predator–prey dynamics of bald eagles and glaucous‐winged

Ecology and Evolution 20191ndash18 emsp|emsp1wwwecolevolorg

Received2August2018emsp |emsp Revised31January2019emsp |emsp Accepted1February2019DOI101002ece35011

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

Predatorndashprey dynamics of bald eagles and glaucous‐winged gulls at Protection Island Washington USA

Shandelle M Henson12 emsp| Robert A Desharnais34emsp| Eric T Funasaki5emsp| Joseph G Galusha6emsp| James W Watson7emsp| James L Hayward1

ThisisanopenaccessarticleunderthetermsoftheCreativeCommonsAttributionLicensewhichpermitsusedistributionandreproductioninanymediumprovidedtheoriginalworkisproperlycitedcopy 2019 The Authors Ecology and EvolutionpublishedbyJohnWileyampSonsLtd

1DepartmentofBiologyAndrewsUniversityBerrienSpringsMichigan2DepartmentofMathematicsAndrewsUniversityBerrienSpringsMichigan3DepartmentofBiologicalSciencesCaliforniaStateUniversityLosAngelesCalifornia4ControlandDynamicalSystemsCaliforniaInstituteofTechnologyPasadenaCalifornia5DepartmentofComputerScienceandMathematicsSulRossStateUniversityAlpineTexas6DepartmentofBiologicalSciencesWallaWallaUniversityCollegePlaceWashington7WashingtonDepartmentofFishandWildlifeOlympiaWashington

CorrespondenceShandelleMHensonDepartmentofMathematicsAndrewsUniversityBerrienSpringsMIEmailhensonandrewsedu

Funding informationUSNationalScienceFoundationDivisionofMathematicalSciencesGrantAwardNumberDMS-1225529andDMS-1407040

Abstract1 Baldeagle(Haliaeetus leucocephalus)populationsinNorthAmericareboundedinthe latterpartof the twentiethcentury the resultof tightenedprotectionandoutlawing of pesticides such asDDTAn unintended consequence of recoverymay be a negative impact on seabirdsDuring the 1980s few bald eagles dis-turbed a large glaucous-winged gull (Larus glaucescens) colony on ProtectionIslandWashingtonUSAintheSalishSeaBreedinggullnumbersinthiscolonyrosenearly50duringthe1980sandearly1990sBeginninginthe1990sadra-maticincreaseinbaldeagleactivityensuedwithinthecolonyafterwhichbeganasignificantdeclineingullnumbers

2 Toexaminewhethertrendsinthegullcolonycouldbeexplainedbyeagleactivitywe fit a LotkandashVolterra-type predatorndashpreymodel to gull nest count data andWashingtonStateeagleterritorydatacollectedinmostyearsbetween1980and2016Bothspecieswereassumedtogrowlogisticallyintheabsenceoftheother

3 ThemodelfitsthedatawithgeneralizedR2=082supportingthehypothesisthatgulldynamicswereduelargelytoeaglepopulationdynamics

4 Point estimatesof themodel parameters indicated approach to stable coexist-enceWithinthe95confidenceintervalsfortheparametershowever110ofbootstrappedparametervectorspredictedgullcolonyextinction

5 Ourresultssuggestthattheeffectsofbaldeagleactivityonthedynamicsofalargegullcolonywereexplainedbyapredatorndashpreyrelationshipthatincludedthepossibilityof coexistencebutalso thepossibilityofgull colonyextinctionThisstudy serves as a cautionary exploration of the future not only for gulls onProtectionIslandbutforotherseabirdsintheSalishSeaManagersshouldmoni-tornumbersofnestsinseabirdcoloniesaswellaseagleactivitywithincoloniestodocumenttrendsthatmayleadtocolonyextinction

K E Y W O R D S

Baldeaglesglaucous-wingedgullsLotkandashVolterramodelpredatorndashpreydynamicsProtectionIslandSalishSea

2emsp |emsp emspensp HENSON Et al

1emsp |emspINTRODUC TION

Afteryearsofdeclinebaldeagle(Haliaeetus leucocephalus)popu-lationsthroughoutNorthAmericareboundedinthelatterpartofthe twentieth century following tightened protection reductionintheuseofleadshotbyhuntersandtheoutlawingofpesticidessuchasDDT(Hipfneretal2012WatsonStinsonMcAllisterampOwens2002)This recoveryhasprovidedoneof thegreat suc-cessstoriesoftheconservationmovement(MillarampLynch2006)NowherehasrecoverybeenmorepronouncedthaninthePacificNorthwestofNorthAmericawhereinlandwaterwayssuchastheSalishSeaColumbiaRiverandscoresofsmallerlakesandstreamsprovide ideal perching hunting and nesting opportunities forthese raptors (Elliott ElliottWilson Jones amp Stenerson 2011StinsonWatson ampMcAllister 2001Watson 2002Watson etal2002)

Anunintendedconsequenceofbaldeaglerecoveryhasbeenthenegativeimpactonseabirdswhicharealreadystressedbyoverfish-inggillnettingandhabitatdestruction(AtkinsampHeneman1987BlightDreverampArcese2015)Althoughpopulationsofsomesea-birdsmaybedeclining tohistoric levels (Elliottetal2011) localpopulationsofseabirdssuchascommonmurres(Uria aalge)maybethreatened (ParrishMarvierampPaine 2001)Numbersof salmonand other fish traditionally eaten by wintering bald eagles haveplummeted in recent years affecting eagle survival and possiblyresulting in their shift to other food sources (Elliott et al 2011)Seabirdsalwayshaveformedpartofthedietofeagles(Stalmaster1987)but increasingnumbersofeaglesandconcurrentprey fishshortageshaveresultedinincreasedeagleforagingonwaterfowlacauseforconcernamongornithologists(Elliottetal2011Hipfneretal2012MoulampGebauer2002Parrishetal2001SullivanHazlittampLemon2002VenneslandampButler2004WhiteHeathampGisborne2006)Potentialimpactsofbaldeaglepopulationsonmarine food-web structure appear to be due to resident eaglesratherthanoverwinteringeaglesandtheratesatwhichtheycon-sumeseabirdsasprey(HarveyGoodampPearson2012)

Bald eagles can impact seabirds both directly and indirectly(Hipfner et al 2012Parrish et al 2001) Themostobviousdi-recteffect is thekillingandeatingofadults juvenilesandeggs(DeGangeampNelson 1982HaywardGalusha ampHenson 2010HaywardGillettAmlanerampStout1977)Aseconddirecteffectistheextraexpenditureofenergyneededfornestingorfeedinginthepresenceofeagles(Hensonetal2012Parrishetal2001)An indirect effect results when disturbances displace breedingadults fromtheirnestsandexposeunprotectedeggsandyoungtootherpredators(Haywardetal2010)Asecondtypeofindi-recteffect involves changes indistributionpatterns in responseto thepresenceofeaglesForexampledivingwaterbirds in theStraitofGeorgiamovedawayfrominshorewatersanddabblingducksformedlargeraggregationsinshoreandweremorevigilantin response to increased eagle presence (Middleton Butler ampDavidson2018)

From1900totheearly1980sbreedingpopulationsofglaucous-wingedgulls (Larus glaucescens)markedly increased in theGeorgiaBasinoftheSalishSeaBritishColumbiaBy2010populationshaddeclined to about 50 of peak levels (Blight et al 2015 Sullivanetal2002)AstudythatincorporatedmoresouthernareasoftheSalishSeaalsoreportedoveralldeclinesfrom1975to2007(Bower2009) Protection IslandWashington located in the southeasternStrait of Juan de Fuca and centrally positioned in the Salish Sea

TA B L E 1 emspObserveddata

YearGull Nests Protection Island

Occupied Eagle Territories WA

Observed Eagles Protection Island

1980 3796 105

1981 126

1982 4068 138

1983 168

1984 4726 206

1985 231

1986 250

1987 4958 268

1988 309

1989 5045 369

1990 403

1991 4551 445

1992 468

1993 5189 493 9

1994 547 8

1995 558 19

1996 594 16

1997 4278 582 16

1998 666 24

1999 26

2000 17

2001 673 12

2002 2472 23

2003

2004 2925

2005 840 38

2006 2281

2007

2008 2830

2009 3018

2010 2495

2011 2364

2012 2093

2013 1850

2014 1589

2015 1832

2016 2512

emspensp emsp | emsp3HENSON Et al

has functionedasabreedingcenter formarinebirdssinceat leastthe 1940s (Power 1976) A large glaucous-winged gull colony hadbecome established by the early 1960s (Richardson 1961) TodayRhinoceros auklets (Cerorhinca monocerata) glaucous-wingedgullspigeon guillemots (Cepphus columba) andharbor seals (Phoca vitu‐lina) breed there in large numbersAdult auklets gulls and guille-motsaswellastheeggsandchicksofgullsandtheafterbirthsandpupsofsealsallserveasfoodfornestingandvisitingeagles(CowlesGalushaampHayward2012Hayward2009Haywardetal2010)

During the 1980s few eagle disturbances of the gull colonyonProtection IslandsViolet Pointwere noted and from1980 to1993 gull nest numbers increased by 37 (3796ndash5189 Table 1)Beginninginthe1990showeveradramaticriseinbaldeagleactiv-ityoverandwithinthecolonywasobserved(GalushaampHayward2002Haywardetal2010)withasignificantdeclineinnumbersofbreedinggullsatthesite(Cowlesetal2012)Baldeaglesconstitutetheonlysignificantsourceofinterspecificpredationonthegullsinthiscolony(Haywardetal2014)ThedeclineoftheVioletPointgullpopulationbeganabout1990 (Table1)slightly later thandeclinesfortheSalishSeagenerally(Blightetal2015)butotherwiseVioletPointtrendsparalleledthosereportedfortheregionAlthoughsys-tematiccountsofgullsnestingontheupperplateauofProtectionIslandhavenotbeenmadenestsarenowabsentfromseveralareasthatoncecontainednestinggullsandnestinghasnotexpandedintootherareasoftheisland(JLHaywardunpublishedobservations)

ThedynamicsoftheVioletPointgullcolonybegtwoquestionsFirstistheobserveddeclinecausedatleastinpartbyeagleactiv-itySecond are this andother seabirdpopulationsmerelydeclin-ing to historic levels or are their fates less certain In this studywe usemathematicalmodeling techniques to investigatewhetherthedynamictrendsinnumbersofgullnestsontheVioletPointcol-onycanbeexplainedbythedynamicsinnumbersofoccupiedeagleterritories inWashingtonState a proxy for numbersof eaglesonProtectionIslandandwhetherthereisanapproachtostablecoex-istenceforgullsandeagles

2emsp |emspMATHEMATIC AL MODEL

21emsp|emspClassic LotkandashVolterra predatorndashprey model

IntheirclassicpaperonCanadalynx(Lynx canadensis)andsnowshoehare(Lepus americanus)populationsEltonandNicholson(1942)used100-yearrecordsfromtheHudsonBayCompanyonthenumbersofpeltspurchasedfromtrappersTheclassicpredatorndashpreycyclesoftheoreticalecology(Figure1a)oftenillustratedwithlynx-haredataareproducedbytheLotkandashVolterrapredatorndashpreyordinarydiffer-entialequationmodel(Henson2012Lotka1925Volterra1926)

HeretheldquoprimerdquodenotesthederivativewithrespecttotimeGandE refer tonumbersordensitiesofprey (gulls in this context) and

predators (eagles)agt0 is the per capita growth rate of the preypopulation in theabsenceof thepredatorsbgt0 is thepercapitadeclinerateofthepredatorpopulationintheabsenceofpreyα gt 0 isthepredationrate(theprobabilityperunittimethatagivenpreyindividualwillbetakenbyagivenpredator)andβgt0istheconver-sionrateofpreyintopredators

Theclassicpredatorndashpreymodel (1)hastwomajordeficienciesFirstthepreypopulationgrowsexponentiallywithoutboundintheabsenceofpredatorsandsecondthepredatorpopulationdeclinesexponentially to extinction in the absence of the prey Neither ofthesescenarios is feasible inmostecologicalcommunitiesbecausepopulation growth is always eventually bounded by self-limitationandpredatorsusuallycanswitchpreyandhencedonotdeclinetoex-tinctionwiththeremovalofasinglepreyspeciesThisistrueforbaldeagleswhichareconsideredopportunisticforagers(Buehler2000)

22emsp|emspGull‐eagle predatorndashprey model

ToexaminetherelationshipbetweentheVioletPointgullcolonyandeagleactivityintermsofapredatorndashpreyinteractionwemodifiedthe LotkandashVolterra predatorndashprey model (1) to include a multiplepreybaseforeaglesandself-limitationtermsforbothgullsandea-glesInparticularweusedtheLotkandashVoterra-typeordinarydiffer-entialequationmodel(Ricklefs1990)

where GandE arethenumbersofgullandeaglepairsrespectivelyratherthanindividualanimalsasintheoriginalLotkandashVolterramodelHerergt0andsgt0aretheinherentpercapitagrowthratesforgullsandeagles respectively at smallpopulationsizes and rKgt0andsCgt0aretheirratesofself-limitationTheparameterα gt0denotesthepredation rateof eaglesongulls andβ is the conversion rateofgullsintoeaglebirthsIntheabsenceoftheotherspecies(whenα = β=0) each species grows logistically with carrying capacitiesKgt0forgullsandCgt0foreagles

Model (2) does not predict sustained predatorndashprey cyclesrather it predicts only equilibrium dynamics Derivations of theequilibriaandstabilityformodel(2)areshowninAppendixAherewesimplysummarizethepossibilities

Model(2)hasfourequilibriumstatestheextinctionequilibrium(00)inwhichbothspeciesareabsentanequilibrium(K0)inwhicheaglesareabsentandgullsareattheircarryingcapacityKanequi-librium(0C)inwhichgullsareabsentandeaglesareattheircarryingcapacityC andacoexistenceequilibrium (GE)withgullandeaglenumbersgivenby

Therearetwomaindynamicalternatives

(1)G =aGminusGE

E =minusbE+GE

(2)G = rGminus

r

KG2minusGE

E = sEminuss

CE2+GE

(3)G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs


bull If r gt αC thenbothG and E arepositive inEquation (3) and sothecoexistencestate(3) isbiologicallyfeasibleInthiscasetheequilibria(00)(K0)and(0C)areunstableandthecoexistenceequilibrium(GE)isstableThisequilibriumiseitherastablespiralorastablenodeThatisgullsandeagleseitherapproachtheco-existenceequilibriumthroughdampedpredatorndashpreyoscillations(Figure1b)orelsetheyapproachequilibriuminanonoscillatoryfashion In the lattercaseearly transientdynamicsmayresem-blepredatorndashpreyoscillationsbuttheoscillationsdonotpersist(Figure1c)

bull Ifr lt αCthecoexistenceequilibrium(3)isnotbiologicallyfeasi-ble (becausetheequilibriumnumberofgullsG isnegative)Theequilibria(00)and(K0)arestillunstablebuttheequilibrium(0C)inwhichgullsareabsentandeaglesareatcarryingcapacityC isnowstableThatisgullsapproachextinctionwhereaseaglesapproachtheircarryingcapacityCEarlytransientdynamicsmayresemblepredatorndashpreyoscillationsbeforegullseventuallygoex-tinct(Figure1d)

ThebiologicalinterpretationofthesealternativesisthefollowingThenumber r is the inherentnet reproductiverateofgullsandthenumberαC istherateatwhichgullsaretakenbyCeaglepairsIftheinherentnetreproductiverateofgullsislargerthantherateatwhichgullscanbetakenbyCeaglepairsthengullsandeaglesbothsurviveandapproachapositivecoexistenceequilibriumIfhowevertheinher-entnetreproductiverateofgullsissmallerthantherateatwhichgullscanbetakenbyCeaglepairsthengullsgoextinctandeaglesapproachtheircarryingcapacityC

3emsp |emspMATERIAL S AND METHODS

31emsp|emspGull nest count data for Violet Point Protection Island

Weusedglaucous-wingedgullnestcountdatacollectedbetween1980and2016atalargebreedingcolonyonVioletPointProtectionIslandNational Wildlife Refuge Washington (48deg07prime40PrimeN 122deg55prime3PrimeW)whichliesattheeasternendoftheStraitofJuandeFucaintheSalishSea(Figure2)TheVioletPointcolonyispopulatedbyglaucous-wingedgulls andglaucous-wingedgulltimeswestern gull (L occidentalis) hybrids(Bell19961997)Mostof thesehybrids resembleglaucous-wingedgullsmorethanwesterngulls(MegnaMoncrieffHaywardampHenson2014MoncrieffMegnaHaywardampHenson2013)hencewerefertothesebirdscollectivelyasglaucous-wingedgulls

Gullcountsfrom1980through2014werecarriedoutinsquadfashionasdescribedinGalushaVorvickOppandVorvick(1987)Alineofhumancountersspacedatdistancesappropriatefornestdensity and visibility moved forward over the colony with eachcounter tallyingall nestsbetweenherselforhimself and thenextcounterAfteradistanceof20ndash30mcountertalliesweresummedandrecordedandtheprocesswasrepeateduntiltheentirecolonywascoveredThe2015and2016countsweremadebymappingthepositionofeachnestwithArcGISDesktop10usingdatacollected

byaTrimble6000SeriesGPSTable1containsthefollowingcor-rectionsandadditionsfrompreviouslypublishedvalues(a)CountspublishedbyGalushaetal(1987)inadvertentlyomittedsomesec-tionsofthecolonythathadbeencountedandweaddedcountsforthesesections(b)countsfor2008ndash2010reportedbyCowlesetal(2012)didnotincludecountsofnestsborderingthewestshoreofthemarina(Figure2)absentbefore2008whichwenowadded(JGGalushaunpublisheddata)(c)countsfor2011ndash2015arenewlyreportedand(d)countsfor20132015and2016includeestimatesofuncountednestsborderingthewestshoreofthemarinaderivedfromlinearinterpolationbasedonthe2008ndash2012and2014countsforthatarea

32emsp|emspOccupied eagle territory data for Washington State

Weobtaineddataforthenumberofbreedingterritoriesoccupiedbybald eagles eachyear from1980 to1998 inWashingtonStatefromWatsonetal (2002)Weobtainedeagleoccupancydata for2001 and 2005 from the Wildlife Resource Data System of theWashingtonDepartmentofFishandWildlifeOlympiaWashington(Table1)

33emsp|emspCounts of nesting and non‐nesting eagles on Protection Island

Weobtainedannualmaximumnumbersofsubadultandadultbaldeagles observed simultaneously on Protection Island from 1993

F I G U R E 1 emspPredatorndashpreydynamics(a)Classicpredatorndashpreycycles(b)Coexistenceapproachedthroughdampedoscillations(c)Coexistenceapproachedinanonoscillatoryfashion(d)Extinctionofprey


through2002(Table1)fromHaywardetal(2010)Eaglecountsfor2005were fromunpublishedobservations (J LHaywardunpub-lisheddata)madeinthesamewayasthosepreviouslypublishedinHaywardetal(2010)

34emsp|emspWashington eagle territories as a proxy for eagles on Protection Island correlation analysis and Poisson regression

Numbersofoccupiedeagleterritories inWashingtonStatewerelargerandrelatively lessnoisy thannumbersofeaglesobservedonProtection IslandHence formodel fittingwewished tousea scaled version of the statewide eagle data as a proxy for theProtectionIslandeagledataTodeterminewhetherwecouldusethenumbersofoccupiedeagleterritoriesinWashingtonStateasaproxyforeagleactivityonProtectionIslandweperformedacor-relationanalysisonthenumberofeaglesobservedonProtectionIslandandthenumberofoccupiedeagleterritoriesinWashingtonState for the eight years these data overlapped (1993ndash19982001 2005 Table 1)We considered a strong positive correla-tion (ρge060) if significant at the 005 level a justification forusingtheproxyinfurtheranalysesBecauseourdependentvari-ableinvolvedcountdataweusedPoissonregressiontoobtaintheproxy equation that predicts the number of eagles observedonProtectionIslandasafunctionofthestatewidenumberofeagleterritoriesWeusedtheglmfitfunctioninMatlabreg(MathWorkstradeR2012a)withdispersiontoobtainthePoissonregressionequationrelatingthetwoquantitiesThedispersionparameterisestimated

F I G U R E 2 emspLocationofProtectionIslandandtheVioletPointglaucous-wingedgullcolony(enclosedbywhitepolygon)atthesoutheastendoftheStraitofJuandeFucaWashingtonUpperlinemapscreatedwithSimpleMapprandaerialphotographofislandcreatedwithGoogleEarthPro

Protecon Island

Washington

Idaho

Oregon

Brish Columbia

ProteconIsland

NStrait of

Juan de Fuca

1 kilometer

TA B L E 2 emspPointestimatesfortheparametersinmodel(2)andtheinitialconditionswith95confidenceintervalsEstimatesfortheequilibriawith95confidenceintervals

Estimate 95 CI

Parameter

G0 3663 (28474422)

E0 1067 (10181121)

r 02327 (0104406642)

K 7395 (618610444)

α 00002417 (0000146000005857)

s 01809 (0168201935)

C 8231 (76888890)

β 1876times10minus23 (2652times10minus632082times10minus13)

Equilibrium

G(GullNests) 1072 (01981)

E (EagleTerr) 8231 (76888890)

PIEagles 3494 (28584459)


in order to increase the P-values appropriately if the data areoverdispersed

35emsp|emspModel parameterization

Ecologists must consider several factors when fitting theoreti-calmodels to time series data Populations are subject to bothdemographic and environmental stochasticity resulting in ldquopro-cess noiserdquo (Dennis Ponciano Lele Taper amp Staples 2006)Inaccuracies in theestimatesofpopulationdensitiesalso resultin measurement error (Carpenter Cottingham amp Stow 1994)Althoughmethodshavebeendevelopedtodealwiththesefac-tors(egValpineampHastings2002)thedatademandsforthesemethodsare sometimesprohibitive Inparticular thedatausedhereraisechallengesformodelparameterestimationTherearegaps in the gull and eagle time series and in some years dataare not available for both species The sample sizes differ forgulls and eagles and the numbers for the two species are dif-ferent inmagnitude These difficulties preclude the application

of techniquesbasedonautoregressive timeseries thatunderliemanyof themethods and software commonly inuse (Bolker etal2013)

Toestimatetheparametersinmodel(2)wefirstscaledthestatevariablesGandEaswellasthedatabydividingbytheobservedstandard deviations σg and σe respectively (that is G=G∕120590g andE=E∕120590e)toscalethedatatocomparablemagnitudesItfollowsthatthederivativesareG =G∕120590gandE =E∕120590eandsoonecanrewritemodel(2)intermsofthescaledvariables

To fit model (4) to the scaled data we used the ode45 dif-ferentialequationsolver inMatlabreg toproducepredictedmodeltrajectoriesfrom1980to2016Wetreatedthe(scaled)initialcon-ditions g0ande0asparameterstobeestimatedGivenavectorofparameter estimates120579=

(g0e0rK120572sC120573

) we computed residuals

onthelogscaletoaccountforenvironmentalstochasticitywhich

(4)G = rGminus

r120590g

KG2minus120572120590eGE

E = sEminuss120590e

CE2+120573120590gGE

F I G U R E 4 emspHistogramsoftheequilibriaforbothspeciesbasedon2000bootstrappedparameterestimates

F I G U R E 3 emspObserveddataandmodelpredictionsObserveddata(symbols)andpredictionsofmodels(2)and(11)from1980to2080(curves)arenumbersofoccupiedbaldeagleterritoriesinWashingtonState(solidcircleslightsolidcurve)numbersofbaldeaglesobservedatProtectionIslandWashington(opencirclesdashedcurve)andnumbersofglaucous-wingedgullnestsattheVioletPointcolonyonProtectionIsland(trianglesdarksolidcurve)The95confidenceintervalsforfittedinitialconditionsandpredictedequilibriaaremarkedwithverticallinesontheleft-andright-handsidesofthegraphrespectively


is approximately additive on the log scale (Cushing CostantinoDennis Desharnais amp Henson 2003 Dennis Munholland ampScott1991)

where gt(120579)andet(120579)arepredictedvaluesobtainedbynumericallyin-tegratingmodel(4)fromyear1980to2016usingparametersθtheparametersg0ande0asinitialconditionsandtheobservedstandarddeviationsσgandσeforgullsandeaglesrespectivelyWeobtainedbestfitparameters120579byminimizingthesumoftherootmeansquares(RMS)oftheresiduals

asafunctionofθwherengandnearethenumberofresidualsforgullsandeaglesrespectivelyusingthefminsearchdownhillsearchalgorithminMatlabreg

The fittingmethoddescribedabove isbasedonanumberofconsiderationsFirst thetwotimeseriesarenotpaired inmanyyears estimates are available for only one of the two speciesTherefore we cannot view the data for every year as a tradi-tional bivariate observationwhichwould allow a traditional sumofsquarederrorsandwealsocannotuseone-steppredictionsincomputingresidualsSecondeachspecieshasdifferentnumbersofobservationssothesumofsquaredresidualsmustbescaledbythenumberofobservationsOtherwisetheparameterestimateswouldbiasthespecieswiththemostobservationsThirdwecan-notestimateparametersusingseparateRMSvaluesbecausetheequationsarecoupledsothefitofonespeciesaffectsthefitoftheotherFourththeoverallmagnitudesofthetwospeciesdiffer

hencewescaledthedatabythestandarddeviationssothatthetwotermsintheRMSequationwouldbecommensurateandtheparameter estimateswould not be biased in favor of fitting thespecieswithoverallhighernumbers

WeperformeddiagnosticanalysesofthegullandeagleresidualstocheckforindependenceandnormalityWeplottedtheresidualsasafunctionoftimeandexaminednormalquantilendashquantileplotsfordeparturesfromnormalityWecomputedfirst-andsecond-orderautocorrelationsofthegullandeagleresidualsthatwereseparatedbyoneandtwoyearsrespectivelyandtestedthesecorrelationsforsignificanceWealsocomputed theShapirondashWilk test statistic fornormality(ShapiroampWilk1965)

36emsp|emspGoodness‐of‐Fit

WeusedageneralizedR2tocheckthegoodnessoffitofthescaledmodel(4)tothescaleddata

HereRMSM is the fitted rootmean square usingmodel (4) as thepredictorandusingEquation(5)tocomputetheresidualswhereasRMSTisthesumoftherootmeansquaresusingthecentraltendency(mean)ofthedataasthepredictorandusingthefollowingequationtocomputetheresidualsinEquation(6)forRMST

Wealsocomputedtheadjustedgoodness-of-fitRA2by

where p=8isthenumberofestimatedmodelparametersIngen-eralRA

2issmallerthanR2becauseittakesintoaccountthenum-berofestimatedparametersandpenalizesthegoodnessoffitasp increasesAversionofEquation(9)isusedinmultipleregressionmodelsbutinthatcaseoneusesminus1insteadofminus2(Zar2009)Herewehavetwomeans(forgullsandeagles)insteadofonemeansowe reduce thedegreesof freedombyonemoreunit Equations(7)and (9)representaldquogeneralizedrdquocoefficientofdetermination(Anderson-Sprecher1994)notthetraditionalvalueusedinlinearregression

37emsp|emspConfidence intervals for parameters

Onceadeterministicmodelhasbeenfittedtopopulationtimeseriesdatabootstrappingmethodscanbeusedtoobtainconfidence in-tervalsfortheestimatedparameters(DennisDesharnaisCushingHensonampCostantino 2001 Falck Bjornstadamp Stenseth 1995)We randomly sampledwith replacement from themodel residu-als12⋯ ngand12⋯ ne tocreatesetsofsurrogate residuals

(5)120574t= ln

(Gt

)minus ln

(gt(120579)

)

120576t= ln(Et

)minus ln

(et(120579)

)

(6)RMS()=

ngsumt=1

t2

ng+

nesumt=1

t2

ne

(7)R2=1minusRMSM

RMST

(8)120574t= ln

(Gt

)minusmean

[ln(Gt

)]

120576t= ln(Et

)minusmean

[ln(Et

)]

(9)RA2=1minus

ng+neminus2

ng+neminuspminus2

(1minusR2

)

F I G U R E 5 emspScatterplotofthe2000estimatesofr versus α withthecoexistencecutoffconditionr = αCforthepointestimateofCappearingasadashedlineEstimatesbelowthatlineleadtogullcolonyextinctionThedottedlinesarer = αCusingthelowerandupper95CIboundsforparameterCTheareabetweenthedottedlinescouldbeconsideredanldquouncertaintyparameterregionrdquoforcoexistence


lowast1lowast2⋯ lowast

ngandlowast

1lowast2⋯ lowast

neThetimeorderoftheresidualswas ig-

noredwhensampling(Dennisetal2001Falcketal1995)Thesurrogateresidualswereusedtocreatesurrogatedata

For each surrogate data set we estimated point parametersusing the method explained in section 36 This process was re-peatednS=2000timesusinganindependentrandomsamplingoftheoriginalresidualsforeachiterationIfthefminsearchalgorithmdidnotconvergetoasolutionwithin1000functionalevaluationsthesestepswererepeatedforanewsetofsurrogatedataThisoc-curredatarateof216The lackofconvergence insomeofthebootstraprealizationswasduetothefact thatthetimeseriesarerelativelyshortandconsequently theoverallnumberof residualsissmallfrequentlyleadingtosetsofresampledresidualsthatnega-tivelyimpacttherateofconvergencefortheminimizationalgorithmWe independently repeated the analyses several times with onlytrivialvariationsintheresultstoverifythat2000repetitionswereadequateand that thenonconvergentbootstrap realizationswerenotaproblem

This procedure yielded a set of bootstrapped parameter esti-mateslowast

1lowast2⋯ lowast

ns that should reflect the variation onewould see

in thebest fitparametersassumingthemodel (4) isvalidandtheobservedresidualsfromthemodelarerandomeffectswithnoauto-correlationorcross-correlation

The95confidenceintervalsforthepointparameterestimateswereobtainedbyrankingtheparameterestimatesforthesurrogatedatasetsandcomputingthe25thand975thpercentiles(Dennisetal2001)

4emsp |emspRESULTS

Numbers of eagles observed on Protection Island were stronglypositively correlated with numbers of occupied eagle territoriesinWashington (ρ=086p=0006n=8)Poisson regressionpro-ducedtherelationship

(FigureA1)withsignificantslopecoefficient (p=00057)butnon-significantintercept(p=042)Theestimateddispersionparameterwas123indicatingasmallamountofoverdispersion

For the fittedpoint estimatesof theparameters (Table2) therelationshipr gt αCholdsindicatingthatthecoexistenceequilibriumis stableParameterβ is effectively equal to zeroThus theeaglepopulation is predicted to grow logistically without dependenceonthegullpopulationThemodelpredictsthatgullnestsonVioletPointwill equilibrate at 1072 in contrast to the estimated carry-ingcapacityofK=7395nestingpairs(Table2)EagleterritoriesinWashington are predicted to equilibrate at 823 territories which

is equal to thepredicted carrying capacityC (Table2) The good-ness-of-fits indicated that themodel explains at least 77 of thevariabilityinthedata(R2=0819andRA

2=0772)Atthepredictedequilibriumof823eagleterritoriesEquation(11)predictsanequi-libriumof35eaglevisitorsonProtectionIslandFittedmodelpredic-tionsfortheyears1980ndash2080areshowninFigure3

Ouranalysisofthemodelresidualsforgullsandeagles(Equation5) reveals no evidence of significant violations of our model as-sumptions The first- and second-order autocorrelation values forgulls were g

(1)=04580 (n=8 p=02538) and g

(2)=minus02975

(n=15 p=02815) The autocorrelation values for eagles weree

(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17

p=02817) Neither of the ShapirondashWilk test statistics for gullsandeaglesweresignificantWg=09875 (p=09134) forgullsandWe=09863 (p=09862) for eaglesHowever thepowerof thesetests is limited due to small sample sizes Time series and normalquantilendashquantileplotsofthemodelresidualsappearinFigureA2

A scatterplotmatrix of the bootstrapped parameter estimatesexcludingβwhichwasalwaysclosetozero isshowninFigureA3The diagonal plots are histograms showing the distribution of the2000estimatesforeachparameterNoneofthesehistogramssug-gestunusualpropertiesforthedistributionssuchashighskewnessormultiplemodesTheoff-diagonalscatterplotsshowthepairwiserelationshipsbetween theparameters for the2000estimatedpa-rametervectorsTheseplotscanrevealstrongorunusualdependen-ciesbetweenparameterestimatesasisthecaseforparametersr and αThissuggeststhatlargeestimatesofgullpopulationgrowthratescoincidewithlargeestimatesofgullpredationratesandviceversa

In134ofcasesthevectorofparameterestimatespredictedgullcolonyextinctionPredictedequilibriawerederivedforeachofthe 2000 bootstrapped parameter vectors and plotted for bothspecies(Figure4)Thescatterplotofthe2000estimatesshowsr versus αwiththecoexistencecutoffconditionr = αCforthepointestimateofCappearingasadashedline(Figure5)Estimatesbelowthat line leadtogullcolonyextinctionThedotted linesare r = αC usingthelowerandupper95CIboundsforparameterCTheareabetweenthedotted linescouldbeconsideredanldquouncertaintypa-rameterregionrdquoforcoexistence

Althoughourdeterministicmodelcannotpredicta time toex-tinctionwecancomputetheamountoftimeittakesgullnumberstofallbelowathresholdinthosecases(267of2000)forwhichthebootstrappedparametervectorspredictextinctionIfwedefinethethresholdas10of theestimatedcarryingcapacity forgulls theestimatedmeanfortheyearinwhichextinctionoccursis2039witha95confidenceintervalof(20222059)Forathresholdof5ofKthemeanis2073witha95confidenceintervalof(20402122)

5emsp |emspDISCUSSION

Wehavedemonstratedastrongdynamicrelationshipbetweenthebald eagle population inWashington State and numbers of glau-cous-winged gull nests on Protection Islands Violet Point colony

(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701

(WAEagle Territories

)


This relationship exhibits a LotkandashVolterra-type dynamic that atthepointestimatesof theparameterspredicts long-termcoexist-enceandequilibriumforthetwospeciesThemodeldoesnotpre-dictpredatorndashpreyoscillationsasdepictedoriginallybyLotka(19251932) and Volterra (1926) It is notable however that the modelpredictsgullcolonyextinctionforsomeparameterswithinthe95confidenceintervalsaboutthepointestimates

Thecarryingcapacityestimate forWashingtoneagleoccupiedterritories(823)iscurrentlyapproximatelyrealizedandperhapshasbeenexceededTherearenowmorethan1000nestingterritoriesinWashingtonStatealthoughthenestsarenotallactiveduringthesame years Themost recent comprehensive survey results showthat in2001therewere923territoriescheckedand705foundtobeoccupiedandin2005thenumbersincreasedto1158territoriescheckedand893occupied(JWWatsonunpublisheddata)

EagleshavenestedonProtectionIslandsinceatleastthe1920s(CowlesampHayward2008)andoneortwoeagletswereraisedfromanestlocatedontheislandduringmanyyearssincetheearly1980s(Hayward et al 2010) Large numbers of transient eagles are at-tractedtothe islandeachbreedingseasonForexampleon4July200453eagleswerecountedduringaboattriparoundtheisland(Neil Holcomb US Fish andWildlife Service volunteer personalcommunication)ThenumberofeaglespresentonProtectionIslandtypicallypeakseachyearduringthesecondweekofJulywhengullchicksarehatchingandsealafterbirthsanddeadpupsaremostabun-dant(Haywardetal2010)Transientadultssometimesarechasedbacktothemainlandbyadultresidents(unpublishedobservations)

The impactofbaldeagleson seabirdshasbecome increasinglyapparent as populations have recovered and some marine fishpopulations on which they feed have declined (Anderson BowerNysewanderEvensonampLovvorn2009AndersonLovvornEslerBoyd amp Stick 2009 Stick amp Lindquist 2009 Therriault Hay ampSchweigert2009)AlongthewestcoastofNorthAmericabaldea-gleshavebeen implicatedasbeingresponsiblefordeclines in localpopulations of common murres (Uria aalge Parrish et al 2001Hipfner Morrison amp Darvill 2011) double-crested cormorants(Phalacrocorax auritusChatwinMatherampGiesbrecht2002HarrisWilson amp Elliott 2005) pelagic cormorants (P pelagicus Chatwinetal2002Harrisetal2005CarterHebertampClarkson2009)greatblueherons(Ardea herodiasVenneslandampButler2004)west-erngrebes(Aechmophorus occidentalisBower2009)andglaucous-wingedgulls(Haywardetal2010Sullivanetal2002)Thedeclineshavebeendramatic insomeplacesForexamplea131-kmstretchalong thecoastofOregon formerly supportedmore than380000breeding pairs of commonmurres but successful reproduction bythesebirdstodayisvirtuallynonexistentEntirecolonieshavebeenabandonedMurresthatremainincoloniesharassedbybaldeaglestypicallygiveuponthebreedingprocessbeforecompletingthenest-ingseason(Hipfneretal2012)SimilareffectsonseabirdshavebeennotedinNorthernEuropewherepopulationsofwhite-tailedeagles(H albicilla)havereboundedfromdeclines(Hipfneretal2012)

Thedeclineinnumbersofglaucous-wingedgullsonVioletPointProtection Island paralleled declines and eventual extirpation of

double-crestedandpelagiccormorantsnestingonProtectionIslandTwo colonies containing several hundred pairs of double-crestedcormorantsthrivedonProtectionIslandformanyyearsandasmallpelagiccormorantcolonyexistedthereforseveralyearsbuteagledisturbancesfrequentlycausedcolonyresidentstofleetheirnestsby2007allthreecolonieswerevacantandhaveremainedso(JLHaywardunpublisheddata)Rhinocerosauklets(Cerorhinca monoc‐erata) remain abundant breeders on Protection Island (PearsonHodum Good Schrimpf amp Knapp 2013) although in 2001 theycomprisedthemostcommonremainsbeneathanactivebaldeaglenestontheisland(Haywardetal2010)Ourinformalobservationssuggestthataukletsarepreyeduponmostlyduringpredawnhourswhenaukletsleavetheirnestburrowstoforage

Althoughthemodelpredictscoexistenceatthepointparameterestimatesthe95confidenceintervalsincludethepossibilityofex-tinctionOfthe2000bootstrappedparametervectorsextinctionwaspredictedin134ofthecasesWhenwelimitourpredictiontothe1788cases inwhichtheelementsofthebootstrappedpa-rametertriplets(rαC)areinsidetheir95confidenceintervalsthenin 110 of the cases gull colony extinctionwas predicted Thusforsomeparametervalueswithinthe95confidenceintervalsourmodelpredictsthattheVioletPointgullcolonyonProtectionIslandwilldisappear Ifwedefine the threshold forextinctionas10oftheestimatedcarryingcapacityforgullstheestimatedmeanfortheyear inwhichextinctionoccurs is2039witha95confidence in-tervalof(20222059)Forathresholdof5themeanis2073witha 95 confidence interval of (2040 2122) In fact extinction didoccuronColville Island located33kmnorthofProtection IslandTheColvilleglaucous-wingedgullcolonygrewfrom1273pairs in1963to1808pairs in1975 (AmlanerHaywardSchwabampStout1977ThoresenampGalusha1971)By2000howeveronly~20pairsnestedonColvilleandmorerecentobservationssuggestthatnest-inggullsareabsentfromtheisland(JLHaywardunpublisheddata)Baldeagleswereknowntodisturbandpreyonthesegullsduringthe1970s(Haywardetal1977)althoughitisunknownwhetherthiswasthecauseofcolonyabandonment

Extinction of the Protection Island gull colony would impactthelocalecosysteminavarietyofwaysTheeffectsonvegetationwouldbepronouncedGullsphysicallyaltervegetationintheircol-oniesthroughtramplingdiggingofnestscrapescollectionofnestmaterialanddisturbanceduringboundarydisputestheychemicallyalter the soil through defecation and regurgitation of nondigest-ible componentsof food (Ellis FarintildeaampWitman2006 LindborgLedbetterWalatampMoffett2012SobeyampKenworthy1979)andthedecompositionofadultandjuvenilegullcarcassesonbreedingcoloniescontributesnutrientstothesoil(EmslieampMessenger1991LordampBurger1984)Thusextinctionofthegullcolonywouldre-sult in significant changes in thevegetationand inorganisms thatdependonthatvegetation (SobeyampKenworthy1979)and in theloss of a nutrient subsidy to the waters surrounding ProtectionIsland(Hutchinson1950Leentvaar1967McCollampBurger1976)Extinction alsowould eliminate a significant local food source forbaldeaglesalthoughgullsarenottheironly islandfood(Hayward


etal2010)andbaldeaglesmaygraduallyrelocateinresponsetodwindlinghistoricsourcesofprey(McClellandetal1994)

FactorsotherthanthedirectandindirecteffectsofeaglesalsomayimpactgullpopulationsGullsarescavengersandgullpopulationsin-creaseddramaticallyduringmostof the twentiethcentury (Amlaneret al 1977Duhem Roche Vidal amp Tatoni 2008 KadlecampDrury1968Sullivanetal2002)Closureoflandfillsinthisandotherregionsworldwidehasbeenassociatedwithsharplyreducedgullpopulations(Payo-Payoet al 2015) Indeed the1992closureof theCoupevilleLandfill (Anonymous 2001) a popular feeding site for gulls located19kmnortheastoftheVioletPointgullcolony(Schmidt1986)wasfol-lowedbya10-yeardeclineingullnestcountsonVioletPoint(Figure3)Declinesinforagefishpopulations(Blightetal2015McKechnieetal 2014 Therriault et al 2009)may have played a role in gull de-clinesGullsnestonlyalong theedgesofdunegrass (Leymus mollis)soincreasesincoverbythisplantcouldimpactthesizeofthecolonyDunegrasscoverincreasedfrom25ha(14ofVioletPoint)in1980to66ha(39ofVioletPoint)by2009(Cowlesetal2012)ConsiderableareasuitablefornestinghoweverremainsunoccupiedsuggestingthatdunegrassisnotalimitingfactorIncreasingseasurfacetemperatures(SSTs) thatoccurwithElNintildeoeventsandclimatechangehavebeenimplicatedasafactorthatincreasesgulleggcannibalismanddecreasesgull colony reproductive output (Hayward et al 2014)We do notknowwhethertheeffectsofincreasingSSTsinteractinsomewaywitheagleeffectsonthegullpopulationAlthoughthesevariousconfound-ingfactorswhicharenotincludedexplicitlyinourmodelundoubtedlycontributedtothedeclineingullnumbersitisimportanttonotethatthegulldynamicsneverthelessarewellpredictedbythemodelThissuggeststhateagledynamicsareoneofthemostexplanatoryfactorsinvolvedinthedeclineofgullsonProtectionIsland

Thedatausedhereraisedchallengesformodelparameteresti-mationWhilemanysophisticatedtechniqueshavebeendevelopedto dealwith deficiencies in ecological time series data (eg ClarkampBjoslashrnstad2004Clark2007)ourgoalwas toobtaina reason-ablefitofthemodeltothedataandfocusontheimplicationsofthemodel predictions forwildlifemanagementWe scaledpopulationnumbersbytheirstandarddeviationsandusedthesumoftherootmeansquaresof themodel residuals (Equation6)as theobjectivefunctionforordinaryleastsquares(OLS)minimizationSomeoftheestimation issuescanbemitigated ifoneassumes thatoneof theequationsisdecoupledfromtheotherasisthecasewhenβ=0InAppendixBwepresentasecondmethodbasedonthisassumptionwhereweuseOLSparameterestimationontheunscaledpopulationdataseparatelyforeachspeciesTheresultingparameterestimates(showninTableA1)arenearlyidenticaltothoseinTable2

Anotherissuewiththeparameterestimationandbootstrappingmethods isa lackof independenceofthemodelresidualsThisoc-curs becausewe are directly fitting themodel to time series datausingOLSAnalternateapproachtoparameterestimationistotakeadvantage of transitions between consecutive model states usingconditional leastsquares (CLS)For thismethodoneusesparame-tervaluesandtheobservedpopulationnumbersatagiventimetopredictthepopulationvaluesinthefollowingtimeinterval(seeeg

DennisDesharnaisCushingampCostantino1995)OnerepeatsthisprocedureforalltimestepsThebestsetofparameterestimatesareonesthatminimizethesumofthesquareddeviationsbetweentheobservednumbersandtheone-steppredictionsThisapproachtakesadvantageoftheMarkovassumptionimplicitintheordinarydiffer-ential equation model Future states depend only on the presentnotthepastSincethedatausedforparameterestimationarecon-ditionalone-steptransitionstheldquoobservationsrdquoarebyassumptionindependent If thepopulationdataarenot spacedevenly in timehowever thenonecannotassume that the randomone-stepdevi-ationsare identicallydistributedsincethevariancewill ingeneraldependon the lengthof the time stepAlso if onedoesnothaveobservationsforallthestatespacevariablesatthesametimesthenone-steppredictionsarenotpossibleGiventhatbothsituationsexistforthegullandeagledatawewereunabletousetheCLSmethodMore complex methods such as Bayesian state space modelingandGibbssamplingmightmitigatethesedatadeficiencies(ClarkampBjoslashrnstad2004)NeverthelessourOLSandbootstrappingproce-duresprovidedparameterestimateswithagoodvisualfittothedataandmodelresidualsthatshowednoevidenceofautocorrelationordeviationsfromnormalityThetechniquesweusedarenotspecifictothepredatorndashpreysystemweanalyzedandcouldproveusefulinothersituationswheretheecologicaldataprovidesimilarchallenges

Our parameter estimation and model predictions for gulls onProtectionIslandarebasedonstatewideeagledataThisisbecausethese data are more frequent and consistent over time than theeagleobservationsforProtectionIslandHoweverthePoissonre-gressioncanbeusedtopredicteaglenumbersontheislandbasedonthestatewidedataWerepeatedourparameterestimationpro-cedureusingthepredictedislandeaglenumbersfromthenonlinearEquation (4) The estimated parameter values appear in the TableA1TheoverallfitwasslightlypoorerwithgeneralizedcoefficientsofdeterminationofR2=0789andRA

2=0735Agraphicalcompar-isonof themodel fits for the regression-predictedeaglenumbersversusstatewidedata (FigureA4)suggeststhattheformerunder-estimates observed gull numbers for the time period 1984ndash1997Agraphicalanalysisoftheresidualssupportsthisobservationandsuggests some consistent departures from normality (Figure A5)Moreovertheanalysisfortheregression-predictedeaglenumbersdoesnottakeintoaccounttheerrorassociatedwiththeregressionpredictions which based on Figure A1 could be substantial Forthese reasons we feel that the parameter estimation and modelanalysesbasedonthestatewideeagledataaremorereliable

6emsp |emspCONCLUSIONS

Wehaveshownthatthedynamicsofaglaucous-wingedgullcolonyonProtectionIslandfrom1980ndash2016canbeexplainedbythenum-berofoccupiedbaldeagleterritories inWashingtonwithgeneral-izedR2=082ThissupportsthehypothesisthattheriseanddeclineingullnumbersobservedonProtectionIslandareduelargelytothedecline and recovery of the bald eagle populationWe also have


shown that with 95 confidence the long-term dynamic predic-tionsincludecoexistencebutalsothepossibilitythatthegullcolonywilldisappearasoccurredonColvilleIsland

Thisstudyservesasareminderthatthenecessaryandsuccess-fulmanagementofonespeciescanhavedirectanddramaticeffectsonotherspeciesanditillustratestheuncertaintyofthoseeffectsItservesasacautionaryexplorationofthefuturenotonlyforgullsonProtectionIslandbutforotherseabirdsintheSalishSeaInpartic-ularmanagersshouldmonitorthenumbersofnestsinseabirdcol-oniesaswellastheeagleactivitywithinthecoloniestodocumenttrendsthatmayleadtocolonyextinction

ACKNOWLEDG MENTS

WethankJenniferBrown-ScottLorenzSollmannandSueThomasWashingtonMaritimeNationalWildlifeRefugeComplexforpermis-sion toworkonProtection IslandNationalWildlifeRefugeDerekStinson and Scott Pearson Washington Department of Fish andWildlifefordiscussionsRosarioBeachMarineLaboratoryforlogis-ticalsupportandJonathanCowlesfornumericalexplorationsRADthanksRichardMMurrayattheCaliforniaInstituteofTechnologyforhishospitalitywhileportionsofthisworkwerecompletedthereThisresearchwassupportedbyUSNationalScienceFoundationgrantsDMS-1407040(SMHandJLH)andDMS-1225529(RAD)

CONFLIC T OF INTERE S T

Nonedeclared

AUTHORSrsquo CONTRIBUTIONS

SMHandETFposedthemathematicalmodelRADandSMHconductedstatisticalmodelfittingandtimeseriesanalysisJWWprovidedeagledataJGGandJLHprovidedgulldataAllauthorscontributedtothewritingofthemanuscriptAllauthorsgavefinalapprovalforpublication

DATA ACCE SSIBILIT Y

ThedataareavailableinTable1ofthismanuscript

ORCID

Shandelle M Henson httpsorcidorg0000-0001-8439-7532

R E FE R E N C E S

AmlanerCJHaywardJLSchwabERampStoutJF(1977)Increasesin a population of nesting glaucous-winged gulls disturbed by hu-mansThe Murrelet5818ndash20httpsdoiorg1023073535708

Anderson E M Bower J L Nysewander D R Evenson J R ampLovvorn J R (2009) Changes in avifaunal abundance in a heav-ilyusedwinteringandmigration site inPugetSoundWashingtonduring1966ndash2007Marine Ornithology3719ndash27

Anderson E M Lovvorn J R Esler D BoydW S amp Stick K C(2009)UsingpredatordistributionsdietandconditiontoevaluateseasonalforagingsitesSeaducksandherringspawnMarine Ecology Progress Series386287ndash302httpsdoiorg103354meps08048

Anderson-SprecherR(1994)ModelcomparisonsandR2 The American Statistician48113ndash117

Anonymous (2001) Trash talk waste division heads it upmoves it out Whidbey News‐Times 6 June Accessed onlineon 2 July 2018 httpwwwwhidbeynewstimescomnewstrash-talk-waste-division-heads-it-up-moves-it-out

AtkinsNampHenemanB(1987)ThedangersofgillnettingtoseabirdsAmerican Birds411395ndash1403

BellDA (1996)Geneticdifferentiationgeographicvariationandhy-bridization in gulls of the Larus glaucescens‐occidentalis complexCondor98527ndash546httpsdoiorg1023071369566

BellDA (1997)Hybridizationandreproductiveperformance ingullsof theLarus glaucescens‐occidentaliscomplexCondor99585ndash594httpsdoiorg1023071370471

BlightLKDreverMCampArceseP (2015)Acenturyofchangeinglaucous-winged gull (Larus glaucescens) populations in a dynamiccoastalenvironmentCondor117108ndash120

BolkerBMGardnerBMaunderMBergCWBrooksMComitaL hellip Zipkin E (2013) Strategies for fitting nonlinear ecologicalmodels inRADModelBuilderandBUGSMethods in Ecology and Evolution4501ndash512httpsdoiorg1011112041-210X12044

BowerJL(2009)ChangesinmarinebirdabundanceintheSalishSea1975to2007Marine Ornithology379ndash17

BuehlerDA(2000)Baldeagle(Haliaeetus leucocephalus)InAFPooleampFBGill(Eds)The Birds of North America No 506 A P a F Gill PhiladelphiaPATheBirdsofNorthAmericaInc

CarpenterSRCottinghamKLampStowCA(1994)Fittingpreda-tor-preymodelstotimeserieswithobservationerrorsEcology751254ndash1264

CarterHRHebertPNampClarksonPV(2009)DeclineofpelagiccormorantsinBarkleySoundBritish Columbia Wildlife Afield43ndash32

ChatwinTAMatherMHampGiesbrechtTD(2002)Changesinpe-lagicanddouble-crestedcormorantnestingpopulationsintheStraitof Georgia British Columbia Northwestern Naturalist 83 109ndash117httpsdoiorg1023073536609

ClarkJS(2007)Models for ecological data An introductionPrincetonNJPrincetonUniversityPress

Clark J SampBjoslashrnstadON (2004)Population time seriesProcessvariability observation errors missing values lags and hiddenstatesEcology853140ndash3150httpsdoiorg10189003-0520

CowlesD L Galusha J G ampHayward J L (2012)Negative inter-speciesinteractionsinaglaucous-wingedgullcolonyonProtectionIslandWashingtonNorthwestern Naturalist9389ndash100httpsdoiorg101898nwn11-121

Cowles D L amp Hayward J L (2008) Historical changes in thephysical and vegetational characteristics of Protection IslandWashington Northwest Science 82 174ndash184 httpsdoiorg1039550029-344X-823174

CushingJMCostantinoRFDennisBDesharnaisRAampHensonSM(2003)Chaos in ecology Experimental nonlinear dynamicsSanDiegoCAAcademicPress

de Valpine P amp Hastings A (2002) Fitting popula-tion models incorporating process noise and observa-tion error Ecological Monographs 72 57ndash76 httpsdoiorg1018900012-9615(2002)072[0057FPMIPN]20CO2

DeGangeARampNelsonJW(1982)BaldeaglepredationonnocturnalseabirdsJournal of Field Ornithology53407ndash409

DennisBDesharnaisRACushingJMampCostantinoRF(1995)NonlineardemographicdynamicsMathematicalmodels statisticalmethods and biological experiments Ecological Monographs 65261ndash282httpsdoiorg1023072937060


DennisBDesharnaisRACushingJMHensonSMampCostantinoR F (2001) Estimating chaos and complex dynamics in an in-sect population Ecological Monographs 71 277ndash303 httpsdoiorg1018900012-9615(2001)071[0277ECACDI]20CO2

DennisBMunhollandPLampScottJM(1991)Estimationofgrowthand extinction parameters for endangered species Ecological Monographs61115ndash143httpsdoiorg1023071943004

DennisBPoncianoJMLeleSRTaperMLampStaplesDF(2006)Estimating density dependence process noise and observationerror Ecological Monographs76323ndash341httpsdoiorg1018900012-9615(2006)76[323EDDPNA]20CO2

Duhem C Roche P K Vidal E amp Tatoni T (2008) Effects of an-thropogenic food resources on yellow-legged gull colony size onMediterranean islandsPopulation Ecology50 91ndash100 httpsdoiorg101007s10144-007-0059-z

ElliottKHElliottJEWilsonLKJonesIampStenersonK(2011)Density-dependenceinthesurvivalandreproductionofbaldeaglesLinkages to chum salmon The Journal of Wildlife Management 751688ndash1699httpsdoiorg101002jwmg233

Ellis J C Farintildea J M amp Witman J D (2006) Nutrient transferfrom sea to land The case of gulls and cormorants in the Gulfof Maine Journal of Animal Ecology 75 565ndash574 httpsdoiorg101111j1365-2656200601077x

EltonCampNicholsonM(1942)Theten-yearcycleinnumbersofthelynx inCanadaJournal of Animal Ecology11215ndash244httpsdoiorg1023071358

Emslie SDampMessenger S L (1991)Pellet andboneaccumulationatacolonyofwesterngulls(Larus occidentalis) Journal of Vertebrate Paleontology11133ndash136

FalckWBjornstadONampStensethNC(1995)BootstrapestimateduncertaintyofthedominantLyapunovexponentforHolarcticmicro-tinerodentsProceedings of the Royal Society of London B261159ndash165

GalushaJGampHaywardJL(2002)Baldeagleactivityatagullcolonyand seal rookery on Protection Island Washington Northwestern Naturalist8323ndash25httpsdoiorg1023073536511

GalushaJGVorvickBOppMRampVorvickPT (1987)NestingseasoncensusesofseabirdsonProtectionIslandWashingtonThe Murrelet68103ndash107httpsdoiorg1023073534115

HarrisMLWilsonLKampElliottJE(2005)AnassessmentofPCBsandOCpesticidesineggsofdouble-crested(Phalacrocorax auritus) andpelagic(P pelagicus)cormorantsfromthewestcoastofCanada1970 to 2002 Ecotoxicology14607ndash625

HarveyCJGoodTPampPearsonSF(2012)Topndashdowninfluenceofresidentandoverwinteringbaldeagles(Haliaeetus leucocephalus) inamodelmarineecosystemCanadian Journal of Zoology 90903ndash914

Hayward J L (2009) Bald eagle predation on harbor sealpups Northwestern Naturalist 90 51ndash53 httpsdoiorg1018981051-1733-90151

HaywardJLGalushaJGampHensonSM(2010)Foraging-relatedactiv-ityofbaldeaglesataWashingtonseabirdcolonyandsealrookeryJournal of Raptor Research4419ndash29httpsdoiorg103356JRR-08-1071

HaywardJLGillettWHAmlanerCJampStoutJF(1977)PredationongullsbybaldeaglesinWashingtonThe Auk94375

Hayward J LWeldon LMHenson SMMegna LC PayneBG ampMoncrieff A E (2014) Egg cannibalism in a gull colony in-creaseswithseasurface temperatureCondor11662ndash73httpsdoiorg101650CONDOR-13-016-R11

HensonSM (2012)Phaseplaneanalysis InAHastingsampLGross(Eds)Encyclopedia of theoretical ecology(p538ndash545)BerkeleyCAUniversityofCaliforniaPress

HensonSMWeldonLMHaywardJLGreeneDJMegnaLCampSeremMC(2012)CopingbehaviourasanadaptationtostressPost-disturbance preening in colonial seabirds Journal of Biological Dynamics617ndash37httpsdoiorg101080175137582011605913

HipfnerJMBlightLKLoweRWWilhelmSIRobertsonGJBarrettRThellipGoodTP(2012)UnintendedconsequencesHowtherecoveryofseaeagleHaliaeetusspppopulationsinthenorthernhemisphereisaffectingseabirdsMarine Ornithology4039ndash52

Hipfner JMMorrisonKWampDarvillR (2011)PeregrineFalconsenabletwospeciesofcolonialseabirdstobreedsuccessfullybyex-cluding other aerial predatorsWaterbirds 34 82ndash88 httpsdoiorg1016750630340110

HutchinsonGE(1950)Surveyofcontemporaryknowledgeofbiogeo-chemistry3ThebiogeochemistryofvertebrateexcretionBulletin of the American Museum of Natural History9634

KadlecJampDruryWH(1968)StructureoftheNewEnglandherringgullpopulationEcology49645ndash676httpsdoiorg1023071935530

Leentvaar P (1967) Observations on guanotrophic environmentsHydrobiologia29441ndash489

LindborgVALedbetterJFWalatJMampMoffettC(2012)Plasticconsumption anddietof glaucous-wingedgulls (Larus glaucescens) Marine Pollution Bulletin64 2351ndash2356 httpsdoiorg101016jmarpolbul201208020

LordWDampBurgerJF(1984)Arthropodsassociatedwithherringgull(Larus argentatus)andgreatblack-backedgull(Larus marinus)carriononislandsintheGulfofMaineEnvironmental Entomology131261ndash1268

LotkaAJ(1925)Elements of physical biologyBaltimoreMDWilliamsandWilkins

LotkaAJ(1932)ThegrowthofmixedpopulationsTwospeciescom-petingforacommonfoodsupplyJournal of the Washington Academy of Science22461ndash469

McClellandBRYoungLSMcClellandPTCrenshawJGAllenH L amp SheaD S (1994)Migration ecology of bald eagles fromautumn concentrations inGlacierNational ParkMontanaWildlife Monograph1251ndash61

McCollJGampBurgerJ(1976)ChemicalinputsbyacolonyofFranklinrsquosgulls nesting in cattailsAmerican Midland Naturalist96 270ndash280httpsdoiorg1023072424068

McKechnie I Lepofsky D Moss M L Butler V L Orchard T JCouplandGhellipLertzmanK(2014)Archaeologicaldataprovideal-ternativehypothesesonPacificherring(Clupea pallasii)distributionabundance and variability Proceedings of the National Academy of Sciences of the United States of America111E807ndashE816

MegnaLCMoncrieffAEHaywardJLampHensonSM (2014)EqualreproductivesuccessofphenotypesintheLarus glaucescens‐occidentaliscomplexJournal of Avian Biology45410ndash416

Middleton H A Butler R W amp Davidson P (2018) Waterbirdsalter theirdistributionandbehavior in thepresenceofbaldeagles(Haliaeetus leucocephalus) Northwestern Naturalist9921ndash30

Millar J G amp Lynch D (2006) USDI Endangered and ThreatenedwildlifeandplantsRemovingthebaldeagleinthelower48statesfromthelistofendangeredandthreatenedwildlifeUSDIFishandWildlifeServiceFederal Register71(32)8238ndash8251

MoncrieffAEMegnaLCHaywardJLampHensonSM (2013)Mating patterns and breeding success in gulls of the Larus glau‐cescens‐occidentalist complexProtection IslandWashingtonUSANorthwestern Naturalist9467ndash75

Moul IEampGebauerMB (2002)Statusofthedouble-crestedcormo-rant inBritishColumbiaBCMinistWaterLandandAirProtectionBiodiversityBranchVictoriaBCWildlWorkingRepNoWR-10536p

Parrish J K Marvier M amp Paine R T (2001) Direct and indi-rect effects Interactions between bald eagles and commonmurres Ecological Applications 11 1858ndash1869 httpsdoiorg1018901051-0761(2001)011[1858DAIEIB]20CO2

Payo-PayoAOroD Igual JM Jover L Sanpera Camp TavecciaG (2015)Populationcontrolofanoverabundantspeciesachievedthrough consecutive anthropogenic perturbations Ecological Applications252228ndash2239httpsdoiorg10189014-20901


Pearson S FHodumP JGoodTP SchrimpfMampKnappSM(2013)Amodelapproachforestimatingcolonysizetrendsandbab-itatassociationsofburrow-nestingseabirdsCondor115356ndash365

PowerEA (1976)Protection Island and the Power familyUnpublishedmanuscriptJeffersonCountyWAHistoricalSocietyArchives

Richardson F (1961) Breeding biology of the rhinoceros auklet onProtection Island Washington Condor 63 456ndash473 httpsdoiorg1023071365278

Ricklefs R E (1990) Ecology 3rd ed (p 497) New York NYW HFreemanandCompany

SchmidtDF (1986)Thenumbersdistributionandbehaviorofglau-cous-wingedgulls(Larus glaucescens)atasanitarylandfillMSthe-sisWallaWallaCollege42p

ShapiroSSampWilkMB(1965)Ananalysisofvariancetestfornor-mality (complete samples) Biometrika 52 591ndash611 httpsdoiorg101093biomet523-4591

SobeyDGampKenworthyJB(1979)TherelationshipbetweenherringgullsandthevegetationoftheirbreedingcoloniesJournal of Ecology67469ndash496httpsdoiorg1023072259108

StalmasterMV(1987)The bald eagleNewYorkNYUniverseBooksStickKCampLindquistA(2009)2008 Washington State Herring Stock

Status Report OlympiaWAWashington Department of Fish andWildlifeFishProgramFishManagementDivision

Stinson DWWatson JW ampMcAllister K R (2001)Washington state status report for the bald eagle(p92)FishandWildlifeOlympiaWashingtonDepartment

SullivanTMHazlittSLampLemonMJF(2002)Populationtrendsofnestingglaucous-wingedgullsLarus glaucescensinthesouthernStraitofGeorgiaBritishColumbiaThe Canadian Field‐Naturalist116603ndash606

TherriaultTWHayDEampSchweigertJF(2009)Biologicalover-viewand trends inpelagic forage fishabundance in theSalishSea(StraitofGeorgiaBritishColumbia)Marine Ornithology373ndash8

Thoresen A C amp Galusha J G (1971) A nesting population studyof some islands in thePuget Sound areaThe Murrelet52 20ndash23httpsdoiorg1023073534522

Vennesland R G amp Butler R W (2004) Factors influencinggreat blue heron nesting productivity on the Pacific Coast ofCanada from 1998 to 1999Waterbirds 27 289ndash296 httpsdoiorg1016751524-4695(2004)027[0289FIGBHN]20CO2

VolterraV(1926)Variazioniefluttuazionidelnumerodrsquoindividuiinspe-cieanimaliconviventiMemoria Della Reale Accademia Nazionale Dei Lincei231ndash113

Watson J W (2002) Comparative home ranges and food hab-its of bald eagles nesting in four aquatic habitats in westernWashington Northwestern Naturalist 83 101ndash108 httpsdoiorg1023073536608

WatsonJWStinsonDWMcAllisterKRampOwensTE (2002)PopulationstatusofbaldeaglesbreedinginWashingtonattheendofthe20thcenturyJournal of Raptor Research36161ndash169

WhiteAFHeathJPampGisborneB(2006)Seasonaltimingofbaldeagleattendanceandinfluenceonactivitybudgetsofglaucous-wingedgullsinBarkleySoundBritishColumbiaWaterbirds29497ndash500httpsdoiorg1016751524-4695(2006)29[497STOBEA]20CO2

ZarJH (2009)Biostatistical analysis5thedUpperSaddleRiverNJPrenticeHall

How to cite this articleHensonSMDesharnaisRAFunasakiETGalushaJGWatsonJWHaywardJLPredatorndashpreydynamicsofbaldeaglesandglaucous-wingedgullsatProtectionIslandWashingtonUSAEcol Evol 2019001ndash18 httpsdoiorg101002ece35011


APPENDIX AModel Analysis

EQUILIBRIA

Theequilibriaofmodel(2)arefoundbysettingthederivativetozeroineachdifferentialequationfactoringtheresultingsystemofalgebraicequationstoobtain

andsolvingforthestatevariablesGandEThisproducesorderedpairs(GE)thatsatisfybothequationssimultaneouslyOnecancheckthatthereareexactlyfourpossibleequilibriumpairs(00)(K0)(0C)and(GE)with

BIOLOG IC ALLY FE A SIBLE EQUILIBRIA

Anequilibrium(GE)isbiologicallyfeasibleifbothGge0andEge0Theequilibria(00)(K0)and(0C)arethereforealwaysbiologicallyfeasi-bleItisstraightforwardtocheckthatthecoexistenceequilibrium(GE)isbiologicallyfeasibleifandonlyifrgeαCIfequalityholdsthatisifr = αCthen(GE)collapsestothe(0C)equilibriumWesaythat(GE) is positiveifandonlyif

ThusthecoexistenceequilibriumispositiveifandonlyiftheinherentgrowthraterofthegullpopulationisgreaterthantherateatwhichgullscanbetakenbyCeaglepairs

G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C

F I G U R E A 1 emspPoissonregressionrelationshipbetweennumberofbaldeaglesonProtectionIslandandoccupiedbaldeagleterritoriesinWashingtonStatefromEquation(11)inthemaintext


S TABILIT Y OF THE EQUILIBRIA

ThestabilityofeachequilibriumpairisdeterminedbytheprocessoflinearizationAcomprehensiveoverviewoflinearizationandstabilityanalysisisfoundinHenson(2012)LinearizationinvolvescomputingtheJacobianmatrix

attheequilibriumpairandfindingitseigenvaluesIfbotheigenvaluesofJacobianmatrixarenegativerealnumbersthentheequilibriumiscalledastablenodeandnearbysolutionsapproachitinanonoscillatoryfashionIftheeigenvaluesareacomplexconjugatepairwithnegativerealpartthentheequilibriumiscalledastablespiralandnearbysolutionsapproachitinanoscillatoryfashionIfatleastoneoftheeigenval-uesispositiveoriftheeigenvaluesareacomplexconjugatepairwithpositiverealpartthentheequilibriumisunstableAttheequilibrium(00)theJacobianmatrixhastwopositiveeigenvaluesλ = randλ = shencetheequilibriumsolution(00)isalways

unstableAtequilibrium(K0)theJacobianhasonepositiveeigenvalueλ = s + βKandonenegativeeigenvalueλ=minusrThereforetheequilib-riumsolution(K0)isalsounstableAtequilibrium(0C)theJacobianhasoneeigenvaluethatisalwaysnegativeλ=minussandoneeigenvaluethatcanbeeitherpositiveor

negativeλ = r minus αCIfcondition(A1)holdsthesecondeigenvalueispositivesotheequilibrium(0C)isunstableNotethatthecoexist-enceequilibriumsolution(GE)ispositiveonlyif(0C)isunstableAtthecoexistenceequilibrium(GE)theeigenvaluesare

If(A1)holdsthenbotheigenvaluesarenegative(ifreal)orhavenegativerealpart(ifcomplex)Thusifthecoexistenceequilibrium(GE)isposi-tivethenitisstableWhetheritisastablenodeorstablespiraldependsontheparametervalues

J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=

minus[(rminusC)+ (s+K)

]plusmn

radic[(rminusC)+ (s+K)

]2minus4(rminusC)(s+K)

(1+

KC

rs

)

2(1+

KC

rs

)

F I G U R E A 2 emspTimeseriesplotsofmodelresidualsfor(a)glaucous-wingedgullsand(b)baldeaglesNormalquantile-quantileplotsofmodelresidualsfor(c)glaucous-wingedgullsand(d)baldeagles


SUMMARY

Ifr gt αCthenallfourequilibriumpairsarebiologicallyfeasibleTheequilibria(00)(K0)and(0C)areunstableandthecoexistenceequilib-rium(GE)ispositiveandstableThesystemwillapproachcoexistenceeitherwithorwithoutdampedoscillationsdependingonwhethertheeigenvaluesarecomplexorrealIfr lt αCthecoexistenceequilibriumisnotbiologicallyfeasibleTheequilibria(00)and(K0)areunstableandtheequilibrium(0C)is

stableInthiscasethegullpopulationwillgoextinctandtheeaglepopulationwillapproachitscarryingcapacity

APPENDIX BGraphical Analyses of Model ResidualsWeconductedagraphicalanalysisofthemodelresidualsdefinedinEquation(5)Theseresidualsarethelog-scaledeviationsfromthepre-dictedgullandeaglenumbersobtainedusingthemodel(4)withthepointestimatesoftheparametersandtheobservedvaluesafterdividingbytheirrespectivestandarddeviationsThefirsttwopanelsofFigureA2showtheresidualsplottedasafunctionoftimeThereisnoevidence

Parameter or Quantity

Coupled least squares

Decoupled least squares

Using regression‐predicted eagle numbers for PI

g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735

TA B L E A 1 emspPointparameterestimatesandcoefficientsofdeterminationforthethreemethodsofparameterestimation(PI=ProtectionIsland)

F I G U R E A 3 emspScatterplotmatrixofthe2000bootstrappedparameterestimatesexcludingβwhichwasalwaysclosetozero


ofsystematicdeparturesfromzeroalthoughtheresidualsforgullsaresmallerintheearlieryearsThelasttwopanelsofFigureA2shownormalquantilendashquantileplotsofthegullandeagleresidualsThedashedlineisthenormaldistributionexpectationandthesolidlineisthereferencelineconnectingthefirstandthirdquartilesThereisnoevidenceintheseplotsoflargesystematicdeviationsfromnormality

APPENDIX CParameter Estimation for Decoupled EquationsForthefullmodel(2)inthemaintexttheparameterβ istheconversionrateofgullsintoeaglebirthsOurestimateofβ=1876times10minus23 is effectivelyzeroThismightbeexpectedsinceweareusingstatewidedataforeagleswhilegullpopulationnumbersarelocaltoProtectionIslandIfweassumeapriorithatβ=0thenwecandecoupletheequationsforeaglesfromtheequationforgullsandestimatetheparametersfortheeaglepopulationseparatelyThepredicteddensitiesfortheeaglepopulationcanthenbeusedtofindbestfittingparameterestimatesforgullsTwoadvantagesofthisapproachare(a)thepopulationdensitiesdonotneedtobescaledbythestandarddeviationstoarriveatcommensuratenumbersforeaglesandgullsand(b)thereisonelessparametertobeestimated

F I G U R E A 4 emspTimeseriesplotsofobserved(circles)andmodel-predictednumbersfor(a)glaucous-wingedgullsand(b)baldeaglesbasedontheregression-predictedeaglenumbersforProtectionIslandThedashedcurveinpanelAisthepredictionforgullsbasedonthestatewidedata

F I G U R E A 5 emspTimeseriesplotsofmodelresidualsfor(a)glaucous-wingedgullsand(b)baldeaglesbasedontheregression-predictedeaglenumbersforProtectionIslandNormalquantilendashquantileplotsofmodelresidualsfor(c)glaucous-wingedgullsand(d)baldeaglesComparetoFigureA2


WeusedthedecoupledequationapproachtoobtainasetofparameterestimatesforcomparisontotheestimatesinTable2Eagledensitiescanbecomputeddirectlyusingtheclosedformsolutionofthelogisticequation

Wecomputedresidualsbetweenthepredictedandobservedeaglenumbersonalogscale

Weobtainedparameterestimatesfore0sandCbyminimizingthesumofthesquaredvaluesfortheseresidualsWethensubstitutedequa-tion(A2)intothedifferentialequationforthegullpopulationandusednumericalintegrationtoobtainpopulationpredictionsgtforgullsWecomputedresidualsonthelogscale

andestimatedtheparametersg0rKandαbyminimizingsum

2t

TableA1liststheparameterestimatesandcoefficientsofdeterminationforthestatisticalmethodfromthemaintextandtheonepre-sentedhereFortheapproachinvolvingthedecoupledequationswescaledtheobservedandpredictedvaluesusingtheobservedstandarddeviationsforthegullandeagledataandcomputedRMSasgiveninEquation(6)WeusedthistocomputeR2andRA

2asdescribedinthemaintextForEquation(9)thenumberofestimatedparameterswasp=8forthecoupledleastsquaresapproachandp=7forthemethodpresentedhereTheparameterestimatesforthetwomethodsarenearlyidenticalTheyhadequivalentgoodness-of-fitsbasedonthecoefficientofdeter-

minationR2HoweversincethedecoupledmodelhadonelessparameterithasaslightlyhigheradjustedcoefficientofdeterminationRA2

(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)

(A3)t= ln(Et)minus ln

(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)


1emsp |emspINTRODUC TION

Afteryearsofdeclinebaldeagle(Haliaeetus leucocephalus)popu-lationsthroughoutNorthAmericareboundedinthelatterpartofthe twentieth century following tightened protection reductionintheuseofleadshotbyhuntersandtheoutlawingofpesticidessuchasDDT(Hipfneretal2012WatsonStinsonMcAllisterampOwens2002)This recoveryhasprovidedoneof thegreat suc-cessstoriesoftheconservationmovement(MillarampLynch2006)NowherehasrecoverybeenmorepronouncedthaninthePacificNorthwestofNorthAmericawhereinlandwaterwayssuchastheSalishSeaColumbiaRiverandscoresofsmallerlakesandstreamsprovide ideal perching hunting and nesting opportunities forthese raptors (Elliott ElliottWilson Jones amp Stenerson 2011StinsonWatson ampMcAllister 2001Watson 2002Watson etal2002)

Anunintendedconsequenceofbaldeaglerecoveryhasbeenthenegativeimpactonseabirdswhicharealreadystressedbyoverfish-inggillnettingandhabitatdestruction(AtkinsampHeneman1987BlightDreverampArcese2015)Althoughpopulationsofsomesea-birdsmaybedeclining tohistoric levels (Elliottetal2011) localpopulationsofseabirdssuchascommonmurres(Uria aalge)maybethreatened (ParrishMarvierampPaine 2001)Numbersof salmonand other fish traditionally eaten by wintering bald eagles haveplummeted in recent years affecting eagle survival and possiblyresulting in their shift to other food sources (Elliott et al 2011)Seabirdsalwayshaveformedpartofthedietofeagles(Stalmaster1987)but increasingnumbersofeaglesandconcurrentprey fishshortageshaveresultedinincreasedeagleforagingonwaterfowlacauseforconcernamongornithologists(Elliottetal2011Hipfneretal2012MoulampGebauer2002Parrishetal2001SullivanHazlittampLemon2002VenneslandampButler2004WhiteHeathampGisborne2006)Potentialimpactsofbaldeaglepopulationsonmarine food-web structure appear to be due to resident eaglesratherthanoverwinteringeaglesandtheratesatwhichtheycon-sumeseabirdsasprey(HarveyGoodampPearson2012)

Bald eagles can impact seabirds both directly and indirectly(Hipfner et al 2012Parrish et al 2001) Themostobviousdi-recteffect is thekillingandeatingofadults juvenilesandeggs(DeGangeampNelson 1982HaywardGalusha ampHenson 2010HaywardGillettAmlanerampStout1977)Aseconddirecteffectistheextraexpenditureofenergyneededfornestingorfeedinginthepresenceofeagles(Hensonetal2012Parrishetal2001)An indirect effect results when disturbances displace breedingadults fromtheirnestsandexposeunprotectedeggsandyoungtootherpredators(Haywardetal2010)Asecondtypeofindi-recteffect involves changes indistributionpatterns in responseto thepresenceofeaglesForexampledivingwaterbirds in theStraitofGeorgiamovedawayfrominshorewatersanddabblingducksformedlargeraggregationsinshoreandweremorevigilantin response to increased eagle presence (Middleton Butler ampDavidson2018)

From1900totheearly1980sbreedingpopulationsofglaucous-wingedgulls (Larus glaucescens)markedly increased in theGeorgiaBasinoftheSalishSeaBritishColumbiaBy2010populationshaddeclined to about 50 of peak levels (Blight et al 2015 Sullivanetal2002)AstudythatincorporatedmoresouthernareasoftheSalishSeaalsoreportedoveralldeclinesfrom1975to2007(Bower2009) Protection IslandWashington located in the southeasternStrait of Juan de Fuca and centrally positioned in the Salish Sea

TA B L E 1 emspObserveddata

YearGull Nests Protection Island

Occupied Eagle Territories WA

Observed Eagles Protection Island

1980 3796 105

1981 126

1982 4068 138

1983 168

1984 4726 206

1985 231

1986 250

1987 4958 268

1988 309

1989 5045 369

1990 403

1991 4551 445

1992 468

1993 5189 493 9

1994 547 8

1995 558 19

1996 594 16

1997 4278 582 16

1998 666 24

1999 26

2000 17

2001 673 12

2002 2472 23

2003

2004 2925

2005 840 38

2006 2281

2007

2008 2830

2009 3018

2010 2495

2011 2364

2012 2093

2013 1850

2014 1589

2015 1832

2016 2512

















(1)G =aGminusGE

E =minusbE+GE

(2)G = rGminus

r

KG2minusGE

E = sEminuss

CE2+GE

(3)G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs




















Protecon Island

Washington

Idaho

Oregon

Brish Columbia

ProteconIsland

NStrait of

Juan de Fuca

1 kilometer


Estimate 95 CI

Parameter

G0 3663 (28474422)

E0 1067 (10181121)

r 02327 (0104406642)

K 7395 (618610444)

α 00002417 (0000146000005857)

s 01809 (0168201935)

C 8231 (76888890)


Equilibrium


E (EagleTerr) 8231 (76888890)

PIEagles 3494 (28584459)








(g0e0rK120572sC120573



(4)G = rGminus

r120590g


E = sEminuss120590e

CE2+120573120590gGE


















(5)120574t= ln

(Gt

)minus ln

(gt(120579)

)

120576t= ln(Et

)minus ln

(et(120579)

)

(6)RMS()=

ngsumt=1

t2

ng+

nesumt=1

t2

ne

(7)R2=1minusRMSM

RMST

(8)120574t= ln

(Gt

)minusmean

[ln(Gt

)]

120576t= ln(Et

)minusmean

[ln(Et

)]

(9)RA2=1minus

ng+neminus2

ng+neminuspminus2

(1minusR2

)




ngandlowast

1lowast2⋯ lowast





1lowast2⋯ lowast




4emsp |emspRESULTS







(1)=04580 (n=8 p=02538) and g

(2)=minus02975


(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17







(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701


)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)

















(1)G =aGminusGE

E =minusbE+GE

(2)G = rGminus

r

KG2minusGE

E = sEminuss

CE2+GE

(3)G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs




















Protecon Island

Washington

Idaho

Oregon

Brish Columbia

ProteconIsland

NStrait of

Juan de Fuca

1 kilometer


Estimate 95 CI

Parameter

G0 3663 (28474422)

E0 1067 (10181121)

r 02327 (0104406642)

K 7395 (618610444)

α 00002417 (0000146000005857)

s 01809 (0168201935)

C 8231 (76888890)


Equilibrium


E (EagleTerr) 8231 (76888890)

PIEagles 3494 (28584459)








(g0e0rK120572sC120573



(4)G = rGminus

r120590g


E = sEminuss120590e

CE2+120573120590gGE


















(5)120574t= ln

(Gt

)minus ln

(gt(120579)

)

120576t= ln(Et

)minus ln

(et(120579)

)

(6)RMS()=

ngsumt=1

t2

ng+

nesumt=1

t2

ne

(7)R2=1minusRMSM

RMST

(8)120574t= ln

(Gt

)minusmean

[ln(Gt

)]

120576t= ln(Et

)minusmean

[ln(Et

)]

(9)RA2=1minus

ng+neminus2

ng+neminuspminus2

(1minusR2

)




ngandlowast

1lowast2⋯ lowast





1lowast2⋯ lowast




4emsp |emspRESULTS







(1)=04580 (n=8 p=02538) and g

(2)=minus02975


(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17







(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701


)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)




















Protecon Island

Washington

Idaho

Oregon

Brish Columbia

ProteconIsland

NStrait of

Juan de Fuca

1 kilometer


Estimate 95 CI

Parameter

G0 3663 (28474422)

E0 1067 (10181121)

r 02327 (0104406642)

K 7395 (618610444)

α 00002417 (0000146000005857)

s 01809 (0168201935)

C 8231 (76888890)


Equilibrium


E (EagleTerr) 8231 (76888890)

PIEagles 3494 (28584459)








(g0e0rK120572sC120573



(4)G = rGminus

r120590g


E = sEminuss120590e

CE2+120573120590gGE


















(5)120574t= ln

(Gt

)minus ln

(gt(120579)

)

120576t= ln(Et

)minus ln

(et(120579)

)

(6)RMS()=

ngsumt=1

t2

ng+

nesumt=1

t2

ne

(7)R2=1minusRMSM

RMST

(8)120574t= ln

(Gt

)minusmean

[ln(Gt

)]

120576t= ln(Et

)minusmean

[ln(Et

)]

(9)RA2=1minus

ng+neminus2

ng+neminuspminus2

(1minusR2

)




ngandlowast

1lowast2⋯ lowast





1lowast2⋯ lowast




4emsp |emspRESULTS







(1)=04580 (n=8 p=02538) and g

(2)=minus02975


(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17







(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701


)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)








(g0e0rK120572sC120573



(4)G = rGminus

r120590g


E = sEminuss120590e

CE2+120573120590gGE


















(5)120574t= ln

(Gt

)minus ln

(gt(120579)

)

120576t= ln(Et

)minus ln

(et(120579)

)

(6)RMS()=

ngsumt=1

t2

ng+

nesumt=1

t2

ne

(7)R2=1minusRMSM

RMST

(8)120574t= ln

(Gt

)minusmean

[ln(Gt

)]

120576t= ln(Et

)minusmean

[ln(Et

)]

(9)RA2=1minus

ng+neminus2

ng+neminuspminus2

(1minusR2

)




ngandlowast

1lowast2⋯ lowast





1lowast2⋯ lowast




4emsp |emspRESULTS







(1)=04580 (n=8 p=02538) and g

(2)=minus02975


(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17







(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701


)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)
















(5)120574t= ln

(Gt

)minus ln

(gt(120579)

)

120576t= ln(Et

)minus ln

(et(120579)

)

(6)RMS()=

ngsumt=1

t2

ng+

nesumt=1

t2

ne

(7)R2=1minusRMSM

RMST

(8)120574t= ln

(Gt

)minusmean

[ln(Gt

)]

120576t= ln(Et

)minusmean

[ln(Et

)]

(9)RA2=1minus

ng+neminus2

ng+neminuspminus2

(1minusR2

)




ngandlowast

1lowast2⋯ lowast





1lowast2⋯ lowast




4emsp |emspRESULTS







(1)=04580 (n=8 p=02538) and g

(2)=minus02975


(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17







(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701


)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)



ngandlowast

1lowast2⋯ lowast





1lowast2⋯ lowast




4emsp |emspRESULTS







(1)=04580 (n=8 p=02538) and g

(2)=minus02975


(1)=01577 (n=18 p=05319) and e

(2)=minus02771 (n=17







(10)Glowastt= gt

(120579)exp

(120574lowastt

)

Elowastt= et

(120579)exp

(120576lowastt

)

(11)ln(PI Eagles

)=05074+0003701


)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)























ACKNOWLEDG MENTS



Nonedeclared





ORCID


R E FE R E N C E S






















































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)































































EQUILIBRIA






G

(rminus

r

KGminusE

)=0

E

(sminus

s

CE+G

)=0

G=sK

(rminus120572C

)

120572120573KC+ rsand E=

rC(s+120573K

)

120572120573KC+ rs

(A1)rgt120572C









J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)








J(GE)=

⎡⎢⎢⎢⎢⎣

rminus2r

KGminusE

minusG

E sminus2s

CE+G

⎤⎥⎥⎥⎥⎦

=


]plusmn



(1+

KC

rs

)

2(1+

KC

rs

)



SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)


SUMMARY








g0 3663 3663 3748

e0 1067 1069 2119

r 02327 02338 05031

K 7395 7376 4913

α 00002417 00002419 0007187

s 01809 01803 01333

C 8231 8254 6500

β 1876x10-23 mdash 1580x10-7

G 1072 1076 3498

E 8231 8254 6502

R2 819 819 789

RA2 772 779 735













2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)











2t




(A2)et=

C

1+(Cminuse0

e0

)exp (minusst)


(et(e0sC)

)

(A4)t= ln(Gt

)minus ln

(gt(g0rK)

)

predator–prey dynamics of bald eagles and glaucous‐winged

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