A high-resolution modeling study on diel and seasonal vertical migrations of high-latitude copepods

Contents lists available atScienceDirect

Ecological Modelling

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l

A high-resolution modeling study on diel and seasonal vertical migrations of high-latitude copepods

Kanchana Bandara

, Øystein Varpe

, Rubao Ji

, Ketil Eiane

aFacultyofBiosciencesandAquaculture,NordUniversity,8049,Bodø,Norway

bTheUniversityCentreinSvalbard,9171,Longyearbyen,Norway

cAkvaplan-niva,FramCentre,9296,Tromsø,Norway

dWoodsHoleOceanographicInstitution,Redfield2-14,WoodsHole,MA02543,USA

a r t i c l e i n f o

Articlehistory:

Received20September2017 Receivedinrevisedform 12December2017 Accepted12December2017

Keywords:

Verticalmigration Seasonality Phenology Optimizationmodel Geneticalgorithm Habitatchoice

a b s t r a c t

Despitedielandseasonalverticalmigrations(DVMandSVM)ofhigh-latitudezooplanktonhavebeen studiedsincethelate-19thcentury,questionsstillremainabouttheinfluenceofenvironmentalseason- alityonverticalmigration,andthecombinedinfluenceofDVMandSVMonzooplanktonfitness.Toward addressingthese,wedevelopedamodelforsimulatingDVMandSVMofhigh-latitudeherbivorouscope- podsinhighspatio-temporalresolution.Inthemodel,auniquetimingandamplitudeofDVMandSVM anditsontogenetictrajectoryweredefinedasaverticalstrategy.Growth,survivalandreproductive performancesofnumerousverticalstrategieshardwiredtocopepodsspawnedindifferenttimesofthe yearwereassessedbyafitnessestimate,whichwasheuristicallymaximizedbyaGeneticAlgorithmto derivetheoptimalverticalstrategyforagivenmodelenvironment.Themodelledfoodconcentration, temperatureandvisualpredationriskhadasignificantinfluenceontheobservedverticalstrategies.

Underlowvisualpredationrisk,DVMwaslesspronounced,andSVMandreproductionoccurredear- lierintheseason,wherecapitalbreedingplayedasignificantrole.Reproductionwasdelayedbyhigher visualpredationrisk,andcopepodsthatspawnedlaterintheseasonusedthehigherfoodconcentrations andtemperaturestoattainhighergrowth,whichwasefficientlytradedoffforsurvivalthroughDVM.

Consequently,thetimingofSVMdidnotchangemuchfromthatpredictedunderlowervisualpreda- tionrisk,butthebodyandreservesizesofoverwinteringstagesandtheimportanceofcapitalbreeding diminished.Altogether,thesefindingsemphasizethesignificanceofDVMinenvironmentswithelevated visualpredationriskandshowsitscontrastinginfluenceonthephenologyofreproductionandSVM,and moreoverhighlightstheimportanceofconductingfieldandmodelingworktostudythesemigratory strategiesinconcert.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Verticalmigrationisacommonbehaviorofmanyzooplankton taxa.Basedontheperiodicity,verticalmigrationsofhigh-latitude zooplankton are classified into diel and seasonal components, whichhavebeenstudiedsincethelate-19thcentury(reviewedin Russell,1927;Cushing,1951; Banse,1964).Theshort-termdiel verticalmigration(DVM) hasa periodicityofup to24h,and is understoodasastrategythattradesoffgrowthpotentialtoreduce themortalityrisk imposedby visualpredators(Lampert,1989;

∗Correspondingauthorat:FacultyofBiosciencesandAquaculture,NordUniver- sity,8049,Bodø,Norway.

E-mailaddresses:kanchana.bandara@nord.no,kanchanabandara@live.com (K.Bandara).

Ohman,1990;LooseandDawidowicz,1994).Thelong-termsea- sonalverticalmigration(SVM)hasaperiodicityofuptooneyear, andreflectsadaptationstoseasonalextremitiesoffoodavailabil- ity(HeadandHarris,1985;Hindetal.,2000;Bandaraetal.,2016), temperature(Hirche,1991;AstthorssonandGislason,2003)and predationrisk(Kaartvedt,1996;Bagøienetal.,2000; Varpeand Fiksen,2010).Ineithercase,sincebothDVMandSVMcanalter feeding,growth,survivalandreproduction,andultimatelyaffect fitness(Aidley,1981;Alerstametal.,2003;Cresswelletal.,2011;

Litchmanetal.,2013),thesemigratorystrategiesaretermedverti- calstrategies(Bandaraetal.,2016).

Empiricalknowledgeonzooplanktonverticalstrategieslargely comesfromstudyingthedynamic verticalpositioningofpopu- lationsinawatercolumn,andareoftenrathercoarseinspatial (vertical)andtemporalresolution(Pearre,1979).Thiscanunder- minethekeyconceptthatsuchmigrationsareindividualresponses https://doi.org/10.1016/j.ecolmodel.2017.12.010

0304-3800/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

Table1

Someendogenousandexogenouscuesthatarebelievedtoproximatelyorultimatelyregulatedielandseasonalverticalmigrationsofmarineandfreshwaterzooplankton.

Literaturedonotcomefromanexhaustivereviewandonlyserveasexamples.

Cue DVM SVM

Temperature McLaren(1963),Enright(1977) Hirche(1991),HeathandJónasdóttir(1999),

AstthorssonandGislason(2003) Light(absoluteorrelativeirradiancefromsun,

moon,stars,orauroraborealis,photoperiod, spectralquality,polarizationetc.)

Clarke(1933),Gliwicz(1986),Frankand Widder(1997),Bergeetal.(2009),Båtnesetal.

(2015),Cohenetal.(2015),BianchiandMislan (2016),Bozmanetal.(2017)

Sømme(1934),Ussing(1938),Milleretal.

(1991)

Dissolvedoxygen Devol(1981),Bianchietal.(2013) –

Waterdepth,transparencyandUVradiation Rhodeetal.(2001),Williamsonetal.(2011), Ekvalletal.(2015)

DupontandAksnes(2012)

Tides,currentsandadvectivetransport Hardy(1935),Wroblewski(1982),Kimmerer andMcKinnon(1987)

Bergeetal.(2012),Irigoien(2004)

Foodavailability HardyandGunther(1935),HuntleyandBrooks

(1982),George(1983),JohnsenandJakobsen (1987)

Herman(1983),Hindetal.(2000),Headand Harris(1985),Bandaraetal.(2016)

Visualandtactilepredation ZaretandSuffern(1976),Iwasa(1982),Ohman (1990),Bollensetal.(1992),Looseand Dawidowicz(1994)

Kaartvedt(1996),Kaartvedt(2000),Daleetal.

(1999),Bagøienetal.(2000),VarpeandFiksen (2010)

Bodysize,ontogenyandpigmentation ZaretandKerfoot(1975),Uyeetal.(1990), Haysetal.(1994),DaleandKaartvedt(2000)

Østvedt(1955),Hindetal.(2000)

Nutritionalstateandlipidreserves FiksenandCarlotti(1998),Sekinoand Yamamura(1999)

VisserandJónasdóttir(1999),Thorisson(2006)

Endogenousrhythmsandinternalbiological clocks

CohenandForward(2009),vanHarenand Compton(2013)

CarlisleandPitman(1961),Milleretal.(1991), Hirche(1996)

tocertaincuesorstimuli andnot a propertyofthepopulation (Zink,2002),andmaycomplicatetheunderstandingoftherela- tionshipsbetweenverticalstrategiesandenvironmentalvariables (seeTable1forexamples).Moreover,sincedielandseasonalver- tical migrations occuron differentspatial and temporal scales, studyingthesemigrationstogetherinthefieldinadequateres- olution remains a major challenge. Althoughnovel optical and acousticmethodsofin-situobservationofferasolutiontosomeof theseproblems(e.g.Basedowetal.,2010;Sainmontetal.,2014b;

Bozmanetal.,2017;Darnisetal.,2017),long-termdeployment andaccuratelyresolvingtheidentityofthemigrantsremainaskey challenges.

Mechanisticmodelsofferanalternativemeansofstudyingzoo- planktonverticalstrategiesinhigherresolution.Modelsrelatedto DVMusuallyencompassthehighestspatial(≤1m),temporal(≤1h) andbiological(=individual)resolution(e.g.FiksenandGiske,1995;

EianeandParisi,2001;Liuetal.,2003;BurrowsandTarling,2004;

HansenandVisser,2016).Modelsrelated toSVMand diapause (i.e.hibrnation in deeper waters,e.g. Hirche, 1996)encompass thesame biologicalresolution,but areusuallycoarseinspatio- temporalresolution.Here,thetimeintervalsrangefrom1hto1d andverticalspatialelementsareusuallyresolvedtoeitherabso- lutedepthunits(e.g.1mbins)orsegregatedhabitats(e.g.Fiksen andCarlotti,1998;Milleretal.,1998;Hindetal.,2000;Ji,2011;Ji etal.,2012;Sainmontetal.,2015;Banasetal.,2016).Thechoice ofa coarserspatio-temporalresolution ofthesemodelsreflects thebroaderspaceandtimescalesatwhichtheSVManddiapause occurs.Thiscontrastingspatio-temporalscalemakesitdifficultto harborlifetimedynamicsofDVMtobesimulatedinSVMmod- elswithoutsignificantlyincreasingcomputertime.Consequently, mostmodelsthatsimulateSVMtendtoeitherfully(e.g.Hindetal., 2000)orpartly(i.e.ofyoungerdevelopmentalstages,e.g.Fiksen andCarlotti,1998)disregardDVM.However,thevalidityofsuch simplificationsarequestionable,giventhegeographicallyandtax- onomicallywidespreadnatureofzooplanktonDVMbehaviorand itsontogeneticpatterns(HuntleyandBrooks,1982;Huangetal., 1993;OsgoodandFrost,1994;Hays,1995).Itisthusinterestingto investigatewhethertheextrabiologicalinformationresultingfrom modelingDVMandSVMinconcertisaworthytrade-offfortheele- vatedcomputertime.Ifso,suchmodelsmayleadtoimprovements ofthecurrentunderstandingabouthowenvironmentalseasonal-

Table2

Evolvable(soft)parametersoptimizedinthemodel.Thefirstsixareproxiesthat definetheverticalstrategy.Verticalstrategiesofcopepodsspawnedindifferent timesoftheyear(tB)areoptimizedusingtheGA.

Term Definition Range Interval Unit

␣ Lightsensitivity parameter

0–Imaxa 1 ␮molm⁻²s⁻¹

ˇ Size-specificityoflight sensitivityparameter

0–10 1 dim.less

Growthallocation parameter

0–1 0.01 dim.less

ı Seasonaldescent parameter

0–1 0.01 dim.less

Overwinteringdepth 1–500 10 m

ε Seasonalascent parameter

0–1 0.01 dim.less

tB Timeofbirth^b 1–8760 1 h

aTheupperlimitof␣changeswiththemaximumsurfaceirradianceofthemodel environment,i.e.Imax=1500␮molm⁻²s⁻¹forEnvironment-L,1300␮molm⁻²s⁻¹ forEnvironment-Mand1100␮molm⁻²s⁻¹forEnvironment-H(cf.Fig.1).

bTimeofbeingspawned.

ityshapesupverticalstrategies,andthemeansofwhichthelatter influenceslifehistoriesofhighlatitudezooplankton.

Inthisstudy,wepresentamodelofzooplanktonverticalstrate- gies.Themodeloperatesinahigh-latitudesettingandsimulates bothDVMandSVMofaherbivorouscopepodwithanannuallife cycleinhighspatial(vertical)andtemporalresolution.Usingthis model,weaimtoinvestigatetheinfluenceofenvironmentalvari- ablesonverticalstrategies,andhowverticalstrategiesaffectfitness andphenologyinseasonalenvironments.Wefurtherdiscusshow short-termbehavior(DVM)influencesandinteractsinthelonger- termandshape-updifferentlifehistorycomponentsofcopepod strategies.

2. Materialsandmethods

Although the model is not strictly individual-based, it is described following the Overview,Design concepts and Details (ODD)protocol(Grimmetal.,2006,2010)toimprovereproducibil- ity.

Fig.1.Themodelleddynamicsofirradianceincidentontheseasurface(hourlyestimates;a,d,g),temperature(b,e,h)andfoodavailability(c,f,i,expressedasChlorophyll-a biomass)inthethreemodelenvironments.SeeAppendixA1inSupplementarymaterialforadetailedcomparison.

2.1. Purpose

The purpose of the model is to investigate the bottom-up andtop-downinfluencesofenvironmentalvariability(i.e.irradi- ance,temperature,food-availabilityandpredationrisk)onvertical strategiesofahigh-latitudeherbivorouscopepod,andtounder- stand the influences of vertical strategies on its fitness and phenology.

2.2. Entities,statevariablesandscales

Themodelconsistsofthreeentities:verticalstrategies,model organismand themodelenvironment. Verticalstrategiesdefine thetiming, amplitudeand theontogenetic trajectoriesof DVM andSVM,andaredescribedusingsixevolvable(soft)parameters (Table2).Thesearehardwiredtothemodelorganism,i.e.copepods spawnedindifferenttimesoftheyear.

Themodelorganismisahypotheticalherbivoroussemelparous femalecopepod(hereafter,thecopepod)withanannuallifecycle thatresemblesCalanusfinmarchicusandC.glacialisintermsofbody size,behaviorandlifehistorystrategies(Conover,1988).Thesetwo speciesoftendominatethecopepodbiomassintheNorthAtlantic andmostEurasiansub-Arcticand Arcticseasandshelves(Falk- Petersenetal.,2009).Theirlifecycleconsistsofanembryonicstage (egg),sixnaupliarstages(NI–NVI),fivecopepoditestages(CI–CV) andanadult.Eggsthatarereleasedinnear-surfacewatersinthe springusuallydevelopintoCIVorCVstagestowardtheendofthe productiveseason.Asfurtherdevelopmentistypicallyconstrained bythedurationoftheproductiveseasonandseasonalpeaksof visualpredationrisk,CIVs andCVs descendintodeeperwaters andremaininastateofdiapause/dormancywithminimalphys-

iologicalactivity(Hirche,1996).Overwinteringstagesascend to near-surfacewatersastheprimaryproductioncommencesinthe followingyear,moltintoadultsandstarttoreproduce(Conover, 1988; Varpe,2012).The lifecycle ofthetwo species isusually completedwithinoneyearinmostsub-ArcticandArcticlocations (Falk-Petersenetal.,2009;Daaseetal.,2013),withinwhichreside themodelenvironmentsofthisstudy.

Themodelrunsinthree500-mdeepartificialseasonalenvi- ronments thatrepresentthree high-latitudelocationsalong the southernandsoutheasterncoastofNorway(60–70^◦N).Theseenvi- ronments do notpoint tospecificgeographiclocations,but the modelledenvironmentaldynamicswereadoptedfromfieldmea- surementsfromtheaboveregion(AppendixA1inSupplementary material).Thebaselinemodelsimulation(hereafter,thebasicrun) runsinEnvironment-L,representingthelowerend(ca.60^◦N)ofthe selectedgeographicalrange.Here,themodelledirradiance,tem- peratureand foodavailability arehighlyseasonalandvertically structured(Fig.1a–c),butareassumedconstantbetweenyears.The irradianceincidentontheseasurfacefollowstheglobalclear-sky horizontalirradiancemodelofRobledoandSoler(2000),andpeaks atca.1500␮molm⁻²s⁻¹(Fig.1a,AppendixA1inSupplementary material).Theseasurfacetemperaturereachesamaximumof18^◦C inthesummer(e.g.Bagøienetal.,2000),anddistributesevenly inthesurfacemixedlayer(Fig.1b).Belowthis,thetemperature decreaseswithdepthandconvergestoaminimumof4^◦Catca.

100m(e.g.IngvaldsenandLoeng,2009).Thepelagicproductive seasonextendsca.180days,withachlorophyll-apeakat8mgm⁻³ in mid-April(Fig.1c: Sakshaugetal.,2009; Daaseetal.,2013).

WemanipulatedtheenvironmentalparametersofEnvironment-L toformulatetwoadditionalartificialenvironments:Environment- M(ca.65^◦N,Fig.1d–f)and Environment-H(ca.70^◦N,Fig.1g–i),

Fig.2.Themodeloverview.VerticalstrategiesthatdefinethetimingandamplitudeofDVMandSVMarehardwiredtocopepodsbornindifferenttimesoftheyear.Growth, survivalandreproductionofthesecopepodsaresimulatedinaseasonalenvironmenttoderiveafitnessestimatethatisheuristicallymaximizedbytheGAtoderivethe optimalverticalstrategy,timeofbirthandseveralassociatedlifehistorytraitsemergingfromthemodel.Dashedlinerepresentstheindirectdependencyofthefitness estimateongrowth(Section2.6.4).

representingthemid-pointandthehigherendoftheselectedgeo- graphicalrange(AppendixA1inSupplementarymaterial).

Copepods are characterized by six states: vertical location (depth),structuralbodymass,energeticreserve,reproductiveout- put(fecundity),survivorshipanddevelopmentalstage.Themodel hasatemporalcoverageofanannualcycleandaunidimensional (vertical)spatialcoverageof500m.Thetimeandspaceconsistof 1hand1mdiscreetintervals.

2.3. Processoverviewandscheduling

Ateach timestep,themodelfollowsverticalstrategieshard- wiredtocopepodsbornindifferenttimesoftheyearandsimulates theirgrowth,survivalandreproduction.Statevariablesareupdated simultaneously.Verticalstrategies are evaluatedusing afitness functionbased on theexpected survival and reproductive per-

formances.ThefitnessisheuristicallymaximizedusingaGenetic Algorithm (GA, Holland, 1975)toestimate theoptimalvertical strategyandoptimaltimeofbirthforagivensetofenvironmental conditions(Fig.2).

2.4. Designconcepts 2.4.1. Basicprinciples

Thehighspatialandtemporalresolutionimplementedinthe modelallowbothDVMandSVMtobesimulatedovertheentire annuallife cycleofthecopepod.Carlottiand Wolf(1998)have implementedasimilarconstruct,buttheSVMoftheirmodelwas constrainedbyfixingthetimingofascentanddescenttomatchthe fieldobservationsoftheregionofinterest.Incontrast,thetiming andtheamplitudeofDVMandSVMofourmodelareflexibleand allowedtoevolveaccordingtotheenvironmentalconditions.To

Table3

Emergentpropertiesofthemodel.ThetimingandamplitudeandofDVMandSVMaltogetherformstheverticalstrategyofacopepod.

Trait/attribute Units Description

Timeofbirth Dayoftheyear Timeofbeingspawned

Surfacetime h UnifiedestimaterepresentingthetimingofDVM,i.e.the

stage-specificmeanno.ofhoursperdayoccupiedinwaters withhighestgrowthpotential(usuallythesurfacewaters)

Amplitudeofdielverticalmigration m Theverticalrangecorrespondingtotheabove

Timeofseasonaldescentandascent Dayoftheyear SeparateestimatesrepresentingthetimingofSVM(ascentand descent)

Amplitudeofseasonalverticalmigration m Overwinteringdepth

Bodymassatseasonaldescent ␮gC Structuralandenergeticreservemassattheonsetofdiapause

Onsetofeggproduction Dayoftheyear –

Fecundity No.ofeggs No.ofeggsproducedduringthelifetime

Breedingmodeindex dim.less Proportionofcapitalbreedingeggs(0=pureincomebreeding,

1=purecapitalbreeding)

Foodlimitationindex dim.less Stage-specifictotalno.hourswithfood-limitedgrowth(Eq.

(3))asafractionofstageduration(0=nofoodlimitation, 1=totalfoodlimitation)

Developmenttime d Fromeggtoagivenstage

Longevity d Durationofthelifecycle,frombirthtodeath

achievethislevelofflexibility,weusedmultipleevolvableprox- iesto representvertical migration (Table2). Thisresulted in a complexseven-dimensionaloptimizationproblemthatcanbeeffi- cientlysolvedusingheuristictechniques(ZanakisandEvans,1981).

Asevolutionaryalgorithmsprovideanefficientmeansofsolving multi-dimensionaloptimizationproblems(Deb,2001;Eibenand Smith,2003),weusedaGAastheoptimizationplatformofthis model.Further,toincreasetheprecisionoftheevolvableparam- etersandthatofthebehavioralstrategies andlifehistorytraits ensued(Fig.2),weusedaGAvariantwithfloatingpointrepresen- tation(i.e.aReal-CodedGeneticAlgorithm,Davis,1989;Lucasius andKateman,1989;Herreraetal.,1998).

Thestrategy-orientedconstructofthismodelcontrastsclassic individual-basedmodelsofzooplanktonlifehistoryandbehaviorin twomainways:first,tradingoffofbiologicalresolution(strategies vs.individuals)toaccommodatehigherspatio-temporalresolution, andsecond,thelackofpopulation-levelresponsessuchasdensity dependence.Asaresult,modelledverticalstrategiesdonotinter- actwitheachotherandshownoquantitativefeedbackswiththe modelenvironment(e.g.impactofgrazingonfoodconcentration anddurationoftheproductiveseason).

2.4.2. Emergence

Thebehavioralstrategiesandlifehistorytraitsemergingfrom themodelarepresentedinFig.2anddescribedinTable3.

2.4.3. Adaptationandsensing

Copepodsaresensitivetotheirinternalstates(i.e.structural body mass, mass of the energetic reserve and developmental stage)andexternalstimuli(i.e.irradiance,temperature,foodcon- centrationand depth). Altogether,thesedeterminethesize- or stage-specificpatternsofgrowth,metabolism,reproductionand verticalbehavior(Section2.6).

2.4.4. Objectives

Themodelusesafitnessestimatethatevaluatestheexpected reproductionandsurvivalperformancesrenderedbydifferentver- ticalstrategies(Section2.6.4).

2.4.5. Predictionandstochasticity

Theverticalsearchpatternofcopepodbehaviorisbasedona semi-stochasticpredictivealgorithm(Section2.6.2.2andAppendix A2inSupplementarymaterial).Stochasticityplaysacentralrolein themodelinitialization(Section2.5)andselection,recombination andmutationoperatorsoftheGA(Section2.6.4).

2.4.6. Observations

Foragivenmodelenvironment,themodelproducesheuristic estimatesoftheoptimalverticalstrategyandoptimaltimeofbirth, alongwitharangeofassociatedlifehistorytraits(Fig.2,Table3).

2.5. Initialization

ThemodelinitializeswithseedingofN(=10⁶)eggsatrandom timesoftheyeartorandomdepths(<50m)ofthewatercolumn.

Eachseedrepresentsanembryonicstageofacopepodwithaspe- cificverticalstrategy,whichisdeterminedbyrandomlyassigning valuestotheevolvableproxies.Theranges(bounds)andresolu- tionsoftheseproxiesarelistedinTable2.

2.6. Submodels

2.6.1. Growthanddevelopment

We modelledsomaticgrowthin Carbonunits(␮gC)accord- ingtothegrowthmodelofHuntleyandBoyd(1984)(Eqs.(1)–(8) below),using a Chlorophyll-a/Cratio of 0.030(Båmstedt etal., 1991; Sakshauget al.,2009).This growthsub-modelwasused duetoitssimplicityand generalapplicability,whichareshown byitsutilitytomodelseveraldifferentcopepodtaxawithvarying bodysizesrepresentingawiderangeofgeographicallocations(e.g.

Robinson,1994;FiksenandGiske,1995;Romanetal.,2000).Defi- nitionsandunitsofallthetermsdescribedhencefortharelistedin Table4.

For ambientfoodconcentrations (F: ␮gCml⁻¹)abovea spe- cificsaturationconcentration(f),growthisfood-independent,and occursatamaximumrate(GT:␮gCind⁻¹h⁻¹)dependentonlyon theambienttemperature(T)as;

(GT)_i,t,z=

G_max

t,z·W_i,t (1)

Here,irepresentsindividual,ttimeandzisdepth,whereG’max

(␮gCmgdrymassh⁻¹)isthemaximum temperature-dependent mass-specificgrowthrate,assumingaCarbon:drybodymass(W, mg)ratioof0.40(HuntleyandBoyd,1984),definedas;

t,z=0.903·exp (0.110·Tt,z) (2) Iftheambientfoodconcentrationdropsbelowthesaturation concentration,thegrowthoccursataratelimitedbyfoodavail- ability(G_F)as;

(GF)_i,t,z=a·bt,z·W_i,t^nt,z·Ft,z−k·W_i,t^mt,z (3)

Table4

Definitions,valuesandunitsofthetermsusedinthemodel.

Term Definition Value/formula Units

a Assimilationcoefficient 0.70^b –

bt,z Clearancecoefficient Eq.(4)^a mlmgdrymassh⁻¹

E Eggdevelopmentparameter 717^e,f –

fi,t,z Saturationfoodconcentration Eq.(8)^a ␮gCml⁻¹

Ft,z Ambientfoodconcentration Section2.2 ␮gCml⁻¹

(G´ımax)t,z Maximummass-specificgrowthrate Eq.(2)^a ␮gCmgdrymassh⁻¹

(GF)i,t,z Food-limitedgrowthrate Eq.(3)^a ␮gCind⁻¹h⁻¹

(GT)i,t,z Nonfood-limitedgrowthrate Eq.(1)^a ␮gCind⁻¹h⁻¹

Hi,t,z Survivorship Eq.(15) –

i Individual – –

I´ıt,z RemappedIt,z 0.9≥I´ı≥0.1 –

It,0 Irradianceincidentonseasurface AppendixA1inSupplementarymaterial^c ␮molm⁻²s⁻¹

It,z Downwellingirradianceatdepthz Eq.(9) ␮molm⁻²s⁻¹

j Developmentalstage 0–12(Egg–Adult) –

Ki,t Scalarforvisualpredationrisk 1>K>0 –

kt,z Respiratorycoefficient Eq.(5)^a ␮gCmgdrymassh⁻¹

(Mn)t,z Non-visualpredationrisk Section2.6.2.1 –

(Ms)i,t,z Starvationrisk Eq.(12) –

mt,z Exponent(respiration) Eq.(6)^a –

(Mv)i,t,z Visualpredationrisk Eq.(10) –

N No.ofinitialseeds 1,000,000 –

nt,z Exponent(clearance) Eq.(7)^a –

Ri Fecundity Eq.(13) no.ofeggs

t Time 1–8760 h

Tt,z Ambienttemperature Section2.2 ^◦C

Ui,t Cruisingvelocity Eq.(11) mh⁻¹

(Wc)i,t Structuralmass – ␮gC

WE Uniteggmass 0.55^d ␮gC

Wi,t Drybodymass(assuming40%C) – mg

(Wq)i,t Catabolizedstructuralmass(proportiontothe maximumlifetimestructuralmass)

0≥Wq≥0.5 –

(WR)i,t,z Matterallocatedforeggproduction – ␮gC

(Ws)i,t Storage(energeticreserve)mass – ␮gC

Wx Stage-specificcriticalmoltingmass Table5 ␮gC

z Depth 0–500 m

˚ TerminationconditionoftheRCGA Section2.6.4 –

Lightattenuationcoefficient 0.06^g m⁻¹

ω Parameterforweighingfitness 0or1 –

˝i Fitness Eq.(14) –

aHuntleyandBoyd(1984).

b FiksenandGiske(1995).

c RobledoandSoler(2000).

d CalculatedfromSalzen(1956).

eCampbelletal.(2001).

f Jietal.(2012).

gEianeandParisi(2001).

wheretwotermsoftheright-handsideoftheequationreferto theassimilationandrespiratoryratesrespectively.Theassimila- tioncoefficient(a)isassumedtobeconstant(Table4),butHuntley andBoyd(1984)foundthatthecoefficientsofclearance(b)and respiration(k), andtheexponents(nandm)varywithambient temperatureas;

bt,z=1.777·exp (0.234·Tt,z) (4) kt,z=0.375·exp (0.0546·Tt,z) (5) nt,z=0.671·exp (0.0199·Tt,z) (6) m_t,z=0.858·exp (−0.008·T_t,z) (7) AtthepointwhereFreachesf,Eqs.(1)and(3)balanceout,and thefbecomes;

f_i,t,z=

t,z·W_i,t+kt,z·W_i,t^mt,z

a·bt,z·W_i,t^nt,z (8) Thisgrowthsub-modelisnotapplicabletothefirsttwonau- pliistages,which do not feed(Fig.3a, Marshall andOrr, 1972;

Mauchline,1998).Forsimplicity,weassumedthegrowthofNIand NIIstagestooccuratatemperature-dependentrate(Eqs.(1)and

(2)).Thegrowthofearlydevelopmentalstages(NI–CIII)issolely allocatedtothebuildingupofstructuralmass(Wc,␮gC,Fig.3a,b andTable5).

TheembryonicdevelopmentfollowsaBˇelehrádektemperature function(Campbelletal.,2001;Jietal.,2012).Thepost-embryonic development(from stagejtoj+1)occursonly ifWc exceeds a stage-specificcriticalmoltingmass(Wx,␮gC,Table5).However, forsimplicity,wedidnotmodelthedependenceofmoltingprocess onthephysiologicalstate(Nivaletal.,1988)andthelimitationof growthbytheexoskeleton(Mauchline,1998).

2.6.2. Survival

2.6.2.1. Predation risk.Visual (v) and non-visual (n) predators inducemortality,which is estimated asa probability following EianeandParisi(2001)as;

It,z=I_t,0·exp

− ·z

(9) whereI_zandI₀areirradianceatdepthzandsurfaceatagiventime, and (=0.06m⁻¹)istheattenuationcoefficientfordownwards directedirradianceinthewatercolumn.Weremappedirradiance (I)between0.1–0.9(I)sothatvisualpredationriskisnotnullified evenatthelowestlevelsofirradiance,andthecopepodhassome chanceofsurvivalevenathighestlevelsofirradiance.

Fig.3.Simplifiedphysiologicalpathwaysmodelledinthisstudy.Somelifestagesaregroupedtogetherduetotheirsimilaritiesinenergyallocationpatterns(a–f).Starvation (highlightedinred)triggerscatabolicpathwaysmarkedinred.TandFareBooleanvaluestrueandfalse.isthegrowthallocationparameter(Table2).Acomparative summaryisgiveninTable5.

The detection efficiency of visually orientating planktivores increaseswiththesizeoftheirprey(BrooksandDodson,1965;

Battyetal.,1990).Forsimplicity,wemodelledthesize-dependent visualpredationriskusingalinearmodel,assumingthatthelargest developmental stage is ca. 10 times more vulnerable to visual predatorscomparedtothesmallestdevelopmentalstage(Fig.4a, Table5,DeRobertis,2002).Thiswasimplementedusingthescalar K(1>K>0)as;

(M_v)_i,t,z=I_t,z ·K_i,t (10)

The initial value of K (i.e. K value at the embryonic stage, range=1×10⁻⁴–1.5×10⁻²)wasdecidedsothatitproduceshourly estimatesofvisualpredator-inducedmortality.

Weassumedthemortalityriskcausedbynon-visualpredators (non-visualpredationrisk,Mn)tobe1%ofthemaximumvisual predationriskandconstantovertimeanddepth(EianeandParisi, 2001).

2.6.2.2. Dielverticalmigration.ThecopepodmayperformDVMto tradeoffgrowthpotentialtominimizethevisualpredationrisk.We usedthephotoreactivebehaviorasaproxytoestimatethetim- ingandamplitudeofDVM(e.g.Kerfoot,1970;CarlottiandWolf, 1998).Here,␣,anevolvablelightsensitivityparameter(Table2) wasusedtodefineanirradiancethresholdabovewhichinduces anegativephototaticresponseintheverticalswimmingbehavior (Båtnesetal.,2015;Cohenetal.,2015).Atanygiventime,thecope- podoccupiesadepthwithanirradiancelevel(I_t,z)below␣.From aseriesofpossibledepthbinsthatsatisfytheIt,z<␣condition,we assumedthatthecopepodsearchesandoccupiesthedepththat maximizesitsgrowthpotential.Forsimplicity,wefurtherassumed thatinternal state-dependentfactors,suchashungerandsatia- tionhaveanegligibleinfluenceonthemodelledDVM.Thevertical searchpatternwaspredictedusingabiasedrandomwalkalgorithm (Codling,2003,AppendixA2inSupplementarymaterial),assum- ingthatthecopepodisneutrallybuoyantandverticallymovesin thewatercolumnatamaximumvelocity(hereaftercruisingveloc-

Table5

Developmentalstages,theirmaximumstructuralbodymasses(Wx)andstage-specificvariabilityinseveralbiologicalprocessesmodelledinthisstudy(cf.Fig.3).Dashes indicatenon-applicability.

Stage Wx(␮gC) Feeding Structuralgrowth EnergeticReserve Respiration Swimming Eggproduction

Egg 0.55 – – – – – –

NI 0.55 – x – x x –

NII 0.68 – x – x x –

NIII 0.91 x x – x x –

NIV 1.84 x x – x x –

NV 2.72 x x – x x –

NVI 3.92 x x – x x –

CI 6.01 x x – x x –

CII 9.84 x x – x x –

CIII 17.58 x x – x x –

CIV 36.42 x^a x^d x x^c x^d –

CV 110.03 x^a x^d x x^c x^d –

Adult 332.27 x – x^b x x x

WxvaluesresemblethosepublishedforC.finmarchicusandC.glacialisbyBåmstedtetal.(1991);andCampbelletal.(2001).

aFeedingceasesduringdiapause.

b Doesnotallocatesurplusgrowthtodeveloptheenergeticreserve,butinheritthereservesfromitsdevelopmentalprogression.

c Reducesduringdiapause.

d Notrelevantduringdiapause.

Fig.4.Relationshipsof(a)visualpredationriskscalar,(b)cruisingvelocity,(c)light sensitivityparameterand(d)thetotalbodymassofthecopepodwithitsstructural mass(Wc).Thecruisingvelocity(U)modelwasfittedusinglaboratoryandfield estimatesofCalanusspp.fromHardyandBainbridge(1954),GreeneandLandry (1985)andHeywood(1996)(pointsinpanelb).Thedifferentlinearmodelsforˇ, thatscalethelightsensitivityparameter(␣)areoptimizedinthemodel(Table2).

Thelowerandupperborderoftheshadedpolygon(paneld)representthetotal bodymassforgrowthallocationparameter()=0and1respectively.

ity,U).Weusedseveralstage-specificcruisingvelocityestimatesof Calanusspp.availableintheliterature(Fig.4b),andrelatedthose tobodymassas;

U_i,t=8.0116·(Wc)^0.4531_i,t (11)

We consideredthesize-or stage-specificvariability ofDVM asaresponsetosize-dependentvisualpredationrisk(Zaretand Kerfoot,1975;Uyeetal.,1990;Haysetal.,1994;EianeandOhman, 2004)andmodelleditbyscalingthelightsensitivityparameter(␣) withthebodymass(Wc).Asdataonthelightsensitivityofyounger developmentalstages(NI–CIII)ofCalanusspp.israre,wecouldnot deriveageneralrelationshipbetweenWc and␣.Toaddressthis, wedefinedanevolvableparameterˇthatdescribesthesizespeci- ficityof␣,which,atitsmaximum(ˇ=10)downscales␣oftheadult femaleto10%ofthat oftheegg/NI (Fig.4c).Highertrajectories thanˇ=10werenotused,asitwasshowninthetrialrunsthatthe modelalwaysconvergesonˇ<10evenathighestlevelsofvisual predationrisk.

2.6.2.3. Energystorage. CIVandCVstagescanallocateaspecific fractionfromsurplusgrowth(evolvablegrowthallocationparam- eter:,Table2)tobuildupanenergyreserve(Fig.3c)thatpossesses amaximumsizeof70%ofthetotalbodymass(Fig.4d,Fiksenand Carlotti,1998).

2.6.2.4. Seasonalverticalmigration. Similartomosthigh-latitude marinezooplankton,whichdescendtodepthsduringtheunpro- ductivepartoftheyear(reviewedinConover,1988;Hagenand Auel,2001;Falk-Petersenetal.,2009),thecopepodmayperform SVM. We usedthe stateof theenergetic reserve asa proxy of timingoftheSVM(cf.VisserandJónasdóttir,1999).Here,thecope- poddescendstoaspecificdepth(evolvableoverwinteringdepth ,Table2)whenthestoresaccountforaspecificfractionofthe totalbodymass(evolvableseasonaldescentparameter:ı,Table2).

Uponreachingtheoverwinteringdepth,thecopepodremainsstag- nant ata diapause state(Hirche,1996)with itsmetabolicrate reducedby90%fromthatundernormalconditions(Fig.3d,Table5, Pasternaketal.,1994;Varpeetal.,2007).Theoverwinteringperiod terminates when a specific fraction (evolvable seasonal ascent parameter:ε,Table2)oftheenergeticreserveisexhausted.After theoverwinteringperiod,surplusgainsarenotallocatedtodevelop furtherenergeticreserves,butmaybeusedforstructuralgrowth andreproduction(Fig.3eandf,Table5).

Fig.5.(a–c)Mechanismofweighingfitness.Fitnessofacopepodismultipliedbyabinaryweightω=0ifitseggproductionseason(tD−tR)doesnotoverlapthetimeofbirth (tB,May1inthisexample,denotedbyablackdot)andviceversa.(d)Simplifiedworkflowofinitializationandoptimizationstepsofthemodel.Theinitialsetofstrategies entertheoptimizationloopaftergoingthroughthefirstlifecyclesimulation(LS1).TheGAoptimizessevenevolvable(soft)parameters(Table2)byrepeatedlyapplying selection,recombinationandmutationoperatorsuntilaterminationcondition(ϕ)issatisfied.TandFareBooleantrueandfalseconditions.No.ofstrategies(i.e.sizeofthe GA-population,Nor2N)ateachoperationisindicatedtotheright.

2.6.2.5. Metabolism. Thebasalmetaboliccostrelateswiththebody massandambienttemperature,expressedask·W^minEq.(3)(terms asdefinedaboveandinTable4).Themetaboliccostofzooplankton verticalmovementscanaccountfor0–300%ofthebasalmetabolic demand(Vlymen,1970;FouldsandRoff,1976;Morrisetal.,1985;

DawidowiczandLoose,1992).Forsimplicity,weassumedthecost ofverticalmovementtobe150%ofthebasalmetaboliccost(mid- pointoftheaboverange).Thisadditionalcostissubtractedfrom thegrowthEqs.(1)or(3).Theenergyreserveisusedtobalancethe metabolicdemandsthatcannotbesustainedunderlowambient foodconcentrations(Fig.3c–f).

2.6.2.6. Starvation risk. Whenenergy reservesare depleted,the metabolicdemands thatcannotbebalanced byfoodintakeare metbycatabolizingstructuralbodymass(Fig.3b–f).Thiselevates themortalityriskduetostarvation(starvationrisk,Ms),whichis definedasaprobabilitythatincreasesasalinearfunctionofcatab- olizedstructuralmassas;

(Ms)_i,t=2·

Wq

i,t (12)

Here,Wq isthecatabolizedstructuralmass expressedasapro- portion of the maximum structural mass prior to structural catabolization.Wqcanreachamaximumof0.5,duringwhichMs

peaksfollowingEq.(12),andthecopepoddiesaccordingtheChos-

sat’srule(Chossat,1843),whichpositsthatstarvinganimalsmay catabolizeabouthalfoftheirbodyweightbeforedeath.Irrespective oftheageofthisgeneralizedrule,ithasbeenusedasaconstraint instarvationstudiesofmanyvertebrateandinvertebratetaxa(e.g.

Threlkeld,1976;Spencer,1997;Costello,1998;Loosetal.,2010).

2.6.3. Reproduction

Weassumedthatsomaticgrowthceasesafterthefinalmolt,and alladultsbecomesexuallymatureataconstantstructuralbody mass(Fig.3f,Table5).Energeticinputtoeggproductionmaybe sourcedfromfoodintake(incomebreeding)orallocatingaspe- cificamountofmatter(C)equivalenttothemaximumgrowthrate (G_T:Eqs.(1)and(2))fromtheremainingenergeticreserve(capital breeding,cf.Varpeetal.,2009).Thefecundity(R)fromthetimeof sexualmaturity(tR=timeoffinalmolt)toagiventimehorizon(tX) isestimatedusingthematterallocatedtoeggproduction(W_R)and theuniteggmass(W_E=0.55␮gC)as;

R_i=

tX

tR

(W_R)_i,t,z WE

(13)

Fig.6.PredictedoptimalverticalstrategyandassociatedgrowthandreproductiveperformancesofthecopepodinthebasicrunatEnvironment-L(cf.Fig.1a–c).Thesurface time(a)isthestage-specificmeanno.ofhoursperdaythatthecopepodoccupiesfood-richsurfacewaters,andamplitude(b)isitsverticalrange.Paneldcomparespredicted developmenttimes(excludingoverwinteringduration)tothoseestimatedforC.finmarchicusandC.glacialisfollowingBˇelehrádekfunctionsparameterizedbyCampbelletal.

(2001)andJietal.(2012).WcandWsrefertostructuralbodymassandsizeoftheenergeticreserverespectively.

2.6.4. Fitnessfunctionandoptimization

Toevaluatetheperformanceofaverticalstrategy,wederiveda fitnessestimate(˝)asafunctionofsurvivorshipandfecundityas;

˝_i=

tB

H_i,t,z·R_i,t,z

·ω (14)

Here,ωisaweightthatadjustsfitness(seebelow)andHisthe survivorship,i.e.theprobabilityofsurvivalfrombirth(t_B)toagiven timehorizon(tX)estimatedasafunctionofvisual,non-visualand starvationrisks(Mv,MnandMs)as;

H_i,t,z=

tX

t_B

1−

(M_v)_i,t,z+(Mn)_t,z+(Ms)_i,t

(15)

Theterm˝ technicallyresembles thenetreproductiverate (e.g.Stearns,1992),andisusedinsomeoptimizationmodels(e.g.

KiørboeandHirst,2008)butmaynotbarethesameinterpreta- tiongiven thestrategy-oriented constructof this model.When themodelpredictsanoptimalverticalstrategyandtimeofbirth

foraparticularenvironment,wecanassumethatthosepredicted optimashouldpersistfromonegenerationtothenextiftheenvi- ronmentremainsconstant.Ifacopepod’sspawningperiodlasts fromtimetR totD (timeofdeath)weassumedthatit produces aseriesofoffspringwiththesameverticalstrategy,butbornat differenttimesoftheyear(rangingfromt_R tot_D).However,only theoffspringwithatimeofbirthmatchingthatofthemothercan representtheentireevolvable(soft)parametersetofthemother, and guarantee its persistence from one generation to another (Fig.5a–c).Therefore,weadjustedthefitnessusingabinaryweight (ω)bysetting ω =0ifthecopepod’sspawningseasondoesnot overlapitstimeofbirth(Fig.5aandb)andviceversa(Fig.5c).

We used a Real-Coded Genetic Algorithm (RCGA) to derive heuristicestimatesofoptimalverticalstrategyandtimeofbirth thatmaximizesfitnessinagivenmodelenvironment(Fig.5d).In theRCGA,sixproxiesofverticalstrategiesandthetimeofbirth of thecopepod that those are hardwired to(Table2) are con- sideredas genesona singlechromosome.TheRCGA beginsby selectingamatingpoolofNchromosomes(=parents,i.e.Nverti- calstrategiesseededindifferenttimesoftheyear)fromtheinitial

Fig.7. Graphicalsummaryofthesensitivityanalysis.Modelparametersandenvironmentalvariablestestedforsensitivityarepresentedontheverticaldimension,andthe model-predictedoptimaoftimeofbirthandverticalstrategy,andtheassociatedfitnessonthehorizontaldimension.+/−:25%increase/decreaseintheparametervalue, E/D:15-dearlier/delayedandS/L:15-dshorter/longerscenariosregardingtiminganddurationoftheproductiveseason(seeAppendixA3inSupplementarymaterial).

seedsusingabinary(two-way)deterministictournamentselection (GoldbergandDeb,1991;MillerandGoldberg,1995).Genesoftwo randomlyselectedparentsfromthematingpoolarerecombined throughblendcrossoverfollowingtheBLX-␣method(Eshelman andSchaffer,1993),whichproducestwooffspring(recombinants).

Genesoftherecombinantsaremutatedataprobabilityof0.02by randomreplacement(uniformmutation:EibenandSmith,2003;

Haupt and Haupt, 2004).The populationof strategies resulting fromtheseoperations comprises ofN parents,whose fitnessis knownandNoffspring,whosefitnessisnotyetknown.Parents withuniquegenecombinationsareselectedtoconstructalibrary (hereafter,thereferencelibrary),whichisupdatedateachitera- tion.Eachoffspringiscomparedwiththoseinthereferencelibrary toassesstheirfitness.Fitnessoftheoffspringwithsimilargene combinationtothoseinthelibraryareassignedin-situ,whilethe restgoesthroughthelifecyclesimulationtodeterminefitness(LS- 2inFig.5d).Oncethefitnessofall2Nindividualsareknown,N survivorsareselectedfollowingaround-robin(all-play-all)tour- namentofsize10(Hariketal.,1997;EibenandSmith,2003).This processisrepeatedforaminimumof100iterations,andterminated whenthemeanfitnessofthepopulationshowsnoimprovement for25consecutiveiterations(␾inFig.5d,EibenandSmith,2003).

2.7. Programming,executionandanalysisofthemodel

WeusedRversion3.3.1(RCoreTeam,2016)andRStudiointe- grateddevelopmentenvironment(IDE)version1.0.136(RStudio Team,2016)alongwiththehigh-performancecomputingpackages Rcpp(Eddelbuetteletal.,2011)andbigmemory(Kaneetal.,2013) toconstruct,simulateandanalyzethemodel.

AbasicrunwasperformedintheEnvironment-Lusingdefault valuesformodelparameters(Table4).Inordertotesttheinfluence of model parameters and environmental variables on model- predictedverticalstrategiesandfitness,weperformedasensitivity analysisfollowing(JørgensenandBendoricchio,2001).Here,we calculatedasensitivityscore(Sx)as;

Sx=(XBR−XM)/XBR

(PBR−PM)/PBR

(16) whereXisthepredictedmodeloutputofthebasicrun(X_BR)and themodifiedrun(XM)foragivenchange(±25%)ofinputparam- etervaluebetweenthebasicrun(P_BR)andthemodifiedrun(P_M).

Wetestedthesensitivityofverticalstrategiesandfitnessfor13dif- ferentinputparameters(AppendixA3inSupplementarymaterial).

Fortheconvenienceofinterpretationoftheseresults,wepresented thesensitivityscoresunderthreecategories:no-sensitivity(Sx=0), lowsensitivity(0<Sx≤3)andhighsensitivity(Sx>3).Finally,we