03yrformalesandfemalesrespectivelyalsousingPaulysmodelFurthermoreasses - PDF document

1 0.3yrformalesandfemales,respectively,als
0.3yrformalesandfemales,respectively,alsousingPauly'smodel.Furthermore,assessmentsarebasedonanestimated0.3yrforbothsexes(ArancibiaandCubillos,1993;Cubillosetal.,1994).Thisestimateofnaturalmortality,forsexescombined,wasalsoobtainedusingPauly'smodel(CubillosandArancibia,1992).NaturalmortalityoftheChileanhakehasbeenestimatedwithasinglemethod,i.e.themodelofPauly(1980).Otherempiricalmodelshavenotbeenexploredtoestimateitsnaturalmortality,norhastherebeenanyevaluationoftheprecisionofestimates.AnotherimportantconsiderationisthatthegrowthandsurvivalofmaleandfemaleChileanhakediffer(AguayoandOjeda,1987;Aguayo,1994).Inthe®sherythesexratiois1:1atrecruitment,buttheproportionoffemalesincreaseswithage(ArancibiaandCubillos,1993).Inaddition,femaleshavealongerlife-spanthanmales,suggestingthatthenaturalmor-talityratecouldbelowerforfemalehake.ThemainpurposeofthispaperistodeterminetheChileanhakeusingempiricalmodels,aswellastoevaluatetheprecisionoftheestimatesforbymeansofaparametricbootstrap.Also,wewishtoexaminethestatisticalsigni®canceofsexualdifferencesinChileanhakenaturalmortalityrates.2.Materialsandmethods2.1.EmpiricalmodelsandmethodsforestimationofMPauly(1980)developedarelationshipbetweennaturalmortalityrateandgrowthparametersandtemperatureusingdatafrom175®shstocks.Theresultingregressionmodeliswhereisthenaturalmortalityrate,vonBertalanffygrowthparameters,thegrowthcoef-ficient(yr)andthetotalasymptoticlength(cm),respectively;istheannualaveragetemperatureofthehabitat(C).ThegrowthparametersusedtoapplythismodelareobtainedfromAguayoandOjeda(1987)(Table1).RikhterandEfanov(1976)foundtorelatetoageof50%maturity()expressedinthefollowing FortheChileanhake,wasestimatedusingthelengthat50%maturity()usingthefollowingequation: ln1 whereisthevonBertalanffyestimateofageatwerepreviouslyde®ned.ThegrowthparametersusedtoapplythismodelwereobtainedfromAguayoandOjeda(1987)andthesizeat50%maturityoffemales(Table1).Weassumedthatisthesameformalesandfemalesbecausetherearenotanyestimatesofsizeat50%maturityformalesofChileanhake.Hoening(1983)foundanempiricalrelationshipbetweenthetotalmortalityrates()andmaximumobservedages()ofseveralspeciesof®sh,mol-luscsandcetaceans.Mostofthedatapertaintounexploitedorlightlyexploitedstocks.Hoeningcal- Table1LifehistoryparametersfortheChileanhak

2 e,thestandarderroroftheparametersisshown
e,thestandarderroroftheparametersisshowninbracketSymbolDefinitionandunitsMalesFemalesAuthorsAsymptoticlength(cm)57.4(1.15)78.1(3.64)AguayoandOjeda(1987)Growthcoefficient(yr)0.288(0.021)0.153(0.019)AguayoandOjeda(1987)Ageat0(yr)0.232(0.011)0.782(0.238)AguayoandOjeda(1987)Sizeatfirstmaturity(cm)±36±37.9AlarconandArancibia(1993)±39.4BalbontõnandFisher(1981)±40Poulsen(1952)±47Furet(1982)L.A.Cubillosetal./FisheriesResearch42(1999)147±153 culatedthefollowingleastsquareslinearregressiononthelogtransformeddata:982lnWedonotusethemaximumobservedagesoffemalesandmalesofChileanhake.Instead,weuseanestimationforbasedontheassumptionthatmaximumagecanbeestimatedwhen®shreach95%oftheasymptoticlength(Taylor,1958,1960;Pauly,1980),i.e. ln1 0:95L1L1t0‡ 2.2.VarianceestimationprocedureThebasisfortheproceduresuggestedhereisaformoftheparametricbootstrapresamplingmethodofEfron(1982,1985),whichisevaluatednumericallybymeansoftheMonteCarloapproach.Forthisprocedure,anumberofequallylikelyalternativenaturalmortalityvalues()aregeneratedusingaparametricformoferrordistribution,where...,andisarelativelylargenumber100).Inthispaper3000wasusedforeachmethodofestimation.Theparametricbootstrap(PB)methodestimateoftheassociatedvarianceisthen where Estimatesofconfidenceintervalscanbeobtainedbyapercentilemethod,whichinvolvesorderingthesetToimplementtheprocedureofvarianceestimate,thestandarderrorofallestimatesoftheinputpara-meters,aswellasthematrixofthecorrelationofthevonBertalanffygrowthparameters,weretakenintoaccount(Tables1,2and3).First,onlyerrorintheestimatesoftheinputparametersweretakenintoaccount.After,predictionerrorfromtheregressionmodelwasincorporatedbyusingtheassumptionsdescribedinTable2.Weassumedacoef®cientofvariationof10%forthecoef®cientsofthemodelofRikhterandEfanov,andHoeningbecausetheseauthorsdidnotcommunicateadditionalstatisticfromthe®ttedmodeltothedata.Inthisattempttotakeintoaccountpredictionerrorfromtheempiricalmodels,wealsoassumedthatthereisnotcovariancebetweenthecoef®cientsofeachmodel.Therefore,theestima-tionofvarianceforwillbeconditionaltotheassumptionsdone. Table2AssumptionsofmodelanddataelementsusedinthevarianceestimationprocedureofthenaturalmortalityratefortheChileanhake,andthedistributionandprecisioninformationusedintheparametricbootstrapE

3 lementDescriptionErrorintheestimatesofth
lementDescriptionErrorintheestimatesoftheinputparametersGrowthparametersGrowthparametersnormallydistributed,meanandstandarderrorasinTable1andcorrelationsbetweengrowthparametersasinTable3.InputtoEqs.(1),(3)and(5).MeanannualtemperatureofthehabitatUniformlydistributedbetween9Cand14C.InputtoEq.(1).Sizeat50%maturityUniformlydistributedbetween36and47cmoftotallength.Sizeat50%maturityformalesequaltothatoffemalesinTable1.InputtoEq.(3).PredictionerrorfromtheempiricalmodelsPauly'smodelCoefficientsofthemodelnormallydistributed,meanasinEq.(1),i.e.0.2790,0.6543and0.4634,andCVof24%,11%and18%,respectively(Table5inPauly,1980).RikhterandEfanov'smodelItisassumedthatcoefficientsofthemodelarenormallydistributed,modelmeanasinEq.(2),i.e.1.521,0.72and0.155,andanassumedCVof10%forallcoefficients.Hoening'smodelCoefficientsofthemodelnormallydistributed,meanasinEq.(4),i.e.1.44and0.982,andanassumedCVof10%forallcoefficients.Coefficientofvariation(CV)SE/mean;CVmeasuresuncertaintyandisinverselyrelatedtoprecision.L.A.Cubillosetal./FisheriesResearch42(1999)147±153 3.Results3.1.EstimatesofMbasedonerrorintheinputparameterstotheempiricalmodelsTheaverageof3000equallyprobableandalter-nativevaluesofobtainedusingthemodelofPauly(1980)was0.44and0.26yrformalesandfemales,respectively.Thecoef®cientsofvariationwere7%formalesand10%forfemales,whilethe95%con®dencelimitswerebetween0.38and0.49formales,andbetween0.21and0.32forfemales(Table4(a)).FromthemethodofRikhterandEfanov(1976),theaverageestimatesofare0.38yrbothmalesandfemales,butmaleshaveahighercoef®cientofvariationthanfemales.The95%con-®dencelimits,however,weresimilar(Table4(a)).InapplyingthemethodofHoening(1983),theaveragefrom3000alternativeandequallyprobablevaluesof0.43and0.24yrformalesandfemales,respectively.The95%con®dencelimitswerebetween0.39and0.47forthemales,andbetween0.19and0.28forfemales(Table4(a)).3.2.EstimatesofMbasedonbotherrorintheinputparametersanderrorinthepredictionfromtheregressionConsideringtheassumptiontoincludepredictionerrortothevarianceestimationprocedure,theaverageof3000alternativeandequallyprobablevaluesfor0.46yrformalesand0.28yrfemalesinthecaseofthePauly'smodel.However,thecoef®cientsofvariationwere37%and41%for Table3MatrixofcorrelationforthevonBertalanffygrowthparametersoftheChileanhakeasestimate

4 dbynon-linearregression(datafromAguayoan
dbynon-linearregression(datafromAguayoandOjeda,1987)MalesFemales1.01.00.9391.00.9711.00.7200.8781.00.7830.8871.0 Table4Summaryof3000alternativeandequallyprobablevaluesforthenaturalmortalityrateoftheChileanhake(Merlucciusgayi)usingempiricalPauly(1980)RikhterandEfanov(1976)Hoening(1983)MaleFemaleMaleFemaleMaleFemaleOnlyerrorintheestimatesoftheinputparametersMean0.440.260.380.380.430.24Median0.440.260.380.380.430.24SD0.030.030.070.060.020.02CV0.070.100.170.140.050.10Min0.340.180.270.260.350.15Max0.530.350.530.520.510.32Percentiles2.50%0.380.210.280.300.390.1997.50%0.490.320.500.490.470.28Botherrorintheestimatesoftheinputparametersand``predictionerror''Mean0.460.280.390.390.450.25Median0.430.260.380.380.430.24SD0.170.120.110.090.120.08CV0.370.410.270.240.270.33Min0.150.080.110.130.190.08Max1.561.040.760.750.980.64Percentiles2.50%0.220.120.210.230.250.1297.50%0.880.560.600.590.720.44L.A.Cubillosetal./FisheriesResearch42(1999)147±153 malesandfemales,respectively(Table4(b)).TheaverageobtainedfromthemodelofRikhterandEfanov(1976)was0.39yrformalesandfemales,butthecoef®cientofvariationincreaseto27%and24%formalesandfemales,respectively.SimilarresultswereobtainedfromthemodelofHoening(1983)(Table4(b)).Itmustbementionedthatthepredictionerrorsobtainedhereforeachmodelareconditionalupontheassumptionabouttheuncertaintyinthecoef®-cientsofthemodels,particularlyfortheRikhterandEfanov(1976)andHoening(1983)models(Table2).ThesemodelsweremoreprecisethatthemodelofPaulybecauseacoef®cientofvariationof10%wasassumedfortheirconstants(Table4(b)).4.DiscussionPascualandIribarne(1993)evaluatethepredictivepowerofthemostcommonlyusedempiricalmodelsand®ndthantheforecasterrorsoftheestimatesofthenaturalmortalityaresubstantial.Theauthorsproposethatthepredictivepowerofanempiricalmodelmustbeevaluatedwithadequatemethodssincethecriterionbasedonthecoef®cientofdeterminationonlydescribesthefoundrelationships.Vetter(1988)indicatesthatthemethodsthatuseparametersofthelifehistorypresenttwoadvantages:(a)requireaminimalquantityofdata,and(b)areusefulindemonstratingtrendsbetweenspeciesandinthedevelopmentoftheecologicaltheory.Neverthe-less,duetothefactthattheygenerateonlyasimpleestimateofforanygivengroupof®sh,theyarenoteffectivetogeneratepreciseestimatesof,ortodeterminetheexistence

5 orextentoftrendsandvaria-bilityinforagiv
orextentoftrendsandvaria-bilityinforagivenstock.Furthermore,theextra-polationswillnotbebetterthantheparametersusedtoestimatethevaluesofusedintheregressions.Therealmortalityratesandtheirvariabilityarestillverypoorlyknown(Vetter,1988).Inestimatingsexualdifferencesin,onlywhenerrorintheinputparameterswasconsidered,themethodsofPauly(1980)andHoening(1983)pro-ducedestimatesformalesandfemalesthatweresigni®cantlydifferent.Infact,whenpredictionerroranderrorintheestimatesoftheinputparameterswereincludedinthevarianceestimationprocedure,naturalmortalityestimateswerenotsigni®cantlydifferentbetweenmalesandfemalessince95%con®dencelimitsoverlap.Byconsideringtheunderlyingassump-tionsmadetoincludetheerrorofpredictionfromthemodels,thevariancesobtainedfortheRikhterandEfanov(1976)andHoening(1983)estimatorscouldbeunderestimated.Unfortunately,thelastauthorsdidnotincludethestandarderrorforthe®ttedmodelstothedata.Infact,thevarianceandcon®dencelimitsforcouldhavebeenobtainedusingananalyticalmethod.However,thealternativeandequallyprob-ablevaluesobtainedthroughtheMonteCarlopara-metricbootstrapseemsinprinciplebemoreef®cient.Ontheotherhand,consideringtheunderlyingassumptionsintheestimateofwithempiricalmodelscertainaspectsshouldbetakenintoaccount.Iftheageat50%maturityisthesameformalesandfemales,butifisdifferent,thenthemethodofRikhterandEfanov(1976)shouldnotbeusedtodetermineformalesandfemalessincetheresultsofthemodelwillbethesame.ThemethodofPauly(1980)cangeneratereason-ableestimates(accurate),buttheestimatesofnotveryprecise.Furthermore,thecon®denceintervalcouldbeasymmetric,withtheupperboundofthecon®denceintervalbeingmuchgreaterthatthelowerboundduetothelognormaldistributionoftheerror(Fig.1).InthecaseofthemodelofHoening(1983),weassumedthatthemaximumage()of®shoccursatthepointwhere95%oftheasymptoticlengthisreached.Thisassumptionmaybevalidonlyinlimitedcircumstances.Infact,some®shcanveryrapidlyreachasymptoticlengthbeforemaximumage(Beverton,1963).FortheChileanhake,themaximumagewasestimatedtobe10.2yrformalesand18.8yrforfemales.Atpresent,themaximumobservedagesinthecatchesare10±11yrformalesand13±14yrforfemales(Payaetal.,1992;Aguayo,1994).Varianceestimationisanimportantaspect,sincenormallythenaturalmortalityformalesandfemalesestimatedbyempiricalmodelsisnotevaluat

6 edforsigni®cance.Also,anestimateofnatur
edforsigni®cance.Also,anestimateofnaturalmortalityshouldalwayshaveaccuracyandacorrespondingnarrowcon®denceband.Accurateandpreciseesti-matesofarerequiredtodeterminethe®shingmortalityrate()becauseahighvarianceontheestimateofcouldprecludetheestimateofasig-ni®cantorwouldresultinahighlyinsig-L.A.Cubillosetal./FisheriesResearch42(1999)147±153 ni®cantvalue.Thepotentialerrorinempiricalesti-matesofanditsimplicationsforstockassessmentsand®sheriesmanagementshouldbetakenintocon-sideration.AcknowledgementsWearegratefultoVidarWespestadforhisadviceandreviewofanearlyversionofthepaper.Also,wearegratefultoananonymousreviewerforthecom-ments.Thisresearchwas®nancedbyFIPNo.95-15.Wewishtothanktothe``ConsejodeInvestigacatioPesqueraoftheFondodeInvestigationPesqucra(FIP)deChile''byfacilitatingthepublicationoftheresults.ReferencesAguayo,M.,Robotham,H.,1984.DinamicapoblacionaldemerluzacomuMerlucciusgayigayi)(Gadiformes-Merlucci-dae).Invest.Pesq.(Chile)31,17±45.Aguayo,M.,1994.BiologyandfisheriesofChileanhakes(M.gayiM.australis).In:Alheit,J.,Pitcher,T.J.(Eds.),Hake,biology,fisheriesandmarkets,FishandFisheriesSeries,vol.15,Chapman&Hall,London.Aguayo,M.,Ojeda,V.,1987.EstudiodelaedadycrecimientodemerluzacomuMerlucciusgayigayiGuichenot,1848)(Gadiformes-Merluccidae).Invest.Pesq.(Chile)34,99±112.n,R.,Arancibia,H.,1993.TalladeprimeramadurezsexualyfecundidadparcialenlamerluzacomuMerlucciusgayigayi(Guichenot,1848).Cien.Tec.Mar,CONA16,33±45.Arancibia,H.,Cubillos,L.,1993.EvaluaciondelstockdemerluzaMerlucciusgayi)deChilecentro-surenelperõ1975±1991,poranalisisdepoblacionvirtual.Invest.Mar.,Valparaõso21,23±41.n,F.,Fisher,W.,1981.CiclosexualyfecundidaddelaMerlucciusgayi,enlacostadeChile.Rev.Biol.Mar.Valparaõso17(3),285±334.Beverton,R.J.H.,1963.Maturation,growthandmortalityofclupeidandengraulidstocksinrelationtofishing.Rapp.P.-V.Reun.Cons.Int.Explor.Mer154,44±67.Cubillos,L.,Arancibia,H.,1992.EvaluaciondelrecursomerluzaMerlucciusgayi)delazonacentro-surdeChileporsdeReducciondeStock.BiologõaPesquera21,15±19.Cubillos,L.,Sobarzo,P.,Arancibia,H.,1994.Analisisretro-spectivodelaevaluaciondemerluzacomuMerlucciusgayiutilizandoanalisissecuencialdelapoblacionsintonizado.aPesquera23,19±30.Efron,B.,1982.Thejacknife,thebootstrapandotherresamplingplans,vol.38.SocietyforI

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