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
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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¶meters.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
ℎ𝛾𝑤`a&ab(3)295
296
with𝛷 = 2𝜋𝜎𝛽ba`exp[ 𝑞 − 𝑛 &𝜎&/2)].Thescalingexponent𝑛 − 2 − 𝑞 ≈ −2.05isin297
accordancewithobservations(BoudreauandDickie1992).Theequilibriumsolutionfor298
eachspeciesspectrumis:299
𝑁 𝑤,𝑊 ∝ 𝜅𝑊&`abam:n𝑤a`an 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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Fulton, E.A., Link, J.S., Kaplan, I.C., Savina-Rolland, M., Johnson, P., Ainsworth, C., 688Horne, P., Gorton, R., Gamble, R.J., Smith, A.D.M., and Smith, D.C. 2011. 689Lessons in modelling and management of marine ecosystems: The Atlantis 690experience. Fish Fish. 12(2): 171–188. doi: 10.1111/j.1467-2979.2011.00412.x. 691
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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
= 𝑟F𝑤`a% 𝜅(𝑤) − 𝑁w(𝑤) − 𝜇L.� 𝑤 𝑁�(𝑤) M14
Carryingcapacity 𝜅 𝑤 = 𝜅w𝑤a3 M15
Trait-basedmodel
Maximum
recruitment
𝑅TUV.A = 𝜅N𝜅 𝛼𝑓Fℎ𝑤F` − 𝑘J𝑤FK 𝑊A
&`abam:n𝑤Fa`an𝛥𝑊A M16
Physiological
mortality𝑎 =
𝑓Fℎ𝛼𝑓Fℎ − 𝑘J
𝛽&`aba% 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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39
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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40
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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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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