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PAGE 1 MATHEMATICALMODELINGOFPACIFICPINKSALMON ( ONCORHYNCHUSGORBUSCHA )POPULATIONDYNAMICS BY TANIAHARRISON AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts UnderthesponsorshipofDr.NecmettinYildirim Sarasota,Florida May,2012 PAGE 2 Acknowledgments Thisthesiswouldnothavebeenpossiblewithoutthehelpofmanywonderful people.Iwouldliketothankmythesissponsor,Dr.NecmettinYildirim,forhis dedication,support,andpatiencethroughoutthethesiswritingprocess.Iwouldlike tothankDr.EiriniPoimenidouforgivingmeguidanceduringmyyearsatNew Collegeandforconvincingmetonishmyundergraduateeducationhere. Iamgratefultoallofmyfamilyandfriends.Especiallytomywonderfulparents forgivingmelifeandforteachingmehowtolearn.Tomysister,forgivingmesomuch happiness.ToMelissaCampbellforbeingmyroommateandbestfriendthroughthe upsanddownsofcollegelife.ToAndrewSmallwoodforbeingamarvelousfriend andskipperandforintroducingmetocommercialsalmonshing.Andlastbutnot least,manythankstomyanc,JamelLister,forprovidingmewithlove,courage, andawonderfullife. TaniaHarrison ii PAGE 3 MATHEMATICALMODELINGOFPACIFICPINKSALMON ( ONCORHYNCHUSGORBUSCHA )POPULATIONDYNAMICS TaniaHarrison NewCollegeofFlorida,2012 Abstract Pacicpinksalmon( Oncorhynchusgorbuscha )aresmallanadromousshwith abrieflifecycle.Atagetwotheyreachsexualmaturityandreturntotheirnatal streamstospawnandthendie.Asmanycommunitiesdependonthelargevolume ofpinksalmonreturningtoPrinceWilliamSound,Alaska,itisimportanttodevelop preseasonforecaststoestimatethenumberofsalmonreturningtospawn.Inthis thesiswedevelopanumberofdeterministicdi erentialequationmodelstodescribe thepinksalmonlifecycleandtopredictthesizeoftheadultpopulationreturning eachyear.Thedevelopmentofthemodelsissequential.Eachnewmodeladdsa degreeofcomplexityandimprovestheaccuracy,eitherincapturingrealdataor inrepresentationofthebiologicalsystem.Thee ectsofthespringzooplankton bloomandtheprey-switchinghypothesisonmortalityofjuvenilepinksalmonhave beenwelldocumented.Therefore,weincorporatethesefactorsintoourforecasting models,signicantlyimprovingtheirpredictivecapabilities. NecmettinYildirim iii PAGE 4 Contents 1Introduction1 1.1BiologyofPinkSalmon.......................... 4 1.1.1EggStage(Stage1)........................ 6 1.1.2AlevinStage(Stage2)...................... 7 1.1.3FryStage(Stage3)........................ 7 1.1.4OceanStage(Stage4)...................... 8 1.1.5SpawningStage(Stage5).................... 9 1.2PrinceWilliamSoundandthePinkSalmonHatcheries........ 11 2MathematicalModelingofPinkSalmonPopulationDynamics17 2.1PinkSalmonPopulationData...................... 17 2.2DevelopmentoftheMathematicalModels............... 27 2.2.1TheSalmonLifeCycleinTwoStages.............. 27 2.2.2TheSalmonLifeCycleinTwoStageswiththeE ectsofZooplankton.............................. 30 2.2.3TheSalmonLifeCycleinTwoStageswiththeE ectsofZooplanktonandPollock....................... 42 2.2.4TheSalmonLifeCycleinFiveStageswiththeE ectsofZooplanktonandPollock....................... 47 3Discussion53 Bibliography ....................................... 60 iv PAGE 5 1Introduction Pinksalmon (Oncorhynchusgorbuscha) areanadromousshinhabitingthePacicNorthwest.Thespeciesischaracterizedbyitsstrictlytwoyearlifecycle,large abundance,andsmallsize.LikeotherPacicsalmonoids,pinksalmonperformincrediblemigrationsfromstreamstotheoceanasjuvenilesandbacktotheirnatal streamsasadults.Theyformlargeschools,bothasfryleavingcoastalwatersandas returningadults.ThepopulationsofpinksalmonthatspawninstreamsandestuariesofPrinceWilliamSound,Alaskaformanintegralpartoftheregion'secosystem andareaninvaluableresourcetothelocalcommunities.Juvenilesalmonareprey fornumerousbirds,shes,andmarinemammals.Likewise,thesummersalmonrun ofadultsreturningtospawnprovidesavaluablefoodsourceforlargershesandmarinemammals.PinksalmonhavebeenshedcommerciallybycommunitiesinPrince WilliamSoundsincethelate1800sandtothisday,theindustryremainsoneofthe majorsourcesofincomeforresidents. PreseasonforecastsareusedtopredictthenumberofpinksalmonthatareestimatedtoenterPrinceWilliamSoundeachyear.Biologists,canneries,hatcheries, andshermenusetheseforecastsforplanningandmanagementpurposes.TheAlaska DepartmentofFishandGame(ADFG)usesrunsizepredictionstosetpreliminary harveststrategiesandcanneriesrelyonthemtoestimatemarketpriceandtodeterminethenumberofcanneryworkers,tenders,andprocessingshipsthatwillbe neededfortheupcomingseason.Unfortunately,preseasonforecastsarefrequently unreliable[14].Forexample,indeterminingthepredictionforwildpinksalmonreturnsof2012,noneoftheADFGmodelsproducedameanabsolutepercenterror below100%.Pinksalmonforecastsaredi culttoconstructduetothelargevariabilityinpinksalmonsurvival,theirsensitivitytoenvironmentalchanges,andtheir shortlifecycle. CurrentlytheAlaskaDepartmentofFishandGameusessevendi erentmethods 1 PAGE 6 ofvaryingcomplexitytopredictthewildstockreturns.Eachyear,thesemodels aretestedandtheonewiththelowesterror(usuallymeasuredwithmeanabsolute percenterror)isusedtoproducetheforecast.From1997-1999,theADFGusedlinear regressionsofadultproductionversusbroodyearescapementindexforwildpink salmonforecasting[19].Theescapementindexisobtainedfromweeklyarialsurveys andprovidesanestimateforthenumberofspawningadults[19].Fromthisdata,the numberofemergentfrycanbeapproximated.Adultproductionofwildsalmonis givenbythetotalreturnminusthehatcheryreturn.Distinctmarkingsonhatchery raisedsalmonhelpbiologistsdistinguishwildstockfromthoseofhatcheryorigin.For agivenabroodyear,theADFGusesthelinearregressionstopredicttheabundance ofreturningadults.Thismethodassumesthattrendsinpinksalmonpopulationsare linearandthattheabundanceofbroodstockisdirectlycorrelatedwiththeabundance ofreturningadults.Italsoassumesthatthee ectsofenvironmentalchangescanbe modeledimplicitly.Inorderforthemodeltobeaccurate,environmentalconditions duringtheyearsusedforttingmustbesimilartothoseoffutureyears.Whilelinear regressionforecastingissimple,itisnote ectiveatpredictingsuddenincreasesor decreasesinrunsizes. TheADFGhaveadaptedlinearregressionmodelstoincludeenvironmentalfactors suchasseawatertemperatureandmajorclimaticshifts.Inpractice,thesemodels havebeenrarely,ifever,used.Errorsfromttingthesemodelsareusuallyhigher thanthosefromsimplermodels.Thispoorttingcanresultfromnumerousfactors. Whilemoreelaboratemodelsarebetterabletodescribethesystem,thecomplexity alsoincreasesthenumberofparametervaluesthatmustbedetermined.Furthermore ecologicalrelationshipsbetweensystems,suchaspinksalmonpopulationsandclimate shifts,arenotfullyunderstoodandaredi culttomodel. Haeseker etal. [16]retrospectivelyevaluatedseveraldiscreteforecastingmodels. Thegoalofthisresearchwastodeterminewhichtypesofmodelswerebestableto 2 PAGE 7 capturethedynamicsofadultpopulationsofpinksalmon.Thesevenmodelsranged incomplexityfromabasictimeseriestovariationsoftheRickermodel.Eachmodel wasappliedto43pinksalmonstocksoveranaveragespanof18years.Totestthe accuracyoftheforecasts,eachmodelwasretrospectivelyanalyzedwithrespectto real,historicaldata.ThesimplestmodelwasR(yr-2),atimeseriesmodelinwhich theforecastwastakentobethereturnforthepreviouscorrespondingbroodyear plusaresidualerrorterm.Thisnavemodelfollowedgeneraltrendsinoddandeven yearlineagesofapinksalmonpopulation,yetitwasnotbeabletopredictsudden changes.Themostadvancedmodel,KF,wasanupdatedRickermodelinwhich theintrinsicgrowthrateconstant"a"wasreplacedwithaKalmanlter.AKalman lterisarecursivealgorithmthatusesweightedaveragesofpreviouslyestimated valuestoproducenewestimates.ThereforethegrowthrateintheRickermodelwas permittedtovaryovertime.Thisenabledthemodeltotracklargescalechanges insalmonreturns.Thetwomodelswiththeleastcorrelationinrawerrorswerethe R(yr-2)modelandtheKFmodel[16].Thus,thesemodelswerecombined,forming aneighthmodel,KF+R(yr-2).Theresultsofthisstudyindicatedthatalthoughno modelwasmostsuitableforexplainingallpinksalmonstocks,navemodelswithfew parametersperformedbetterthanexpected.Morecomplicatedmodelssuchasthe KF+R(yr-2)wereoveralllesssuccessfulthanthebasicRickermodel.Thiswas,in part,duetothefactthatpinksalmonreturnsdemonstratelargeuctuationsona yeartoyearbasisratherthanonalongtermbasis[16]. Inmostofthemodelsdiscussedabove,environmentalconditionswereincluded implicitly.Willette etal .[2]conductedastudyontheenvironmentalfactorsa ecting survivalofjuvenilehatcherypinksalmon.Thisresearch,conductedoveraspanofve years,wasintendedtodeterminemajorpredatorsofjuvenilesalmonandtoinvestigatethreehypotheses:(i)alternativepreyforpredators ( prey-switchinghypothesis); (ii)salmonforagingbehavior(refuge-dispersionhypothesis);and(iii)salmonsizeand 3 PAGE 8 growth(size-refugehypothesis)[2].Theresearchconsistedofacombinationofeld studies,dataanalysis,andmathematicalmodeling.Itwasdeterminedthatinordertounderstandthemortalityofjuvenilesalmon,bothtop-downe ects(e ectsof highertrophiclevels)andbottom-up(e ectsoflowertrophiclevels)mustbeconsidered.Theprimarypredatorsofjuvenilepinksalmonarefacultativeplanktivoresand piscivores.Herringandwalleyepollockarethemostsignicantpredatorsduetotheir largenumbers.Althoughtheyprefertofeedonzooplankton,whentheabundanceof zooplanktonislow,theywillswitchtofeedingonpinksalmonfry.Thisphenomenon isknownasthe prey-switchinghypothesis .Therefuge-dispersionhypothesisdescribes pinksalmondispersionfromnearshorehabitats.Asjuvenilesmature,theymoveo shoreinsearchofzooplanktonprey.Inthiso shoreenvironmentjuvenilepinksalmon aremorevulnerabletopredation,yetthepredationonfryisinverselyproportional totheirsize.Thereforetheriskofpredationmustbeweighedagainstthebenetsof anabundanceoffoodandlargerbodysize.Thesize-refugehypothesisfocusesonthe relationshipbetweenthesizeofjuvenilesalmonandtheirvulnerabilitytopredators. Whilepredationonfryisgenerallysizedependent,thisrelationshipvariesdepending onthetypesofpredators. Beforedevelopingmathematicalmodelsforthepinksalmonlifecycle,wemust rstunderstandthebiologyofthespeciesandtheecologyoftheenvironmentthat itinhabits.Inthisthesiswedevelopfourdeterministicmathematicalmodelsof increasingcomplexityinordertopredictthesizeoftheadultpopulationofpink salmonreturningtoPrinceWilliamSoundeachyear. 1.1BiologyofPinkSalmon Pinksalmon (Oncorhynchusgorbuscha) arethemostabundantofthevePacic salmonspeciesandtheyhavetheshortestlifespan.Adultsalmondieshortlyafter spawning.Asaresult,oddandevenyeargenerationsarealmostcompletelyisolated 4 PAGE 9 fromoneanother,thuspreventinggeneow[17].Theirlifecyclecanbedividedinto vestages:egg,alevin,fry,ocean,andspawning.Thegurebelowgivesatimeline forthesalmonlifecycle. Sept Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Life Cycle of Prince William Sound Pink Salmon Ocean Stage Return to Natal Streams to Spawn and Die Egg Stage Alevin Fry Stage Migrate to Ocean Figure1:ThechartaboveshowsthetwoyearlifecycleofPrinceWilliamSoundPink Salmon.Notethatthereissignicantvariabilityinthedurationofeachstage.The coloredsectionsmerelymarkestimatesthatcanvarybyamonthormore. 5 PAGE 10 1.1.1EggStage(Stage1) Femalesdigshallownestsingravelstreambedsandlayroughly1500-2000 eggsinseveralclutches[1].Theeggsarefertilizedbyatleastonemaleandthen buriedunderathinlayerofgravelwithaickofthefemale'stail.Soonafter,both parentsdie.Theeggsremainunderthegravelshelter,knownasa"redd",from falluntilwinter.Itisimperativethattheeggsremainundisturbed,abovefreezing, oxygenated,andsurroundedbyanadequateowofwater[1].Inthisstage,salmon areextremelyvulnerable.Evenaslightdisturbanceintheenvironmentcankill thousandsofeggs.Mortalityduetopredationisalsohigh.Astheeggsenterthe eyedstage(asseeninFigure(2)below)thesalmonbegintodevelopspecialstructures inthehead,knownasotoliths.Theseareneededforbalanceandauditorysenses[18]. Theyforminlayersmadeupofproteinandcalciumcarbonate[5].Thethicknessand spacingofthelayersvarydependingonwatertemperature[5].Otoliths,muchlike theringsintreetrunks,canprovideinformationaboutpinksalmonpopulationsand theenvironmentalconditionspresentduringtheirearlylifestages. Figure2:Pinksalmoneggsareshowninthegureabove.Notethattheeggsinthis photographhavealreadydevelopedeyes.Theyhavereachedwhatisknownasthe eyed-stage,whichtypicallyoccursinOctoberofeachyear.Thephotographisfrom theUSFish&WildlifeServicewebsite,"CyberSalmon"[12]. 6 PAGE 11 1.1.2AlevinStage(Stage2) TheeggshatchbetweenDecemberandFebruaryofeachyear[1].Theexact timeframeisprimarilydependentonwatertemperature.Alevinareabout2.5cm long,translucent,andhavealargeyolksacattachedtotheirbellies.Asketchofan alevinisgiveninFigure(3).Theyremaininthegravelreddsuntilspring,feeding exclusivelyo oftheiryolksac.Asintheeggstage,alevinhaveahighmortalitydue tochangesintheirenvironmentandpredation. Figure3:Thegureaboveisasketchofapinksalmonalevin.Whenalevinemerge fromeggsinwinter,theyareextremelysensitivetolightandremainhiddenbeneath thegravelduringthisstage[5].ThedrawingisfromtheUSFish&WildlifeService website,"CyberSalmon"[12]. 1.1.3FryStage(Stage3) Oncetheyolksachasbeenabsorbed,salmonenterthethirdphaseoftheirlife cycle,thefrystage.Theybeginmigratingdownstreaminearlyspring.Fryleave thereddsatnightand,untiltheybegintoformschools,theyavoidlight[1].Most frydonoteatonthejourneydownstream,howeversomehavebeenobservedto occasionallypreyonsmallaquaticinsectsandlarvae.PinksalmoninPrinceWilliam Soundgenerallydonotspawnfarupstream,thusthejourneyforthefryisrelatively short.Duringtheirexodus,juvenilesalmonimprintontheirnatalstreams.Little isknownaboutsalmonimprinting,butitisbelievedtobeolfactoryandmayoccur multipletimesalongasalmon'sjourneytotheocean. Fortherstfewmonths,fryremainclosetoshore,migratingseawardinbursts. ThespringzooplanktonbloominPrinceWilliamSoundisextremelyimportanttothe survivalofpinksalmonduringthisstage.Itisavaluablesourceoffoodand,bythe 7 PAGE 12 prey-switchinghypothesis,o ersprotectionfromfacultativeplanktivorepredators[2, 17].Asjuvenilesalmonmature,theymigratefurtherfromPrinceWilliamSound.By latesummerorearlyfalltheyare6-12mileso shore[1].Asketchofapinksalmon fryisgiveninFigure(4). Figure4:Apinksalmonfryisshownabove.Whenthefryemerge,theyrangein lengthfrom3.2-3.8cmandweighabout0.3g[1].Theygrowrapidlyoverthenext severalweeks,increasinginweightbysixfold[1].ThedrawingisfromtheUSFish &WildlifeServicewebsite,"CyberSalmon"[12]. 1.1.4OceanStage(Stage4) Adultsalmonremainatseauntiltheyarereadytoreturntotheirnatalstreams. Oceanstagesalmonarebrightsilverwithagreenishblueback.Theirbacksandtails arespottedwithblackandthesespotsbecomemorepronouncedasthesalmonage. Figure(5)showsadrawingofanoceanstagepinksalmon.Duringthisstage,pink salmonmigrationpatternsvarywidelybetweenpopulationsandregions[1].Theirdiet consistsofplankton,smallcrustaceans,crustaceanlarvae,squid,andsmallshes. Relativelylittleisknownaboutthehabitsofoceanstagepinksalmon.Theyare di culttotrackduetotheirlargenumbers,farrangingmigrations,highmortality, andsmallsize.Whilenumerousspeciesofseals,dolphins,andwhalespreyonpink salmon,theratesofpredationarehardtoestimateasdistinguishingpinksalmon fromothersalmonspeciesinstomachcontentsisdi cult[17]. 8 PAGE 13 Figure5:Anoceanstagepinksalmonisshowninthisgure.Thesalmoncontinueto growthroughouttheiroceanresidencetoanaverageweightof2-4kg[1].Thedrawing isfromtheUSFish&WildlifeServicewebsite,"CyberSalmon"[12]. 1.1.5SpawningStage(Stage5) Atagetwo,salmonreachsexualmaturityandbeginreturningtocoastalwaters. Whentheyre-enterPrinceWilliamSound,apercentageofthepopulationisharvested forthecommercialshery.Thosethatmakeitpassedtheshingnetsandreach protectedbaysandstreamsmakeupwhatisknownas"escapement".Aerialsurveys andsonarsh-countersplacedinmajorstreamsareusedtoestimateescapement. Forhatcheryraisedsalmon,aseparateharvestisperformedatthebeginningofthe seasonforcostrecoveryandattheendoftheseasonforbroodstock.Costrecovery sharesoldtoprocessorstoraisemoneyforthehatcheriesandbroodstockprovides eggsandmiltforthenextgenerationofsalmon. Aspinksalmonenterthespawningstagetheyundergoseveralphysicalchanges[17]. Whennearingfreshwater,salmon'sbacksdarkenandtheirsilverscalesdull.Females turnolivegreenwithsplotchesofdarkercoloralongtheirsides.Malesoftendevelop hookednosesandlongerjaws.Bothsexesdevelophumpsontheirbacks,asseenin Figure(6). 9 PAGE 14 Figure6:Thegureaboveshowsamalepinksalmoninthespawningstage.The femaleshavealesspronouncedhumpandareusuallysmaller.Thedrawingisfrom theUSFish&WildlifeServicewebsite,"CyberSalmon"[12]. Aspinksalmonmigratebacktotheirnatalstreams,theirconsumptionoffood decreases.Ithasbeenhypothesizedthatadultpinksalmonre-enteringPrinceWilliam Soundmayoccasionallypreyonemergingjuvenilesalmon[3].Thee ectsofthis phenomenonareunknown.However,givenaverylargereturn,predationbyreturning salmoncouldpossiblybesignicant[3].Uponenteringfreshwater,pinksalmoncease tofeed[1].Theirsensesweakenandasaconsequence;theyswimveryclosetothe shoreline,nearlytouchingtherocks.Females'internalorgansatrophyaseggsacs growlargeintheirplace[17].Thephysicalchangesduetotheirreturntofreshwater aredrasticandbythetimesalmonspawn,theyarenearlydead. Littleisknownabouthowsalmonperformtheirmigrationbacktotheirnatal streams.Likelyfactorsincludeolfactorysenses,seawatersalinitylevels,watertemperature,andcurrents[1,3].Ofthereturningpopulation,malesusuallycomprisethe largestpercentageofearlyrecruits.Thus,estimatesoffemalepercentageareoften usedtoindicatewhenarunisnearlyover.Pinksalmonmigratetonearshorewaters, schoolatthemouthsofstreams,andeventuallyndtheirwaytotheirnatalspawning ground.UnlikethelengthymigrationsofotherPacicsalmon,PrinceWilliamSound pinksalmondonotventurefarupstreambeforespawning. 10 PAGE 15 Pink salmon return to their natal streams i n the late summer and early fall to spawn, and die. Salmon eggs incubate in gravel nests until they hatch, during the winter months. Alevin remain beneath the gravel until early spring Fry leave the gravel nests and travel downstream to Prin ce William Sound, where they remain until late summer Juvenile salmon leave Prince William Sound and swim to the ocean. At age two pink salmon reach sexual maturity and migrate back to Prince William Sound the in summer. Life Cycle for Prince William Sound Pink Salmon Figure7:Thediagramabovegivesasummaryofthemajoreventsineachlifestage ofPrinceWilliamSoundpinksalmon. 1.2PrinceWilliamSoundandthePinkSalmonHatcheries PrinceWilliamSoundisarichbodyofwateronthesoutherncoastofAlaska.It isafjord-typeenvironmentwithasubarcticclimate[4].Asitisonthenorthernedge ofacoastaltemperaterainforest,theregionisalsocharacterizedbyheavyprecipitationandauniqueecologicalinteractionbetweenfreshwater,estuarine,andsaltwater environments[15]. 11 PAGE 16 Figure8:ThegureaboveshowsPrinceWilliamSoundinrelationtotherestof NorthAmerica.TheimagewasgeneratedusingGoogleEarthsoftware[22]. PrinceWilliamSoundisdividedintoelevengeographicaldistricts:Unakwik, Coghill,Eshamy,Northern,Northwestern,Eastern,Southwestern,Southeastern,Montague,CopperRiver,andBeringRiver.TheseareseenbelowinFigure(9). 12 PAGE 17 Figure9:AboveisamapofPrinceWilliamSound,Alaska.Theblacklineindicates theboundarybetweenPrinceWilliamSoundandtheGulfofAlaska.Thepurple linesdenetheboundariesoftheelevendistricts.Thecoloredmarkersindicatethe locationofthePWSACpinksalmonhatcheries.RedmarkstheWallyNoerenberg Hatchery,bluemarkstheCanneryCreekHatchery,andgreenmarkstheArminF. Koernighatchery.ThismapwasobtainedfromtheAlaskaDepartmentofFishand Gamewebsite[13]. Theshingindustryprovidesamajorsourceofincomeandawayoflifeforthe residentsofPrinceWilliamSound.Intheearly1970s,wildstocksofpinksalmon wereseverelydepletedandtheshingindustrywasstruggling.Lowreturnsduring theseyearswerepossiblycausedbymismanagementandovershing[6].Atthistime, thePrinceWilliamSoundAquacultureCorporation(PWSAC)wasfoundedandthe proposaltointroducenon-prothatcheriestotheregionwasaccepted[5].In1976, theArminF.Koernighatchery(AFK),locatedinSawmillBayintheSouthwestern District,wasthersthatcherytoreleasefry.TheAlaskaDepartmentofFishand GameFisheriesRehabilitation,Enhancement,andDevelopmentDivisionfounded theCanneryCreekHatchery(CCH)in1978inUnakwikInlet.Tenyearslaterthe hatcherywastakenoverbyPWSAC.In1985,theWallyNoerenbergHatchery(WNH) 13 PAGE 18 wasfoundedinLakeBayintheCoghilldistrict[5].CurrentlyAFK,CCH,andWNH arealloperatedbyPWSAC.TheValdezFisheriesDevelopmentAssociation(VFDA) runstheSolomonGultchHatcheryinValdez.Thishatchery,locatedintheEastern district,wasfoundedin1982[5].Severalotherhatcheriesforcoho,sockeye,andchum salmonexistinthisregion. HatcheriesboostthenumberofpinksalmonreturningtoPrinceWilliamSound bycreatingasanctuaryforeggsandalevin.Protectionduringthersttwostages ofasalmon'slifecyclegreatlyincreasestheprobabilityofsurvivalofanindividual egg.Duringtheeggstage,watertemperatureintheincubationtankiscarefully adjustedtoproduceaspecialpatterninthesalmonotoliths.Eachhatcheryhas itsownuniquepatternofrings,makingitpossibletodetermineapinksalmon's origin.Thesemarkingsarealsouseddistinguishwildsalmonfromthoseofhatchery stock.Theyareaninvaluabletoolinthemanagementofhatcherystocksandinthe protectionofwildpopulations. Figure10:Thegureaboveshowstwodi erentpinksalmonotoliths.Thegureon theleftshowstheotolithofasalmonfromtheCanneryCreekHatchery.Thegure ontherightistheotolithofasalmonfromtheWallyNoerenbergHatchery.These imagesarefromthePrinceWilliamSoundAquacultureCorporationwebsite[5]. Pinksalmonarereareduntiltheyreachthefrystage.Theyareoftenheldin pensnearthehatcheriesuntilthefryarereadytobereleasedintotheSound(see Figure(11))[5]. 14 PAGE 19 Figure11:Thegureshowspensforraisingsalmonfry.Thephotographisfromthe PrinceWilliamSoundAquacultureCorporationwebsite[5]. Wildsalmonnaturallyhavetheabilitytooptimizethetimingoftheirentryinto PrinceWilliamSoundtocoincidewiththezooplanktonbloom[11].Ontheother hand,thetimingofthereleaseofhatchery-raisedfryisdeterminedbyaveragebody sizeandestimatesofzooplanktonabundance.Anindexofzooplanktonabundanceis approximatedfromnetsamplestakennearthehatcheries[23,4].Whenlevelsappear toapproachapeak,thefryarereleasedeitheratthehatcheryorremotelyintoother partsoftheSound.Inaremoterelease,fryareplacedintanksofwatertakenfrom thefry'sdesiredfuturespawningground.Theyarethentransportedtoarelease location.Therearefewremotereleasesperformedbecausereturningsalmonoften stray.Thisislikelyduetoimproperimprinting[3]. Thehatcherieshavehadasubstantiale ectonthesalmonsheryinPrince WilliamSound.Therunsizeshaveincreasedsignicantlysincetheywererstintroducedandinrecentyears60-80%ofyearlypinksalmonreturnshavebeenofhatchery origin[6].Despitethesuccesstotheshery,somebiologistsarguethatthehatcheries havecreatedunnaturallylargesalmonrunsthatdraintheecosystemandinterfere withwildstocks.However,thetotale ectsofthehatcheriesontheecologyofPrince 15 PAGE 20 WilliamSoundandonthewildpinksalmonpopulationsareundetermined. 16 PAGE 21 2MathematicalModelingofPinkSalmonPopulationDynamics Forhatcherypinksalmon,theinitialtwomonthsafterreleaseisaparticularly vulnerablestageintheirlifecycle.Survivalduringthisperiodcanbeindicativeofthe sizeofthereturningadultpopulation[2,24].Therefore,itislogicaltoseparatethe populationofpinksalmonintotwostages:fryandadult.Heretheterm"adult"refers totheocean-stageofthesalmonlifecycle.Itdoesnotcorrelatewithsexualmaturity, whichisreachedafterabouttwoyears.Thedeathratefortheadultstageistakento beconstantasthereislittledataavailableforoceanstagepinksalmon.Becausethe frystagehasthehighestdeathrate,wefocusedondeterminingthefactorsa ecting juvenilesalmonmortality.Indevelopingthemodels,aseparatesetofparametersis chosenforoddandevenyearsateachhatchery.Becausethehatcheriesarelocated indi erentregionsofPrinceWilliamSound,thelengthofafry'sjourneytothe oceancanvary.Forexample,fryfromCanneryCreekHatcherytravelmuchfurther thanthosefromArminF.KoernigHatchery.Asoddandevenyearpinksalmon aregeneticallydistinct,itispossiblethattheirpopulationsexhibitdi erentsurvival rates. Modelsimulationsareproducedin Matlab bynumericallysolvingsystemsofdifferentialequations.Inallofthesimulations,timeismeasuredinmonthsafterthe fryrelease,timet=0.Theadultreturnissampledattimet=15,theestimated numberofmonthsafterreleaseforsalmontoreturntoPrinceWilliamSound(see Figure(1)).Thedataisenteredinto Matlab fromthePWSACdatabaseandthen normalizedbydividingbothreturnandreleasebythereleasepopulationsize. 2.1PinkSalmonPopulationData Currentandhistoricaldataforhatcherypinksalmonstockshasbeencom17 PAGE 22 piledbythePrinceWilliamSoundAquacultureCorporation(PWSAC)[5].While somedataisavailablefromthemid1970s,inthisthesisalldataistakenfromat most1980topresent.Usingmorerecentdatainthemodelsbetterreectsthe changesthathavetakenplaceoverthepastfewdecades.Forexample,thevolume ofshreleasedbythehatcherieshasgreatlyincreased,therearemorecommercial shingvesselsinPrinceWilliamSound,andthevesselshavebecomelargerandmore e cient.Inaddition,usingdatafromthepast20yearsbetterrepresentsrecentclimaticandoceanicshifts.Studiesofhistoricalclimatedatahaveshownthatthere havebeenmajorshiftsinclimateandoceanicconditionsin1925,1947,1977,and 1989[20].Whiletheseregimeshiftshaveyettobefullyexplained,theymaybedue tothePacicDecadalOscillation,aclimaticpatternofwarmingandcooling.The phasescanlast20-30yearsandmayhavesubstantiale ectsontheecosystemsofthe NorthernPacic[21]. ThedatausedinthisthesisisobtainedfromthreeofthePWSAChatcheries.The SolomonGultchhatchery(runbytheVFDA)isnottakenintoconsiderationasit doesnotpublishreleaseandreturndata.Forallyearssince1980,thenumberoffry releasedbythehatcheryandthenumberofadultsreturninghavebeenreportedfor eachofthePWSAChatcheries.Thereleaseandreturnpopulationsizesforthethree PWSAChatcheries(AFK,WNH,andCCH)areplottedinFigure(12),Figure(14), andFigure(16),respectively.NotethattheWallyNoerenbergHatchery(WNH)was foundedin1985,thusforyears1980through1984nodataisavailable.Inorderto determinehowtoscalethedata,weexaminedthereleaseandreturndataforeach ofthehatcheries. 18 PAGE 23 5 10 15 20 25 0 2 4 6 8 10 12 14 16 18 20 Population size [fry release] (x10 7 ) Population size [adult return] (x10 6 ) 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 Figure12:Thegureaboveshowsthepopulationsizeofpinksalmonreleasedfrom theAFKhatcheryversusthepopulationsizeofreturningadults.Eachdatapointis labeledwithitscorrespondingbroodyear. 19 PAGE 24 0 5 10 15 20 0 2 4 6 8 10 12 14 16 Populaton size [fry release] (x10 7 ) Populaton size [adult return] (x10 6 ) y =0 45 x +0 4 Figure13:ThisgureshowsalineartofthedatafromAFKhatchery.Thelinear tisgivenbytheequation y =0 45 x +0 4 .Thedatapointsplottedinthisgure aboveareexactlythoseinFigure(12). FromFigure(13),itisclearthatthereisapositivecorrelationbetweenthenumber offryreleasedandthenumberofadultsreturningovertheyears.However,this correlationdoesnotnecessarilyholdifwedividethegraphintogroupsbasedon thesizeofreleaseandexaminethevariabilitywithineachgroup.Forexample,we canpartitionthereleasesintothreegroups,low(from5 10 7 to10 10 7 ),average (from10 10 7 to15 10 7 ),andhigh(from15 10 7 to20 10 7 ).Inthelowrangewe seeanegativecorrelationbetweenthesizeofthereleaseandthesizeofthereturn. Thereisapositivecorrelationinboththeaverageandhighreleasegroups.However, thereisalsohighvariabilityinreturnsforthesethesereleasesizes.Whilethereisa 20 PAGE 25 weakcorrelationbetweenthesizesofthefryandadultpopulations,thereareclearly othermajorfactorsinvolvedindeterminingthesizeofthereturningadultpopulation. Thesefactorsmayincludeenvironmentalconditionsandavailabilityoffood.From Figure(12)wecanseethatmostoftheextremelylowreturnstotheAFKhatchery occurredduringtheearly1990s.Thesepoorreturnswerepossiblyaresultofthe ExxonValdezoilspillof1989thatimpactedthePrinceWilliamSoundecosystemfor severalyearsaftertheincident[25].However,researchinvestigatingthee ectsofthe oilspillonpinksalmonisoftenconictingandtheissueiscontroversial[25]. 0 5 10 15 20 25 0 2 4 6 8 10 12 14 16 18 20 Population size [fry release] (x10 7 ) Population size [adult return] (x10 6 ) 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 1995 Figure14:Thegureaboveshowsthepopulationsizeofpinksalmonreleasedfrom WNHplottedagainstthepopulationsizeofreturningadults.Eachdatapointis labeledwithitscorrespondingbroodyear. 21 PAGE 26 0 5 10 15 20 25 0 2 4 6 8 10 12 14 16 18 20 Population size [fry release] (x10 7 ) Population size [adult return] (x10 6 ) y =0 0014 x +6 7 Figure15:ThisgureshowsalineartappliedtotheWNHdata,seeFigure(14). Thelineartisgivenbytheequation y =0 0014 x +6 7 ThelineartintheplotshowninFigure(15)hasaslopeof0.0014.Therefore thecorrelationbetweenthesizeofthefryreleaseandthesizeoftheadultreturnis statisticallyinsignicant.FromFigure(14),weobservethatthethreelowestreturn yearswere1990,1991,and1993.Howeverreleasesizeswerelargerthanaverage.This indicatesthatperhapsecologicalconditionsduringthoseyearsloweredthecarrying capacityforjuvenilesalmoninPrinceWilliamSound.Asaresult,therelativelylarge releasessu eredhighmortalityandproducedasmallreturningadultpopulation.The poorreturnsforthoseyearsareconsistentwithcorrespondingdatafromtheAFK hatchery. 22 PAGE 27 0 5 10 15 20 25 0 2 4 6 8 10 12 14 16 18 20 Population size [fry release] (x10 7 ) Population size [adult return] (x10 6 ) 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 Figure16:Thegureaboveshowsthepopulationsizeofpinksalmonreleasedfrom CCHasafunctionofthepopulationsizeofreturningadults.Eachdatapointis labeledwithitscorrespondingbroodyear. 23 PAGE 28 0 5 10 15 20 25 0 2 4 6 8 10 12 14 16 18 20 Population size [fry release] (x10 7 ) Population size [adult return] (x10 6 ) y =0 47 x 0 21 Figure17:Inthisgurealineartwastakenofthefryreleaseandadultreturndata fortheCCH.Thelineartisgivenbytheequation y =0 47 x 0 21 .Thedatapoints plottedinthegureaboveareexactlythoseinFigure(16). AsseeninFigure(17),theCanneryCreekHatcherydatashowspositivecorrelation betweenreleaseandreturnpopulationsizesofpinksalmon.Forfrypopulations smallerthan 10 10 7 ,thedatapointsarewellmatchedbythelineart.Forfry populationslargerthan 10 10 7 ,thedatapointsarescatteredinaverticaldistribution thatrangesfromabout 1 10 6 to 14 10 6 .Thusforreleasepopulationsof 10 10 7 to 15 10 7 ,thecorrelationbetweenreleaseandreturnisnotevident.Thesereleases areroughlythesamesize,yettheycorrespondwithtwodistinctgroupsofreturns: high(6 10 6 to14 10 6 )andlow( 1 10 6 to 4 10 6 ).FromFigure(16)weseethat thedatapointswithstrongpositivecorrelationconsistprimarilyofthehatchery's earlyyears:broodyears1980-1990(notethatthe1993releasealsofallswithinthis 24 PAGE 29 category).ThesereleaseswerethesmallestfromanyPWSAChatcheryinPrince WilliamSound.Itislikelythatduringthe1980sthenumberoffryreleasedwas consistentlybelowcarryingcapacity.Thus,increasingthenumberoffryproduced higheradultreturns.OnceCCHbeganproducingmorethan 10 7 fry,itislikelythat thesizeofthereturningadultpopulationbecamemoredependentonenvironmental factorsthanonthethesizeoftherelease.NotethatCCHproduceslessfrythan eitherWNHorAFK.ItisalsolocatedfurthestfromtheGulfofAlaska(seeFigure(9)), givingtheCCHfryalongerjourney.Itisunknownifthesefactorsarethecausefor thedistinctrelease/returnpatternfromthishatchery. ForWNHthevariationinthenumberoffryreleaseddoesnothavesignicant e ectsonsurvivalrates.CCHandAFKshowslightlymorecorrelation,yettheerrors fromthelinearttingarelarge.Thusfromanalysisofthehatcherydata,itisreasonabletoassumethatthenumberoffryreleaseddoesnothaveaconsistentlysignicant e ectonthenumberofreturningadults.Forsimplicity,alloftherelease/returndata isnormalizedbydividingthenumberofreturningadultsbythenumberoffryreleased.Byexaminingpercentreturnweareabletoconsideronlysurvivalrateand ignorediscrepanciescausedbydi erencesinthenumberoffryreleased. 25 PAGE 30 Table1:ThistableshowsPWSACyearlypinksalmonpercentagereturnsforbrood years1980-2007.Theaveragepercentreturnandstandarddeviationaregiveninthe linebeloweachhatchery'sabbreviatedname.Theredentriesindicatehigherthan averagereturns.Theblueentriesindicatelowerthanaveragereturns.WNHentries foryears1980-1984areNaNasthehatcherywasnotfoundeduntil1985.Thisdata wasobtainedfromPWSAC[5]. BY RY AFKreturn WNHreturn CCHreturn Total 4.99 2.82 5.80 3.81 4.39 2.41 5.06 3.01 1980 1982 7.36 NaN 3.59 5.48 1981 1983 5.31 NaN 3.37 4.34 1982 1984 3.20 NaN 5.15 4.18 1983 1985 6.55 NaN 8.31 7.43 1984 1986 4.79 NaN 2.34 3.57 1985 1987 6.77 8.75 3.83 6.45 1986 1988 5.23 5.09 0.53 3.62 1987 1989 2.46 2.98 5.99 3.81 1988 1990 4.31 8.57 4.46 5.78 1989 1991 4.97 5.18 6.22 5.46 1990 1992 2.13 0.98 1.21 1.44 1991 1993 1.41 0.92 0.63 0.99 1992 1994 1.57 3.54 6.75 3.95 1993 1995 1.06 1.51 6.20 2.92 1994 1996 1.63 3.15 5.14 3.31 1995 1997 6.42 3.65 4.30 4.79 1996 1998 14.05 8.03 5.35 9.14 1997 1999 8.25 9.13 5.87 7.75 1998 2000 5.19 6.57 4.96 5.57 1999 2001 3.37 6.19 1.60 3.72 2000 2002 5.16 4.40 1.14 3.57 2001 2003 4.53 16.80 5.98 9.10 2002 2004 3.57 2.26 2.04 2.62 2003 2005 5.80 8.36 9.90 8.02 2004 2006 3.97 4.84 2.30 3.70 2005 2007 9.87 8.89 5.38 8.05 2006 2008 3.41 11.32 7.80 7.51 2007 2009 7.43 2.38 2.53 4.11 26 PAGE 31 2.2DevelopmentoftheMathematicalModels 2.2.1TheSalmonLifeCycleinTwoStages ThepinksalmonpopulationinPrinceWilliamSoundcanbemodeledbyan extremelysimple,two-dimensionalsystemofordinarydi erentialequations.This modelassumesaconstantdeathrateforeachofthetwolifestages:juvenileand adult. F A Figure18:Adiagramshowingthepinksalmonlifecycleintwostages.Thefry populationdecayseitherbydeathorbymaturationtotheadultstage.Theadult populationdecaysonlybydeath. FromthediagraminFigure(18),wecanconstructthefollowingmodel. dF dt = ( k 1 + k 2 ) F (1) dA dt = k 2 F k 3 A Inthismodel,Eq.(1), F representsthefrypopulationand A representstheadult population.Thefryeitherdieo duetoaconstantdeathrate k 1 ormatureata constantrate k 2 totheadultstage.Thepopulationsurvivingtotheadultstage dieataconstantrate k 3 .Asthefrystageshouldspantheinitial60daysofmarine residence,theparametersforfrydeathrateandfrymaturationratemustbebounded toreectthis.Attime t =2 ,wetakethepopulationoffry F tobe1%oftheinitial population F 0 .Thedi erentialequationforthefrypopulationislinear,separable, 27 PAGE 32 andindependentofthedi erentialequationfortheadultpopulation.Therefore,we caneasilycalculatethesolutionforthefrypopulation(seeEq.(2)below). F = F 0 exp ( k 1 + k 2 ) t (2) Solvingthisequationfor( k 1 + k 2 )yields: k 1 + k 2 = ln(100) 2 2 3 Thisprovidesanumericalestimateforthecombinedvalueeofparameters k 1 and k 2 Inadditionthedeathrateconstantforfry, k 1 ,mustbelargerthanthatforadults, k 3 Forttingthismodeltorealdata,weuse Matlab's built-infunction lsqcurvefit todetermineoptimalvaluesfortheparameters k 1 k 2 ,and k 3 .Thefunction lsqcurvefit isusedforsolvingnonlinearleastsquaresproblems,particularlydata-tting.Givena dataset,afunction,andasetofinitialestimatesforparametervalues, lsqcurvefit choosesparametervaluestominimizetheleastsquareerrorsbetweenthefunction andthegivendata.Forthismodel,asetofinitialparametersisrandomlychosenbycalling Matlab's built-inpseudorandomnumbergenerator, rand .Theinput functionfor lsqcurvefit isthesolutiontoEq.(1)(obtainedusing ode15s ),andthe realdataisgivenbyoddandevenyearreturnsforeachhatchery.Thedi erential equationsolver ode15s isabuilt-in Matlab functionforsolvingsystemsofordinary di erentialequationsusingthebackwarddi erentiationformula,animplicit,linear, andmultistepmethod. FortheevenyearreturnstotheAFKhatchery,theparametersarechosentobe k 1 =1 2950 k 2 =1 4449 ,and k 3 =0 1656 28 PAGE 33 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 time (months) pink salmon population percentage fry adult Figure19:ThegureaboveshowsthesolutionofEq.(1)forparametervalues k 1 = 1 2950 k 2 =1 4449 ,and k 3 =0 1656 andrealdata.Thecircularmarkersattime t =15 indicatetheevenyearreturnstotheAFKhatcheryforevenbroodyears 1980-2006. ThemodelgivenbyEq.(1)isadeterministicmodel.Usingfryreleasedatascaled to1andxedparametervalues,Model1producesthesameoutputforallyears. Theoutputofthemodelcanonlybevariedbychangingtheselectionofdataused fortting.InFigure(19),allevenyearsreturnstotheAFKhatcheryfrom19802007areusedtotModel1.Environmentalconditionsshapethecarryingcapacity ofPrinceWilliamSoundandthesurvivalofpinksalmon,yettheyarenotexplicitlyincludedinthemodel.Insteadtheyarereectedinthereturnsofpastyears. Selectingaspanofyearswithsimilarenvironmentalpatternsforttingthemodel canimprovethepredictionforaparticularyear.Inthismanner,large-scaleshifts canbetracked.However,givenasuddenenvironmentalperturbation,thismodelis incapableofcreatinganaccurateprediction. Asspeciedbytheboundsonparameters,thefrypopulationisnegligiblysmall aftertwomonths. 29 PAGE 34 2.2.2TheSalmonLifeCycleinTwoStageswiththeE ectsofZooplankton ResearchhasconsistentlyshownthatthesurvivalofpinksalmonfryinPrince WilliamSoundisdependentonzooplanktonabundance[2,4,7,8,9,23,24].During thespring,surfacestraticationofthewatercolumnandmorehoursofsunlight createprimeconditionsforasuddenincreaseintheabundanceofzooplankton.At thepeakofthisbloom,abundanceofzooplanktoncanbe50-timesthatofpre-bloom levels[11].Thedynamicsofplanktonbloomscandi eryear-to-yeardependingon weather,temperature,currents,andnumerousotherenvironmentalfactors.Blooms canvaryinduration,intensity,timing,andspeciescomposition[8,9].Afterreachinga peak,thezooplanktonbiomassinPrinceWilliamSounddecreasesduetomaturation, predation,andenvironmentalfactors.Figure(20)showsthegeneralshapeofthe zooplanktonbloom. 30 PAGE 35 Figure20:Thegureaboveisaplotshowingzooplanktonabundanceasafunction oftime.ThedataintheplotisexperimentaldatacollectedinPrinceWilliamSound forthreeconsecutiveyears[26]. WeuseEq.(3)todescribehowthezooplanktonprolechangesovertime. z ( t )= + K exp ( t ) 2 (3) InEq.(3)above,theamplitude K ,duration ,andtiming ofthebloomcanbe adjusted,aswellasthezooplanktonbasalorpre-bloomlevel .Althoughtheshape ofthezooplanktonbloomisgenerallynotsymmetric(seeFigure(20)),forsimplicity itisassumedthatitcanbemodeledassuch.Below,Figure(21),Figure(22),and Figure(23)demonstratehowchangingeachofthethreeparameters( K )can 31 PAGE 36 altertheshapeofthefunction, z ( t ) zooplankton abundance z(t) t Increasing K Figure21:Thegureaboveshowshowthemodeledzooplanktonprolechangesfor variousvaluesof K ,rangingfrom1to4.Theothertwoparametersremainxed. Figure(21)demonstratesthatincreasing K increasestheheightofthepeakofthe bloom.Theabundanceofzooplanktonuctuatesfromyeartoyear,yetthisdi erence ismuchlessdramaticthanthechangeinabundancefrompre-bloomtobloomlevels. Herethepre-bloomlevelsarearbitrarilytakentobe =0 1 .Inaccordancewith observeddata,themodeledzooplanktonbloomamplitude K ispermittedtovaryon ayeartoyearbasisbyatmostvefold.Avalue K =5inthemodelwouldcorrelate withalargebloomanda50-foldincreasefrompre-bloomlevels. 32 PAGE 37 zooplankton abundance z(t) 2 0 2 0 t Increasing Figure22:Thegureaboveshowshowthetimingofthezooplanktonbloomchanges forvariousvaluesof .Fromtoptobottomthevaluesof are-2,0,and2.The othertwoparametersremainxed. Increasing shiftsthezooplanktonbloomtotheright(seeFigure.(22)).Note that canbepositiveornegativebecausetime t =0istakentobethetimeofthe fryrelease.Juvenilesalmonsurvivaliscriticallydependentonzooplanktonduring thefry'sinitial1-2monthsatsea.Fryreleasednearthepeakofthebloomenteran environmentrichinfood,thustheirchancesofsurvivalandtheirgrowthratesare muchhigherthanthoseoffryreleasedbeforeorafterthebloom[7,9].Insomeyears thereisasecondbloomduringthefall[8].Itwasnotnecessarytotakeintoaccount thepossibilityofafallbloombecauseinthismodel,zooplanktondirectlyimpactonly thejuvenilesalmonstages,whichoccurduringtherstfewmonthsafterrelease. 33 PAGE 38 ! 4 2 0 2 4 0 zooplankton abundance z(t) t Increasing Figure23:Thegureaboveshowshowthezooplanktonfunctionchangeswithrespect tothedurationparameter, .Theothertwoparametersremainxed.Forsmall valuesof ,thezooplanktonprole z ( t ) isbriefwithasteepriseanddecline.As increases,thebloomconditionsareprolonged. As decreases,thedurationofthebloomshortens.Although istheonly parameteradjustedinthegureabove,changesinthedurationofthebloomusually coincidewithchangesinthepeakheight.Bloomscantypicallybeclassiedaseither briefandintenseorprolongedandlowalthoughothervariationsmayoccur. Predationonzooplanktonbyfry,pollock,andotherorganismsisimplicitlydemonstratedinthemodel'sdeclineafterthepeakofthebloom.Byadjustingthepeak, timing,anddurationofthebloomforeachyear,thesurvivalrateofpinksalmon changescorrespondingly.Intheextremecases(eithernozooplanktonoranunusuallylargeabundanceofzooplankton)anysmallchangeinthebloomwouldnotgreatly a ectthefry.Large,longlastingbloomsthatpeakshortlyafterthereleaseofthe frywillsignicantlyincreasethesurvivalrateofthefry.Therefore,thepresenceof azooplanktonbloomwillnegativelya ectthedeathrateofthejuvenilepopulation, asseeninthediagrambelow,Figure(24). 34 PAGE 39 F A z(t) Figure24:Aboveisadiagramrepresentingthepopulationofpinksalmonwiththe e ectsofzooplankton.Therelationshipbetweenfryandadultisthesameasinmodel (1).However,herethedeathrateofthefrydependsonthetiming,duration,and amplitudeofthezooplanktonbloom. Fromthediagramabove,wecanconstructthefollowingmodel.Here z ( t ) refers tothezooplanktonfunctiongivenbyEq(3).Asthezooplanktonbloomismodeled withatime-dependentfunctioninsteadofadi erentialequation,thedimensionof thenewmodeldoesnotincrease. dF dt = k 1 1+ # 1 z ( t ) m + k 2 F (4) dA dt = k 2 F k 3 A Asintherstmodel,inEq(4)thevariables F and A representfryandadult populations,respectively.Theparameters k 1 ,k 2 ,and k 3 havethesamephysical meaningasinEq(1)althoughthevaluesoftheseconstantsareslightlydi erent. Anadditionalparameter # 1 isneededtoadjustthee ectofzooplanktononthefry population.AsinEq(1),thematurationrateconstantandfrydeathrateconstant areboundedtoreectthedurationofthefrystage.Theboundsonthedeathrate constantfortheadultstagearethesameasthoseusedinEq(1).Thedecayrateofthe fryisdeterminedby k 1 # 1 z ( t ) ,and m .Theformofthisparameterfunctionischosen 35 PAGE 40 sothatintheabsenceofazooplanktonbloom( z ( t )=0 ),Model2wouldresemble Model1,inwhichthefrydieataconstantrate.Asthebloomapproachesoptimal conditions,thedeathrateparameterforthejuvenilepinksalmongrowssmaller.In theequationforthefrypopulation,thezooplanktonfunctionisraisedtoapower m Thisparametercontrolsthemodel'ssensitivitytochangesinzooplanktonabundance. Increasing m steepensthee ectofthezooplanktonfunctionforanarrowerrangeof K ,asisdemonstratedinFigure(25).Bysettingtimetconstantandvarying K ,we candenethezooplanktonfunctionintermsof K : z ( K ) .Changingtheamplitude ofthezooplanktonbloomallowsustoobservehowthedeathrateparameterchanges fordi erentpowers m 0 1 2 3 4 5 0 2 4 6 8 10 12 K death rate due to zooplankton k 1 1+ 1 z ( K ) k 1 1+ z 2 ( K ) k 1 1+ z 3 ( K ) Figure25:Thegureshowsthezooplankton-dependentcomponentofthefrydeath rateparameterasafunctionoftheamplitudethespringbloom( K ).Inthisequation, k 1=12 # 1 =1.4,and =0.Intheequationforzooplankton, =1,time t =1, and K rangesfrom0to5. FromFigure(25),itisapparentthatbyadjustingthesizeofthezooplankton bloom,itispossibletoobtainawiderangeofdeathratesforpinksalmonfry.If m 36 PAGE 41 islargerthan1,smallchangesinveryloworveryhighrangesof K ,donotchange thee ectofzooplanktonsignicantly.However,foracertainrangeof K ,thechange inthee ectofzooplanktonispronounced.Notethatwhen K =0,thezooplankton function z ( K ) isnonzeroduetothebasallevelofzooplankton, .Becauseofthis, theparameterfunctionsintermsof z z 2 ,and z 3 donothavethesamevalueat K = 0. ForthePrinceWilliamSoundpinksalmon, m ischosentobe2.Becauseitis possibletomatchthespanofhistoricaldataandobtaingoodresultswith m =2,there isnoneedtousehigherpowersof z(t) .Themodelparametersare k 1 =12 k 2 =0 6 k 3 =0 1 # 1 =1.4,and n =2 .Theseareobtainedbyadjustingtheparametersby handtogeneratereasonableoutputswithintheaforementionedbounds. Theadditionofzooplanktonintroducesavariablecomponentintothemodel.For agivenyear,apredictioncanbemadewithrespecttotheavailabilityoffoodfor juvenilepinksalmon.Giventhesameinitialpopulationofpinksalmon,thepercentageofreturningadultscanbemadetodi erbyabout140%simplybyadjusting z ( t ) .UnlikeinModel1,environmentalchangesfromyeartoyeararereectedin thezooplanktonproleandsubsequentlyinthepinksalmonpredictions.Consistent datadescribingyearlyzooplanktonprolesisnotavailable,yetwecanassumethat pinksalmonreturnscorrespondtoabundanceofzooplankton. Thepeakheight( K )ofthespringbloomistakentobetherelativeabundance ofzooplankton.Adjustingtheparameter K producesdi eringreturnsizes.We representlowzooplanktonbloomswith K =1 ,averagebloomswith K =2 ,and aboveaveragebloomswith K =3 5 .ThetotalreturnstoPWSAChatcheriescan dividedintothreegroups:low,average,andhighreturns.Therangeforeachgroupis determinedbyevenlydividingthewholeinterval(0.99,9.14)intothreeevensegments. Thethreeclassicationsofrelativeabundanceofzooplanktonaretakentocorrespond withthreegroupsizesofpinksalmonreturns,asgivenbyTable(2). 37 PAGE 42 Table2:ThetablebelowshowsthetotalreturnstoPWSAChatcheriesdividedinto threeintervals:low,average,andhigh.Thegroupingsarenotbasedexplicitlyon ecologicalfactors. Low 0.99 1.44 2.62 2.92 3.31 3.57 3.57 3.62 3.70 Average 3.72 3.81 3.95 4.11 4.18 4.34 4.79 5.46 5.48 5.57 5.78 High 6.45 7.43 7.51 7.75 8.02 8.05 9.10 9.14 Thefollowingthreeguresshowhowthemodeloutputchangesfordi erentzooplanktonbloomproles.Themodelappearstotthethreedi erentgroupsofreturns almostperfectly.However,ifweadjustthescaleofthey-axisitisclearthatthere arestillsignicantdiscrepanciesbetweentherealdataandthemodelpredictions(see theinsetsinFigure(26),Figure(27),andFigure(28)). 38 PAGE 43 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 time (months) pink salmon percentage 0 2 4 6 8 10 12 14 16 0 1 2 3 4 zooplankton 14 15 0 4 8 fry adult Figure26:AboveistheModel2prediction,givenazooplanktonprolewithanabove averageamplitude.Here K =3 5 =0,and =1.Thisplotrepresentsahigh returntoPrinceWilliamSound.Thecircularmarkersindicatethegroupofhigh returns,asgiveninTable(2). InFigure(26)thezooplanktonbloomhasalargerthanaveragepeakheight,with averagetimingandduration.Theseconditionsarereectedinthemodel'sprediction ofanaboveaveragereturn.Asexpected,thejuvenilepopulationdecaystozero betweentherstandsecondmonthatseaandslightlylessthanhalfofthefryreach theadultstage. 39 PAGE 44 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 time (months) pink salmon percentage 0 2 4 6 8 10 12 14 16 0 1 2 3 4 zooplankton 15 0 2 4 6 time fry adult Figure27:Aboveisthemodelprediction,givenazooplanktonprolewithanaverage amplitude.Here K =2 =0,and =1.Thisplotrepresentsanaveragesizereturn toPrinceWilliamSound.Thecircularmarkersindicatethegroupofaveragereturns, asgiveninTable(2) Inthegureabove,Figure(27),themodelshowsanaveragezooplanktonprole inwhichthefryarereleasedatthepeakofthebloom.Thefrypopulationdecays justbeforetwomonths,asexpected. 40 PAGE 45 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 time (months) pink salmon percentage 0 2 4 6 8 10 12 14 16 0 1 2 3 4 zooplankton 15 0 2 4 fry adult Figure28:Aboveisthemodelprediction,givenazooplanktonprolewithabelow averageamplitude.Here K =1 =0,and =1.Thisplotrepresentsabelow averagesizereturntoPrinceWilliamSound.Thecircularmarkersindicatethegroup oflowreturns,asgiveninTable(2) Figure(28)showstheModel2outputforalowzooplanktonprole.Notethat thejuvenilesalmonpopulationappearstodecaytozerobeforeonemonth.InEq(4), thetotaldecayrateforthefrypopulationisgivenbythesumofthedeathratedue tozooplanktonandthematurationrate.Forlowzooplanktonblooms,thedeath rateparameterisverylarge,causinganunrealisticallyquickdecay.Inreality,alow zooplanktonbloomwouldcausethejuvenilesalmontomatureataslowerrateand thereforebevulnerabletopredationforalongerperiodoftime.Ourmodelfailsto expressthisphenomenon,yettheerrorisasmallone.Themodeldoescorrectlyshow thatlowerzooplanktonlevelsdecreasethepercentageoffryreachingadulthoodand subsequently,thepercentageofadultsreturningtoPrinceWilliamSound. 41 PAGE 46 2.2.3TheSalmonLifeCycleinTwoStageswiththeE ectsofZooplanktonandPollock WhileModel2isabletodescribethedynamicsofjuvenilesalmonpopulationsin termsoftheirfoodsource,itdoesnottakeintoaccountthee ectsofpredators.The relationshipbetweenzooplankton,pollock,andjuvenilesalmoncanbedescribedby theprey-switchinghypothesis[2,7].Includingwalleyepollockinthemodeliscritical becausepollockfunctionbothasacompetitorforafoodsourceandasapredator[7]. Forlowlevelsofzooplankton,itisexpectedthatthelossofjuvenilesalmonwill bemuchhigherwhenthee ectofpollockisadded.However,whenthelevelsof zooplanktonarehigh,thelossofjuvenilesalmonshouldbeonlyslightlye ectedby thepresenceofpollock.Pacicherringalsodemonstratethesameprey-switching behaviorbuttheyhavenotbeenincludedinthismodel.Intheyearsfollowingthe ExxonValdezoilspilltheherringpopulationinPrinceWilliamSoundhasbeenlow[8]. Forsimplication,onlythee ectsofpollockareconsideredhere. A F z(t) P Figure29:Aboveisadiagramrepresentingthelifecycleofpinksalmonwiththe e ectsofzooplanktonandpollock.Theadditionofpollockhasapositivee ecton thedeathrateofthefry.However,aspollockprefertofeedonzooplankton,the presenceofalargespringbloomweakensthea ectofpollockonfry.Whenthe abundanceofzooplanktonislow,pollockfeedheavilyonjuvenilesalmon.Therest ofthemodelrepresentsthesamerelationshipsseeninthepreviousdiagram(see Fig(24)). 42 PAGE 47 Fromthediagramabovewecanconstructthefollowingmodel.AsinEq(4), z ( t ) isthezooplanktonfunctiongivenbyEq(3). dF dt = k 1 1+ # 1 z ( t ) m + k 2 + k 3 P 1+ # 2 z ( t ) n F (5) dA dt = k 2 F k 4 A Inthismodel,Eq(5),weincludepredationduetoaconstantpopulationofpollock P .Thepositivea ectofpollockonthefrydecayrateisdampenedbytheinclusion zooplankton( # 2 z ( t ) m )inthedenominator.Intheabsenceofzooplankton,thefry populationdecayswithrespecttopollockataconstantrateof k 3 .Itispossibleto replicatetheprey-switchingofpollockbetweenfryandzooplanktonwiththemodel. Alargespringbloomwillcausethedecayrateduetopollocktobenegligible.The otherparameters ( k 1 # 1 k 2 )arethesameasforthemodelgivenbyEq(4).Thedeath rateconstantfortheadultpopulationis k 4 .Thezooplanktonfunctionisraisedto thepowerof m inthedeathrateparameterduetozooplanktonandtothepower of n inthedeathrateparameterduetopollock.AsinModel2,thisisnecessaryin ordertoaugmentthesensitivityoffrytothee ectsofzooplanktonandtoincrease therangeofpossiblevaluesforthedeathrates. Theparametersforthethirdmodelareestimatedbyatrialanderrorprocessof manuallyadjustingparametervaluesuntilreasonablegraphsandoutputsareproduced.Todetermineestimatesforparametervalues,certainboundsaredened.The pollockpopulationistakentobeaconstantthatcanbeadjustedeachyear.The pollocklifecycleismuchmorecomplexthanthatofpinksalmonandspansroughly 15years.Themodelrunsforaboutoneyearandpollockonlya ectsalmonpopulationsduringthersttwomonths.Thereforeitisnotnecessarytoproduceatime dependentmodelforpollock.Foragivenyear,ifthepollockpopulationisknownto 43 PAGE 48 belargerorsmallerthanaverage,theparameter P canbeadjustedaccordingly.The boundsonthematurationrateconstant k 2 arethesameasforthepreviousmodels. TheboundsondeathrateduetozooplanktonarethesameasthoseinModel2, Eq(4).ParametervaluesfortheAFKhatcheryaregiveninthefollowingtable. Table3:ThetablebelowgivestheparametervaluesforModel3. Parameters Values k 1 6 k 2 0.6 k 3 7 k 4 0.1 # 1 1.4 # 2 1.6 m 2 n 2 44 PAGE 49 0 0.5 1 1.5 2 t Death rate parameter (pollock predation) D ( t )= Pk 3 1+ 2 z ( t ) 2 P = 0.5 P = 1 P = 1.5 Figure30:Thegureaboveshowshowthedeathrateparameterduetopollock predationonjuvenilepinksalmonchangesovertimeforthreedi erentpopulation sizesofpollock( P ).Thezooplanktonproleisgivenby =0, K =2,and =1, andrepresentsanaveragebloomwithapeakattime t =0 FromFigure(30)wecanobservethea ectofpollockonthedeathrateparameter ofjuvenilepinksalmon.Asthepollockpopulationdecreases,thereislesspredation onfry,allowingahigherpercentageoftheinitialfrypopulationtoreachtheocean stage.ForPrinceWilliamSound, P =1 istakentobeanaveragepopulationof pollock.Itisclearthatadjusting P canproducevaryingreturnsizes.However,one themaingoalsofincludingpollockinthemodelistodemonstratetheprey-switching hypothesis.Wetestourmodelagainsttheprey-switchinghypothesisbyexamining howthevalueofthedeathrateparameterduetoaxedpollockpopulationchanges withrespecttozooplanktonabundance.ThisisdemonstratedinFigure(31). 45 PAGE 50 0 1 2 3 4 5 0 1 2 3 4 5 6 7 zooplankton abundance (K) death rate parameter (pollock predation) D ( K )= Pk 3 1+ 2 z ( K ) n Figure31:Aboveisaplotofthedeathrateparameterduetopollockpredation asafunctionof K ,theamplitudeofthezooplanktonfunction.Thedeathrate parameterforpollockpredation, D ( K ) ,isgivenby D ( K )= Pk 3 1+ 2 z ( K ) n where z ( K ) isthezooplanktonfunctionattime t =0 asafunctionofK.Here # 2 =1 6 n =2 =0 =1 ,and P =1 Bytheprey-switchinghypothesis,weexpecttoseeadecreaseinthedeathrate asthezooplanktonpopulationincreases.Aspollockprefertoconsumezooplankton, evenaslightincreasein K causesasignicantdecreaseinthedeathrateoffrydue topredation.Whenlevelsofzooplanktonareaboveathresholdlevel,pollockhave asu cientsupplyoffood.Undertheseconditions,smallchangesin K havelittle e ectonthiscomponentofthefrydeathrate. Figure(32)showsanexaggeratedplotofFigure(31). 46 PAGE 51 0 1 2 3 4 5 0 1 2 3 4 5 6 7 zooplankton abundance (K) effect of pollock predation on fry D ( K )= Pk 3 1+ 2 z ( K ) n No pollock predation Pollock predation Figure32:Thegureaboveemphasizesthepreyswitchinghypothesis.Asin Figure(31),thedeathrateparameterforpollockpredation, D ( K ) ,isgivenby D ( K )= Pk 3 1+ 2 z ( K ) n .Here # 2 =1 6 n =20 =0 =1 ,and P =1 Themodel'ssensitivitytothethee ectsofzooplanktonisdrasticallyincreasedby makingtheparameter n large.InFigure(32),theswitchbetweenpollockpredation andnopollockpredationcanbeseenclearly.Here, K =1 isthethresholdvalueat whichpollockshiftfromfeedingonfrytofeedingonzooplankton. 2.2.4TheSalmonLifeCycleinFiveStageswiththeE ectsofZooplanktonandPollock Inthetimeseriessimulationsforeachofthepreviousmodels,theadultpopulationsappearimmediatelyafterthefryhavebeenreleased.Thisisunreasonableas salmonmaturegradually.Inaddition,thissuggeststhatsomefryreachtherelatively safe'oceanstageinstantaneously.However,itisknownthatsalmonremainparticu47 PAGE 52 larlyvulnerableduringtheirinitial60daysinthewild.Furthermore,duringtherst twomonthsinPWStheirvulnerabilitydecreasesastheymatureandgrow[2].Adelay intheappearanceoftheadultpopulationisneededtocapturethesephenomena.By dividingthefrystageintoseveralsubstageswecanreecthowjuvenilesmatureto adultsalmoninstagesoveraperiodoftime(seethediagraminFigure(33)). F F 1 F 2 F 3 A z(t) P z(t) P z(t) P z(t) P Figure33:Aboveisadiagramrepresentingthepinksalmonlifecycleinvestages withthee ectsofzooplanktonandpollock.Notethatthezooplanktonbloom,representedby z ( t ) ,andthepollockpopulation P arethesameforeachofthefourfry stages, F F 1 F 2 ,and F 3 Asystemofdi erentialequationsrepresentingdi erentintermediatestagesof growthwasusedtorepresentatimedelayedappearanceofvariouslifestagesof BluenTuna[10].Thismethodcanbeappliedtopinksalmonpopulations. dF dt = k 1+ m 1 1+ # 1 z ( t ) m + m 2 P 1+ # 2 z ( t ) n F (6) dF 1 dt = k 1 F k 2 + m 1 1+ # 1 z ( t ) m + m 2 P 1+ # 2 z ( t ) n F 1 dF 2 dt = k 2 F 1 k 3 + m 1 1+ # 1 z ( t ) m + m 2 P 2(1+ # 2 z ( t ) n ) F 2 48 PAGE 53 dF 3 dt = k 3 F 2 k 4 + m 1 1+ # 1 z ( t ) m + m 2 P 2(1+ # 2 z ( t ) n ) F 3 dA dt = k 4 F 3 k 5 A InEq(6),thefrystagehasbeenseparatedintofourarbitrarysubstages(see Figure(33)).Thedi erentialequationforeachofthesestagescloselyresemblesthe equationforthefrystageinEq(5).Thefryreleasedfromthehatchery( F )mature tosuccessivefrysubstages F 1 F 2 F 3 ,atratesof k 1 k 2 ,and k 3 respectively.The deathrateforeachfrystageisdividedintotwocomponents:thedeathratedueto zooplanktonandthedeathrateduetopollockpredation.Theseparameterfunctions aretakenfromEq(5).Asjuvenilesalmongrowlargerinsizetheyarelessvulnerable topredationbypollock.Therefore,thedecayratesduetopredationfor F 2 and F 3 populationsarehalved.Theequationfortheadultpopulation( A )hasthesameform asinthepreviousthreemodels. BecausethesubstagesofFarechosenarbitrarilyanddonotreectanydistinct biologicallifestages,thematurationrates, k 1 k 2 k 3 ,and k 4 ,canbetakentobethe same.Thisgreatlysimpliesthemodelbyreducingthenumberofparameters.In thepreviousmodelsthereisonlyonefrystage.Maturationrateconstantsanddeath rateconstantsareobtainedbyadjustingtheparametersbyhandwithinreasonable bounds.BecausethefrystageisdividedintofourequalsubdivisionsinModel4,each maturationrateconstantshouldberoughlyfourtimestherateconstantinModel3. Likewise,thedeathrateconstantscanbeestimatedtobeaboutonefourthofthose inModel3.Morecarefuladjustmentoftheseestimatesisneededtotthemodelto data.Valuesfor # 1 # 2 m ,and n aretakenfromModel3.Therateconstantsfor Model4forevenyearreturnstotheAFKhatcheryaregiveninthetablebelow. 49 PAGE 54 Table4:Model4ParametervaluesforsimulationofevenyearreturnstotheAFK hatchery. Parameters DescriptionoftheParameters Values k 1 MaturationrateconstantfromFtoF 1 2.4 k 2 MaturationrateconstantfromF 1 toF 2 2.4 k 3 MaturationrateconstantfromF 2 toF 3 2.4 k 4 MaturationrateconstantfromF 3 toA 2.4 k 5 Deathrateconstantforadultsalmon 0.1 m 1 Deathrateconstantforfrystage 2 m 2 Deathrateconstantduetopollockpredation 3 # 1 Adjuststhee ectofzooplankton 1.4 # 2 Adjuststhee ectofzooplankton 1.6 m Powerofzooplanktonfunction 2 n Powerofzooplanktonfunction 2 50 PAGE 55 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 time (months) pink salmon percentage 0 1 2 0 0.1 0.2 0.3 F F 1 F 2 F 3 A Figure34:ThegureaboveshowstheModel4predictionforazooplanktonprole givenby K =2, =0,and =1.Theinsetshowsthefoursubstagesoffryandthe adultlifestage. FromtheinsetinFigure(34),wecanseethatModel4isabletocapturethedelay intheappearanceoftheadultpopulation.Thefrysubpopulationsdieo beforethe secondmonthafterrelease.Asexpected,theadultstageofthepopulationreachesa peakabout 1 5 monthsaftertheinitialfryrelease.Notethatthepeaksoftheboth F 2 and F 3 subpopulationsarelowerthanthepeakoftheadultpopulation.Thedecay ratesforthefrystagesaremuchhigherthanthedecayratefortheadultpopulation. Thereforetheadultpopulationisabletoattainahigherpeakbeforedecayingtozero thaneither F 2 or F 3 Thetransitionfromfrytoadultstagesoccursgraduallyoveraspanoftwomonths. Thefrystagehasahighdeathratewhichisdependentonzooplanktonandpollock. 51 PAGE 56 Theadultstagehasalow,constantdeathrate.Withouttheinclusionofadelay, weobservethatsomeofthejuvenilesmaturetotheadultstageimmediatelyafter thefryarereleased.Therefore,thisportionofthepopulationisnota ectedbythe higherdeathrateofthefrystage.Thisisunrealisticandincreasespotentialerror inthemodel.Withadelay,thepercentageoftheinitialfrypopulationimmediately maturingtotheadultstageisdrasticallyreduced. WhileModel4doesnotnecessarilyhavehigherpredictivecapabilitiesthanModel 3,itdoesrepresentthebiologyofpinksalmonmoreaccurately. 52 PAGE 57 3Discussion Inthisthesiswedevelopedfourdeterministicmodelstoexplainthelifecycleof pinksalmonandtopredictfuturereturns. First,wedevelopedasimplemodelthatformedthebasisforsubsequent,more complicatedmodels.InModel1,weshowedthatpinksalmonpopulationscanbe describedwithatwodimensionalsystemofdi erentialequations.Themodelwas abletocapturedataforyearswithreturnsofaveragesize.Becausetheparameter valuesandinitialconditionswerexed,themodelalwaysproducedthesameoutput yeartoyear.Themeanabsolutepercentageerror(MAPE)canbeusedtotestthe successofamodel'sabilitytotdata.Meanabsolutepercenterroriscalculated usingtheformulabelow. MAPE = 100 s s # i =1 $ $ $ $ R i M i R i $ $ $ $ Herethenumberofdatapointsbeingttedis s ,theactualvalueofdatapoint i is R i ,andthemodeloutputis M i TheMAPEcanbeusedtodeterminethepredictive capabilitiesofModel1,Eq(1).Usingdatafromallbroodyearsfrom1980-2007for theAFKhatcheryandparameterssettok 1 =1.2950,k 2 =1.4449,andk 3 =0.1656, theMAPEofthismodelis49.52%.Howevermorecarefulselectionofthespanof yearsusedforttingcanresultinalowerMAPE.Inparticular,asenvironmental changesareimplicitinModel1,usingtimespanswithsimilarclimaticpatternstot Eq(1)couldproducebetterresults.Asthelastmajorclimaticshiftoccurredin1989 [20],wechosetouseonlyevenyearsfrom1990to2006.TheMAPEfromthist is35.92%withparameters k 1 =1 2097 k 2 =1 4625 ,and k 3 =0 170 7.Furthermore, excludinganomalies(exceptionallyloworhighreturns)couldalsoimprovethemodel's t.Forexample,whenweomittedthelowest(year1992)andhighest(year1996) 53 PAGE 58 percentreturnsfromtheevenbroodyears,theMAPEbecame18.69%.Inthist,the parameterswereadjustedslightlyto k 1 =1 274 9, k 2 =1 4025 ,and k 3 =0 1836 .In theexamplesabove,theMAPEwascalculatedusingthesamedatapointsasthose usedtotthemodel.Whenthemodelttedwithevenyearswasappliedtothe oddyeardata,theerrorincreased.Usingparametersforevenyears k 1 =1 2950 k 2 =1 4449 ,and k 3 =0 1656 withoddyeardataproducedaMAPEof64.40%.As demonstratedbefore,thiserrorcanbeloweredbyimprovingtheselectionofdata setsusedfortting.WhilethemodelgivenbyEq(1)issimplistic,itispossibleto produceareasonablepredictionforyearswithreturnsthatdonotdeviatetoofar fromaverage. WesoughttoimproveModel1byincludingaparameterthatcanbevariedfrom yeartoyear.Fromanalysisofavailabledataandevaluationofotherstudiesweconcludedthatecologicalconditionsarekeytodeterminingthesizeofthereturningadult population.Anumberofstudieshaveindicatedthattheabundanceofzooplankton isofparticularimportancetojuvenilesalmonpopulations[2,4,7,8,9,23,24].The springzooplanktonbloomdeterminesboththeavailabilityoffoodandthethreat duetopredationbypollock.Thezooplanktonbloomcanbemodeledwithasimple functioninwhichthepeakheight,duration,andtimingofthebloomcanbeadjusted manuallytotthezooplanktonproleforaparticularyear.Toincludethee ectsof thespringbloomonpinksalmon,weassumedthatthedeathrateconstantinModel 1wasdependentonthezooplanktonfunction.Giventhesameinitialfrypopulation, Model2,canproducedi erentreturnsbasedonchangingecologicalconditions. InordertocompareModel1andModel2,theMAPEforeachwascomputed usingtotalreturnstoallofthePWSAChatcheriescombined.TheMAPEforModel 1,usingparameters k 1 =1 5958 k 2 =1 4124 ,and k 3 =0 1526 ,was50.44%.In Model2,thelowreturnswerepredictedwith K =1 ,theaveragereturnswith K =2 andthehighreturnswith K =3 5 .Theotherparameterswerexedat k 1 =12 54 PAGE 59 k 2 =0 6 k 3 =0 1 # 1 =1.4and m =2 .Byusingthismethodofgroupingreturns withbloomproles,theMAPEforModel2,became23.07%.Thiswaslessthanone halfoftheMAPEofModel1.Usingthreedi erentzooplanktonprolestodescribe thereturndatagreatlyenhancesthetofthemodelanditspredictivecapabilities. Withmoredataontheyearlyzooplanktonabundance,itislikelythattheforecast couldbeimprovedfurther. ThedevelopmentofModels3and4wasdrivenbythedesiretoimprovethemodels' abilitytodescribethepinksalmonlifecycle.Themodelsaddresstwoissuesthatwere notincludedinModel2:thee ectofpredatorsandthetimingofthetransitionfrom thefrystagetotheadultstage.Model3includedthee ectsofpredationonfrybya constantpollockpopulation.Foreachyear,givenanindexofthepollockpopulation andazooplanktonprole,theadultreturncouldbepredicted.However,themain achievementofModel3wastheimplicitdemonstrationoftheprey-switchingbehavior ofpollockbetweenfryandzooplankton. Model4introducedatimedelayintheappearanceoftheadultpopulation.Includingatime-delaybetterdescribesthebiologicalsystemanditcanreducetheerror duetofryenteringtheadultstagetoosoon. Nomodelisabletoperfectlydescribeadynamicbiologicalsystem.Whilethe modelsdevelopedinthisthesisareabletocaptureawiderangeofdata,theirsuccessis limitedtoourabilitytomeasurethemagnitude,shape,andtimingofthezooplankton bloomandourabilitytoaccuratelydeterminevaluesofparameters.Inapractical setting,Model2islikelytobethemoste ective.Ithastheabilitytotrackyear toyear,zooplankton-dependentuctuationsinthepinksalmonpopulationyetitis stillarelativelysimplemodel.Itcouldbefurthersimpliedbyxingthetimingand durationofthezooplanktonbloom.Thenonewouldonlyneedtoknowarelative zooplanktonabundanceindexinordertogenerateageneralzooplanktonprole. Withmoreresearchonthedynamicsofspringzooplanktonbloomsandtheir 55 PAGE 60 inuenceonpinksalmon,themodelscouldbefurtherrened.Inparticular,consistent yeartoyeardataonzooplanktonabundanceneareachhatcherywouldbeextremely valuable. 56 PAGE 61 Appendix Matlab codeforModel1 %%%%%%%%%%%%%%%%%%%%%%Driverfunction%%%%%%%%%%%%%%%%%%%%%% %Realdata EYF=[11111111111111]; EYA=[0.07360.03200.04790.05230.04310.02130.01570.0163... 0.14050.05190.05160.03570.03970.0341]; OYF=[11111111111111]; OYA=[0.05310.06550.06770.02460.04970.01410.01060.0642... 0.08250.03370.04530.05800.09870.0743]; TYF=[1111111111111111111111111111]; TYA=[5.484.344.187.433.576.453.623.815.785.461.440.99... 3.952.923.314.799.147.755.573.723.579.102.628.023.70... 8.057.514.11]/100; select=1; cc='m'; %%Thispartisforfitting ifselect==1 par0=5*rand(1,3); LB=[11e-21e-3]; UB=[221]; options=optimset('Display','iter','TolFun',1e-8); xdata=ones(1,length(EYA)); [EstFVal]=lsqcurvefit(@myfitfun,par0,xdata,EYA,LB,UB,options,EYF) fori=1:length(EYA) 57 PAGE 62 init=[TYF(i)0]; tspan=[015]; [TY]=ode15s(@mymodelfit,[tspan],[init],[],Est); subplot(2,1,1),plot(T,Y(:,1),cc) subplot(2,1,2),plot(T,Y(:,2),cc) holdon i plot(15,EYA(i),'og') pause clf end end %%Thispartisforplotting ifselect==2 fori=1:length(EYA) par0=[1.29501.44490.1656]; init=[EYF(i)0]; tspan=[016]; [TY]=ode15s(@mymodelfit,[tspan],[init],[],par0); plot(T,Y(:,1),'r',T,Y(:,2),'-b','LineWidth',2) xlabel('time(months)','fontsize',16) label('pinksalmonpopulationpercentage','fontsize',16) set(gca,'linewidth',3,'fontsize',16) axis([-0.116-0.021.1]) holdon %axis([-.216-1.2102]),holdon i 58 PAGE 63 plot(15,EYA(i),'.g','markersize',20) pause clf end end %%%%%%%%%%%%%%%%%%%%%%myfitfunfunction%%%%%%%%%%%%%%%%%%%%%% functionF=myfitfun(par0,xdata,EYF) tspan=[015]; fori=1:length(EYF) init=[EYF(i)0]; [TY]=ode15s(@mymodelfit,tspan,[init],[],par0); F(i)=Y(end,2); end %%%%%%%%%%%%%%%%%%%%%%mymodelfitfunction%%%%%%%%%%%%%%%%%%%%%% functiondy=mymodelfit(t,y,par0) k1=par0(1); k2=par0(2); k3=par0(3); Km=2; Vm=2; F=y(1); A=y(2); dF=-(k1+k2)*F; dA=k2*F-k3*A; dy=[dFdA]'; 59 PAGE 64 References [1] Bonar,S.A,G.B.Pauley,andG.L.Thomas.(1989). Speciesproles:lifehistoriesandenvironmentalrequirementsofcoastalshesandinvertebrates(Pacic Northwest-pinksalmon). U.S.FishWildl.Serv.Biol.Rep.82(11.88).U.S.Army CorpsofEngineers,TREL-82-4. [2] Willette,T.M.,Cooney,R.T.,Patrick,V.,Mason,D.M.,Thomas,G.L.and Scheel,D.(2001). Ecologicalprocessesinuencingmortalityofjuvenilepink salmon(Oncorhynchusgorbuscha)inPrinceWilliamSound,Alaska .Fisheries Oceanography,10,p14-41. [3] PersonalconversationwithDr.SteveMo tt,CordovaAK,29June2011 [4] R.TedCooney.(1993). AtheoreticalevaluationofthecarryingcapacityofPrince WilliamSound,Alaska,forjuvenilePacicsalmon .FisheriesResearch,18(1-2), p77-87. [5] PrinceWilliamSoundAquacultureCorporation.N.p.,09/08/2011. PAGE 65 [8] Vaughan,S.L.,Mooers,C.N.K.andGay,S.M.(2001). Physicalvariabilityin PrinceWilliamSoundduringtheSEAStudy,1994-98 .FisheriesOceanography, 10.p58-80. [9] Cooney,R.T.andK.O.Coyle.(1995). TheRoleofZooplanktoninthePrince WilliamSoundEcosystem .ExxonValdezOilSpillRestorationProjectAnnual Report(RestorationProject95320-H),InstituteofMarineScience,Universityof AlaskaFairbanks,Fairbanks,Alaska99775-1080. [10] Bowen,Esther,M.Hoerner,C.Kontur.(2010). PopulationDynamicsofWesternAtlanticBluenTuna:ModelingtheImpactsofFishingusingDi erential Equations. UniversityofChicago. [11] Cooney,RT.,Coyle,K.O.,Stockmar,E.andStark,C.(2001). Seasonalityin surface-layernetzooplanktoncommunitiesinPrinceWilliamSound,Alaska FisheriesOceanography,10.p97109. [12] CyberSalmon.U.S.FishandWildlifeService,FairbanksFishandWildlifeField O ce.2009 PAGE 66 [16] StevenL.Haeseker,RandallM.Peterman,ZhenmingSu&ChrisC.Wood (2005). RetrospectiveEvaluationofPreseasonForecastingModelsforPink Salmon .NorthAmericanJournalofFisheriesManagement,25:3p897-918 [17] Heard,WilliamR.(1991 )."LifeHistoryofPinkSalmon."PacicSalmonLife Histories .Ed.CornelisGrootandEd.L.Margolis.Vancouver:UBCPress,p121230. [18] UnitedStates.AlaskaDepartmentofFishandGame.MarkLab,Otolith& ThermalMarkingLaboratory. PAGE 67 ectearlymarinecarbonsourcedependency. ProgressinOceanography.77. p.194-202.Web.22Jan.2012 [25] Rice,Stanley.D.,Thomas,R.E.,Heinz,R.A.,Wertheimer,A.C.,Murphy,M.L., Carls,M.G.,Short,J.W.,andMoles,A.(2001). Synthesisoflong-termimpactsto pinksalmonfollowingtheExxonValdezoilspill:persistence,toxicity,sensitivity, andcontroversy. USNationalMarineFisheriesService,AukeBayLaboratory, ExxonValdezOilSpillRestorationProject99329,FinalReport,Juneau,Alaska. Web.23April2012. [26] Eslinger,DavidL.,Cooney,R.T.,Mcroy,P.C.,Ward,A.,Kline,T.C.,Simpson,J.W.,Allen,J.R.(2001). Planktondynamics:observedandmodelledresponsestophysicalconditionsinPrinceWilliamSound,Alaska. Fisheries Oceanography,10.p.81-96 63 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||