Rufener etall 2017 Bayesian prediction in data poor fisheries

ContentslistsavailableatScienceDirect

Ecological

Modelling

j o u r n al ho me p ag e :w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l

Bayesian

spatial

predictive

models

for

data-poor

fisheries

Marie-Christine

Rufener

_,

_Paul

_Gerhard

_Kinas

_,

_Marcelo

_Francisco

_Nóbrega

_,

Jorge

Eduardo

Lins

Oliveira

a_{LaboratóriodeEstatísticaAmbiental,InstitutodeMatemática,EstatísticaeFísica,UniversidadeFederaldoRioGrande,AvenidaItláliakm8,Carreiros,}

CEP:96201-900,RioGrande,RioGrandedoSul,Brazil

b_{LaboratóriodeBiologiaPesqueira,DepartamentodeOceanografiaeLimnologia,UniversidadeFederaldoRioGrandedoNorte,AvenidaCosteiras/n,Mãe}

Luiza,CEP:59014-002,Natal,Brazil

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received1March2016

Receivedinrevisedform6January2017 Accepted20January2017

Keywords:

Bayesiangeostatisticalmodels

IntegratedNestedLaplaceApproximations StochasticPartialDifferentialEquations Essentialfishhabitats

Fisheriesecology

a

b

s

t

r

a

c

t

Understandingthespatialdistributionandidentifyingenvironmentalvariablesthatdriveendangered fishspeciesabundancearekeyfactorstoimplementsustainablefisherymanagementstrategies.Inthe presentstudyweproposedhierarchicalBayesianspatialmodelstoquantifyandmapsensitivehabitats forjuveniles,adultsandoverallabundanceofthevulnerablelanesnapper(Lutjanussynagris)presentin thenortheasternBrazil.Datawerecollectedbyfishery-unbiasedgillnetsurveys,andfittedthroughthe IntegratedNestedLaplaceApproximations(INLA)andtheStochasticPartialDifferentialEquations(SPDE) tools,bothimplementedintheRenvironmentbytheR-INLAlibrary(http://www.r-inla.org).Ourresults confirmedthattheabundanceofjuvenilesandadultsofL.synagrisarespatiallycorrelated,havepatchy distributionsalongtheRioGrandedoNortecoast,andaremainlyaffectedbyenvironmental predic-torssuchasdistancetocoast,chlorophyll-aconcentration,bathymetryandseasurfacetemperature.By meansofourresultsweintendedtoconsolidatearecentlyintroducedBayesiangeostatisticalmodelinto fisheriesscience,highlightingitspotentialforestablishingmorereliablemeasuresfortheconservation andmanagementofvulnerablefishspeciesevenwhendataaresparse.

©2017ElsevierB.V.Allrightsreserved.

1. Introduction

Fisheries action as well as environmental fluctuations may inducemanychangesinfishstocks,whicharecommonlyrelated totheirabundance,sizeand spatialdistribution(Haddon,2001; King,2007).Inthissense,detectingthemainenvironmental fac-torsdrivingnaturalabundancefluctuationandunderstandinghow theseassociationsvaryoverspaceandtimearekeyconceptsin ecologyandinfisheriessciences(Rossetal.,2012).

The links betweenfish stock dynamics and its surrounding environment aretherefore of fundamental importancein order toimprovesustainablefisheries’management andconservation strategies(Babcoketal.,2005;Valavanisetal.,2008).Thewayin whichwemayaccessthespecies-environmentrelationshipsare commonlytreatedinthespecializedliteratureasSpecies

Distri-∗ Correspondingauthorat:LaboratóriodeEstatísticaAmbiental,Institutode Matemática,EstatísticaeFísica,UniversidadeFederaldoRioGrande,AvenidaItlália km8,Carreiros,CEP:96201-900,RioGrande,RioGrandedoSul,Brazil.

E-mailaddresses:machris55@hotmail.com(M.-C.Rufener),paulkinas@furg.br (P.G.Kinas),marnobrega@hotmail.com(M.F.Nóbrega),jorgelins@ufrnet.br (J.E.LinsOliveira).

butionModels(SDMs),andarefrequentlyanalyzedinthecontext ofstatisticaltoolstoevaluateaspeciesdistributionwithrespect to environmental variables (Franklin, 2010). The main goal of SDMsreliesontheprediction,identificationand understanding ofaspeciesspatialdistribution,andmaybeconsideredasthose habitats where itfulfills someof itsbiological process,suchas reproduction,spawningandfeeding.Whendealingwithmarine species,andparticularlywithfishes,theoutputprovidedbythe SDMsdesignatethetermEssentialFishHabitat(EFH),which con-stituteareasthatpromotethefishesmostfavorablehabitatsfor spawning,feeding,orgrowthtomaturity.

Overthepastthreedecadesahugeefforthasbeenspentinthe developmentofpowerfulstatisticalmodelstoexploremore real-isticscenariosregardingthespecies-environmentalrelationships. ArtificialNeuralNetworks(ANN,e.g.,SPECIES),MaximumEntropy (ME,e.g.,MAXENT),ClimaticEnvelops(CE,e.g.,BIOCLIM), Classifi-cationandRegressionTrees(CART,e.g.,BIOMOD)andregression modelssuchasGeneralizedLinearandAdditive(Mixed)Models (GLM/GLMM/GAM/GAMM)areamongthemanymethodological toolsthathavebeenproposedformodellingthespecies distribu-tion(Franklin,2010;GuisanandThuiller,2005).

http://dx.doi.org/10.1016/j.ecolmodel.2017.01.022 0304-3800/©2017ElsevierB.V.Allrightsreserved.

Althoughtheseapplicationsareconsideredrobust,theyusually addressonlyexplanatorymodelsthatsimplyaimtoverifythe rela-tionshipbetweentheresponsevariable(e.g.presenceorabundance ofthespecies)andsomeenvironmentalpredictors(e.g.bathymetry andchlorophyll-aconcentrationinmarinecontext),without con-sideringexplicitlythespatialcomponent(Ciannellietal.,2008). Thismayresultinapoorcharacterizationofthespeciesresponse toenvironmentalfactorsbyunderestimatingthedegreeof uncer-taintyinitspredictors(Latimeretal.,2006).

Consideringthatthemarineenvironmentisanextremely het-erogeneousdomain,andthatagivenmarinespecieshasbiological andecologicalconstraints,it iscommonly notedthat biological resources like most fish species present a gregarious distribu-tion. Hence, ignoring spatial autocorrelation violates the main assumptionof classical inference,which assumesthat thedata areindependentandidenticallydistributed,andthuscanleadto biasedestimates.Spatialcorrelationshouldthereforebe consid-eredsincespeciesaregenerallysubjecttosimilarenvironmental factors(Mu ˜nozetal.,2013).Suchspatialmodelsallowconstructing powerfulpredictivemodelsthatnotonlyprovidetheestimation ofprocessesthatinfluencespeciesdistribution,butalsopromote thepossibilityofpredictingtheiroccurrenceinunsampledareas (Chakrabortyetal.,2010).

Additionally,itisadvantageoustoincorporateBayesian infer-enceinpredictivemodels,giventhatitispossibletointegrateall typesofuncertaintiesusingexclusivelytheprobabilityasitsmetric. Combiningtheuncertaintyinthedata(expressedbythelikelihood) withextra-data information (expressed by prior distributions), posteriorprobabilitydistributionsarebuiltforallunknown quan-titiesofinterestusingBayes‘theorem(KinasandAndrade,2010).

HierarchicalBayesianModels(HBMs)areverysuitableforsuch situations,astheyallowtointroducesequentiallythe uncertain-tiesassociatedwiththeentirefisheryphenomenon,aswellasa spatialrandomeffectintheformofaGaussianrandomfield(GRF) (Cosandey-Godinetal.,2015).TraditionallyHBMsreliedon sim-ulationtechniquessuchasMarkovChainMonte Carlo(MCMC). However, with increased model complexity the computational timerequiredtoapproximatetheposteriordistributionsbecame unfeasible.Tosidestepthislimitation,Rueetal.(2009)proposed analternativenumericalcomputationtoobtainposterior distribu-tions,calledIntegratedNestedLaplaceApproximations(INLA),and whichiscurrentlyimplementedintheRenvironmentbytheR-INLA package(http://www.r-inla.org).Ratherthanusingstochastic sim-ulationtechniques,INLAusesnumericalapproximationsbymeans oftheLaplaceoperator,whichrevealedtobemuchfaster,flexible andaccuratethanMCMCwheneverapplicable.

ThankstoIllianetal.(2013)andMu ˜nozetal.(2013), spatio-temporalmodelswereintroducedtotheecologicalcommunityin ordertofitpointprocessandpoint-referenceddatathroughthe INLAapproach.Inregardtomarineecologyandfisheriesresearch, morestudieshaveslowlyemergedsince thenusingexclusively INLAfor spatial andtemporal purposes (Cosandey-Godinet al., 2015;Mu ˜nozetal.,2013;Paradinasetal.,2015;Penninoetal.,2013, 2014;Quirozetal.,2015;Roosetal.,2015;Wardetal.,2015;Bakka etal.,2016;Damasioetal.,2016;Paradinasetal.,2016;Pennino etal.,2016).

However,mostofthesestudiesreliedon“data-rich”fisheries, wherecertainlyanyquantitativemethodwouldhavehadgood per-formance.Ontheotherhand,indevelopingcountriessuchasBrazil, fisheriestendtobepoorly documentedand inadequately man-agedduetolackofresearchfundingformonitoringandanalysis (Honeyetal.,2010).Conventionalanalyticalfisheriesstock assess-menttoolsthat demandbigdatasets maynotbeapplicablein “data-poor”situationslikethese.Therefore, flexibleandreliable statisticaltoolswithgoodperformancealbeitlimitedinformation areparamount(Bentley,2014).

Inordertoexpandtheuseofsuchtoolsindatalimitedfisheries, wewilldemonstratetheflexibilityandusefulnessoftheBayesian modellingapproachforsomeimportantfisheriesissues,suchas delimitingfishstocksintoagegroupingsandevaluatingthespatial distributionoftheseagegroupings.Specifically,ourmain objec-tivesare:(i)predictabundanceand agegroupingsofthetarget lanesnapper(Lutjanussynagris)alongafractionofthenortheastern coastofBrazil;and(ii)identifyenvironmentaldriversthataffect lanesnapper’sabundanceandso,provideimportantinsightsofits spatialdistribution.

Thispaperisorganizedasfollows:firstlywedescribethe impor-tanceofourstudysubject,how themaindatasetwasobtained andhowweachieveagegroupingsusingBayesianlogistic regres-sion.Thereafter,weapplyhierarchicalBayesianspatialmodelsinto EFH’s,anddiscuss theentiremodellingproceduresusedin this study.Finally,wedescribeanddiscussourresults,outlining oppor-tunitiesforfuturespatialfisheriesmanagement.

2. Material&methods

2.1. Casestudy

Asacasestudy,wemodelledthespatialoccurrenceofLutjanus synagris(Linnaeus,1758), popularlyknownaslanesnapperand whichisconsideredoneofthemostimportantfishingresource caughtwithintheLutjanidaefamily(Luckhurstetal.,2000).Lane snappersinhabitavarietyofhabitatsfromcoastalwaterstodepths upto400m,oftenoccurringincoralreefsandvegetationonsandy bottoms(Carpenter,2002),andarewidelydistributedthroughout thetropicalregionofthewesternAtlantic,fromNorthCarolinato southeasternBrazil,includingtheGulfofMexicoandtheCaribbean Sea(McEachranandFechhelm,2005).

Givenitshighcommercialandrecreationalvalue,thisspecies isoneofthemainstaysofartisanalfisheriesnotonlyinVenezuela andtheCaribbeanSea,butalsoinnortheasternBrazil(Gómezetal., 2001;Luckhurstetal.,2000;Rezendeetal.,2003).Accordingto Lessaetal.(2009),catchesoflanesnapperinnortheasternBrazil havebeenrecordedsincethelate1970swhereitissufferingstrong fishingpressurewhich,despitetheirhighabundance,isleadingto itsdeclineoverthepastfewdecades.

In general, most information about L. synagris relies on its biology,whereasitspopulationdynamicsandspatialdistribution remainpoorlyunderstood(Cavalcanteetal.,2012;Freitasetal., 2011,2014).Therefore,knowingtheirspatialdistributionand iden-tifyingenvironmentalvariablesthatdrivetheirabundancearekey factorstoimplementsustainablefisherymanagementstrategies. 2.2. Studyareaanddatasurvey

Situated inthenortheast of Brazil,theRio Grandedo Norte state (RN) is located in an important coastline transition zone which abruptlychanges itsdirection fromsouth-north to east-west.BetweenJuly2012andJune2014,aboutthreeexperimental fisheriesweremonthlyconductedbyfishingvesselsofthe arti-sanalfleetswhichoperatewithbottomgillnetsalongtheRNcoast. Throughoutthisperiod,126fishingeventswerereportedwhose depthsrangedfrom 5to50m(Fig.1).Biologicalsampling was recordedalongwithextrainformationforeachfisheryincluding geolocation(latitudeandlongitude),bathymetry(m),seasurface temperature(◦C),distancetocoast(km),month(fromJanuaryto December),gillnetlength(m)andheight(m),aswellasitssoaking time(h).

Itisnoteworthythatdatacollectedfromartisanalfleetswould usuallycharacterizeaclearexampleofpreferentialsampling,since fleetsare commerciallydriven tocatchtargetspecies hotspots.

Fig.1.(A)Studyareahighlightingthesamplestations(dots);(B)Triangulationused tocalculatetheGMRFfortheSPDEapproach.Thedotsarereferringifthespecies waspresent(red)orabsent(black)duringthesamplestations(Forinterpretation ofthereferencestocolourinthisfigurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Hence,thepreferredsampledfishinglocationstendtoberepeated producingfishery-biaseddata(fishery-dependentdata).Ifthisis ignoredduring thestatisticalmodelling process,it mayleadto biasedestimates(DiggleandRibeiro,2007;Penninoetal.,2016). Theexperimentaldesignofthisstudyavoidsthisproblemby cover-ingtheentirestudyareawithregulardistancesbetweenthefishing bidsandwithoutrepeatingthem.Thus, weclaim thatthe fish-ingeffortisstochasticallyindependentfromthesamplingstations, resulting in spatial fishery-unbiased data (fishery-independent data).

Allcapturedindividualswereweighted(g)andmeasuredforits totallength(TL,cm).Attheendofeachfishingevent,a representa-tivesubsamplewasrandomlyselectedandtakentothelaboratory. Sexandmacroscopicmaturationstatesweredeterminedforeach specimenaccordingtoVazzoler(1996).

2.3. Bayesianmodellingfordata-poorfisheries

Thissectionaims todescribe theuseof Bayesianmodelsin twoimportantfisheriesissues,namelytheestimationofthesize atwhichaspeciesreachesfirstmaturity(L50)andtheprediction ofaspeciesspatialdistribution.Definingagegroupingsofafish stockbymeansofL50representsamajorchallengeinmostcases, sincealargequantityofdataisneeded.Thishelpsexplainwhy spatialpredictionsforagegroupingsareusuallyscarcein scien-tificreports.Spatialpredictionforabundanceispreferredasthis kindofinformationtendstobemoreaccessible.AppendixAofthe supplementarymaterialsummarizeshowtheestimationofL50is connectedtothespatialmodellingprocedures.

2.3.1. Bayesianestimationofmeanlengthatfirstmaturity

Theknowledgeofpopulationparametersisessentialfor mon-itoring fisheries dynamics since theyare potential exploitation indicators(King,2007).Amongtheseparameters,thesizeatfirst maturation(L50)stands out, which corresponds tothe average length at which 50% of the individuals reach sexual maturity (Vazzoler,1996).OneofthemainobjectivesofdeterminingL50 istodelimittheyoung(meanlength<L50)andadultstocks(mean length_≥L50).

Ofthesubsamplestakentothelaboratory,89individualsoflane snapperwereanalyzed(TableD.1ofsupplementaryAppendixD). TheL50iscommonlycalculatedusingalogisticregression,whose parameters(ˇ0 andˇ1)aretraditionallyestimatedbyMaximum Likelihoodmethod(KarnaandPanda,2011).However,whenthe sampleissmall,thismethodgivesbiasedestimatesandconfidence limitscannotbeadequatelycalculated(Peduzzietal.,1996). There-forewereliedonBayesianinferencetoestimateL50(DollandLauer, 2013).

Initially,klengthclasseswereestablishedforthesetofdata, wherethetotalnumberofindividualsinsizeclassi(ni)andthe numberofmatureindividuals(Yi)wereregisteredforeachclass (TableD.1ofsupplementaryAppendixD).Wedefinedas“mature” allindividualsthatwereatleastinstageC.Thenumberofmature individualsbysizeclasswasmodelledaccordingtoalogistic regres-sionasdescribedbyKinasandAndrade(2010),whoseprobability ofanindividualoflength-classi(fori=1,2,...,k)beingmature assumesabinomialdistribution:

Yi∼Binomial(ni,pi) g (pi)=log







wherepidenotestheprobabilityofanindividualofthei-th length-classbesexuallymature;g(p_i)representsthelogitlinkfunction; ˇ0isthelogitprobabilitythatanindividualwiththemeanlength ¯x issexuallymature;ˇ1istheaverageincreaseinthelogitofpifor eachcentimeteraddedtolength;xidenotestheaveragelengthof sizeclassi,andkrepresentsthenumberoflengthclassesthatwas establishedforthedataset.

EstimatesofL50wereprovidedfromtheequation[(−ˇ0/ˇ1)+¯x]. Whereastheparametersofthelogisticregressionwereestimated byaBayesianframework,apriorsensitivityanalysisforits param-eterswaspreviouslyconducted.Fourdifferentprioralternatives wereevaluated:aGaussianN(0,100),aGaussianN(0,10000),a heavy-tailedStudent(5degreesoffreedom)andaCauchy distri-bution(withscale2.5).Sincetheresultingposteriordistribution forL50hadnegligibleeffectbetweeneachoftheevaluated alter-natives (Fig.D.1 ofSupplementaryAppendix D), weassumed a Normalvaguepriordistributionforthemeanandthevariancefor theparameters(ˇ0∼N(0,100);ˇ1∼ (0,100)).

The posterior probability distributions were simulated by meansofMarkovChainMonteCarlo(MCMC)methodsusingthe R2jagspackage(SuandYajima,2014).Wesimulatedthreedifferent situations,wheretheparameterswerederivedfromgroupedsexes (females+males−case1)orseparatedsexes(malesonly−case2; femalesonly−case3).ForeachofthesecasesweranthreeMCMC chainssimultaneouslyforatotalof60,000iterations,wherethe first20,000iterationswerediscardedasaburn-inperiodandeach 20thstepwasstored(thinning)inordertoreduceautocorrelation. The chains’ convergence was assessed using conventional graphicalmethodssuchasiterationandautocorrelation graphs, andbyconvergenceindicator ˆRwhichdenotestheratioof vari-ancebetweenandwithinchains(Gelman,1996).Finally,aBayesian hypothesistestwasperformedtoassesswhethertherewere sig-nificantdifferencesbetweentheposteriordistributionsoffemale

L50andmaleL50.Forthispurpose,theposteriordistributionofthe differenced=L50male−L50femalewasobtainedandevaluatedwith respecttotheprobabilityratiop(d≥0)/p(d<0);withratiosaway fromone(e.g.,largerthan5orlowerthan1/5)suggesting rele-vantdifferences.Theposteriormedianwastakenaspointestimate ofL50,andusedascutofftodelimitthelanesnapperstocksinto youngandadultgroupings.

2.3.2. Bayesianspatialmodellingforabundanceandage groupings

WeusedhierarchicalBayesianspatialmodelsinorderto esti-mateand predictoverall abundance(case1),number of adults (case2)andnumberofjuveniles(case3)ofL.synagriswithrespect toseveralenvironmentalpredictors.Asabundanceindexweused catch-per-unit-effort(CPUE),whichwasdefinedaslanesnapper’s totalcatch(g)weightedbythefishingeffort.Fishingeffortwas definedasgillnetarea(m2_{)multipliedbysoakingtime(h).}

SimilartoaGLMapproach,theresponsevariableYiisassumed tohaveaprobabilitydistributionthatbelongstotheexponential family,withmeani=E (Yi) linkedtoastructuredadditive pre-dictorithroughalinkfunctiong (· ) suchthatg (i)=␩i:Wherei istheindexforeachsamplingstatio

␩i=ˇ0+ M





fl(

vli

whereiistheindexforeachsamplingstation,nrepresentsthetotal numberofsamplingstations,iisthelinearpredictoreitherfor CPUEorforcountdata,ˇ0isascalarrepresentingtheintercept,M denotesthetotalnumberoflinearcovariates,␤misthecoefficient whichquantifiestheeffectofsomecovariatesXmontheresponse, Lreferstothetotalnumberofnon-linearcovariates,andfl(· ) are functionsdefinedvlforasetofcovariates.

Thef (_{· ) terms}canbeusedeithertorelaxthelinearityofthe covariates(e.g.smootheffects),ortointroducerandomeffects(e.g. spatialand/ortemporal effect)(Rueet al.,2009).Inthepresent studyweusedthef (· ) totestsmootheffectsforsomepredictors andtoincludeaGaussianMarkovRandomField(GMRF)(See sup-plementaryAppendixCfordetails),whichconcernsthespatially structuredrandomeffect(W).

TomodelCPUEdata,wetestedbothGammaandlog-Normal distributionsastheyarecommonlyusedtomodelsuchkindof fisheriesdata(VenablesandDichmont,2004).Becauseneitherof thesetwodistributions cancontainzeros,we addeda constant equalto10%oftheCPUEsmediantoalldataprevioustomodel fitting.Withrespecttothecountmodelsforadultsandjuveniles, weexpectedtoobserveahighamountofzerosastherecouldbe spatialsegregation.Thus,fourdifferentdistributionsweretested: Poisson(P),zero-inflatedPoisson(ZIP),NegativeBinomial(NB)and zero-inflatednegative-binomial(ZINB).Also,anoffsetwasusedin thecountdatamodelsinordertoaccountforthefishingeffort.

Toevaluatelane snappershabitatsuitabilitywithrespectits abundanceandagegroupings,weusedmonthorseasonandsix environmentalpredictors:seasurfacetemperature(SST,◦C), dis-tancetocoast(km),bathymetry(m),rugosity(index),slopeofthe seabed(◦)andchlorophyll-aconcentration(mg/m3_)(See supple-mentaryAppendixB).

Bayesianparameterestimates and predictionin theform of marginal posterior distributions were obtained throughout the INLAapproach. Default priorswereassigned for allfixed-effect parameters as recommended by Held et al. (2010), which are approximationsofnon-informativepriorsdesignedtohavelittle influenceontheposteriordistribution.Forthespatialcomponent (W)weusedtheSPDEapproach(SeesupplementaryAppendixCfor details).AsrecommendedbyLindgrenandRue(2013),

multivari-ateGaussiandistributionswithmeanzeroandspatiallystructured covariancematrixwereassumedforthespatialcomponent.

Forallmodels,variableselectionproceededbyamanually for-wardstepwiseentry.Inordertocomparethegoodness-of-fitof thesemodels,weusedtheDevianceInformationCriterion(DIC) whichisequivalenttotheAkaikeInformationCriterion(AIC)but bettersuitedforHBMs(Spiegelhalteretal.,2002), andwhichis alsodirectlycomputedbyR-INLA.Additionally,asusedbyMu ˜noz etal.(2013)andPenninoetal.(2014),weevaluatedthelogarithmic scoreofConditionalPredictiveOrdinate(LCPO)asameasurement ofpredictivepower.LowervaluesforbothDICandLCPOare indica-tiveofbetterfitandpredictivepower,respectively.Abest(and parsimonious)modelwaschosenbasedonacombinationoflow valuesforLCPOandDIC,containingonlyrelevantpredictors;i.e., thosepredictorswith95%credibilityintervalsnotcoveringzero.

Finally,selectedmodelswerealsoevaluatedforgoodness-of-fit accordingtostandardgraphicalchecks,suchasobservedversus predictedscatterplotsandresidualsQuantile-Quantileplots.Since linearity is expected between theobserved and predicted val-ues,Pearson’scorrelationcoefficient()wascalculatedandtested. Modelswith≥0.7andp-value≤0.05wereconsideredacceptable andthususedforfinalprediction.

3. Results

3.1. Lengthatfirstmaturity&agegrouping

Graphical evaluation of the logistic models revealed good convergence of the chains for all cases. Fig. 2 shows thejoint distribution of the logistic regression parameters for grouped andseparatedsexes.Despiteallthreecaseshadareasonablefit, groupedsexesrevealedaslightlybetterfitasthejointdistribution ofmodel’sparameterswasalmostentirelyconcentratedwithinthe rightupperquadrantwhere␤1>0(Fig.2).

Table1givesbasicstatisticsummariesoftheposterior distribu-tionforthelogisticmodels.Agoodconvergencewasalsoobserved bythenumericaloutputs,whose ˆRvalueswereallsituatedcloseto 1.0.WithrespecttheL50estimation,bothmeanandmedianvalues eitherforfemalesandmaleswereverysimilar(Table1)(Fig.3A andB).

The Bayesian hypothesis test pointed out that there was no relevant difference between the L50 of males and females (p(d_≥0)/p(d<0)=1.32)(Fig.3D).Baseduponthisinformation,we decidedtousetheL50estimationderivedfromthegroupedsexes modelanduseditsmedian(25.17cm)tosplitthesampledlane snapper’sintojuvenilesandadultsforthemodelsofthenext sec-tion.

3.2. Dataoverview

Among the 126 sample stations, lane snapper occurred at 83 (66.14%)whose distance tocoastranged from2.8 to41km (mean=11.58km; sd=7.86) and depths from 5.1 to 52.7m (mean=16.8m; sd=8.85). The total weight caught during this periodwas259.808kg,whereasCPUErangedfrom0to6.12g/m2_*h (mean=0.35g/m2_{*h;sd=0.85).Totallengthofallspecimenvaried} from18.5cmto46cm(mean=30.1cm;sd=3.92).Accordingtothe adoptedL50wecountedatotalof101juvenilesand505adults indi-viduals,therebyevidencingthatcatchesweremostlycomposedby adultindividuals (83.3%).Whereasadultsoccurredinalmostall samplestations(81),juvenilesoccurredin36samplestationonly. 3.3. Modelselection

Amongallenvironmentalpredictors,onlybathymetryand dis-tancetocoastwerehighlycorrelated(␳>0.7),andthereforenever

Fig.2. Posteriorjointdistributionoflogisticregressionparametersforfemales(A),males(B)andgroupedsexes(C).

Table1

Summaryofparametersandtheirassociatedstatisticsresultedfromthelogisticmodel(Sd=standarddeviation;CI95%=95%credibleinterval).

Sex Parameters Mean Median Sd CI95% Rˆ

Q0.025 Q0.975 Females ␤0 0.835 0.820 0.387 0.112 1.633 1.001 ␤1 0.230 0.227 0.103 0.041 0.450 1.001 L50 25.332 25.870 12.757 16.835 29.002 1.063 Deviance 16.728 16.163 2.004 14.749 22.280 1.001 Males ␤0 0.306 0.297 0.331 −0.322 0.978 1.001 ␤1 0.128 0.125 0.080 −0.022 0.291 1.001 L50 25.375 25.072 77.786 10.546 37.669 1.068 Deviance 24.368 23.758 2.111 22.37 29.831 1.001 Females& Males ␤0 0.457 0.456 0.238 −0.010 0.927 1.002 ␤1 0.161 0.159 0.061 0.044 0.285 1.002 L50 24.653 25.174 5.996 18.701 28.094 1.038 Deviance 35.062 34.443 2.026 33.075 40.667 1.001

Fig.3. Posteriordistributionhistogramsoffemales(A),males(B),females&males(C)andthedifferencebetweenmalesandfemales(D)totallength(TL)simulations.(For interpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

usedtogetherduringthemodellingprocedures. Severalmodels withdifferentprobabilitydistributionsweretestedaccordingto datanature.Wealsotestedmodelsthatincludedquadraticterms and/orsmoothingeffectswhich,however,didnotshowany rele-vantfitimprovement.

TableD2(seesupplementaryAppendixD)summarizesthemost relevantmodelsregardingdifferentcombinationsof environmen-talpredictorsforlanesnappersCPUE.LCPOandDICdidnotdiffer greatlyamongmodelsofeachdistribution.However,DICdiffered significantly between models withdifferent distributions, with Gammamodelsalwayshavingabetterfit.Thebestmodelincluded onlydistancetocoastandthespatialeffect(model1,TableD.2of supplementaryAppendixD).

With respect to models for adult and for juvenile individ-uals, both agree on the NB distribution when LCPO and DIC wereanalyzedsequentially(SeeTableD.3and D.4,respectively, in supplementaryAppendix D).The bestselectedNBmodelfor adults count data included as relevant predictors distance to coast,chlorophyll-a concentrationand thespatialeffect (model 8,TableD.3ofAppendixD),whereasforjuvenilescountdatathe selectedmodelcontainedseasurfacetemperature,bathymetryand the spatial effectas relevant predictors (model 7, TableD.4 of supplementaryAppendixD).Posteriorsummarystatisticsforall parameterofeachselectedmodelareshowninTable2.

Fig.4denotes2typesofmodelsfitevaluation.Predictedversus observedCPUE,numberofadultsandnumberofjuvenilesare

dis-Fig.4.Observedversuspredictedvaluesforlog(CPUE)(A),numberofadults(C)andnumberofjuveniles(E).Therightpanelsshowquantile-quantileplotsforeachevaluated case.

Table2

Summaryofthemarginalposteriordistributionformodelparametersprovidedby theselectedmodelforeachconsideredcase.Thehyperparameters␸Gand␸NBare

theprecisionparametersofthegammaandnegativebinomialobservations, respec-tively,and␶and␬representsthevarianceandscalingparameterofthespatialeffect, respectively.

Responsevariable Parameters Mean Sd CI95%

Q0.025 Q0.975 CPUE (model1) (Intercept) −1.957 0.224 −2.410 −1.525 DC 0.572 0.204 0.170 0.976 ␶ −5.576 0.593 −6.796 −4.472 ␬ 3.996 0.458 3.140 4.937 (km) 0.053 0.020 0.015 0.109 G 0.961 1.441 0.709 1.274 Adults (model8) (Intercept) −8.222 0.255 −8.762 −7.756 DC 0.951 0.264 0.441 1.487 CHLa 0.650 0.308 −0.059 1.279 ␶ −6.002 0.757 −7.549 −4.582 ␬ 4.281 0.529 3.284 5.359 (km) 0.040 0.018 0.009 0.091 NB 1.322 0.455 0.658 2.421 Juveniles (model7) (Intercept) −10.330 0.834 −12.135 −8.617 SST −0.702 0.318 −1.368 −0.113 B 0.615 0.296 0.077 1.250 ␶ −4.066 1.160 −6.114 −1.596 ␬ 2.438 0.992 0.332 4.743 (km) 0.233 0.242 0.018 1.371 NB 0.748 0.435 0.218 1.858

playedintheleftpanelsofFig.4.Asitcanbenoticed,forallthree casesthepredictedversusobservedvalueswerepositivelyand sig-nificantlycorrelated(0.45<<0.89,p<0.05)(Fig.4A,CandE).Q-Q plotsshowedreasonablenormaldistributionfortheresidualsof theselectedCPUEmodel.Someoutlierswerepresentedfor juve-nilesandadults,suggestingthatsomeobservedcountsaremuch higherthanthemodelwouldpredict(rightpanelsofFig.4B,D andF).TheCPUEmodelwasabletopredictfairlynon-zero val-ueswhilevaluesaroundzerowerehighlyoverestimated(Fig.4A). Thus,togetherwiththerespectiveQ-Qplotbothfiguresclearly sug-geststhatourdataissplittedintotwodata-sets,oneregardingthe zerosandanotherregardingnon-zerovalues.Foradultsand juve-niles,bothmodelswereabletopredictwithmoreaccuracysmall tomediumvalues(Fig.4CandE).

3.4. Predictionofessentialfishhabitatforthelanesnapper

Whencombiningboththefixedeffectsandthemodeloutput fromtherandomfield,wecaneasilydisplaythemeanandstandard deviationoftheposteriormeanofthelinearpredictoraswellfor therandomfielditself.Fig.5displaysmeanpredictionandstandard deviationmaps.Accordingtothelinearpredictormaps(Fig.5A, CandE),CPUE,adultsandjuvenileshadsimilarpatchy distribu-tionsalongtheentirecoast,differingsolelyontheirspatialextent. Specifically,itwaspossibletoidentifyfourmajorhotspots,twoof themlocatedontheeastcoastandtwo onthenorthcoast.The standarddeviationmapsforalllinearpredictors(Fig.5B,Dand

Fig.5. Posteriormean(leftpanels)andstandarddeviation(rightpanels)ofthepredictivedistributionoflanesnappersCPUE(A,B),numberofadults(C,D)andnumberof juveniles(E,F).Smallpanelsrepresentthemeanandstandarddeviationofthespatialrandomeffectfortheirrespectiveresponsevariable.Itisworthmentioningthatall mapsarelog-scaled.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

F),asexpected,showedseveralsmallerpatchesoflowerstandard deviationvalues,whichcorrespondtoareaswheredatawere sam-pled.Comparatively,higherstandarddeviationsweremoreorless constantalongtheentiredomaininnon-sampledareas.

Thesmallerpanels in each figurereflect thespatialrandom effectandindicatestheintrinsicvariabilityofthelanesnappers dis-tributionwhenremovingallcovariables.Itmayreflectotherhidden factorsthatwerenotaccountedforinthemodels,suchasbiotic pro-cesses(e.g.,competitionandpredation)andabioticcharacteristics (e.g.,seasurfacesalinityandcurrents).Again,forallthree consid-eredcasesthepatternsofmeanspatialrandomeffectbehavedin asimilarway,withahighernumberofaggregationspotsalong thecoastandwhichdifferedonlyintheirspatialextent(smaller panelsofFig.5A,CandE).Regardingthestandarddeviationofthe spatialcomponent,allthreecaseshadsimilardistributionpatterns asobservedforthestandard deviationoftheirrespectivelinear predictors,withsmallerandhighervaluescorrespondingto sam-pledandnon-sampledareas,respectively(smallerpanelsofFig.5 B,DandF).

4. Discussion

4.1. Biologicaldiscussion:sizeatfirstmaturity

Thefirstpartofourresultaimedtoestimatethesizeoffirst maturity for L.synagris. Although differencesin this parameter

amongsexesarecommonlyreportedforlanesnappersfrom dif-ferentregions(Aiken,2001;Figuerolaetal.,1998;Freitasetal., 2014;Luckhurstetal.,2000;Manickchand-Dass,1980),ourresults indicatedotherwise.Thusanoverallsizeforbothsexeswas esti-matedatlengthsof25.17cm.Ourestimationagreedwiththose fromCavalcanteetal.(2012)alsoconductedintheRNstate,which reportedmediansizesof25.7cmforgroupedsexes,wellwithinthe posterior95%credibilitybounds.

4.2. Modeldiscussion

The Bayesian spatial modelling analysis yielded some clear resultswithrespecttotheenvironmentalpreferencesofadults, juvenilesandtheiroverallabundance(CPUE).Despitemost mod-elsfittedwelltothedata,someofthemsuggestedthattheywere inappropriate(producingbyNAand–Inf.inTableD.2–D.4of sup-plementaryAppendixD).Moreover,whenattemptedtothecount models(TableD.3andD.4ofsupplementaryAppendixD),itseemed thatthenumberofadultsandjuvenilesalwaysfittedbetter(lower DICs)accordingtoPoissonoritsZero-inflatedversionwhen com-pared to thealternative models belonging to otherprobability distributions.Nevertheless,thesemodelsalsodisplayedtheworst predictivemeasures(higherLCPOs).Thiscouldbeexplainedsince extremelowvaluesofCPOareindicativeofoutliersandinfluential observations.Thus,thelowerCPOvaluesthehigherLCPOvalues

which,inturn,revealthatthesemodelparameterswerebiased andconsequentlyinappropriateforpredictionpurposes.

Regarding the modelsfor CPUE, we concluded that besides thespatialeffect,onlydistancetocoastwasstatisticallyrelevant toexplain the variability in CPUE (Table D.2 of supplementary AppendixD).TableIIrevealedapositiverelationshipbetweenthe responseandthefixedeffect,indicatingthattheCPUEofL.synagris increasestowardoffshoreareas.Theremainingparametersarethe hyperparametersthatspecifytherelevanceofthespatialeffectin themodel.

R-INLAusuallyprovidesthesimplestinternalrepresentationof theseparameters,namelylog(␶)=␪1 andlog(␬)=␪2.However,it ismorenaturaltoconstructtheseparametersasafunctionofthe spatialcorrelationrange(␳),since␶and␬havejointinfluenceon themarginalvariancesoftheresultingspatialfield.Forpractical purposes,weexposedallhyperparameters,butmainlyfocusedon ␳asithasamoreintuitiveinterpretation.Inthisway,themean valueof(␳)was0.053whichrepresentsthedistanceatwhich cor-relationisreducedtoapproximately0.13.Theposteriormeanof theprecisionforthegammaobservations()was0.961.Itisworth notingthattheselectedCPUEmodelsuggestedthatourdatawould haveprobablybeenbetterfitted,ifwehadconditionedthezero values.Thus,ifwehadmodelledourCPUEaccordingaGamma hur-dlemodelasdonebyQuirozetal.(2015)withPeruviananchovies biomass,thenwemaybecouldhavehadmorepreciseestimations andpredictions.

Withrespecttothecountmodelsappliedtonumberofadults andjuveniles,we wereabletoaffirmthat theyrespond differ-entlytodistinctenvironmentalpredictors.Distancetocoastand chlorophyll-aconcentrationwerestatisticallyrelevanttoexplain thevariation inthenumber ofadults(Table D.3of supplemen-tary Appendix D). Furthermore, Table 2 revealed that all fixed effectshadalsopositivecoefficients,indicatingthatthenumber ofadultsincreasestowardoffshoreareasandhigher chlorophyll-aconcentration.Numberofjuveniles,instead,showedadifferent responsepattern,whereseasurfacetemperatureandbathymetry werestatisticallyrelevant(TableD.4ofsupplementaryAppendix D).AccordingtoTable2,itwasfoundthatnumberofjuveniles increasestowardareaswithlowerseasurfacetemperatureaswell astowardoffshoreareas(higherbathymetry).

Concerningthehyperparametersfromboth countmodels,it wasnotedthatjuvenilesshowedslightlyhighervaluesforthemean varianceandsmallervaluesforthescalingparameterwhen com-paredtoadults(Table2).Themeanspatialcorrelationrange(␳)for adultsandjuvenileswere0.04and0.233,respectively(Table2). ContrastingthesenumberswiththatfromCPUE(0.053),itseems thatdespiteanoverallweakspatialcorrelationinallcases,itis apparentlystrongerforjuveniles.

Itisworthmentioningthateachselectedmodeldiscussedin thissectionwasalsotestedwithoutspatialeffect(seepenultimate modelsinTableD.2toD.4ofsupplementaryAppendixD)asthey hadsmallspatialcorrelationranges.Thesemodelsusedtohave higherDICsand LCPOswhen compared totheremaining mod-els,whichrevealsthatthespatialeffect,despitesmall,wasindeed relevantinallthreecases.

4.3. Ecologicaldiscussion

ThroughtheHBMswewereabletoinvestigatetherelationship betweenthelifestagesofL.synagrisandenvironmentalpredictors, andthereforequantifyanddefinesuitablehabitatsaccordingto rel-evantpredictors.Althoughallthreeconsideredcaseshadslightly differentenvironmentalpreferences,theyshowedsimilarspatial distributions.Specifically,distancetocoastandbathymetrywere themainpredictorsthatdrovethespatialdistributionofL.synagris, indicatingpreferencetowardsoffshoreareas.Bymeansofour

pre-dictionmaps,weidentifiedfourmainhotspotslocatedinoffshore areas,whichalsocuriouslycorrespondtothemajorreef,beachrock andsandcomplexespresentintheRNcoast(Vitaletal.,2010).

Moreover,regardingparticularlythedifferencesbetweenadults and juveniles of L. synagris, they showed specific preferences for chlorophyll-a concentration and sea surface temperature, respectively. Both predictors are commonly related to marine productivity, and may be used as potential indicators of ther-mal and productivity-enhancing fronts (Valavanis et al., 2008). Duetohighernutrientavailabilityandsunlightincidence,higher chlorophyll-aconcentrationoccurmainlyincoastalareasandin upwelling fronts, the latter being characterized by cold water temperatures.Reefs,inparticular,areareasofhighprimary produc-tivity,andthussupporthighertrophicfoodwebsfromwhichadults andjuvenilesofL.synagrisalsobenefit(Crosslandetal.,1991).

Althoughjuvenilesappearedtobemorewidelydistributedthan theadults,theirspatialdistributionmostlyoverlappedwiththatof adults.Inthissense,ourresultsweresomewhatsurprisingsince a spatialsegregationbetweenadultand juvenilesis commonly reportedforL.synagris(RodríguezandPáramo,2012)and,toa widerextent,formarinefauna(Gillanderset al.,2003).The lit-eratureusuallyreports theoccurrenceofjuvenilelanesnappers overmuddybottomsneartomarineestuariesandseagrassbeds, whereasadultstendtopreferconsolidatedbottomstowards off-shoreareas (Doncel and Paramo, 2010;Rodríguez and Páramo, 2012).

However,webelievethatourresultsprobablyreflectmorea limiteddefinitionofjuvenilesinthiswork.Sincethecatcheswere predominatelycomposedbyadults,thisstudyrevealedthatthe fishinggearusedinthisworkhadaclearsizeselectivityforthe L. synagris populations. Among the few catches of juveniles, it wasobservedthattheyalreadyhadrelativelylargesizes measur-ingatleast18.5cm.Whereastheterminologyjuvenilesisusually appliedtoindividualswithsizesshorterthan10cm(Franksand VanderKooy,2000; Lindemanet al., 1998;Mikulas and Rooker, 2008;PimentelandJoyeux,2010),weconcludedthatindividuals denotedasjuvenilesinthisstudyshouldbebetterregardedas sub-adults.Thus,whatwe mightbeobservingis,in fact,thespatial segregationamongadultandsub-adultsofL.synagris.

5. Concludingremarks

Asetofstatisticalapproacheswasusedinordertoextendthe scopeofadata-poorfishery. BymeansofBayesianmodelsand GeographicInformationSystems(GIS)schemesweprovidedsome novelinsightsofthepotentialspatialdistributionforabundance anddifferentlifestagesofthevulnerablelanesnapper.However, ourcasestudyattemptedtopresentanemergingSDMtool,rather thanfocusingsolelyontraditionalbiologicalandecological discus-sions.

Marineecosystemsconstitutedynamicareas,wherefisheries experimentsarealmostimpossibleandvulnerabletoseveralerror sourcesassociatedwithobservations,samplingprocedures,model structureandparameters.Inthissense,itisconvenienttoapply Bayesianinferenceintofisheriesmodellingprocedures,sincethe posteriordistributionitselfisadynamicprocessinitiallyshaped accordingtoourpriorbeliefs,whichturnsoutarelevantmodel stabilizingfactorindata-poorsituationsandadaptsitself automat-icallyasweacquiremoreandmoredata.

ThehierarchicalBayesianspatialmodelsproposedinthisstudy areextremelypowerfulandsuitperfectlyinfisheriesscience,by quantifyingboththespatialmagnitudeandthedifferentsources ofuncertainty.Spatialpredictions,therefore,becamemuchmore accurateandconsistentwithreality.Furthermore,asweare deal-ingwithinaBayesianframework,posteriorpredictivedistribution

mapssuchasthosedefiningtheprobabilityofthemostfavorable areasforconservation,would beantagonisticwithintraditional frequentistconceptsofprobabilitybutareofenormousvaluefor fisheriesmanagement.

However,itisnoteworthythatthemodelspresentedin this studyare limited not onlyin space butalso in time. Thus,the fittedandpredictedmodelsrevealedonlyasnapshotofthe eco-logicalprocess.Sincefisheriesaredynamicinspaceandtime,we encouragetheuseofbotheffectsastheymayimproveevenmore thewantedrealism.Bayesianspatio-temporalmodelswerealready appliedinsomeimportantfisheriesissues,suchasdiscardand by-catchproblems(Cosandey-Godinetal.,2015;Penninoetal.,2014). Moreover,whendealingwithfishery-dependentdata,itis impor-tanttoaccountfordistinctsourcesofbiasesasthosearerelatedto fisher’sbehavior,andwhichhasalsobeenintroducedrecentlyby Roosetal.(2015)andPenninoetal.(2016).

Ourstudy,demonstratedoncemorehoweasyitisnowadaysto accountforrandomeffectsinHBMswhenperformedthroughthe INLAmethodology.R-INLArevolutionizedthewaywemay per-formeasilyinotherwisesophisticatedBayesianinference,sinceits interfaceissimilartotheconventionalglmfunctioninRandthus wedonothavetowriteminutelythemodelsasneededinBUGS andJAGS,althoughMCMCisstillaslightlymoreflexibleapproach. Besidesthis,R-INLAiscontinuouslyevolvingandgreatlyextending thescopeofBayesianmodelsforappliedscientists.UsingR-INLA enablesustofitcomplexmodelsatconsiderablelessertimeand withmoreaccuracywhencomparedtostandardMCMCmethods. Finally,it isworthmentioningthat R-INLAcanalsobeused tofitmodelsthatdonothavenecessarilyspatialand/ortemporal componentsinitsstructure.Rueetal.(2009)providesomeother examples,whichincludegeneralizedlinearandadditive(mixed) models,dynamiclinearmodels,survivalmodels,spline smooth-ingandsemiparametricregressions,amongmanyothers.Inthis way,wealsoencouragetheuseofR-INLAwhenperforming eco-logicalBayesianmodellingwithconventionalGLMsorGAMs,since thepackage’sinterfacedisplayssimilaritywithalreadyexistingglm andgamtoolsinR.Atleast,wehaveshownthat,evenfor “data-poor”fisheries,thismodellingapproachhasgoodperformance.

Acknowledgments

AuthorswouldliketogratefullythankDr.MariaGraziaPennino forhervaluabletechnicalsupportintheusageofR-INLA.Thefirst authorisalsogratefultotheBrazilianNationalResearchCouncil (CNPq)thatprovidedfinancialsupportduringherM.Sc.Research, whichwasdevelopedundertheguidanceofthesecondauthor. Finally,allauthorswishtothanktoallfishermen,researchersand (under)graduatestudentswhichcontributedwithbothfieldand laboratoryworks,andthereforeenabledthisstudy.

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttp://dx.doi.org/10.1016/j.ecolmodel.2017. 01.022.

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