowthcurves.Ourmodelalsopredicts how the curvatures of mortality and fecundity functionsrelatetomaximumlifespanandmaximumbodysize (Figure3c,d).Predictions from our model about variation in energy-allocationpatterns can therefore be connected to data on growth curves,bodysizes,andage.Forexample,MyersandDoyle (1983)usedamodelsimilartoourstoreconstructmortalitycurvaturesfromdataon thegrowthand reproductive successondifferent fish species.As the curvatures ofmortality and fecundity functions ultimatelydepend on ecological and physiological constraints on life-historyevolution,variationinthesefunctionscanbeassumedtovarywithfactors,thatinfluencetherelevantconstraints.Itwould,forexam-ple,beinterestingtostudywhetherlifehistoriesvaryalonggradi-entsofsiblingcompetition(cf.Stockley&Parker,2002)ordependonreproductiveconstraints(cf.Shine,1988)inlinewithourmodelpredictions. In sum, even if some components of ourmodel maybe hard tovalidate, themultitude of connections betweenmodelpredictionsandaccessibledatamakeusbelievethattherearemanyways themodel predictions usefully generate empirically testablehypothesesaboutpatternsofdiversityinlifehistories.

Onespecificpossibilityforfutureresearchderivesfromourresultthatthepropertiesofbang–bangtypesandtypeswithcontinuousin-creaseinreproductiveallocationdependverydifferentlyonvariationincurvatureparameters.Forexample,withsettingsasinFigure3c,d,therewouldbea strongpositivecorrelationbetweenmaximum lifespanandmaximumbody size among randomly sampledbang–bangtypes,butnotamongnonbang–bangtypes.Itwouldbeinterestingtoinvestigatewhethersuchpatternscanbeobservedinempiricaldata(cf. Blueweiss etal., 1978;Hendriks, 2007) by grouping species ac-cordingtolife-historytypeandseeingifcorrelationsbetweenlifespanandbodysizedifferbetweenthesegroups.

Ourinvestigationheregoesbeyondthetraditionalperspectiveofdividingallocationpatterns into those leading toeitherdeterminate

     |  11JOHANSSON et Al.

or indeterminate growth. Our findings thereby offer a conceptualfoundation for studying intermediatecases, enabling the systematicexplorationofricherandmorenuancedvariationinlifehistories.Ourresults alsoprovide linksbetweenecological andphysiological con-straintsandtheselife-historytypes.Byestablishinganewandwiderscopefortestablepredictions,wehopetheseresultswillinspirecon-tinuedresearch intounderstanding the fullvariationof lifehistoriesencounteredinnature.

ACKNOWLEDGEMENTS

WethankDanielFalsterandJörgenRipaforhelpfulcommentsonear-lierversionsofthemanuscript.J.Johanssongratefullyacknowledgessupport by the Evolution and Ecology Program of the InternationalInstitute for Applied Systems Analysis (IIASA) and by the SwedishResearchCouncil(2015-00302toJ.Johansson)andMarieSklodowskaCurieActions,Cofund,ProjectINCA600398.Å.Brännströmgratefullyacknowledges support fromTheSwedishResearchCouncil and theSwedish Research Council Formas. J.A.J. Metz benefitted from thesupportfromthe“ChaireModélisationMathématiqueetBiodiversitéof Veolia Environnement-Ecole Polytechnique-Museum Nationald’HistoireNaturelle-FondationX.”U.Dieckmanngratefullyacknowl-edges support by the Sixth Framework Program of the EuropeanCommission,theEuropeanScienceFoundation,theAustrianScienceFund,theAustrianMinistryofScienceandRe-search,andtheViennaScienceandTechnologyFund.

CONFLICT OF INTEREST

Nonedeclared.

AUTHOR CONTRIBUTIONS

JJ,ÅB,andUDinitiatedthestudyanddesignedthemodel.JJanalyzedthemodel, conceived the synthetic framework, andwrote the firstdraftofthemanuscript.JAJMprovidedthemarginal-valueargumentsandanalysesoffitnessreturns.Allauthorsdiscussedtheresultsandimplicationsandcontributedsubstantiallytorevisions.

ORCID

Jacob Johansson http://orcid.org/0000-0002-0018-7018

REFERENCES

Aksnes, D. L., & Giske, J. (1993). A theoretical model of aquatic vi-sual feeding. Ecological Modelling, 67, 233–250. https://doi.org/10.1016/0304-3800(93)90007-F

Andersson,M., & Iwasa,Y. (1996). Sexual selection. Trends in Ecology & Evolution,11,53–58.https://doi.org/10.1016/0169-5347(96)81042-1

Bell,G.(1980).Thecostsofreproductionandtheirconsequences.American Naturalist,116,45–76.https://doi.org/10.1086/283611

Benedetti,M.G.,Foster,A.L.,Vantipalli,M.C.,White,M.P.,Sampayo,J.N.,Gill,M.S.,…Lithgow,G.J.(2008).Compoundsthatconferthermal

stressresistanceandextendedlifespan.Experimental Gerontology,43,882–891.https://doi.org/10.1016/j.exger.2008.08.049

Bertsekas,D.P.(1987).Dynamic programming: Deterministic and stochastic models.EnglewoodCliffs,NJ:PrenticeHall.

Blueweiss, L., Fox,H.,Kudzma,V.,Nakashima,D., Peters,R.,&Sams, S.(1978).Relationshipsbetweenbodysizeandsomelifehistoryparam-eters.Oecologia,37,257–272.https://doi.org/10.1007/BF00344996

Calow, P. (1979). The cost of reproduction – a physiological approach.Biological Reviews of the Cambridge Philosophical Society, 54, 23–40.https://doi.org/10.1111/j.1469-185X.1979.tb00866.x

Calow,P.,&Woollhead,A.S.(1977).Relationshipbetweenration,repro-ductive effort and age-specific mortality in evolution of life-historystrategies–someobservationsonfreshwatertriclads.Journal of Animal Ecology,46,765–781.https://doi.org/10.2307/3639

Charnov, E. L. (1976). Optimal foraging: The marginal value the-orem. Theoretical Population Biology, 9, 129–136. https://doi.org/10.1016/0040-5809(76)90040-X

Charnov,E.L.(1993).Life history invariants: Some explorations of symmetry in evolutionary ecology.Oxford,UK:OxfordUniversityPress.

Cohen,D. (1971).Maximizing finalyieldwhengrowth is limitedby timeor by limiting resources. Journal of Theoretical Biology,33, 299–307.https://doi.org/10.1016/0022-5193(71)90068-3

Dieckmann,U.,Heino,M.,&Parvinen,K. (2006).Theadaptivedynamicsoffunction-valuedtraits.Journal of Theoretical Biology,241,370–389.https://doi.org/10.1016/j.jtbi.2005.12.002

Ejsmond,M.J.,Czarnołeski,M.,Kapustka,F.,&Kozłowski,J.(2010).Howtotimegrowthandreproductionduringthevegetativeseason:Anevo-lutionarychoiceforindeterminategrowersinseasonalenvironments.American Naturalist,175,551–563.https://doi.org/10.1086/651589

Feeny,P.(1976).Plantapparencyandchemicaldefence.Recent Advances in Phytochemistry,10,1–40.

Fraenkel, G. S. (1959). The raison d’être of secondary plant sub-stances. Science, 129, 1466–1470. https://doi.org/10.1126/science.129.3361.1466

Gadgil, M., & Bossert, W. H. (1970). Life historical consequences ofnatural selection. American Naturalist, 104, 1–24. https://doi.org/10.1086/282637

Goodman, D. (1974). Natural selection and a cost ceiling on repro-ductive effort. American Naturalist, 108, 247–268. https://doi.org/10.1086/282906

Greene, D. F., & Johnson, E. A. (1994). Estimating the mean an-nual seed production of trees. Ecology, 75, 642–647. https://doi.org/10.2307/1941722

Heino, M., & Kaitala, V. (1999). Evolution of resource allocation be-tween growth and reproduction in animals with indeterminategrowth. Journal of Evolutionary Biology, 12, 423–429. https://doi.org/10.1046/j.1420-9101.1999.00044.x

Hendriks,A.J. (2007).Thepowerofsize:Ameta-analysisrevealsconsis-tency of allometric regressions. Ecological Modelling, 205, 196–208.https://doi.org/10.1016/j.ecolmodel.2007.02.029

Houston, A., & McNamara, J. M. (1999).Models of adaptive behaviour. Cambridge,UK:CambridgeUniversityPress.

Intrilligator,M.D. (1971).Mathematical optimization and economic theory. EnglewoodCliffs,NJ:PrenticeHall.

Janczur,M.K.(2009).Optimalenergyallocationtogrowth,reproduction,andproductionofdefensivesubstancesinplants:Amodel.https://doi.org/10.1371/journal.pone.0089535

King, D., & Roughgarden, J. (1982). Graded allocation between vegeta-tiveandreproductivegrowthforannualplantsingrowingseasonsofrandom length. Theoretical Population Biology, 22, 1–16. https://doi.org/10.1016/0040-5809(82)90032-6

Klinkhamer, P. G. L., Kubo, T., & Iwasa, Y. (1997). Herbivores and theevolutionof the semelparous perennial life-historyof plants. Journal of Evolutionary Biology, 10, 529–550. https://doi.org/10.1007/s000360050040

http://orcid.org/0000-0002-0018-7018http://orcid.org/0000-0002-0018-7018https://doi.org/10.1016/0304-3800(93)90007-Fhttps://doi.org/10.1016/0304-3800(93)90007-Fhttps://doi.org/10.1016/0169-5347(96)81042-1https://doi.org/10.1086/283611https://doi.org/10.1016/j.exger.2008.08.049https://doi.org/10.1007/BF00344996https://doi.org/10.1111/j.1469-185X.1979.tb00866.xhttps://doi.org/10.2307/3639https://doi.org/10.1016/0040-5809(76)90040-Xhttps://doi.org/10.1016/0040-5809(76)90040-Xhttps://doi.org/10.1016/0022-5193(71)90068-3https://doi.org/10.1016/j.jtbi.2005.12.002https://doi.org/10.1086/651589https://doi.org/10.1126/science.129.3361.1466https://doi.org/10.1126/science.129.3361.1466https://doi.org/10.1086/282637https://doi.org/10.1086/282637https://doi.org/10.1086/282906https://doi.org/10.1086/282906https://doi.org/10.2307/1941722https://doi.org/10.2307/1941722https://doi.org/10.1046/j.1420-9101.1999.00044.xhttps://doi.org/10.1046/j.1420-9101.1999.00044.xhttps://doi.org/10.1016/j.ecolmodel.2007.02.029https://doi.org/10.1371/journal.pone.0089535https://doi.org/10.1371/journal.pone.0089535https://doi.org/10.1016/0040-5809(82)90032-6https://doi.org/10.1016/0040-5809(82)90032-6https://doi.org/10.1007/s000360050040https://doi.org/10.1007/s000360050040

12  |     JOHANSSON et Al.

Kozłowski, J. (1991). Optimal energy allocationmodels – an alternativetotheconceptsofreproductiveeffortandcostofreproduction.Acta Oecologica,12,11–33.

Kozłowski,J.,&Uchmanski,J.(1987).Optimalindividualgrowthandrepro-duction inperennial specieswith indeterminategrowth.Evolutionary Ecology,1,214–230.https://doi.org/10.1007/BF02067552

Kozłowski, J., &Wiegert, R. G. (1986). Optimal allocation of energy togrowth and reproduction. Theoretical Population Biology, 29, 16–37.https://doi.org/10.1016/0040-5809(86)90003-1

Kozłowski,J.,&Ziólko,M.(1988).Gradualtransitionfromvegetativetore-productivegrowthisoptimalwhenthemaximumrateofreproductivegrowthislimited.Theoretical Population Biology,34,118–129.https://doi.org/10.1016/0040-5809(88)90037-8

León,J.A.(1976).Lifehistoriesasadaptivestrategies.Journal of Theoretical Biology,60,301–335.https://doi.org/10.1016/0022-5193(76)90062-X

Lord, J.M., &Westoby,M. (2012).Accessory costs of seed productionandtheevolutionofangiosperms.Evolution,66,200–210.https://doi.org/10.1111/j.1558-5646.2011.01425.x

McNamara,J.M.,Houston,A.I.,Barta,Z.,Scheuerlein,A.,&Fromhage,L.(2009).Deterioration,deathandtheevolutionofreproductiverestraintinlatelife.Proceedings of the Royal Society B,276,4061–4066.https://doi.org/10.1098/rspb.2009.0959

Metz,J.A.J.,Mylius,S.D.,&Diekmann,O.(2008).Whendoesevolutionoptimize?Evolutionary Ecology Research,10,629–654.

Metz,J.A.J.,Staňková,K.,&Johansson,J.(2016).Thecanonicalequationofadaptivedynamicsforlifehistories:Fromfitness-returnstoselectiongradientsandPontryagin’smaximumprinciple.Journal of Mathematical Biology,72,1–28.https://doi.org/10.1007/s00285-015-0938-4

Miller, T. E. X. Tenhumberg, B., & Louda, S. M. (2008). Herbivore-me-diated ecological costs of reproduction shape the life history of aniteroparous plant. American Naturalist, 171, 141–149. http://doi.org/10.1086/524961

Myers,R.A.,&Doyle,R.W.(1983).Predictingnaturalmortalityratesandreproduction-mortalitytrade-offsfromfishlifehistorydata.Canadian Journal of Fisheries and Aquatic Science, 40, 612–620. https://doi.org/10.1139/f83-080

Orton,J.H. (1929).ObservationsonPatella uulgata.Part III.Habitatandhabits.Journal of the Marine Biological Association of the UK,16,277–288.https://doi.org/10.1017/S0025315400029805

Paine, R.T. (1976). Size-limited predation:An observational and experi-mental approach with the Mytilus-Pisaster interaction. Ecology, 57,858–873.https://doi.org/10.2307/1941053

Parker,G.A. (1974).Courtshippersistenceandfemale–guardingasmaletime investment strategies. Behaviour, 48, 157–184. https://doi.org/10.1163/156853974X00327

Parvinen,K.,Dieckmann,U.,&Heino,M.(2013).Function-valuedadaptivedynamicsandoptimalcontroltheory.Journal of Mathematical Biology,67,509–533.https://doi.org/10.1007/s00285-012-0549-2

Perrin, N., Ruedi, M., & Saiah, H. (1987). Why is the cladoceranSimocephalus vetulus(Müller)nota‘bang-bangstrategist’?Acritiqueoftheoptimal-body-sizemodel.Functional Ecology,1,223–228.https://doi.org/10.2307/2389424

Perrin, N., & Sibly, R. M. (1993). Dynamic models of energy alloca-tion and investment. Annual Review of Ecology Evolution and Systematics, 24, 379–410. https://doi.org/10.1146/annurev.es.24.110193.002115

Perrin,N.,Sibly,R.M.,&Nichols,N.K.(1993).Optimal-growthstrategieswhenmortalityandproduction-ratesaresize-dependent.Evolutionary Ecology,7,576–592.https://doi.org/10.1007/BF01237822

Pontryagin,L.S.(1962).The mathematical theory of optimisation processes. Interscience4.

Reiss,M.J. (1989).The allometry of growth and reproduction. Cambridge,UK: Cambridge University Press. https://doi.org/10.1017/CBO9780511608483

Roff,D.(1992).The evolution of life histories: Theory and analysis.NewYork,NY:ChapmanandHall.

Sallabanks,R.(1992).Fruitfate,frugivory,andfruitcharacteristics:Astudyofthehawthorn,Crataegus monogyna(Rosaceae).Oecologia,91,296–304.https://doi.org/10.1007/BF00317800

Satoh,A.,Brace,C.S.,Rensing,N.,Cliften,P.,Wozniak,D.F.,Herzog,E.D.,…Imai,S.(2016).Sirt1extendslifespananddelaysaginginmicethroughtheregulationofNk2homeobox1intheDMHandLH.Cell Metabolism.,18,416–430.https://doi.org/10.1016/j.cmet.2013.07.013.

Schaffer,W.M.,&Schaffer,M.V.(1979).Theadaptivesignificanceofvari-ationsinreproductivehabitintheAgavaceaeII:Pollinatorforagingbe-haviorandselectionfor increasedreproductiveexpenditure.Ecology,60,1051–1069.https://doi.org/10.2307/1936872

Scharer, L. (2009). Tests of sex allocation theory in simultaneouslyhermaphroditic animals. Evolution, 63, 1377–1405. https://doi.org/10.1111/j.1558-5646.2009.00669.x

Shine, R. (1988). Constraints on reproductive investment:A comparisonbetweenaquaticandterrestrialsnakes.Evolution,42,17–27.https://doi.org/10.1111/j.1558-5646.1988.tb04104.x

Sibly,R.,Calow,P.,&Nichols,N.K.(1985).Arepatternsofgrowthadaptive?Journal of Theoretical Biology,112,553–574.https://doi.org/10.1016/S0022-5193(85)80022-9

Sletvold, N., & Ågren, J. (2015). Nonlinear costs of reproduction in along-lived plant. Journal of Ecology, 103, 1205–1213. https://doi.org/10.1111/1365-2745.12430

Stockley,P.,&Parker,G.A.(2002).Lifehistoryconsequencesofmammalsibling rivalry.Proceedings of the National Academy of Sciences of the United States of America,99,12932–12937.https://doi.org/10.1073/pnas.192125999

Svensson,J.-E.(1997).FishpredationonEudiaptomus gracilisinrelationtoclutchsize,bodysize,andsex:Afieldexperiment.Hydrobiologia,344,155–161.https://doi.org/10.1023/A:1002966614054

Taylor, H. M., Gourley, R. S., Lawrence, C. E., & Kaplan, R. S. (1974).Natural selection of life history attributes: An analytical ap-proach. Theoretical Population Biology, 5, 104–122. https://doi.org/10.1016/0040-5809(74)90053-7

Ware, D. M. (1980). Bioenergetics of stock and recruitment. Canadian Journal of Fisheries and Aquatic Science, 37, 1012–1024. https://doi.org/10.1139/f80-129

Wenk,E.H.,&Falster,D.S.(2015).Quantifyingandunderstandingrepro-ductiveallocationschedulesinplants.Ecology and Evolution,5,5521–5538.https://doi.org/10.1002/ece3.1802

Williams,G.C. (1966).Natural selection, the costs of reproduction, anda refinement of Lack’s principle.American Naturalist,100, 687–690.https://doi.org/10.1086/282461

How to cite this article:JohanssonJ,BrännströmÅ,MetzJAJ,DieckmannU.Twelvefundamentallifehistoriesevolvingthroughallocation-dependentfecundityandsurvival.Ecol Evol. 2018;00:1–15. https://doi.org/10.1002/ece3.3730

APPENDIX A

Derivation of fitness- return functionBelowwecalculatethefitnessreturnsaccordingtoEquation(7)forour optimization problem defined by Equations (1–6). As inMetzetal.(2016),fitnessreturnsr(ũ,t;u)denotetheeffectsonthelifetimereproductionR0ofanindividualthatresultfromchangingutothe

https://doi.org/10.1007/BF02067552https://doi.org/10.1016/0040-5809(86)90003-1https://doi.org/10.1016/0040-5809(88)90037-8https://doi.org/10.1016/0040-5809(88)90037-8https://doi.org/10.1016/0022-5193(76)90062-Xhttps://doi.org/10.1111/j.1558-5646.2011.01425.xhttps://doi.org/10.1111/j.1558-5646.2011.01425.xhttps://doi.org/10.1098/rspb.2009.0959https://doi.org/10.1098/rspb.2009.0959https://doi.org/10.1007/s00285-015-0938-4http://doi.org/10.1086/524961http://doi.org/10.1086/524961https://doi.org/10.1139/f83-080https://doi.org/10.1139/f83-080https://doi.org/10.1017/S0025315400029805https://doi.org/10.2307/1941053https://doi.org/10.1163/156853974X00327https://doi.org/10.1163/156853974X00327https://doi.org/10.1007/s00285-012-0549-2https://doi.org/10.2307/2389424https://doi.org/10.2307/2389424https://doi.org/10.1146/annurev.es.24.110193.002115https://doi.org/10.1146/annurev.es.24.110193.002115https://doi.org/10.1007/BF01237822https://doi.org/10.1017/CBO9780511608483https://doi.org/10.1017/CBO9780511608483https://doi.org/10.1007/BF00317800https://doi.org/10.1016/j.cmet.2013.07.013https://doi.org/10.2307/1936872https://doi.org/10.1111/j.1558-5646.2009.00669.xhttps://doi.org/10.1111/j.1558-5646.2009.00669.xhttps://doi.org/10.1111/j.1558-5646.1988.tb04104.xhttps://doi.org/10.1111/j.1558-5646.1988.tb04104.xhttps://doi.org/10.1016/S0022-5193(85)80022-9https://doi.org/10.1016/S0022-5193(85)80022-9https://doi.org/10.1111/1365-2745.12430https://doi.org/10.1111/1365-2745.12430https://doi.org/10.1073/pnas.192125999https://doi.org/10.1073/pnas.192125999https://doi.org/10.1023/A:1002966614054https://doi.org/10.1016/0040-5809(74)90053-7https://doi.org/10.1016/0040-5809(74)90053-7https://doi.org/10.1139/f80-129https://doi.org/10.1139/f80-129https://doi.org/10.1002/ece3.1802https://doi.org/10.1086/282461https://doi.org/10.1002/ece3.3730https://doi.org/10.1002/ece3.3730

     |  13JOHANSSON et Al.

constantvalueũstartingatt>0forashortdurationoftime,withthedifferencethatwehereconsiderchangesinuthatarenotnec-essarilysmall.Weassumethatthecontrolu,asafunctionoftime,ispiecewisecontinuouslydifferentiable,whichmeansthatouranaly-sisallowsforcontrolsofbang–bangtype(DD)andothertypeswithdiscontinuous increase in reproductive allocation. As the value ofthecontrolatfinitelymanypointsdoesnotmatterfortheanalysis,weassumethatthetimet>0doesnotcorrespondtoadiscontinu-ityofthecontrolu.Weconsiderachangeoftherelativeallocationtoreproduction,u(τ),

forτ ∊[t,t + δ), say, toũ,while leaving the restofuunchanged.WedenotebyΔmthedifferencethatthismakesinmassm,andbyΔPthedifferencethatthismakesinsurvivalprobabilityP.Duringtheperiodwhentheperturbationisactive,betweenagesttot + δ(withudenot-ingtheallocationfunctiontowhichwemakethechange),wehave

The first equation holds as m is differentiable at t and thereforem(τ)=m(t)+O(δ) for τ ∊[t,t + δ). Now, as a direct consequence ofEquation(A1),wehave

Weusetheseintermediateresultstoderiveanexpressionforthefitnessreturn,proceedinginfoursteps.First,wenotethattheimme-diate fitness gain from this strategy change for an individual thatalreadyhassurviveduntiltimetisgivenby

Second,bylinearizingtheintegrandintheexpressionforlifetimereproduction,i.e.,P(t)b(u(t)E(m(t)))inEquation(6),weapproximatethefuturefitnesslossresultingfromthischangeinstrategyas

NotethatweheredividewithP(t)becauseweagainconsiderthefit-nesseffectsforanindividualthathassurviveduntiltimet.Third,wedetermine linear approximationsofΔP(τ) andΔm(τ) for

thetimeperiodaftertheperturbation,i.e.forτ > t + δ.Tothisend,weusethatthedifferentialequationforP(τ)/P(t + δ)withrespecttoτisthesameasthatforΔP(τ),exceptforadifferenceininitialconditions.WedefinêP(τ;t)andm̂(τ;t)by

where ̂Pandm̂arefunctionsofbothτandt.However,weshallfrom now on hide the latter argument. Using Equations (A2) and (A5), we get ΔP(τ)=ΔP(t+δ) ̂P(τ)=−P(t) (q(ũ)−q(u(t))) δ ̂P(τ)+O(δ2) and

Δm(τ)=Δm(t+δ)m̂(τ)=−E(m(t))(ũ−u(t))δm̂(τ)+O(δ2). We also note

thatP(τ)=P(t) ̂P(τ)+O(δ)andwewillusethisbelowtoexpressallsur-vivalratesintermsof̂P(τ).Fourthweexpressthefitnessreturnrasthedifferencebetween

theimmediatefitnessgain(Equation[A3])andthefuturefitnessloss(Equation[A4]withΔP(τ) andΔm(τ) replacedby theapproximationsabove),dividedbyδ.Wewillalsoignorehigher-orderterms.Thus,wefind

Inourcomparison(Figure4),wefocusonhowthefitnessreturnsthroughthedifferentvitalratesaffecttheformoftheoptimalalloca-tionu*.Inordertofacilitatereading,werenamethefirstandsecondintegralinEquation(A6),whichbothareindependentofũ,asy1(t,u*)andy2(t,u*)inEquation(7).Atũ(t)=u

∗(t),thetotalreturnr(t)is0when0 < u*(t)<1, nonpositive when u*(t)=0, and non-negative whenu*(t)=1.AsshownbyMetzetal.(2016),itmaybenotedthatfortheoptimalallocationstrategy,thefunctionsy1(t,u*)andy2(t,u*)corre-spondtotheso-calledcostatesfromPontryagin’smaximumprinciplefromoptimalcontroltheory(Intrilligator,1971;Pontryagin,1962).Asinthepresentarticleweaimatprovidingbiologicalinsight,wehaveoptedforabiologicallyinspiredargumentintermsoffitnessreturnsandamarginal-valueconsideration,insteadoffallingbackonthetra-ditionsofmathematicalcontroltheory.

APPENDIX B

Robustness and sensitivity of numerical resultsInordertoevaluatetherobustnessofourapproach,westudyherehowtheshapesoftheoptimalallocationschedulesareaffectedby(1)variationinmodelparameters,(2)explicitfixedcostsofreproduc-tion, and (3) size-dependentmortality.Wemainly consider the ef-fectson thediversityofoptimal types,which ismain focusof thisstudy.

Effects of variation in model parametersWe identify twomajoreffectsof varying theparametersc1,…,c4 andke.First,variation in theseparameterscanmake itoptimal toswitchtoreproductionearlierorlaterinlife.Specifically,earlierin-vestments to reproduction are optimal when mortality increases(increased c1 or c2) or when fecundity or growth efficiency de-creases (increased c3 or c4), all in line with general expectations.Thesechangesdonotnecessarilyaffecttheshapesoftheoptimalallocationschedules,but inextremecases, itmaybecomeoptimalnottogrowatallandinvestallsurplusenergytoreproductionfrombirth onward (e.g., for very high c1). Close to such unrealistic

dΔmdτ

=dm̃dτ

−dmdτ

= (1− ũ)E(m+Δm)− (1−u)E(m)=− (ũ−u) E(m)+O(δ),

Δm(t)=0,

dΔPdτ

=d ̃Pdτ

−dPdτ

=−q(ũ)(P+ΔP)+q(u)P=− (q(ũ)−q(u))P+O(δ),

ΔP(t)=0.

Δm (t+δ)=−E (m(t)) (ũ−u(t)) δ+O(δ2),

ΔP (t+δ)=− (q(ũ)−q (u(t)))P(t)δ+O(δ2). (A2)

(A3)δ [b (ũE(m)) − b (uE(m))](t)+O(δ2).

(A4)−1

P(t)

∞

∫t

[

ΔPb(

uE(m))

+ΔmPb�(

uE(m))

uE�(m)]

(τ)dτ.

(A5)d̂P

dτ=−q(u) ̂P, ̂P(t;t)=1,

dm̂

dτ= (1−u)E�(m)m̂, m̂(t;t)=1.

r(ũ,t;u)=[

b(ũE(m))−b(uE(m))]

(t)

−∞∫t

̂P(τ)[

E(m(t))(

ũ−u(t))

b�(

uE(m))

uE�(m)m̂

+(

q(ũ)−q(u(t)))

b(

uE(m))]

(τ)dτ

=[

b(ũE(m))−b(uE(m))]

(t)−(

ũ−u(t))

E(m(t))∞∫t

[

̂Pb�(

uE(m))

uE�(m)m̂]

(τ)dτ

−(

q(ũ)−q(u(t)))

∞∫t

[

̂Pb(

uE(m))]

(τ)dτ. (A6)

(A1)

14  |     JOHANSSON et Al.

conditions, the model will exhibit a lower diversity of types.Increasingkemayeitherincreaseordecreasefecundityreturnsde-pendingonbodymass(note,e.g.,thatEinEquation(5)willdecreasewithkeifm<1,butincreaseotherwise),suchthatu*(m)decreases(increases) for small (large)bodymasses,potentially resulting inaratherflatu*(m).Ifthelattereffectisstrong,thepotentialfordiver-sitytoarisethroughothermechanismsnaturallydecreases.Second,variationintheparametersc2,…,c4influenceswhetherthe

shape of the optimal allocation schedule is mainly determined byallocation-dependentmortalityorbyallocation-dependentfecundity.Ifc2islargecomparedtoc3andc4,theshapeoftheoptimaltypewillmainlydependonthemortalitycurvaturekq,andintheoppositesitu-ation, the shapewill mainly depend on the fecundity curvature kb. Becauseseveraltypesinourscheme(e.g.,dD,aD,andad)appearbe-causeofaninteractionbetweenthemortalityandthefecundityfunc-tion,wegetlowdiversitywheneffectsfromoneofthesedominateovereffectsfromtheother.Theoverarchingresultfromtheseobservationsisthatthediver-

sityofoptimaltypesmaybeconstrainedifthesegeneralmodelpa-rameterstakeonrelativelylargeorsmallvalues.Extremevaluesoftheparameterspands,whichare introducedbyus tocontrol thephysiologicallimit(Figure1c),mayalsoreducediversityorhaveun-desiredeffects.Asnoted intheResultssection,highvaluesoftheparameterpfavortypeswithaccelerating(a)onsetofreproductionandlowervaluesfavortypeswithdecelerating(d)onset.Inaddition,if p is set to a very low value, fecundity will be a linear function(b(uE)=puE),whichcausestheonsettobediscontinuous(D).Itfol-lowsthatvariation in theparameterpmayreducethediversityofoptimal types if it ismade toosmallor too large.Theparameters

controlshowclosethesmoothminimumfunctionfitsthestandardminimumfunction.Withverylargenegativevaluesofs,thefecun-dityfunctionwillhaveasharptransitionfromalineartoanonlinearsection, which causes numerical instability when determining theoptimalallocationschedule.Whenshasvaluesclosetozero,ontheotherhand,thesmoothminimumfunctionstartsdeviatingalotfromthestandardminimumfunctionand ishencenotameaningfulap-proximation anymore. We also examine the survival probability(P = 10−6)accordingtowhichgrowthisclassifiedas indeterminate.Specifically,weconsideredeffectsonthelineinFigure3athatsepa-rates indeterminate from determinate growth. Decreasing P dis-placesthislineupwardsandtotheleft,butonlyrelativelylittle,asmanyoptimalallocationschedulesinthetop-leftcornerseemnevertoreachfullreproduction(u*hasanasymptotebelow1).IncreasingP,however,canmovethislinearbitrarilyfardownandright,whichisexpected,asorganismthenhavelesstimetoreachfullreproductiveallocation (u=1) before their survival probability falls below thecriticallevel.

Effects of fixed costs reproductionWhile in this study we assume accelerating fecundity functions(kb>1),whichmayrepresentaneedforinitialinvestmentstobemadebeforeenergycanbedivertedintooffspringdirectly,wealsoexploreanalternativewaytomodelsuchsituationsFollowingCharnov(1979),weassumeafixedcostc5forreproductionbyspecifyingtheenergydevotedtooffspringasf(uE)=max(0,uE−c5)andassumingthefe-cundityb(uE)= c3f(uE)kb. As illustrated by FigureB1a,whereweusethetypeadasabaseline,thesecostscausetheonsetofreproductiontobediscontinuous.Whenthefixedcost is increased,reproduction

F IGURE  B1 Robustnesstofixedcostsofreproduction(a)orsize-dependentmortality(b,c).Wevaryparametersinthemodifiedfunctionsforfecundity(a)ormortality(b,c)asdescribedinthetext;otherwise,parametersettingscorrespondtothoseoftypeadinFigure2a,whichservesasareferencetype.(a)Comparedtothereferencetype(graycontinuousline,c5=0),assumingfixedcostsofreproduction(c5>0)causestheoptimalreproductive-allocationscheduletoexhibitadiscontinuousonsetofreproduction(blackcontinuousanddottedlineswithc5 = 0.01 and0.02,respectively)orbeofbang–bangtype(blackdashedline,c5=0.1).(b,c)Comparedtothereferencetype(graycontinuousline,c6=0),size-dependentmortalitycandisplacetheoptimalallocationscheduletotheleft(blackcontinuousline,c6=0.75,km=0.5),totheright(blackdashedline,c6=5,km=1.5),ormakeitu-shaped(blackdottedline,c6=5,km=0.8).Theeffectsofincreasingconstantmortalityby25%with25%comparedtothebaselineareshownforcomparisonasagraydashedline(c6=0.25,km=0).In(b)wesetu=0toillustratetheeffectofthesize-dependentmortalitycomponentonly

(b) (c)(a)

     |  15JOHANSSON et Al.

starts later and the discontinuous jump becomes larger, eventuallymakingabang–bangstrategyoptimal.Thesameoccurswhenapply-ingfixedcoststotheothertypeswithcontinuousonsetinFigure2a.Fixedcostsmodeledinthiswaythushaveeffectssimilartoincreasingkb inourmodel.Notice, forexample, thatwhenmovingtowardtheright inFigure3a,theoptimaltypestypicallyfirstdevelopadiscon-tinuousonsetof reproduction (DaorDd)and then turn intobang–bangstrategies(DD).

Effects of size- dependent mortalityAsafurtherrobustnesstest,weexploretheeffectsofassumingsize-dependent mortality, given its common occurrence in nature (e.g.,Paine, 1976). To this end, we add a size-dependent term toEquation(1), representing increased mortality for juveniles.Specifically, we assume q(u,m)= c1+c2ukq +c6mkm (cf. Sibly etal.,1985).Weexploretheeffectsofvaryingc6andkmontheoptimalal-locationschedulesinFigure2a.Asintheprevioussection,weusethetype ad from Figure2a for reference. The results are neverthelessrepresentativefortheremainingtypesthereaswell.In three scenarios with size-dependent mortality, we observe

relatively small changes in the shapes of the optimal allocationschedules.Firstlywhenmortalityisinitiallymoderateandthende-clines relatively slowly with size (black continuous line inFigureB1b),theoptimalallocationscheduleisdisplacedtotheleft,that is, reproduction occurs earlier (black continuous line inFigureB1c).Thiscanbeinterpretedasaneffectofincreasedaver-agemortality,similartotheeffectofanincreasedsize-independentmortality(graydashedlinesinFigureB1b,c).Secondly,ifmortalityisinitiallyhighandthendropsrelativelyfastwithsize(blackdashedlineinFigureB1b),theoptimalallocationscheduleisdisplacedtothe right, that is, reproduction is delayed (black dashed lines inFigureB1c). This occurs because by growing as fast as possiblewhen young, individuals can avoid the high juvenile mortality.Thirdly,whenkmissufficientlylarge,theoptimalallocationsched-ule remains virtually unchanged (i.e., it is nearly identical to the

graycontinuouslineinFigureB1candthusnotshown).Thisoccursbecause thesize-dependentcomponentofmortalityquicklyvan-ishesastheindividualgrows,anditthusremainsoptimaltogrowfast(lowu*)atsmallsizes.Althoughthesize-dependentmortalityfunctionsdiscussedsofar

(e.g.,theblackcontinuousanddashedlinesinFigureB1b)arequalita-tivelydifferent fromtheconstantmortality function in thebaselinecase(graycontinuouslineinFigureB1b),theireffectsontheoptimalallocationschedulesarecomparabletotheeffectsofrelativelyminorvariationoftheparametersoftheoriginalmodel.Note,forexample,thatincreasingthe(size-independent)mortalityoftheoriginalmodelby25%(i.e.,movingfromthegraycontinuoustothegraydashedlineinFigureB1b)hasalargereffectonthesizeatwhichfullreproductionis reached (u*=1) thananyof the threescenariosdiscussedso far.Also,noneofthesescenarioscausesachangeintheshapeoftheop-timalallocationschedulesaccordingtoourclassificationsystem(i.e.,forthetypeadinFigureB1bandtheremaining11typesinFigure2binthemaintext).Intwootherscenarios,weobservemoredrasticeffectsofincluding

size-dependentmortality inourmodel. Firstly, theoptimal allocationschedulecanbecomeu-shaped(dottedlineinFigureB1b,c).Thisoc-curswhenmortality is initially very high anddrops relatively slowly,such that lowsurvivalprospects initiallymake itoptimal to invest inreproductionfrombirthonwardanditpaysofftogrowonlystartingatlarger sizes. Such a strategy is, however, not biologically feasible, asgrowthwouldstopimmediatelyuponbirth.Secondlytheextremeandunrealisticstrategyofdevotingallenergyto reproduction frombirthonward(u*(m)=1forallm),isoptimalwhenmortalityishighforallsizes(whenc6islargeandkmsmall).Insum,addingamortalitycomponent,thatdecayswithsizeonly

slowly,canleadtoextremesituationsinwhichitisoptimaltoinvestallenergy into reproduction already from birth. However, if mortalitydrops relatively fastwithsizeor isnot too largeoverall,weexpectrelativelymildeffectsontheshapesoftheoptimalallocationsched-ules,andthusalsoonoveralllife-historydiversity.