draft...draft 1 1 the theoretical foundations for size spectrum models of fish 2 communities 3 ken...

Draft

The theoretical foundations for size spectrum models of fish

communities

Journal: Canadian Journal of Fisheries and Aquatic Sciences

Manuscript ID cjfas-2015-0230.R2

Manuscript Type: Review

Date Submitted by the Author: 16-Nov-2015

Complete List of Authors: Andersen, Ken; Technical University of Denmark, National Institute of Aquatic Resources Jacobsen, Nis S; Technical University of Denmark, AQUA Farnsworth, Keith; Queens University, Institute of Global Food Security

Keyword: MARINE < Environment/Habitat, COMMUNITIES < General, ECOSYSTEMS < General, MODELS < General

https://mc06.manuscriptcentral.com/cjfas-pubs

Canadian Journal of Fisheries and Aquatic Sciences

Draft

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Thetheoreticalfoundationsforsizespectrummodelsoffish1

communities2

KenH.Andersen1,NisS.Jacobsen1andK.D.Farnsworth23

4

1CenterforOceanLife,NationalInstituteofAquaticResources(DTU-Aqua),Technical5

UniversityofDenmark,CharlottenlundCastle,DK-2920,Charlottenlund,Denmark6

2InstituteofGlobalFoodSecurity,QueensUniversityBelfast,97LisburnRoad,Belfast7

BT97BL,NorthernIreland,UK8

9

Abstract10

Sizespectrummodelshaveemergedfrom40yearsofbasicresearchonhowbodysize11

determinesindividualphysiologyandstructuresmarinecommunities.Theyarebased12

oncommonlyacceptedassumptionsandhavealowparameterset,whichmakethem13

easytodeployforstrategicecosystemorientedimpactassessmentoffisheries.We14

describethefundamentalconceptsinsize-basedmodelsaboutfoodencounterandthe15

bioenergeticsbudgetofindividuals.Withinthegeneralframeworkthreemodeltypes16

haveemergedthatdiffersintheirdegreeofcomplexity:thefood-web,thetrait-based17

andthecommunitymodel.Wedemonstratethedifferencesbetweenthemodels18

throughexamplesoftheirresponsetofishingandtheirdynamicbehavior.Wereview19

implementationsofsizespectrummodelsanddescribeimportantvariationsconcerning20

thefunctionalresponse,whethergrowthisfood-dependentorfixed,andthedensity-21

dependenceimposedonthesystem.Finallywediscusschallengesandpromising22

directions.23

24

Keywords:Ecosystemapproach,food-web,ecosystembasedfisheriesmanagement 25

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Introduction26

MarinecommunitymodelsrangefromtheoriginalLotka-Volterradifferentialequations27

toextremelycomplicatedend-to-endsimulations(Plagányi2007;Fultonetal.2011).In28

themiddleoftherangewefindsizespectrummodels.Sizespectrummodelsusebody29

sizeofindividualstorepresenttheentirefishcommunityasasizedistribution.The30

relianceofbodysizesimplifythedescriptionofpredator-preyinteractions,individual31

physiologyandvulnerabilitytofishinggear.Thispaperhighlightsoneoftheimportant32

advantagesofthesizespectrumapproach:awell-foundedandunifyingmechanistic33

basisallowingforgreatexplanatorypowerandparsimonioususeofdata.34

35

Size spectrum models are relevant to fisheries science in the context oftheecosystem36

approachtofisheriesmanagement(Pikitchetal.2004).Whilesingle-speciesstock37

assessmentsandimpactassessmentwillcontinuetobeimportantmanagementtools,38

theyneedtobesupplementedbystrategicimpactassessmentsatthelevelofthe39

ecosystem.Suchimpactassessmentsassistthedevelopmentandimplementationof40

strategiclong-termmanagementgoalsfortheecosystem,e.g.,howshouldfishing41

pressurebedistributedovertheentireecosystem?Howdowebalanceexploitationof42

competingfisheriessuchasforagefisheriesandconsumerfisheries?Howdowe43

maximizetheyield(ofbiomassorwealth)oftheentireecosystemwhileminimizingrisk44

offailureorimpoverishcomponentsofthesystemunderenvironmentalchange?To45

answerthesequestions,weneedtoquantitativelyunderstandtherelationshipbetween46

fishingpractice(what,whenandhowmuch)andtheabundanceofspeciesandsizesof47

organismsthroughoutthecommunity.48

49

Sizespectrummodelsareespeciallysuitedtothesequestionsbecausetheyresolvethe50

mostimportaspectsoffishlifehistoryandtrophicecology.Akeycharacteristicoffishis51

thatindividualsgrowthroughseveralordersofmagnitudeinbodysizethroughtheir52

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life.This,combinedwiththestrongrelationshipbetweenbody-sizeandtrophicniche53

(Barnesetal.2008;Gilljametal.2011),meansthatindividualschangetheirtrophic54

nichethroughoutontogeny(WernerandGilliam1984).Suchontogenetictrophicniche55

shiftsmakesitdifficulttoapplytheconventionalfood-webapproach,whereeach56

speciesisdescribedbyasinglemetric(abundanceorbiomass)andaspecifictrophic57

level,tofishcommunities.Therelationbetweenbodysizeandtrophicnichehas58

promptedthehypothesisthatindividualbodysize(ratherthanspeciesidentity)isthe59

primarydeterminantofcommunitystructure(Jenningsetal.2001).60

61

Sizespectrummodelsarebaseduponthelongtraditioninecologyofrecognizingbody62

sizeasacentraltraittodescribeindividuals(Elton1927;Haldane1928;Andersenetal.63

2016a)becauseitcorrelatesstronglywith:metabolism(Kleiber1932;Winberg1956,64

Brownetal.2004),predator-preyrelations(Ursin1973;Barnesetal.2008),encounter65

rates(Acuñaetal.2011),functionalresponses(Ralletal.2012),reproductiveeffort,and66

othervitalrates(Peters1983).Forapplicationtofisheries,bodysizefurthermoreisan67

excellentdescriptorofmesh-sizeregulationsandcharacterizesthevalueofacatch68

(Andersenetal.2015).Finally,distributionsofabundancevs.sizeshowaremarkable69

regularity(SheldonandPrakash1972;Sheldonetal.1977;BoudreauandDickie1992)70

anddeviationsfromthisregularityhasbeenusedtocharacterizeecosystemlevel71

impactoffishing(RiceandGislason1996;Daanetal.2005).Thesizespectrum72

modelingparadigmpromisesa“charminglysimple”(Popeetal.2006)setoftoolswitha73

lowtointermediatecomplexitythatcanbereadilydeployedforagivensystemand74

providequantitativeinformationabouttheecosystemimpactoffishing(Collieetal.75

2014).Thismakesitpossibletoapplythemodelsinsituationswheremorecomplexbut76

alsodata-demandingend-to-endmodelscannotbeemployedeitherbecauseoflackof77

dataormanpowertocalibrateandrunthem.78

79

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Thevarioustypesofsizespectrummodelcanbeviewedasdifferentdevelopmentsof80

thesamecoreconcepts.Wereviewthecommonbasicconceptsbehindsizespectrum81

modelsfocusingonmodelsthatdescribeanentirefishcommunity.Amongthemwe82

recognizethreebroadclassesofdecreasinglevelsofcomplexity.Mostcomplexarethe83

‘food-web’models,socalledbecausetheyexplicitlyrepresentindividualpopulations84

withspecies-specificenergybudgetparametersandpreypreferences,thereby85

quantifyinganetworkoftrophicinteractionsasanexplicitfood-web.Theseare86

simplifiedintothe‘trait-based’modelsbyreducingdifferencesamongthepopulations87

toasinglecontinuousvariablerepresentingatrait(usuallymaturationsize)and88

simplifyingpreyselectiontoafixedpredator-preybodysizeratio.Furthersimplification89

producesthe’community’sizespectrummodels,socalledbecausetheyignore90

differencesamongpopulationstherebyrepresentingthecommunityasasingle91

populationofinteractingindividualsthatdifferonlyintheirbody-size.Weexplainhow92

thesimplermodelscanbederivedfromthemorecomplex,startingfromthefood-web93

andendingwiththecommunitymodel.Furtherwedevelopanalytical“equilibrium”94

solutionstothemodels.Weillustratethemodels’behavior,inparticulartheirresponse95

tofishing,andfinallydiscusschallengesandopenissuesforfurtherdevelopment.96

Conceptsunderlyingsizespectrummodels97

Sizespectrummodelsarefoundedonthreecommonconcepts:First,biomass(and98

equivalentenergy)isconserved,enablingaccountancyofenergyflowsatthe99

communitylevelbasedonindividuallevelprocesses.Second,trophicinteractionsare100

themaindeterminantofcommunitystructureandtheseareforemostdeterminedby101

predator-preysizeratios.Third,theenergybudgetofanindividualisallometrically102

linkedtobodysize,sothatbodysizecanbeusedasakeyidentifieroforganismsand103

theirinteractionswiththecommunity.Thethreemainecologicalprocessesforany104

organismaregrowth,reproductionandmortalityandallthreecanbelinkedtobody105

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sizeinthismodelingframework.Thissimplificationhasgreatstrategicvalueasit106

enablesecologicalmeasuressuchasproductionrateandsizestructure(whichare107

importantforecosystembasedfisheriesmanagement)tobederivedfromrelatively108

littleandaccessibledataaboutphysiologyandlifehistoryinvariants.Theequationsfor109

themodelsusedintheexamplestofollowareprovidedinTable1andparametersin110

Table2.111

112

Thesizespectrum113

Thesizespectrumrepresentsabundanceorbiomassofindividualsasafunctionoftheir114

bodysize.Inthiscontext‘bodysize’usuallymeansbodymassbecauseitisthenatural115

metrictoformulateanenergybudget.116

117

Threesizespectrumrepresentationsarecommonintheliterature(SprulesandBarth,118

thisissue;AndersenandBeyer2006;Rossberg2012):theabundancedensityspectrum,119

thebiomassdensityspectrumandthe“Sheldon”biomassspectrum.Theabundance120

densityspectrum𝑁(𝑤)representsthenumberofindividualsinthebodymassrange121

from𝑤%to𝑤&as 𝑁 𝑤 d𝑤)*)+

,andhereitisreferredtoasthesizespectrumforbrevity.122

Itcanbeconstructedfromobservationsbydividingthetotalnumberofindividualsina123

sizeclassbythewidthofthesizeclassandthereforehasdimensionsofnumbersper124

mass(oftenreferredtoasthe“normalizedsizespectrum”;SprulesandBarth,this125

issue).Thebiomassdensityspectrumisconstructedfromtheabundancedensity126

spectrumbymultiplyingwithbodymass𝑁 𝑤 𝑤(dimensionsbiomasspermass).The127

“Sheldon”biomassspectrum(SheldonandParsons1967)isthebiomassin128

logarithmicallywideclasses,i.e.,thebiomassintherange𝑤to𝑐𝑤where𝑐isaconstant129

largerthanonedeterminingthewidthofthesizeclass.Forexample,the“octave”bin130

usedbySheldonimplies𝑐 = 2,andnormallog10baseimplies𝑐 = 10.Ifweassumethat131

theabundancedensityspectrumfollowsapower-law𝑁 𝑤 = 𝜅𝑤3withintheclassthen132

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thebiomassspectrumcanbefoundas:133

134

𝐵567 𝑤 = 𝑁 𝜔 𝜔d𝜔9)

)= 𝜅

𝑐&:3 − 12 + 𝜆

𝑤&:3 ∝ 𝑤&𝑁 𝑤 ,135

136

where𝜔isadummyvariablefortheintegration.Alltermsexcept𝑤&:3areindependent137

ofsize,hencethebiomassspectrumisproportionaltothenumberdensityspectrum138

multipliedbythebodymasssquared.Formathematicalanalysesthedensityspectraare139

convenientbecauseintegralsoverthesegivetheabundanceandbiomass.For140

presentationpurposestheSheldonbiomassrepresentation𝑤&𝑁(𝑤)isconvenient141

because,atthecommunitylevel,itshowshowthebiomassofpreyisdistributedwith142

size(assumingthatthesizerangeofpreferredpreyisconstant),andonaspecieslevelit143

isproportionaltothecohortbiomass.144

145

Conservationequation146

Thesizespectrumiscalculatedbyconsideringabalancebetweenmortalityandgrowth147

atallbodysizes 𝑤 .Individualsflowintosizeclassesviasomaticgrowthwhilstsome148

arelosttonaturalandfisheriesmortality.ThisbalanceisformalizedbytheMcKendric-149

vonFoersterequation(seeSilvertandPlatt(1978)foraderivation):150

151

𝜕𝑁A 𝑤𝜕𝑡

+𝜕𝑔A 𝑤 𝑁A 𝑤

𝜕𝑤= −𝜇A 𝑤 𝑁A 𝑤 ,(1)152

153

where𝑔A 𝑤 isthegrowthrate(masspertime)and𝜇A 𝑤 themortality(pertime),and154

𝑁A 𝑤 isthesizespectrumofspecies𝑖.Thuseq.(1)scalesfromindividual-level155

processesofgrowthandmortalitytothepopulation-levelsizespectrum.Recruitment156

fromthepopulationflowsintothesizespectrumatthesmallestbodysize(typicallythe157

eggsize)𝑤F.Thisisrepresentedasaboundarycondition:158

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159

𝑔A 𝑤F 𝑁A 𝑤F = 𝑅A,(2)160

161

where𝑅A istherecruitment(numberofrecruitsoreggspertime).Theabovetwo162

equationsaremathematicalformalizationsofamassbalance,andcanbethoughtofas163

thesize-basedversionofclassicsurvivoranalysisusedinage-basedmodels.164

165

Thefollowingoutlinesthecentralassumptionsinthemodelsabouthowgrowth𝑔A ,166

mortality𝜇Aandreproduction(recruitment)𝑅A,arecalculated.Withthepartial167

exceptionofrecruitmenttheseareallcalculatedfromindividuallevelprocessesof168

predator-preyencounterandabioenergeticbudget(Figure1).169

170

TheAndersen-Ursinencountermodel171

Thekeyprocessinthemodelsispredator-preyencountersbetweenindividuals172

governedbyaformalizationofthegeneralrule,biggerfisheatsmallerfish(Andersen173

andUrsin1977).Individualspreferpreyacertainfractionsmallerthanthemselves(M1,174

Table1)(Ursin1973).Theclearancerate(dimensionstime-1)isanincreasingfunction175

ofbodysize(largerfishclearalargervolumeofwaterforpreypertimethansmallfish)176

(M2).Thecombinationofpreference,clearancerateandpreyabundancespecifiesthe177

foodencounterrate(M3,biomasspertime).Intakeuponencounter(“satiation”)is178

representedwithatypeIIfunctionalresponse(M5)asthe“feedinglevel”𝑓A 𝑤 ,i.e.,the179

ratiobetweenconsumptionandmaximumconsumption(M4)(dimensionlessnumber180

between0and1).181

182

Individualenergybudget183

Theenergybudgetdescribeshowconsumedfoodisusedformaintenance,activity,184

growthandreproduction.Consumedfoodisassimilatedandfirstusedforstandard185

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metabolism,widelyrecognizedtobeanallometricfunctionofbodymass:𝑘J𝑤K.Juvenile186

individualsusetheremainingavailableenergyforgrowth,whilematureindividuals187

apportiontheenergybetweengrowthandreproduction(M6-M8).Theexact188

specificationofallocationofenergybetweengrowthandreproductionisnotcrucial.189

Theoneusedhereensuresthatwhenthefeedinglevelisconstant,size-at-agecurves190

resemblesavonBertalanffycurveandthegonado-somaticisindependentofbodysize191

(Hartvigetal.2011).192

193

Reproductionandrecruitment194

Reproductionandrecruitmentrepresentthereproductiveoutputfromtheentire195

population.Thereproductiveoutput𝑅L.A (numberspertime;M9)isdiscountedbya196

reproductiveefficiencyεtorepresentlossesduetoeggmortalityandspawningeffort,197

andusedtocalculatetherecruitment𝑅A .Withinthiscontext,recruitmentreferstothe198

rateofproductionofnewindividualsfromthefertilizedeggstage,butitcouldbedone199

atalaterstageifproperlydiscounted(AndersenandBeyer2015).Ideallythe200

recruitmentisequaltothereproductiveoutput,butmanymodelsapplyastock201

recruitmentrelationship(M10).Thedensitydependenceimposedbythestock202

recruitmentrelationshipavoidsthecompetitiveexclusionbetweenspeciesthat203

otherwisetendstooccur(HartvigandAndersen2013).Thestock-recruitment204

relationshipcontainstwoessentialparameters:the“slope”parameterthatspecifies205

recruitmentatlowpopulationsizesandthemaximumrecruitmentthatspecifiesthe206

populationcarryingcapacity.Theslopeparameterisgivendirectlybytheegg207

productionofthepopulation(AndersenandBeyer2015),butthemaximumrecruitment208

hastobespecifiedseparately.Thisparameterrepresentsalleffectsonthepopulation209

thatarenotexplicitlyrepresentedinthemodel,suchaslimitationsduetojuvenile210

habitatsizethatisknowntolimitsomemarinepopulations(RijnsdorpandLeeuwen211

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1992).Themaximumrecruitmentandpossiblytherecruitmentefficiencyarekey212

parametersforcalibratingamodeltodatafromrealfishstocks(seediscussion).213

214

Mortality215

Threecategoriesofmortalityarerecognized.First,predationmortality(M13)emerges216

fromthetrophicdynamicswithinthesystem;second,intrinsicorbackgroundmortality217

(M12)isusuallyrepresentedasanallometricfunctionofasymptoticsize(Brownetal.218

2004),thoughstarvationmortalitycanbeexplicitlyadded(e.g.,Hartvigetal.2011);219

third,exogenoussourcesofmortality(especiallyfishing)areoftenadded.220

221

Resource222

Theresourcespectrum𝑁N 𝑤 representsfoodotherthanfish.Theresourceisneeded223

forthesmallestindividualswhoarenotyetlargeenoughtobepiscivorousbutitcan224

representanykindoffood:asinglesize-groupofsmallzooplanktonpreyspecies,asize225

spectrumofzooplanktonprey(asinFig.1),orasizedistributionincludinglargerprey,226

e.g.benthicproduction.Theresourcecanbeconstant,inwhichcasethegrowthrateof227

smallfishisfixed,oritcanbemodeleddynamically,e.g.,asasemi-chemostat(M14).The228

semi-chemostatformulationisconvenientbecauseitleadstoaverystabledynamicsof229

theresource.Usinglogisticgrowthresultsinamorepronounceddynamicalresponseof230

theresourcewhichtranslateintostrongerdynamicsofthefishpartofthemodel(de231

Roosetal.2008).232

Modelstypes233

Wenowbrieflydescribehowthesecommonconceptsareusedtocreatesizespectrum234

modelsatthreelevelsofcomplexityanddemonstrateanapproximateanalytical235

solutionfortheequilibrium.236

237

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Food-webmodel238

Inthefood-webmodel,theprocessesM1–M15areinstantiatedwithallparameters239

fromTable2beingspeciesspecific(eitherrepresentingidentifiedspeciesor240

hypotheticalonesmatchingrelevantcriteria),butinpracticesomeparametersare241

usuallycross-speciesconstants,suchastheexponents𝑛, 𝑝and𝑞.Populations,thus242

identifiedasdifferentspecies,interactthroughpredator-preyrelationswithinteraction243

strengthsspecifiedbyaninteractionmatrix,whichrepresentsacombinationofspecies-244

specificpreferencesandencounterprobabilities.Thismatrixcouldbepopulatedwith245

empiricalinteractioncoefficientsderivedfromstomachcontentanalyses,spatial246

overlap(Blanchardetal.2014),oritmayrepresenthypotheticaldistributions--random247

anduniform(everythingeatseverythingelse)interactionnetworksarepopular248

hypotheses.Afullyspecifiedfood-webmodelhas14parametersforeachspecies,plus249

aninteractionmatrix,soforkspeciesthetotalisupto3 + 14𝑘 + 𝑘&parameters.250

251

Trait-basedmodel252

Thetrait-basedmodelrepresentsdifferencesamongspeciesonlybythegoverningtrait253

ofasymptoticsize(Popeetal.2006).Thisassumesthatthemostimportanttraitisthe254

asymptoticsize(or,equivalently,sizeatmaturation),whichembodiesatrade-off255

betweenreproductiveoutputandasymptoticsize.Thetrait-basedmodelis256

conceptuallyderivedfromthefood-webmodelbyassumingthatallparametersinTable257

2arecross-speciesconstantsandbyusingtheoreticalargumentstodetermine𝑅TUVasa258

functionof𝑊(appendixA).Feedinginteractionaresolelydeterminedbyindividual259

size.Thesolutionisthetraitsizespectrum𝑁(𝑤,𝑊)(dimensionsnumberspermassper260

asymptoticmass)describingthejointdistributionofindividualandasymptoticsizes261

(AndersenandBeyer2006).Innumericalimplementationstheasymptoticsizeaxisis262

discretized,typicallyintologarithmic‘bins’groupingspeciesinasymptoticsizeclasses.263

Inpractice,theresultsofthetrait-basedmodelareeffectivelyindependentofthe264

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numberofsimulatedasymptoticsizeclassesoncethisnumberisgreaterthan10.The265

trait-basedmodelisspecifiedwiththe18parametersinTable2.266

267

Communitymodel268

Thecommunitymodelignoresalldifferencesbetweenspeciesandonlyconsiders269

differencesinsize(BenoîtandRochet2004).Itcanbederivedfromthetrait-basedand270

food-webmodelsbyintegratingoveralltrait-classesorsummingoverspecies(Zhanget271

al.2012):272

273

𝑁9 𝑤 = 𝑁 𝑤,𝑊 d𝑊∞

)= 𝑁A(𝑤)

AX%

.274

275

Theintegralonlyrunsfrom𝑤becauseasymptoticsizegroupswith𝑤 < 𝑊doesnot276

contributetothecommunityspectrumatsize𝑤.Thecommunitymodelonlyresolves277

the‘communityspectrum’𝑁9(𝑤).Substantialconsequencesarisebecausethismodelis278

unabletorepresentmaturation,reproductionandrecruitment.Theenergybudget(M6-279

M8)issimplifiedsuchthatgrowthissolelyavailableenergymultipliedbyan“average280

growthefficiency”derivedfromequilibriumtheory(M7b;AppendixA).Because281

energeticlossestoreproductionarenotexplicitlyaccountedfor,themodelwillnot282

reproducevonBertalanffygrowth.Further,mostimplementationsintheliteratureuse283

justalinearfunctionalresponse,ignorestandardmetabolismanduseafixedresource284

(Table3),however,thesesimplificationsarenotsignificantforthecommunitymodel.In285

itsmostcomprehensiveform,thecommunitymodelrequiresonly11parameters(less286

withoutthefunctionalresponseandwithfixedresource).287

288

Equilibriumsolutions289

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Analyticalsolutionstothetrait-basedmodelcanbederivedundertheassumptionthat290

thefeedinglevelisconstant𝑓 𝑤 = 𝑓Fandthatthespectrumisinfinitelylong,i.e.𝑤F =291

0andmax 𝑊 =∞(AndersenandBeyer2006;Hartvigetal.2011).Thisresultsinan292

‘equilibriumcommunityspectrum’:293

294

𝑁9 𝑤 =1𝛷

𝑓F1 − 𝑓F

ℎ𝛾𝑤à&ab(3)295

296

with𝛷 = 2𝜋𝜎𝛽baèxp[ 𝑞 − 𝑛 &𝜎&/2)].Thescalingexponent𝑛 − 2 − 𝑞 ≈ −2.05isin297

accordancewithobservations(BoudreauandDickie1992).Theequilibriumsolutionfor298

eachspeciesspectrumis:299

𝑁 𝑤,𝑊 ∝ 𝜅𝑊&àbam:n𝑤aàn 1 −𝑤𝑊

%a` n/(%a`)

(4)300

301

withthe“physiologicalmortality”𝑎givenin(M17).Thisresultisusedtocalculate302

expectedscalingsolutionstopredationmortality,thescalingofmaximumrecruitment303

withasymptoticsizeusedinthetrait-basedmodels,andtheaveragegrowthefficiency304

inthecommunitymodel(AppendixA).Notethatthesolutionineq.(4)doesnotfulfill305

theboundarycondition(M9);thetotalreproductiveoutputcalculatedfrom(4)willlead306

toalife-timereproductiveoutputlargerthan1andincreasingwithasymptoticsize307

(discussedinHartvigetal.2011andRossberg2012).Thisdiscrepancyhastobe308

resolvedbydensity-dependenteffectswithineachpopulationnotaccountedforin(4),309

andithasbeenusedtorelatetheslopeparameterinstock-recruitmentsrelationshipto310

asymptoticsize(AndersenandBeyer2015).Inthedynamicalmodelstheemergent311

physiologicalmortalitydependsonasymptoticsize,withsmallerspecieshavinga312

smaller𝑎thanlarger(Hartvigetal.2011),inaccordancewithempiricalmeasurements313

ofhowmortalitydependsonasymptoticsize(Gislasonetal.2010).Eventhough(4)is314

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notanexactsolutionoftheentiremodel,itisstillausefulapproximation,as315

demonstratedbyitsabilitytoresolvespeciesdiversity(Reumanetal.2014).316

317

Theequilibriumresultsallrelyonthemetabolicassumptioninherentinthefunctional318

responsewhereconsumptionisproportionalto𝑤`.Ifafunctionalresponseisnotused319

thesolutionfortheexponentofthecommunityspectrumwilldifferfromeq.(3).In320

particularitwillnotdependonthemetabolicexponent𝑛,butratheronthepreferred321

predator-preymassratio𝛽(BenoîtandRochet2004;Dattaetal.2010;Rossberg2012).322

Suchsolutionswillresultinconsumptionratesthatdonotfollowmetabolicscaling.323

324

Parameters325

Parametersareeitherdeterminedfromknowledgeaboutthespecificspecies(forthe326

food-webmodel),orfromcross-speciesinvestigationsoflife-historyinvariants(Table327

2;seeHartvigetal.(2011),AppE.foradetaileddiscussion).Therelativelysmallsetof328

parameterfacilitatesformalinvestigationsofmodelbehaviorundervaryingparameter329

values(Thorpeetal.2015;Zhangetal.2015).330

331

Implementation332

Sizespectrummodelsmaybesimulatedwith“Mizer”,areferenceimplementationinR333

(Scottetal.2014),orwiththematlabcode(seeonlinesupplementary).Forthefood-334

webmodelwehaveusedtheparameterizationfortheNorthSea(Blanchardetal.2014),335

whichuses𝑛 = 2/3asiscustomaryinthefisheriesliterature.Forthetrait-basedand336

communitymodelswehaveused𝑛 = 3/4toconformwith“metabolic”theory(Westet337

al.2001).Theresultsarequalitativelysensitivetothevalueof𝑛aslongasitischanged338

inalltherelationshipsinTable1.Thecommunitymodelhasbeenimplementedasa339

trait-basedmodelwithasingletraitgrouphavingaverylargeasymptoticsize.Asthe340

individualsinthetraitgroupmaturetheirgrowthratedeclines(Fig.2d).Inthiswaythe341

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averagegrowthinthecommunitymodelcorrespondstotheaveragegrowthinatrait-342

basedmodel.Ourimplementationavoidstheneedforanadditionalsenescentmortality343

forthelargestindividualsasiscommonpractice(Lawetal.2009;RochetandBenoît344

2012).345

346

Allsimulationsaresetupwith100logarithmicallyspacedgridpointsonthemassaxis,347

withthefirstgridpointsettoeggsizew0.Eachsimulationwasrunwithatimestepof348

0.25yearuntilconvergence(AppendixB).349

Examplesimulations350

Allthreemodelspredictsizespectra,growthratesandmortality(Figure2).Inthe351

absenceoffishingandtoppredatorsthelargestsizegroups(𝑤 > 10kg)are352

superabundant,i.e.,thatpartofthespectrumisgreaterthanpredictedbythe353

equilibriumsolution.Thisismostpronouncedinthefood-webmodel,andcouldbe354

changedbyincreasingthebackgroundmortality𝑍F.Thesuperabundanceresultsin355

higherpredationmortalityonmedium-sizedindividuals,whichtriggersatrophic356

cascadeandassociatedchangesingrowthandmortalityofsmallerindividuals357

(AndersenandPedersen2010).358

359

Responsestofishing360

Thebehaviorofthemodelsisillustratedbyexaminingtheresponseofthetime-361

averagedsolutiontofishingandtheirdynamicalbehavior.Fishingusingsize-selective362

gearisrepresentedbyaddingafishingmortalitythatdependsonbodysize,and363

possiblyalsoasymptoticsizeorspecies.Weillustratefishingthroughtwoscenarios:a)364

community-widefishingonallspecieswithatrawl-typeselectivitypatternhaving50%365

selectivityat0.05𝑊,andb)abottom-upperturbationwhereforagefisheryisremoved366

fromscenario(a)simulatedbysettingfishingmortalityzeroonallspecieswith𝑊 <367

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200ginthefood-webandtrait-basedmodelsandonindividualswith𝑤 < 200ginthe368

communitymodel(Figure3).Inscenario(a)(communitywidefishing)thereductionof369

largeindividualsinducesatrophiccascadethroughoutthecommunityseenasawavein370

thesizespectrum.Whenforagefishingisremoved(b),thefood-webandtrait-based371

modelspredictanincreaseofforagefishbutrelativelymodesteffectsontherestofthe372

community.Theforagefishhaveahigherbiomassthanintheunfishedsituationdueto373

thepartialreleasefrompredationbyhighertrophiclevelscausedbyfishing.The374

responseofthecommunitymodelisdifferent:itpredictsadeclineinthesize-range375

wherefishingisremoved,whiletheeffectsontherestofthecommunityareweak,asin376

theothermodels.Thisdifferenceinthecommunitymodelsstemsfromitsinabilityto377

representfishing(or,inthiscase,absenceoffishing)onspecificspeciesorlifehistories,378

butonlyonbodysizes.Thisresultemphasizestheimportanceofrepresenting379

individualpopulationsinfisheriesapplications.Thetwoscenariosillustratesthe380

relativeimportanceofmortalityandgrowthtomediatetrophiccascades:inscenario(a)381

thetrophiccascadeismainlymediatedbychangesinpredationmortality,whichleads382

toastrongcascade.Inscenario(b)increasesinforagefishabundancehastwoeffects383

withoppositeconsequences:1)itincreasesgrowthratesoflargerindividuals,but2)it384

alsoincreasescompetitionbetweenjuvenileindividualsoflargerspeciesandforagefish385

leadingtodecreasedgrowthrates.Theendresultisamodesttrophiccascade(Houleet386

al.2013;Jacobsenetal.2015).Theimportanceofcompetitionbetweenforagefishand387

juvenilepredatoryfishinrealecosystemscouldbeanalyzedbycomparingstomach388

contentsofforagefishandjuvenilepredatoryfish.Insummary:thefood-webandtrait-389

basedmodelspredictsimilarresponsetoselectivefishing,whilethecommunitymodel390

failstoresolveeffectsondifferentpopulations.391

392

Dynamicsolutions393

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Thecommunitymodelalsodiffersfromthefood-webandtrait-basedmodelin394

dynamicalbehavior,i.e.,howthesolutionvariesovertime.Allmodelstendtobe395

unstable(oscillateovertime)ifthetrophicoverlapissmall.Thetrophicoverlapis396

determinedbytheratiobetweenthewidthofthesizepreferencefunction,𝜎,andthe397

predator-preysizeratiolog(𝛽):thesmallerthevalueof𝜎/log(𝛽),thesmallerthe398

trophicoverlap,andthemoreunstablethesolutionbecomes(largeroscillations)(Datta399

etal.2011;Zhangetal.2012)(Figure4).Oscillationsinthetrait-basedmodelsarefairly400

modest,butthecommunitymodelispronetounrealisticallystrongnon-linear401

dynamics:thesolutionvariesbyupto10ordersofmagnitude(Figure4bandd).This402

meansthatsomepartsofthespectrumalternatebetweenbeingcompletelydevoidof403

fishandbeingfullypopulated.Itisthereforeevidentthatthenon-linearpropertiesof404

thecommunitymodelarefundamentallydifferentfromthemodelswithlife-history405

diversity,suchasthetrait-basedmodelsorafood-webmodel.Evenifthecommunity406

modelismadelinearlystablewithahightrophicoverlaporadiffusionterm(Dattaet407

al.,2011),thestrongdynamicalresponsewillstillbepresentifthemodelisperturbed408

awayfromtheequilibrium.Thisshouldbekeptinmindifthemodelisusedtosimulate409

thedynamicalbehaviorofmarineecosystems(Zhangetal.2012;Rossberg2013p.273).410

Challengesandopenissues411

Sizespectrummodelsdistinguishthemselvesfromunstructuredmodelsbyresolving412

individualbodysizeasacontinuousstatevariable(bodysize).Towhatextentdo413

individualbodysizeandspeciesidentitydeterminetheecologicaloutcomesof414

communitydynamics?Thethrustofsizespectrummodelinghasbecomeanemphasis415

ontheformer,whilstunstructuredmodelshaveemphasizedthelatter.Somesize416

spectrummodelsrepresentspeciesinteractions,andsomeattemptshavebeenmadeto417

findacommonunderstandingbetweenthetwoperspectives.Notably,afood-webmodel418

withimplicitrepresentationofintra-speciessizestructurewasobtained(Rossbergand419

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Farnsworth2011)to(indirectly)describeinteractionsamongspeciesofdifferentsizes.420

Theimportanceofexplicitlyresolvingthesize-structureofspecies,orwhereitcan421

safelybeignored,iscontextdependentsorequiresspecificandsystematicexploration422

(Jacobsenetal.2015;Woodworth-Jefcoatsetal.2015).423

424

Animportantdifferencebetweenimplementationsofsizespectrummodelsiswhether425

growthisfixedorfood-dependent(Table3).Fixinggrowthsimplifiesmodelsetupand426

calibration.Itisjustifiedbythemodestvariationsingrowthobservedinmarinespecies.427

Fixinggrowth,however,hasconsequenceswhichshouldbeconsideredwhenthemodel428

iscalibratedandresultsareinterpreted.Amodelwithfixedgrowthwillstillresolve429

trophiccascadesmediatedbymortality,butitwillnotresolvecompetition,whichis430

crucialtodescribethephenomenonofovercompensation(DeRoosandPersson2002)431

andmaybeimportanttounderstandtheresponseofthespectrumto,e.g.fishingon432

‘forage’species(Houleetal.2013;Jacobsenetal.2015).Moreimportantly,without433

food-dependentgrowth,themassbalancingbetweengrowthandpredationmortalityis434

broken.Thisrequiresthatcareistakeninthesetuptoensurethatpredationmortalities435

areinthecorrectrangebyadjustingthe“otherfood”compartment(Thorpeetal.2015).436

Havingtoolowpredationmortalities(asinHalletal.2006;Wormetal.2009;Rochetet437

al.2011)willresultinamodelthatisessentiallyasetofweaklycoupledsingle-species438

modelsthusdefyingthepurposeofamulti-speciesmodel.439

440

Size-basedmodelshavebeencharacterizedas“highlyunrealistic”andbeingbasedon441

“unrealisticandevencontradictoryassumptions”(Froeseetal.2015;Andersenetal.442

2016b).Wehopetohavemadeitclearthatthebasicassumptionsarerealisticand443

internallyconsistent.Nevertheless,whilethesize-basedmodelshavematuredtoa444

degreewheretheycanbeappliedtomakeimpactassessmentsoffishingonmarine445

ecosystems,theystillfacechallengesrelatedtodensity-dependence,life-historytrade-446

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offs,terminationatthelargebodysizesend,calibrationprocedure,andnumerical447

implementation,whichmustbeconfronted:eachwillbebrieflydiscussed.448

449

Densitydependence450

Allfood-webmodelsofrealecosystems,i.e.withspecificspecies,requiresomeformof451

densitydependentregulationoftheabundanceofeachspeciestoavoidcompetitive452

exclusion.Notmuchisknownabouttheexactmechanismoftheregulationandhow453

differentmechanismsaffectmodelresults.Thesizebasedinteractionsandtrait454

differencesamongspeciesinsizespectrummodelsprovideinsufficientniche455

differentiationtoavoidcompetitiveexclusion(HartvigandAndersen2013).Additional456

nichedifferentiationmayberepresentedbyarandomspecies-specificinteraction457

matrix,whichcansupportcoexistence(Hartvig2011;Hartvigetal.2011).Other458

commonlyusedmechanismsare(Table3):stock-recruitmentrelationships;fixed459

recruitment;predator-dependentfunctionalresponses,wherebyintakedependsonthe460

densityofcompetitorsaswellasprey(alsousedinEcosim)(Abrams2014);andprey461

switching(MauryandPoggiale2013),wherebyrarepreyarenotattacked(leadingtoan462

emergenttypeIIIfunctionalresponse).Whichofthesemechanismsisthemostcorrect463

representationofeffectsinrealecosystemiscurrentlyunknown:stock-recruitment464

relationsandfixedrecruitmentareinlinewithstandardpracticeinfisheriesscience,465

buthavelittletheoreticalsupport;predator-dependentfunctionalresponsesandprey466

switchingcertainlyoccurtosomeextent,buttheunderstandingiscurrentlytooweakto467

makegeneralstatementsofthestrengthoftheprocesses.Withinstructuredmodels,468

suchassizespectrummodels,thetypeofdensitydependentregulationmayhavea469

profoundimpactonthesolution,boththesizespectrumoftheindividualspeciesand470

therelativeabundancesofspecies(compareFig.4inMauryandPoggiale(2013)with471

Figure2B).Asanexample,wetestedthepredictionfromthetrait-basedmodelagainst472

empiricalobservationsbycomparingtheasymptoticsizedistributionwithobservations473

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fromthreetrawlsurveysintheNorthSea(Daanetal.2005)(Figure5).Eventhoughthe474

comparisondoesnotrejectthemodeleddistribution,morecomparisonswithsimilar475

datafromothersystemsareneededtobuildconfidenceinthepredictedasymptoticsize476

distributions.477

478

Inadditiontoensuringcoexistenceofspecies,theimposeddensitydependencealsoacts479

asacarryingcapacity.Inthefood-webmodelthecarryingcapacity(𝑅TUV)isfoundby480

calibratingtoobservedbiomasses.Thetrait-basedmodel,however,reliesontheoretical481

resultsfromtheequilibriumtheory.Thattheorycomparesfavorablytothecalibrated482

results(Figure6).Nevertheless,theuseofastock-recruitmentrelationshipis483

unsatisfactoryasitintroducesadominatingexternalregulationonthebiomassof484

species.Thismaybiastheresponsetimeofcommunitysizestructure(Fungetal.2013),485

whichisofinterestinconservation.Further,thestock-recruitmentrelationshipmeans486

thatalargeamountofspawnedbiomassissimplylosttounspecifieddensity-dependent487

processes.Thestock-recruitmentrelationshipthereforebreaksthemass-balancing488

whichiscarefullyobservedintheotherprocessesinthemodel(Perssonetal.2014).489

Sincethereisnogenerallyacceptedsolutiontotheproblemofmaintainingcoexistence,490

resultsshouldbeinterpretedinlightoftheassumptionsusedtorepresentdensity491

dependentregulation.Itmustbeemphasizedthatthisproblemiscommontoallfood-492

webmodelsandnotuniquetosizespectrummodels.493

494

Traitsandtrade-offs495

Thetrait-basedmodelassumesthatthemostimportanttraitistheasymptoticsize.Fish,496

however,varyinothertraitsthanasymptoticsize.Thequestionsarethen:whichother497

trait(s)shouldbeincludedinamodeltorepresentobservedvariation?Andhowcan498

suitabletrade-offsbeformulatedandparameterized?Anobvioustraitisactivity.499

Increasedactivitycauseshigherpreyencounterrates(highervalueoftheclearancerate500

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constant,𝛾).Ontheotherhand,higheractivityresultsinincreasedmetabolicratesand501

increasedvulnerabilityduetohigherexposuretopredators.Itispossiblethatinclusion502

ofanactivitytraitwouldmakeitpossibletodistinguishsedentaryfromactivespecies503

withthesameasymptoticsize,suchasanglerfishandscombroids.Suchatraitmaynot504

solvetheproblemofcompetitiveexclusionbecauseitdoesnotleadtosufficientniche505

differentiation.Atraitwhichwouldleadtonichedifferentiationcouldberelatedto506

habitat(Hartvig2011;Zhangetal.2013),i.e.,pelagicvs.benthic(Blanchardetal.2011).507

Inbothcasesmoretheoreticalinvestigationsareneededbutalsoempiricalworkto508

establishandparameterizethetrade-offs.509

510

Closureofthespectrumatlargebodysizes511

Anoverlookedissueisthetermination(closure)ofthemodelspectrumatthelargest512

bodysizes.Closureisusuallyachieved(ratherarbitrarily)bychoosingamaximumbody513

sizeandenforcingsomebackgroundmortalitytokillofthelargestindividuals.514

However,thesizeofthismortalityclearlyinfluencestheresults,inparticularintheun-515

fishedsituation.Inthesimulationspresentedhere(Figure2),thisbackgroundmortality516

isrelativelylow,leadingtothesuperabundanceoflargeindividualscomparedtothe517

equilibriumsolution.However,wedonotknowtherealabundanceofthelargest518

individualsinanunfishedsystem,becausemostsystemsareheavilyperturbed.519

Further,whatisthetheoreticallargestsizeofafish?Whyareteleostfishnotlargerthan520

afewhundredkg?Thereisnophysiologicalmechanisminthemodeltolimitthe521

asymptoticsize,andcurrenttheoreticalunderstandingcanonlyguessatananswerto522

thisquestion(FreedmanandNoakes2002;Andersenetal.2008;Andersenetal.2016a).523

Asatisfactorytheoreticalunderstandingofthefactorslimitingtheuppersizeoffishis524

neededtobolstertheconsistencyofthemodels.525

526

Calibrationtorealsystems527

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Sizebasedmodelscanbecalibratedtorealecosystems,forinstanceonthescaleofa528

continentalshelf(insmallersystemsimmigration/emigrationviolatetheassumed529

populationclosure).Inthetrait-basedmodelscalibrationisachievedbyvaryingsome530

ofthecrucialparameters,suchasthegrowthrateparameterℎandthecarryingcapacity531

oftheresourcetoreproduceobservedaveragegrowthratesofindividualsandeco-532

systemlevelcatchratesofthefishery(Popeetal.2006;Koldingetal.2015).Thenext533

levelofsophisticationistomatchmodeledspeciestoknownspeciescharacteristics534

(Jacobsenetal.2015),andfurthertoincludeaspeciesinteractionmatrix(Halletal.535

2006;Blanchardetal.2014).Inthesecasesthebiomassofeachspecieshastobe536

calibratedbyadjusting𝑅TUV.Anotherimportantparameterthathithertohasbeen537

ignoredisthereproductiveefficiency𝜀.Thisparameterrepresentseggsurvival,which538

islikelytovarysubstantiallybetweenspecies.Introducingyetanothercalibration539

parameter,however,requiresmoredata.Currently,thecalibrationmethodsappliedare540

statisticallysimple.Amoresophisticatedapproachacknowledgesuncertaintyby541

creatinganensembleofplausiblemodels,requiringthemtofulfillgeneralcriteria542

(Thorpeetal.2015).Apossiblefuturedirectioncouldbeintroductionofafullstatistical543

frameworkwheredistributionsofparametervaluesarederivedfromobservationsof544

biomassesandstomachcontentbymaximizingalikelihoodfunction(LewyandVinther545

2004;Spenceetal.2015).Finally,itshouldbekeptinmindthateventhoughamodel546

maybewellcalibratedtocurrentsituationsthereisnoguaranteethatitwillreliably547

predictthefuture.548

549

Numericalsolutionprocedure550

Gainsinaccuracyandspeedofthenumericalsolutionmaybeachievedbymovingto551

moreadvancedmethods.Thestandardmethodisafirst-ordersemi-implicitupwind552

schemethatissimpletoimplement(AppendixB).Thedrawbackofthismethodisthatit553

hasnumericaldiffusivity,isnotveryefficient,possiblyinaccuratefordynamics,andis554

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unabletoresolve“cohortcycles”(deRoosandPersson2001)withdiscontinuitiesinthe555

solutionintheformof“shocks”.Tomoveforward,werecommendlookingintherich556

literaturefromcomputationalfluidmechanicsforinspiration,inparticulartowards557

higherorderfinite-volumetechniqueswithlimiters,whichmaintainspositivityofthe558

solution(Zijlema1996),orspectralmethods(Rossberg2012).Enhancementstothe559

numericalschemecouldbeimplementedtocommonbenefitinthe(openaccess)560

referenceimplementation“Mizer”(Scottetal.2014).561

562

Currentandfutureapplications563

Wehaveshownhowsizespectrummodelscanbeusedtosimulatehowfishingofone564

groupofspeciesaffectstheentiresystemandthatisverydifficulttoachievewith565

alternativemodels.Despiteseveralopenissuesinsize-basedmodeling,asdiscussed566

above,themodelframeworkhasalreadyshownitsuseto:illustratehowfishingdrives567

trophiccascades(AndersenandPedersen2010),simulatetheimpactofrising568

temperatures(Mauryetal.2007;Popeetal.2009),explorethepotentialimpactsof569

climatechangescenariosonfishproduction(Blanchardetal.2012;Woodworth-Jefcoats570

etal.2013;Barangeetal.2014;Lefortetal.2014),describetheindirecteffectof571

ecosystemrecoverystrategies(AndersenandRice2010),quantifytheinteraction572

betweenforageandconsumerfisheryfleets(Engelhardetal.2013;Houleetal.2013)or573

betweenfisheriesandmarinemammals(Houleetal.2015),evaluateecosystemfishing574

strategiesandindicators(Houleetal.2012;Blanchardetal.2014;Jenningsand575

Collingridge2015;Spenceetal.2015;Thorpeetal.2015),evaluatebalancedharvesting576

(Rochetetal.2011;Jacobsenetal.2014;Lawetal.2014)anddescribetheecosystem577

levelyield(Wormetal.2009;Andersenetal.2015).Otherobvioususeswouldbeas578

operatingmodelsinmanagementstrategyevaluations,asthebasisforbio-economic579

evaluationsoffishingontheentirecommunity(Andersenetal.2015),andtofurther580

developingourbasicunderstandingoffishcommunityfunctioning.Inourview,the581

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communitymodelisbestlimitedtotheoreticalworkexaminingthesteady-state582

solutions.Thetrait-basedmodelcanbequicklydeployedindata-poorsituationsand583

makesaflexibletoolforexplorationofcommunity-levelfisheryinteractions.Thefood-584

webtypesofmodelsarewellsuitedtomorespecificfisheriesquestionswhereahigher585

levelofspeciesidentityisneededthanprovidedbythetrait-basedmodel.Besidesthese586

strategicapplications,itistemptingtodeploysizespectrummodelsfortactical587

ecosystembasedmanagement,e.g.,forprovidingadviceonspecificspecies.Weare588

reluctanttoendorsesuchusesbecausewefindthepurelyprocess-orientedframework589

toorigidtoprovideprecisequantitativeinformationonthespecieslevel.Inconclusion:590

thesmallnumberofparameters,thelowcomputationalrequirementsandthesolid591

mechanisticallybasisprovideaframeworkoflowtointermediatecomplexity,highly592

suitedtothestrategicimpactassessmentofpressuressuchasfishingand593

environmentalchangeonmarineecosystems.594

595

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596

Acknowledgements597

WethankJuliaL.BlanchardandAxelRossbergforcommentsonthemanuscript.This598

workwassupportedbytheCentreforOceanLife,aVKRCentreofExcellencesupported599

bytheVillumFoundation.600

601

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Hartvig, M., Andersen, K.H., and Beyer, J.E. 2011. Food web framework for size-712structured populations. J. Theor. Biol. 272(1): 113–122. 713

Houle, J.E., Andersen, K.H., Farnsworth, K.D., and Reid, D.G. 2013. Emerging 714asymmetric interactions between forage and predator fisheries impose 715management trade-offs. J. Fish Biol. 83(4): 890–904. doi: 10.1111/jfb.12163. 716

Houle, J.E., de Castro, F., Cronin, M.A., Farnsworth, K.D., Gosch, K.D., and Reid, 717D.G. 2015. Effects of seal predation on a modeled marine fish community and 718consequences for a commercial fishery. J. Appl. Ecol. doi: 10.1111/1365-7192664.12548. 720

Houle, J.E., Farnsworth, K.D., Rossberg, A.G., and Dave, R. 2012. Assessing the 721sensitivity and specificity of fish community indicators to management action. 722Can. J. Fish. Aquat. Sci. 69(6): 1065–1079. 723

Jacobsen, N.S., Essington, T.E., and Andersen, K.H. 2015. Comparing model 724predictions for ecosystem based management. Can. J. Fish. Aquat. Sci. doi: 72510.1139/cjfas-2014-0561. 726

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Jennings, S., and Collingridge, K. 2015. Predicting Consumer Biomass, Size-730Structure, Production, Catch Potential, Responses to Fishing and Associated 731Uncertainties in the World’s Marine Ecosystems. PLoS One 10(7): e0133794. 732

Jennings, S., Pinnegar, J.K., Polunin, N.V.C., and Boon, T.W. 2001. Weak cross-733species relationships between body size and trophic level belie powerful size-734based trophic structuring in fish communities. J. Anim. Ecol. 70(6): 934–944. 735

Kleiber, M. 1932. Body size and metabolism. Hilgardia 6: 315–353. 736

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Law, R., Plank, M.J., James, A., and Blanchard, J.L. 2009. Size-spectra dynamics 739

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Lefort, S., Aumont, O., Bopp, L., Arsouze, T., Gehlen, M., and Maury, O. 2014. 743Spatial and body-size dependent response of marine pelagic communities to 744projected global climate change. Glob. Chang. Biol. doi: 10.1111/gcb.12679. 745

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870

871

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872

873

Table1.Governingequationsofthefullfood-webmodel.Subscript“i"referstospeciesnumber.874

Encounterandconsumption

Preysizeselection 𝜙𝑤Lwxy𝑤

= exp − ln𝛽A𝑤Lwxy𝑤

&

/(2𝜎A&) M1

Clearancerate 𝑉A 𝑤 = 𝛾A𝑤bwith𝛾A =𝑓FℎA𝛽A

`ab

1 − 𝑓F 2𝜋𝜅�𝜎A M2

Encounteredfood 𝐸A 𝑤 = 𝑉A(𝑤) 𝜃A��

𝜙𝑤Lwxy𝑤

𝑁� 𝑤Lwxy 𝑤Lwxyd𝑤Lwxy

∞

F

M3

Maximumconsumption

rate

𝐼TUV.A(𝑤) = ℎA𝑤`M4

Feedinglevel 𝑓A 𝑤 =𝐸A(𝑤)

𝐸A 𝑤 + 𝐼TUV.A(𝑤) M5

Growthandreproduction

Maturationfunction 𝜓 𝑤 = 1 +𝑤𝜂A𝑊A

a%F a% 𝑤𝑊A

%a` M6

Somaticgrowth 𝑔A 𝑤 = 𝛼𝑓A 𝑤 𝐼TUV.A − 𝑘A𝑤K (1 − 𝜓 𝑤 ) M7a

𝑔 𝑤 = 𝜖 𝛼𝑓 𝑤 𝐼TUV − 𝑘𝑤K (1 − 𝜓 𝑤 ) M7b

Eggproduction 𝑔� 𝑤 = 𝛼𝑓A 𝑤 𝐼TUV.A − 𝑘A𝑤K 𝜓 𝑤 M8a

𝑔� 𝑤 = 𝜖 𝛼𝑓 𝑤 𝐼TUV − 𝑘𝑤K 𝜓 𝑤 M8b

Boundarycondition

Populationegg

production𝑅L.A =

𝜀2𝑤F

𝑁A 𝑤 𝑔� 𝑤 d𝑤��

)� M9

Recruitment 𝑅A = 𝑅TUV.A𝑅L.A

𝑅TUV.A + 𝑅L.A M10

Boundarycondition 𝑁A 𝑤F 𝑔 𝑤F = 𝑅A M11

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Mortality

Backgroundmortality 𝜇F = 𝑍F𝑊A� M12

Predationmortality 𝜇L.A 𝑤Lwxy = 𝜙𝑤Lwxy𝑤

1 − 𝑓� 𝑤 𝑉� 𝑤 𝜃A�𝑁� 𝑤 d𝑤��

)�� M13

Resourcespectrum

Populationdynamics d𝑁w(𝑤)d𝑡

= 𝑟F𝑤à% 𝜅(𝑤) − 𝑁w(𝑤) − 𝜇L.� 𝑤 𝑁�(𝑤) M14

Carryingcapacity 𝜅 𝑤 = 𝜅w𝑤a3 M15

Trait-basedmodel

Maximum

recruitment

𝑅TUV.A = 𝜅N𝜅 𝛼𝑓Fℎ𝑤F` − 𝑘J𝑤FK 𝑊A

&àbam:n𝑤Faàn𝛥𝑊A M16

Physiological

mortality𝑎 =

𝑓Fℎ𝛼𝑓Fℎ − 𝑘J

𝛽&àba% exp2𝑛 𝑞 − 1 − 𝑞& + 1 𝜎&

2 M17

875

876

877

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878

Table2.Symbolsandvaluesofspecies-independentparametersinthetrait-basedandcommunitymodels.879

880

Symbol Explanation Valueandunita

𝑤 Bodyweight g

𝑊 Asymptoticbodyweight g

𝑁(𝑤) Abundancedensityspectrum numbers/gramb

𝛽 Preferredpredator-preymassratio 100c

𝜎 Widthofsizepreference 1.3d

𝜃 Speciespreference 1

𝑞 Exponentforclearancerate 0.8e

𝑓F Initialfeedinglevel 0.6f

𝛼 Assimilationefficiency 0.6

ℎ Factorformaximumconsumption 20g0.25yr-1g

𝑛 Exponentformaximumconsumption ¾h

𝑘J Factorforstandardmetabolism 2.4g0.25yr-1

𝑝 Exponentforstandardmetabolism ¾h

𝜂 Ratiobetweensizeatmaturationand𝑊 0.25(0.01)i

𝜅N Constantformax.recruitment 1.7g

𝜀 Efficiencyofreproduction 0.1j

𝑤F Eggsize 1mgk

𝑍F Factorforbackgroundmortality 2g0.25yr-1l

𝑧 Exponentforbackgroundmortality -0.25m

𝑟F Resourceproductivity 4g0.25yr-1n

𝑤�� Maximumsizeofresource 1go

𝜅� Resourcecarryingcapacity 1012g-1b,g

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aValuesinparenthesesreferstothecommunitymodel

bTheunitsoftheabundancedensityspectrumcouldalsobeexpressedasaconcentration,i.e.,as

numbers/gram/volumeornumbers/gram/area.Inthatcasetheunitsoftheresourcecarryingcapacity

shouldalsobechangedaccordingly.Further,theunitsoftheclearanceratewouldbecomevolumepertime

orareapertimerespectively.

cUrsin1973;Jenningsetal.2001.

dUrsin(1973)finds𝜎 = 1,buthere𝜎isincreasedtorepresentcross-speciesvariation.Inthefood-web

model𝜎 = 1exceptifspecificknowledgeaboutaspeciesexists.

eAndersenandBeyer,2006.

fAssumesthatfishingeneralarenotsatiated(𝑓F < 1)whilealsohaveasignifantsurplusafterassimilation

andstandardmetabolism,i.e,largerthan𝑘J/(𝛼ℎ) = 0.2.Setting𝑓Finthemiddleoftherange0.2…1gives

0.6.SeealsoHartvigetal.(2011),App.E.

gAdjustedtogivesimilarresultstotheNorthSeamodel(Blanchardetal.2014),despitetheuseofdifferent

exponentfor𝑛;seetextandFigure2.

hSeetext.

iBeverton,1992.

jAndersenandBeyer(2013).NotethatthisdiffersfromtheNorthSeamodel(Blanchardetal.2014),

whereavalueof𝜀 = 1wasused.Thiswill(intheNorthSeamodel)leadtooverestimationsof𝐹T�yfor

individualspecies.

kNeuheimeretal.,2015.

lThisvalueleadstoabackgroundmortalityona10kgindividualof0.2yr-1.Theseindividualswillhave

littlepredationmortalityinthemodel,sobackgroundmortalityisthelargestpart.Seealsodiscussion.

mStandard“metabolic”assumption(Brownetal.2004).

nHartvigetal.,2011.

oSettoincludemesoplankton,suchasshrimps.

881

882

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883

Table3Implementationsofsizespectrummodels.Referencesaregiventowhereaparticular884

modelwasfirstformulatedorcalibratedtospecificsystem,butnottoapplicationsofthemodels.885

Thecolumn“growth”referstothefunctionalresponseusedtocalculategrowthrate.886

Reference Growth Reproduction Resource Densitydependence

Food-webmodels

Halletal.2006,

Rochetetal.2011

Fixed Fixed N.a. Stock-recruitment

Hartvig2011,Hartvig

etal.2011

TypeII Dynamic Dynamic Emergent

Houleetal.2012,

Blanchardetal.2014

TypeII Dynamic Dynamic Stock-recruitment

Rossbergetal.2013 TypeII Dynamic Dynamic Emergent

Trait-basedmodels

Popeetal.2006,2009 Fixed Fixed N.a. Stock-recruitment

relationship

Andersenand

Pedersen2010

TypeII Fixed Dynamic Fixedrecruitment

Houleetal.2013;

Jacobsenetal.2014

TypeII Dynamic Dynamic Stock-recruitment

relationship

MauryandPoggiale

2013

TypeII Dynamic Dynamic Switching

Communitymodels

BenoîtandRochet

2004;Blanchardetal.

2009;Lawetal.2009

(1)

TypeI N.a. Fixed Fixedboundary

condition

Blanchardetal.2011 TypeI& Dynamic Fixed Fixedboundary

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887

(1)NotethatLawetal.(2009)usesaslightlydifferentconservationequationthanthe888

othermodels.889

890

II condition

Mauryetal.2007 TypeII Dynamic Dynamic

Thisarticle TypeII N.a. Dynamic Fixedboundary

condition

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891

892

Figure1.Illustrationofthesizespectrumcalculationofforaspecies(thinblackline)fromtheperspectiveof893

anindividualatagivensize.Allprocessesinsidetheovalaroundthefishareindividuallevelprocesses,with894

darkgreyarrowsrepresentingmassflowsandthelightgreyarrowrepresentingmortalitylosses.Thescaling895

fromindividuallevelprocessesofgrowth,eggproductionandpredationarefacilitatedbytheMcKendric-von896

Foersterequation(eq.1)andtheboundarycondition(M11).Theprocessisiterativewiththesizespectrum897

beingneededtocalculatethepopulation-levelmeasuresofreproductionandrecruitment.Thenumbersrefer898

totheequationsinTable1. 899

EncounterM2andM3

AssimilationM4andM5

RespirationM7

MaturationM6

GrowthM7

Eggprod.M8

ReproductionM9

RecruitmentM10

Sizespectrumeq.1+M11

Suitableprey Predators

Individualsize,w

Biom

assspectrum,B

log(w

)

Offspringsize,w0

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900

Figure2.Solutionsofthecommunitymodel(leftcolumn),thetrait-basedmodel(middlecolumn),theNorth901

Seafood-webmodel(rightcolumn)andtheequilibriumsolution(dashedlines)inanun-fishedsituationasa902

functionofbodymass.Top:Biomasssizespectrainthe“Sheldonscaling”depictingbiomassinlog-spacedsize903

groups,showingcommunityspectrum(thick),backgroundspectrum(thickgrey),speciesspectra(thingrey)904

andthetheoreticaveragespectrumfromtheequilibriumsolution(dashed).Middle:growthrates(solid)905

comparedtothemaximumpossiblegrowthratewhenfedadlibitum(dashed;forthefood-webmodelthisisan906

averageoverallspecies).Bottom:averagepredationmortality,(solid),foreachspecies(grey)andfromthe907

equilibriumsolution(dashed).Notethatinthetrait-basedmodelallasymptoticsizeclassesexperiencethe908

samepredationmortalitysincefeedingisonlybasedonsize.909

910

Community model

108

109

1010

1011

1012

1013Sh

eldo

n sp

ectra

(g)

(a)

10-1101103105

Gro

wth

(g/y

ear)

(d)

10-3 10-1 101 103 1050246

Mor

talit

y (y

ear-1

)

(g)

Trait-based model(b)

(e)

10-3 10-1 101 103 105

Body mass (g)

(h)

Food-web model(c)

(f)

10-3 10-1 101 103 105

(i)

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911

912

Figure3.ResponsetofishingonthecommunityfortheNorthSeafood-webmodel(grey),thetrait-basedmodel913

(solidblack)andthecommunitymodel(dashedblack).Sizespectraareshownrelativetotheunfishedsize914

spectrafromFigure2.a)Responsetofishingallspecieswithatrawl-typeselectivitypattern.Selectiononeach915

speciesstartsat0.05𝑊andthefishingmortalityis0.5yr-1.Inthecommunitymodelallindividualslargerthan916

10garefished.b)Changestothefishedcommunityinpanela)whenforagefishingisremoved,i.e.fishingon917

specieswithanasymptoticsizesmallerthan200g(orindividualssmallerthan200gforthecommunity918

model).919

920

a)

Rel

ativ

e sp

ectra

0.10.20.5

125

10

b)

Body mass (g)10-3 10-1 101 103 105

Rel

ativ

e sp

ectra

0.2

0.512

5

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921

Figure4.Illustrationofthetime-dependentsolutioninthecommunitymodel(twoleft-mostcolumns)andthe922

trait-basedmodel(right-mostcolumn).Toprow:averagebiomassspectra(solid)andrangeofvariation(grey923

patch).Bottomrow:thephase-planebetweenapreyof10gandapredatorwithasizeonepredator-preymass924

larger(𝛽10𝑔)illustratedwithverticaldashedlinesinthetoprow.Leftcolumn:communitymodelwithwidth925

ofsizepreference𝜎 = 1.3.Middlecolumn:communitymodelwith𝜎 = 1.Rightcolumn:trait-basedmodelwith926

𝜎 = 1.OtherparametersasinFigs.2and3,exceptthatthelargestsizeoftheorganismsis1000kg.927

928

10-310-1 101 103 105100102104106108

10101012

Shel

don

spec

trum

(g) (a)

103 105 107 109 101110-210-1100101102103104105106107108

Pred

ator

spe

ctru

m (g

-1) (d)

10-310-1 101 103 105

Body mass (g)

(b)

103 105 107 109 1011

Prey spectrum (g-1)

(d)

10-310-1 101 103 105

(c)

103 105 107 109 1011

(f)

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42

929

Figure5.Asymptoticweightspectrum(extendedSheldonhypothesis),showingtotalnumbersofindividualsin930

logarithmicallyspacedasymptoticweightgroupsinanunfished(grey)andfishedsituation(black)fromthe931

trait-basedmodel.Thesimulationresultsarecomparedtodatafromthreedifferenttrawlsurveys(Daanetal.932

2005).Itisassumedthatthetrawlsurveysonlyretainindividualslargerthan10cm.933

934

935

Figure6.Maximumrecruitment,𝑅�n� ,fortheNorthSeamodel(circles)andthetrait-basedmodelwithlog-936

spacesasymptoticsizegroups(line)andwithasymptoticsizegroupswiththesameasymptoticsizesasinthe937

NorthSeamodel(crosses).938

939

W1 (g)102 103 104 105

Num

bers

106

107

108

109

1010

1011

1012

W1 (g)101 102 103 104 105

Rm

ax (n

umbe

rs/y

r)

109

1010

1011

1012

1013

1014

1015

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Appendicesfor:Thetheoreticalfoundationsforsizespectrummodelsoffishcommunities

AppendixA:derivationsofequilibriumsolutions

Thepredationmortalitycanbederivedfromthecommunityspectrum(eq.3)(AndersenandBeyer

2006):

𝜇" = 𝑎𝐴𝑤'(),(𝐴1)

where𝐴 = 𝛼𝑓1ℎ − 𝑘5and𝑎isthephysiologicalmortality(M16).Mortalityscaleswithexponent𝑛 −

1 ≈ −0.25inaccordancewithempiricalobservations(McGurk1986).

Maximumrecruitment

Themaximumrecruitmentinthetrait-basedmodeliscalculatedusingtheequilibriumsolution

𝑁 𝑤1 from(eq.2).Thissolutionprovidesapredictionfortherelativeabundanceofspecies

dependingontheirasymptoticsize.Therecruitmentneededtosupportthisabundanceof

individualsatsize𝑤1isgivenbytheboundarycondition(M11):

𝑅>[email protected] = 𝜅C𝑔(𝑤1)𝑁 𝑤1,𝑊A Δ𝑊A

where𝛥𝑊A = 𝑊A/𝑊A()isthespacingbetweenasymptoticsizegroups,whichisneededtocome

fromthejointsizeandasymptoticsizedistributionin(1)tothesizedistribution(withoutasymptotic

size)neededfortheboundarycondition.Insertingexpressionsforgrowthandabundanceatsize𝑤1

andassumingthat𝑤1 ≪ 𝑊leadstotheapproximationin(M16).Thedimensionlessconstant𝜅C

determinesthestrengthofearly-lifedensitydependentregulationimposedbythestock-

recruitmentrelationshipbysettingtheratiobetween𝑅>?@and𝜅Thisfreeparametereffectively

determinestheratiobetweenpiscivoryandplanktivoryamongthesmallestfish:ahighervalueof𝜅C

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leadstoamoreabundantfishspectrumcomparedtotheresourcespectrum,andthusahigherthe

levelofpiscivory(Houleetal.2013).Forthesimulationspresentedhere,𝜅C hasbeenmanually

regulatedtoensurethatthepredationmortalityof1gindividualsisontheorderof3year-1(Figure

2).If𝜅C isverylarge,piscivorousmortalityishighandearly-lifedensitydependentregulationis

small,leadingtocompetitiveexclusionbetweenthespeciesgroupsuntilonlytwospeciesareleft,a

smallandalargespecies(HartvigandAndersen2013).

Averagegrowthefficiency

Foranindividualattheasymptoticsizethegrowthrateiszeroandthegrowthefficiencyiszero,

whileajuvenileusesallavailableenergyforgrowth.Theaveragegrowthefficiencyusedinthe

communitymodelrepresentsthedistributionofgrowthefficienciesbetweenindividualswiththe

samesize(Andersenetal.2009).Itcanbecalculatedsimplyfromtheconservationequationat

steadystate:

d𝑔𝑁Kd𝑤

= −𝜇"𝑁K

Insertingthedescriptionofgrowthinthecommunitymodelsas𝑔 𝑤 = 𝛼𝐴𝑤'andthemortality

from(A1)givestheaveragegrowthefficiency:

𝛼 =𝑎

2 + 𝑞 − 2𝑛≈ 0.51

TheentiredistributionofthegrowthefficiencyisderivedinZhangetal.(2012).

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AppendixB:numericalsolutionprocedure

Thesolutionisdiscretizedonalogarithmicgridstartingwiththefirstgridpoint𝑤)settotheeggsize

𝑤1,andthefollowinggridpointsas𝑤N = 𝐶𝑤N(),for𝑗 > 1.The“expansionfactor”𝐶 > 1

determinesthenumberofgridpoints.Foreachspecies𝑖,thegrowthandmortalityarecalculated

usingtheprevioustime-step’ssolution,𝑔AS.Nand𝜇AS.N,with𝑡beingthetimestep,atallgridpoints𝑗.

Thediscretizationschemeisfirst-orderimplicitforthetimederivativeandfirst-orderupwindforthe

mass-derivative:

𝑁A.NSU) − 𝑁A.NS

𝛥𝑡+𝑔A.NS 𝑁A.NSU) − 𝑔A.N()S 𝑁A.N()SU)

𝛥𝑤N= −𝜇A.NS 𝑁A.NSU)

with𝛥𝑤N = 𝑤N − 𝑤N().Thismayberewrittenbycollectingtermsof𝑁:

𝑁A.N()SU) −𝛥𝑡𝛥𝑤N

𝑔A.N()S + 𝑁A.NSU) 1 +𝛥𝑡𝛥𝑤N

𝑔A.NS + 𝜇A.NS = 𝑁A.NS ,

whichbynamingthetermsinthetwoparentheses𝐴A.N and𝐵A.N becomes:

𝑁A.N()SU) 𝐴A.N + 𝑁A.NSU)𝐵A.N = 𝑁A.NS .

Thismayrewrittenasanexplicitexpressionfor𝑁A.NSU):

𝑁A.NSU) =𝑁A.NS − 𝐴A.N𝑁A.N()SU)

𝐵A.N,

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whichcanbesolvedasaniterationoncethespectruminthefirstgridcell,𝑁A.)SU),isknown.This

foundusingtheboundaryconditioneq.(2)togive

𝑁A.)SU) = 𝑁A.)S +𝛥𝑡𝛥𝑤)

𝑅A.

Theequilibriumsolutionconvergeswhenmorethan100gridpointsareused(FigureB1).

FigureB1:Influenceofthenumberofgridpointsonthesolutionofthetrait-basedmodelat

equilibrium.A)ThevalueoftheLargeFishIndicator;B)thepredationmortalityofindividuals

withasizeof10g.

0.4

0.42

0.44

0.46

LF

I

0 100 200 300 400

number of grid points

1.8

2

2.2

M2 (

yr-1

)

A

B

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ReferencesforAppendices

Andersen,K.H.,andBeyer,J.E.2006.Asymptoticsizedeterminesspeciesabundanceinthemarinesizespectrum.Am.Nat.168:54–61.

Andersen,K.H.,Beyer,J.E.,andLundberg,P.2009.Trophicandindividualefficienciesofsize-structuredcommunities.Proc.Biol.Sci.276(1654):109–14.doi:10.1098/rspb.2008.0951.

Hartvig,M.,andAndersen,K.H.2013.Coexistenceofstructuredpopulationswithsize-basedpreyselection.Theor.Popul.Biol.89:24–33.doi:10.1016/j.tpb.2013.07.003.

Houle,J.E.,Andersen,K.H.,Farnsworth,K.D.,andReid,D.G.2013.Emergingasymmetricinteractionsbetweenforageandpredatorfisheriesimposemanagementtrade-offs.J.FishBiol.83(4):890–904.doi:10.1111/jfb.12163.

McGurk,M.D.1986.Naturalmortalityofmarinepelagicfisheggsandlarvae:roleofspatialpatchiness.Mar.Ecol.Prog.Ser.34:227–242.doi:10.3354/meps034227.

Zhang,L.,Thygesen,U.H.,Knudsen,K.,andAndersen,K.H.2012.Traitdiversitypromotesstabilityofcommunitydynamics.Theor.Ecol.6(1):57–69.doi:10.1007/s12080-012-0160-6.

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