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Demographic Studies of Plants: Data Limitations and Spatial Sampling Methods

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DEMOGRAPHICSTUDIESOFPLANTS:DATALIMITATIONSANDSPATIALSAMPLINGMETHODSByIANJ.FISKEATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2006

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Copyright2006byIanJ.Fiske

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ACKNOWLEDGMENTSTherearemanyindividualswhohavesupportedmeduringthiswork.First,Iwouldliketothankmyadvisor,EmilioBruna,forgivingmetheopportunitytobecomeanecologistdespitemypriortrainingasanengineerandmathematician.Emiliowasendlesslyencouragingandhumorousduringstressfultimes.IwouldalsoliketothankmycommitteemembersBenjaminBolkerandMadanOliforhelpfuldiscussionsandcommentsonthemanuscriptandforallthatIlearnedintheirquantitativeecologycourses.Inparticular,Chapter1isadirectproductofacommentBenmadeatmyrstcommitteemeeting.IalsothankthestaattheBiologicalDynamicsofForestFragmentsProjectforlogisticalsupportandthemanyeldassistantswhohelpedwithdatacollection.FinancialsupportforthisprojectcamefromtheUniversityofFlorida'sCollegeofAgriculturalandLifeSciencesandNationalScienceFoundationgrantsOISE-0437369andDEB-0309819.Finally,Iamgratefulforthenever-endingpatienceandsupportofbothmyfriendsandfamilyduringthisproject. iii

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. v LISTOFFIGURES ................................ vi ABSTRACT .................................... vii CHAPTER 1INTRODUCTION .............................. 1 2SAMPLINGINTENSITYANDBIASEDESTIMATESOFPOPULATIONGROWTH ................................... 3 2.1Introduction ............................... 3 2.2Methods ................................. 6 2.3ResultsandDiscussion ......................... 9 3SPATIALSAMPLINGMETHODSFORPLANTDEMOGRAPHY ... 18 3.1Introduction ............................... 18 3.2Methods ................................. 20 3.3Results .................................. 24 3.4Discussion ................................ 25 4CONCLUSION ................................ 40 APPENDIXPAPERSREVIEWED ..................... 42 REFERENCES ................................... 47 BIOGRAPHICALSKETCH ............................ 53 iv

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LISTOFTABLES Table page 2{1Resultsfromsamplesizeliteraturereview ................. 17 3{1Minimumsamplingareasfor1%precision ................. 38 3{2Negativeexponentialmodelevaluatedat1000m2 ............. 38 3{3Areassampledintheliterature ....................... 39 v

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LISTOFFIGURES Figure page 2{1Flowchartofsimulations ........................... 13 2{2Heliconiaacuminatatransitionmatrix ................... 14 2{3Relativebiasinestimatesof ........................ 15 2{4Histogramsofthe6000estimatesof .................... 16 2{5Samplesizesfrompublishedstudies ..................... 16 3{1Illustrationofsamplingterminology ..................... 31 3{2Flowchartofsamplingmethodsimulations ................. 32 3{3Minimumsamplingareafor1%precision .................. 33 3{4Logisticmodelofprecision .......................... 34 3{5Negativeexponentialmodelofprecision .................. 35 3{6Samplingmethodsinpublishedstudies. ................... 36 3{7Areassampledinpublishedstudies ..................... 37 vi

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceDEMOGRAPHICSTUDIESOFPLANTS:DATALIMITATIONSANDSPATIALSAMPLINGMETHODSByIanJ.FiskeAugust2006Chair:EmilioBrunaMajorDepartment:WildlifeEcologyandConservationMatrixmodelsarewidelyusedtostudythedynamicsofstructuredpopulationsandhavebeenappliedtobasicandappliedecologicalquestions.Animportantbutmostlyoverlookedissueishowthenumberofindividualssampledandtheeldmethodusedtosamplethoseindividualsaectestimatesoftheasymptoticpopulationgrowthrate,,derivedfrommatrixmodels.Weuseddatafromalong-termstudyofthedemographyoftheAmazonianunderstoryherbHeliconiaacuminataHeliconiaceaetoaddressbothofthesequestions.Toinvestigatehowthenumberofindividualssampledaectstheaccuracyofestimates,wesimulatedtheeectsofsamplingvariabilitywhencollectingdemographicdatabydrawingvitalratesfromappropriateprobabilitydistributions.Becausesamplingvariabilityofdependsuponthenumberofindividualssampled,thedistributionofsamplingeortamongsizeclasses,thevarianceoffecundityrates,andthesurvivalrates,wevariedthesefactorstoinvestigatetheireects.WefoundnopracticallyimportantbiasinestimateswhensamplingH.acuminatavitalrates,independentofsamplesize,varianceoffecundityamongindividuals,anddistributionofsamplesamongstageclasses.However,wefoundsignicantbiasatsmallsamplesizes vii

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whensurvivalwasreducedto0.5forallstageclasses.Furthermore,thisbiaswasexacerbatedwhensamplingwitharealisticJ-shapeddistributionofindividualsamongsizeclasses.Becausebiaswasstrongestwhensurvivalisequalto0.5,theseresultssuggestthattheextentofsmallsamplesizebiasstronglydependsuponthestudyorganismandtheparticularyearthatsamplingoccurs.Areviewofliteraturerevealedthatmanystudieshaveparameterizedmatrixmodelsusingfewerindividualsthansamplesizeswefoundtosuerfromsmallsamplesizebias.Thesecondgoalofthisinvestigationwastodeterminehowthespatialmethodusedtosampleplantsaectstheprecisionofstochasticpopulationgrowthestimates.Wecomparedtherelativeeciencyofthreecommonlyusedmethodstosampleplantsfordemographicstudies:surveyingallplantswithinasingleplot,multiplerandomlyplacedplots,ormultiplesystematicallyplacedplots.Toevaluateeachmethod,wesimulatedsamplingfromtheH.acuminatadata.Systematicsamplingprovidedthemostpreciseestimatesofpopulationgrowthaccordingtoallanalysesweusedtocomparethemethods.Aswiththerstquestion,weconductedareviewoftheplantdemographyliteraturetodeterminehowpublishedstudieshavesampledplants.Althoughourresultssuggestthatsystematicsamplingwasthemostecientmethod,itwastheleastcommonmethodfoundinpublishedstudies. viii

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CHAPTER1INTRODUCTIONMatrixmodels Leslie 1945 ; Lefkovitch 1965 areanimportanttoolthatecologistsusetostudythedemographyofstructuredpopulations.Theyareexible,readilyapplicabletoadiversityoflife-historystrategies,andthereisabroadbodyofliteraturedescribingtheirconstruction,interpretation,assumptions,andlimitationssee Bierzychudek 1999 ; deKroonetal. 2000 ; Wisdometal. 2000 ; Caswell 2001 .Althoughmatrixmodelsarecommonlyusedtoestimatetheasymptoticgrowthrateofapopulationbothdeterministic,,andstochastic,s,increasinglysophisticatedcomputationaltechniquesalsoallowonetoexploretheunderlyingdemographicfactorsinuencingtherateofpopulationgrowth Caswell 2001 ; MorrisandDoak 2002 ; Bruna 2003 .Forexample,elasticityanalysesandotherprospective"approaches Horvitzetal. 1997 ,evaluatehowhypotheticalchangesinvitalrateswouldalter,whileretrospective"techniquessuchasLifeTableResponseExperiments Caswell 1989 2001 decomposeobserveddierencesinintotheactualcontributionsfromindividualdemographicvariables.Theseanalyseshavebecomeincreasinglycommoninappliedsettingsbecausetheycansuggestwhatdemographictransitions,andthereforewhatecologicalprocesses,mightbethefocusofconservationandmanagementstrategies ValverdeandSilvertown 1998 .Despitethepopularityofmatrixmodelsinecologyandconservation,fewstudieshaveinvestigatedhowtocollectdatatoparameterizethesemodelsforstudyingplantsbutsee Gross 2002 ; Doaketal. 2005 ; MunzbergovaandEhrlen 2005 .Thisknowledgegapissurprisingbecausestudydesigninecologyhasgainedmuchattentioninoverthelastfewdecadese.g., Green 1979 ; Scheiner 1

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2 andGurevitch 2001 .Theaccuracyandprecisionofanymodelprojectiondirectlydependuponthequalityoftheestimatedmodelparametersandthequalityoftheseparametersisaresultofhowthedatawerecollected.Thus,agoalofthisworkistoinvestigatethesamplingmethodsofpublishedstudiesofplantpopulationsandtodetermineifworkinplantpopulationbiologycanbeimprovedbyalteringsamplingmethodsorsamplingintensity.InChapter1,weaddresshowsmallsamplesizesaecttheaccuracyofestimatesofpopulationgrowth.Weuselong-termdemographicdataontheAmazonianunderstoryherbHeliconiaacuminata Bruna 2003 tolookatbiasofpopulationgrowthestimateswhenvitalratesareestimatedwithfewindividuals.InChapter2,wecomparethreecommonlyusedspatialsamplingmethodsforstudyingplantswithmatrixmodels:singleplots,multiplerandomlyplacedplots,andmultiplesystematicallyplacedplots.WeusetheH.acuminatasystemtoinvestigatetherelativeprecisionofestimatesofstochasticpopulationgrowththatresultfromthesethreesamplingmethods.Inbothchapters,weconductareviewofpublisheddemographicpapersonplantstocompareourndingswithhowpreviousworkhassampled.

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CHAPTER2SAMPLINGINTENSITYANDBIASEDESTIMATESOFPOPULATIONGROWTH2.1IntroductionMatrixmodelsareconstructedusingdemographicvitalrates MorrisandDoak 2002 ; FrancoandSilvertown 2004 ,suchastheprobabilitythatanindividualsurvivesfromonetimesteptothenextortheprobabilityofgrowingintolargersizeclasses.Althoughtheseratescanbeestimatedinanumberofways,onecommonmethodistotrackthefatesofindividualsofdierentsizesoragesthroughtime HorvitzandSchemske 1995 ; Meagher 1982 .Thisprocedurerequiresestablishingpermanentplots,mark-recaptureprograms,orothermeansoffollowingindividualsfromapopulation.Asmightbeexpected,thenumberofindividualssampledfromeachsizeorageclasscaninuenceestimatesofvitalratesforthosestagesKendall1998,White2000.However,howestimatesofpopulation-levelstatisticssuchasareinuencedbythetotalnumberofindividualsobservedinademographicstudyremainsvirtuallyunexploredbutsee Doaketal. 2005 .Despitethecentralrolethatsamplesizeestimationplaysinthedesignofecologicalexperimentse.g., Green 1979 ; Krebs 1999 ; ScheinerandGurevitch 2001 ,demographershavehadlittleguidanceregardingtheappropriatesamplesizesneededtohavecondenceinestimatesof.Offundamentalimportancetothisissueisthedierencebetweenthepre-cisionandaccuracyofestimatesof Wackerlyetal. 2001 .Preciseestimateshavelowvarianceandwilllikelybesimilarafterrepeatedsampling.Ontheotherhand,accurateestimatesareexpectedtocorrectlyestimatethequantityofinterest,onaverage,overrepeatedsampling.Precisiondoesnotimplyaccuracy, 3

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4 nordoesaccuracyimplyprecision.Inthisstudy,weinvestigatehowsamplesizeaectstheaccuracyofestimatesof.Webeginbyrstreviewingthestatisticalframeworkthatguidestheestimationofvitalrates,withparticularemphasisontheimplicationsofsamplingvarianceandthemathematicaltheoremknownasJensen'sInequality RuelandAyres 1999 .Wethenpresenttheresultsofsimulationsusedtoinvestigatehowthetotalnumberofplantssampledinademographicstudyinuencestheaccuracyofprojectedvaluesoftheasymptoticpopulationgrowthrate,.Finally,wereviewtheecologicalliteratureanddescribetherangeofpopulationsizesusedtoparameterizematrixmodelsofplantdemography.BackgroundDemographictransitionsareoftenmodeledastheproductsofunderlyingvitalrates,andmanyofthesevitalratesaremodeledasbinomialrandomvariables MorrisandDoak 2002 .Thebinomialmodelassumesthefatesofindividualsareindependentandthatallindividualsofagivenstageclasshaveequalprobabilitiesofsurviving,growing,andshrinking.Forexample,theprobabilitythatanindividualinstageclassigrowstothenextstageclassfromonetimesteptothenextistheproductoftheprobabilitythataclassiindividualsurvivessiandtheconditionalprobabilityaclassiindividualgrowsgiventhatitsurvivedgi.Assumingadetectionprobabilityof1,wecanestimatetheseprobabilitieswithsampleproportionse.g.,^si=nit=nit)]TJ/F15 11.955 Tf 12.46 0 Td[(1wherenitisthenumberofindividualsofclassifoundattimet Thompson 2002 .IfE[^si]6=si,thentheestimateofthissurvivalprobabilityisbiased Wackerlyetal. 2001 .Whenanestimateisbiasedandtheresearcherisunawareofit,theresultcanbeprecisebutinaccurateestimatesoftransitionprobabilities.Thesebiasedestimatescanbeeliminatedentirelybysamplingtheentirepopulationofinterest;however,completepopulationsurveysarerarelyfeasible.Alternatively,onecanreducebiasbyusing

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5 techniquesdevelopedinstatisticalsamplingandestimationtheory,suchastheuseofrandomsamplese.g., Mack 2002 .Nevertheless,unbiasedestimatesofvitalratesdonotensureunbiasedestimatesofthepopulationgrowthrate.Becauseisthedominanteigenvalueofthetransitionmatrix,itisbydenitionanonlinearfunctionoftheunderlyingvitalrates Leslie 1945 .AccordingtoJensen'sinequality Jensen 1906 ,amathematicaltheoremthathasbeenincreasinglyappliedinecologicalstudiese.g., Karbanetal. 1997 ; RuelandAyres 1999 ; Inouye 2005 ,anyvarianceinthevitalrateswillresultinbiasedestimatesof.Theamountanddirectionofbiasdependontwofactors:rst,thestrengthofthenonlinearityoftherelationshipbetweenandthevitalratesandsecond,thevarianceofthevitalratesthemselves.Variationofvitalratesarisesfromtwosources.Therstoftheseisprocessvariance,whichisaresultofrealvariationinthepopulationoverspace Stratton 1995 ortime Tuljapurkar 1990 .Thesecondissamplingvariance,whichisaresultofstudyingasampleratherthantheentirepopulation.Methodsfordealingwithprocessvariance,especiallyovertime,arewell-developed LewontinandCohen 1969 .Morerecently,methodshavebeendevisedthatattempttoseparatethesamplingvariancefromthetotalobservedvarianceinmatrixmodels Kendall 1998 ; White 2000 .Here,weconsidertheeectthatthissamplingvarianceofvitalrateshasontheaccuracyofestimates.Thebinomialdistribution,whichassumeshomogeneousvitalratesamongindividualswithinastageclass,maximizesthesamplingvarianceofthevitalrate KendallandFox 2002 .Therefore,thebinomialdistributionisaconservativechoiceforinvestigatingtheeectsofvitalratesamplingvariance.Thesamplingvarianceofthesebinomialvitalratesis:2^si=Var^si=si)]TJ/F22 11.955 Tf 11.956 0 Td[(si ni{1

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6 whereniisthenumberofindividualsinclassiinthesample,andsiisthetruevalueofthevitalrate Thompson 2002 .Forestimatesoffecundity,suchasthenumberofospringanindividualproducesgivenitreproduces,thevarianceis:2^fi=Var^fi=2i ni{2wherefiisthefecundityofindividualsinclassiandfiisthetruevarianceofthefecundityamongindividualsinclassi;niisdenedasabove.Notethatthesesamplevariancesincreaseassamplesizedecreases.Therefore,Jensen'sInequalitywouldleadonetopredictthatsmallsamplesizeswillleadtobiasedestimatesofasaresultofincreasedsamplingvarianceintheestimatesofvitalrates,irrespectiveofwhethertheestimatesofthevitalratesarethemselvesaccurate.2.2MethodsWeusedsimulationmodelstoestimatehowtheaccuracyofprojectionsofvariedwithsamplesize.Thesemodelswerebasedondatacollectedduringalong-termandlarge-scalestudyofplantdemographyconductedatBrazil'sBiologicalDynamicsofForestFragmentsProjectBDFFP;230'S,60W.ThefocalspeciesforthisstudywasHeliconiaacuminata,aperennialherbnativetocentralAmazoniaandtheGuyanas Berry 1991 .Descriptionsofthestudysiteandexperimentaldesigncanbefoundelsewhere Bierregaardetal. 1992 ; BrunaandKress 2002 ; Bruna 2003 .Briey,permanent50mx100mplotswereestablishedin13oftheBDFFP'sreservesinJanuary1998.AllH.acuminataineachplotweremarkedandmappedandthenumberofvegetativeshootseachplanthadwasrecorded Bruna 2003 .Sincetheirestablishmenttheplotshavebeensurveyedannuallytorecordplantgrowth,mortality,andtheemergenceofnewseedlingsi.e.,establishedplantslessthan1yearold.Theplotswerealsosurveyedduringtheoweringseasontorecordtheidentityofreproductiveindividuals.Theanalysispresentedhereisbasedondatafromthe1998-2003surveys;duringthis

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7 timeperiodwemarked,measured,andrecordedthefatesofN=6591plantsN=3842incontinuousforest,N=1688in10-hafragments,andN=1061in1-hafragments.Usingthesedata,wesimulatedasamplingprocessthatmodeledthesamplingvariabilityofvitalratesseealsoFigure 2{1 .Weusedthe1998-1999estimatesofthevitalratesas`true'populationvitalrates;theresultswerenotsensitivetothisdatachoice.Then,weusedoneoftwoprobabilitydistributionstosimulatesamplingeachvitalrate:abetaprobabilitydistributionforthebinomialvitalratese.g.,probabilityofsurvivorship,probabilityofgrowthandthegammadistributionforthecount-basedvitalratesi.e.,fecundity.Becausethebetadistributioniscontinuous,boundedby0and1,andcanbeparameterizedtohaveavarietyofmeansandvariances,itisanappropriatechoiceformodelingestimatesofthebinomialvitalrates MorrisandDoak 2002 .Wechosethegammadistributiontomodelestimatesofaveragefecunditybecauseitisnon-negativeandcanalsobeexiblyparameterized.Weparameterizedboththebetaandgammasamplingdistributionsaccordingtothemethodofmoments,atechniquewhichparameterizesadistributionbyspecifyingitsexpectedvalueandvariance HilbornandMangel 1997 .Todenethesamplingprocess,wesettheexpectedvalueandvarianceofanestimatedvitalrateequaltothetruepopulation'smeanvitalrateandsamplingvariance,respectively.Wedeterminedthesamplingvarianceateachsamplesizewithequations 2{1 and 2{2 .Then,weusedwell-knownmethodofmomentsrelationshipsbetweentheparametersofthedistributionsandtheirexpectedvalueandvariancee.g., Gelman 2004 .Forthebetadistribution,weusedthefollowingrelationshipbetweentheparametersandthemeanandvarianceofthedistribution:=^si^si)]TJ/F22 11.955 Tf 11.955 0 Td[(^si=2^si)]TJ/F15 11.955 Tf 11.955 0 Td[(1{3=)]TJ/F22 11.955 Tf 11.955 0 Td[(^si^si)]TJ/F22 11.955 Tf 11.956 0 Td[(^si=2^si)]TJ/F15 11.955 Tf 11.955 0 Td[(1

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8 whereand^siand2^siarethemeanandvarianceoftheestimateofvitalratesi,respectively.Similarly,tocalculatetheparametersofthegammadistribution,weusedtherelationship:scale=2^fi=^fi{4shape=2^fi=2^fiwhere^fiand2^fiarethemeanandvarianceoftheestimateoffecundityfi,respectively.Accordingtoequation 2{1 ,thevarianceofbinomialvitalrateestimatesismaximizedwhentherateis0.5.Totestforadierenceinbiaswhensurvivalestimatesvarymaximally,wesimulatedwithboththetruesurvivalratesandafterreplacingthesurvivorshipofallstageswith0.5.Similarly,becausethevarianceofestimatesoffecundityisproportionaltotherealvarianceoffecundityamongindividualsequation 2{1 ,wesimulatedwith3levelsoffecundityvariance,whichwedenedintermsofcoecientofvariation,^fi=^fi:0.5,2,and16.Becausethedistributionofindividualsamongsizeclassesmayaectdemographicanalyses Gross 2002 ; MunzbergovaandEhrlen 2005 ,wesimulatedsamplingwithtwodierentdistributionsofindividuals.First,weusedauniformdistributionofindividualsamongsizeclassesi.e.,equalnumbersofindividualsinallsizeclasses.Next,weusedamorerealisticdistribution,knownastheJdistribution."AJdistribution"containsfewerstagei+1individualsthanstagei,suchthatahistogramofstageclassesinasampleisJ-shaped.WeusedtheactualdistributionofclassesobservedintheeldaveragedoverallyearsandsitestocomputetheJdistribution.Foreachvitalrate,weran2000simulationswithpopulationsranginginsizefrom25to200individualsforeachcombinationofsurvivalrealand0.5,fecundityCV0.5,2,and16,andsamplingdistributionrealandequal.Wechose25individualsasthesmallestsamplesizebecausesmallersampleswouldyieldtoofew

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9 individualsperstageclasstoresemblewhatarealstudymightsample.BecauseweusedMonteCarlomethodstosimulatethevitalratesdirectly,fractionalsamplesizesposednoproblem.Ineachrunofthesimulation,wedrewall31vitalrates6binomialvitalratesand5fecunditiesfromtheirappropriatesamplingdistributions,computedasampletransitionmatrixfromthesevitalrates,andthenestimatedasthedominanteigenvalueofthistransitionmatrixFigure 2{2 .Wethenestimatedtheexpectedvalueandstandarddeviationoftherelativebiasofestimates,^)]TJ/F22 11.955 Tf 12.898 0 Td[(=100%,ateachcombinationofsurvivalrates,fecundityCV,distributionofsamplingeort,andsamplesize.AllsimulationswereconductedusingtheRstatisticalcomputingenvironment RDevelopmentCoreTeam 2005 .Toassesstheimportanceofourndings,weconductedareviewofsamplesizesusedintheplantdemographyliterature.WesurveyedthisliteratureusingaWebofSciencesearchonMarch15,2006.Oursearchtermswerecombinationsofmatrixmodel",plant",anddemography."Wealsofollowedtherelevantcitationsofthesepaperstoincreasethenumberofpaperssurveyed.Fromeachstudy,weextractedthenumberofindividualssampledtoparameterizeonetransitionmatrix.Ifastudyestimatedmorethanonematrix,asinamulti-sitestudy,weusedtheaveragenumberofindividualssampledforamatrix.Wealsodeterminedthenumberofstudiesthatdidnotincludeasamplesize.2.3ResultsandDiscussionWedetectednobiaswhensamplingfromtherealHeliconiavitalrates.However,wefoundanappreciablepositivebiasinestimatesofwhenusing50orfewerindividualstoestimatevitalratesfromapopulationwithameansurvivalprobabilityof0.5Figure 2{3 andFigure 2{4 .Inaddition,therealisticJ-shapedsamplingdistributionincreasedthepositivebiascomparedtosamplingequalnumbersofindividualsinallstageclasses.62%32.54SDvs.11.44%

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10 25.40SDrespectivelywhensampling25individuals.Asexpected,largersamplesizesproducedmorepreciseestimatesofinallscenarios.Therefore,theaccuracyofestimatesofcalculatedfromsmallsamplesizesstronglydependeduponthetruevitalratesofthesystem.Specically,increasinguncertaintyinthefatesofindividuals,representedbya0.5probabilityofsurvival,causedtobeoverestimated.OurresultsextendthoseofpreviousstudiesdemonstratingprojectionsofcanbebiasedduetotheeectsofJensen'sInequality.Forinstance, Boyce 1977 showedthatprocessvariance,intheformofstochasticityovertime,biasesestimatesof.Usingananalyticalderivationwith22matrices,hedemonstratedthatisaconcavefunctionofthematrixelementsandisthuspositivelybiased. Daley 1979 extendedBoyce'sanalysistoLesliematricesofanysize,andshowedthatmaybeeitherpositivelyornegativelybiasedifmatrixelementsarestochastic.Althoughthescopeofthesestudieswaslimitedtoprocessvariance, Houllieretal. 1989 usedasimilartheoreticalapproachalongwithMonteCarlosimulationstoexaminebiasresultingfromsamplingvarianceofthematrixelements.UsingbothLesliematricesandmoregeneralUshermodels Usher 1966 ,theyfoundthatbiasinestimatesofwassmall-lessthan0.5%inallbutonescenario.However,theanalysisof Houllieretal. 1989 usedthetransitionelementsofthematrixratherthantheunderlyingvitalrates.Itisimportanttorecognizethatsamplingvariationaectsthevitalrates,andthisdistinctionmayaectsampling.Ouranalyses,whichconsideredamoregeneralLefkovitchmodel,demonstratedthatsignicantoverestimationandlargesamplingvariationofcanresultfromsmallsamplesizes.Wewereabletodeterminethenumberofplantsthathadbeenusedtoparameterizematrixmodelsin52ofthe68paperswecollectedseeAppendixA.Studiesofperennialherbsusedfewerindividualsthanthoseoftreesherbs:

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11 median=215,N=28studies;trees:median=575,N=16studies;shrubs:median=303,N=9studies,Figure 2{5 ,Table 2{1 .Over12%ofthestudiesonperennialherbsusedfewerthan100individualssummedacrossallstageclasses,andonly25%ofstudieswerebasedon500ormoreindividualsrange:30{4963.Oursimulationssuggestthatsamplescontainingasmanyas100individualshaveanexpectedbiasofalmost4%.Thislevelofbiascouldleadtoseverelyinatedestimatesoftimeuntilextinctionandtheprobabilityofpersistence.Wehaveshownthattheamountofbiasinestimatesofthatcanbeattributedtosamplingvariationatsmallsamplesizesisafunctionofthevitalratesofthesystem.Furthermore,thebiasinestimatesalsodependsuponthedistributionofsamplesamongstageclasses.ApossibleexplanationforourndingsisthatH.acuminataindividualsarecharacterizedbylowmortality,lowgrowth,andlowfecundity.Therefore,itspopulationgrowthisespeciallysensitivetochangesinthesurvivorshipofindividualsinthelargerstageclasseswhilebeingrelativelyrobusttochangesinfecundity Bruna 2003 .Thisplant'ssurvival-dominatedsensitivitystructurecouldexplainboththedrasticincreaseinbiasinresponsetohighvarianceofsurvivalestimatesaswellastheindependenceofbiastovarianceoffecundityestimates.Inaddition,theresultsofsimulationsinwhichwevariedthesamplingdistributionprovidefurtherevidencethattherelativesensitivitystructurecandeterminehowthevitalratevarianceaectsbias.SamplingwiththemorerealisticJ-distributionincreasedthesamplingvarianceofthemoredemographically`important'vitalratesoflargerstageclassesanddecreasedthesamplingvarianceoftheless`important'vitalratesofthesmallerones.Ourresultssuggestthatsamplingproportionallytotheelasticitystructureofapopulationmaynotonlyincreasetheprecisionofestimatesof Gross 2002 ,butmayalsoincreasetheaccuracy.However,moreworkisneededtoexplainthe

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12 discrepancybetweenourtheory-backedresultsandtheresultsof MunzbergovaandEhrlen 2005 .Onepotentialexplanationforthepoorperformanceofthismethodpresentedby MunzbergovaandEhrlen 2005 isbecausetheelasticitystructureofastudyorganismisusuallyunknownbeforethedataiscollected.Samplingaccordingtotheelasticitystructuresofsimilarorganismstobeproblematicbecausesimilarorganismsmaydierenoughthatsamplingaccordingthemwouldmistakenlyundersamplecriticalstages.Still,H.acuminata'srelativeelasticitystructureiscommontomanylong-livedplants Silvertownetal. 1993 ; FrancoandSilvertown 2004 andsamplingwiththerealHeliconiavitalrateswasrelativelypreciseandaccurate,evenatlowsamplesizes.Therefore,ourresultssuggestthatestimatesofmayberobusttolowsamplesizesinpopulationsoflong-livedorganisms.However,theseresultsaresensitivetothevitalratesofthestudyorganism.Therefore,weencourageotherstoconductsimilarsimulationswithdemographicdatafromspecieswithdierentlifehistoriessincethesetaxacouldhavedemographicmatriceswithmarkedlydierentcharacteristics.

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13 Figure2{1.Flowchartofsimulationsusedtoestimateamountanddirectionofbiasinestimatesofvitalratesduetosamplingvariance.WeusedempiricalvitalratesfromHeliconiaeldstudiesasthetruevitalratesandthensimulatedsamplingfromappropriateprobabilitydistributionstoapproximaterealsamplingvariation.

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14 0BBBBBB@0s2p2fnes3fnes4fnes5fnes6fnes1s2)]TJ/F22 11.955 Tf 11.955 0 Td[(g2s3)]TJ/F22 11.955 Tf 11.955 0 Td[(g3r3s4)]TJ/F22 11.955 Tf 11.955 0 Td[(g4r4k1)]TJ/F21 7.97 Tf 6.586 0 Td[(4s5)]TJ/F22 11.955 Tf 11.955 0 Td[(g5r5k1)]TJ/F21 7.97 Tf 6.587 0 Td[(5k2)]TJ/F21 7.97 Tf 6.587 0 Td[(5s6r6k1)]TJ/F21 7.97 Tf 6.586 0 Td[(6k2)]TJ/F21 7.97 Tf 6.586 0 Td[(6k3)]TJ/F21 7.97 Tf 6.587 0 Td[(60s2g2)]TJ/F22 11.955 Tf 11.955 0 Td[(h1)]TJ/F21 7.97 Tf 6.587 0 Td[(2s3)]TJ/F22 11.955 Tf 11.955 0 Td[(g3)]TJ/F22 11.955 Tf 11.955 0 Td[(r3s4)]TJ/F22 11.955 Tf 11.955 0 Td[(g4r4)]TJ/F22 11.955 Tf 11.955 0 Td[(k1)]TJ/F21 7.97 Tf 6.587 0 Td[(4s5)]TJ/F22 11.955 Tf 11.955 0 Td[(g5r5k1)]TJ/F21 7.97 Tf 6.587 0 Td[(5)]TJ/F22 11.955 Tf 11.955 0 Td[(k2)]TJ/F21 7.97 Tf 6.586 0 Td[(5s6r6k1)]TJ/F21 7.97 Tf 6.586 0 Td[(6k2)]TJ/F21 7.97 Tf 6.587 0 Td[(6)]TJ/F22 11.955 Tf 11.955 0 Td[(k3)]TJ/F21 7.97 Tf 6.586 0 Td[(60s2g2h1)]TJ/F21 7.97 Tf 6.587 0 Td[(2)]TJ/F22 11.955 Tf 11.955 0 Td[(h2)]TJ/F21 7.97 Tf 6.586 0 Td[(2s3g3)]TJ/F22 11.955 Tf 11.955 0 Td[(h1)]TJ/F21 7.97 Tf 6.587 0 Td[(3s4)]TJ/F22 11.955 Tf 11.955 0 Td[(g4)]TJ/F22 11.955 Tf 11.955 0 Td[(r4s5)]TJ/F22 11.955 Tf 11.955 0 Td[(g5r5)]TJ/F22 11.955 Tf 11.955 0 Td[(k1)]TJ/F21 7.97 Tf 6.587 0 Td[(5s6r6k1)]TJ/F21 7.97 Tf 6.586 0 Td[(6)]TJ/F22 11.955 Tf 11.955 0 Td[(k2)]TJ/F21 7.97 Tf 6.586 0 Td[(60s2g2h1)]TJ/F21 7.97 Tf 6.587 0 Td[(2h2)]TJ/F21 7.97 Tf 6.586 0 Td[(2)]TJ/F22 11.955 Tf 11.955 0 Td[(h3)]TJ/F21 7.97 Tf 6.586 0 Td[(2s3g3h1)]TJ/F21 7.97 Tf 6.587 0 Td[(3)]TJ/F22 11.955 Tf 11.955 0 Td[(h2)]TJ/F21 7.97 Tf 6.586 0 Td[(3s4g4)]TJ/F22 11.955 Tf 11.955 0 Td[(h1)]TJ/F21 7.97 Tf 6.586 0 Td[(4s5)]TJ/F22 11.955 Tf 11.955 0 Td[(g5)]TJ/F22 11.955 Tf 11.955 0 Td[(r5s6r6)]TJ/F22 11.955 Tf 11.955 0 Td[(k1)]TJ/F21 7.97 Tf 6.587 0 Td[(60s2g2h1)]TJ/F21 7.97 Tf 6.587 0 Td[(2h2)]TJ/F21 7.97 Tf 6.586 0 Td[(2h3)]TJ/F21 7.97 Tf 6.587 0 Td[(2s3g3h1)]TJ/F21 7.97 Tf 6.586 0 Td[(3h2)]TJ/F21 7.97 Tf 6.586 0 Td[(3s4g4h1)]TJ/F21 7.97 Tf 6.586 0 Td[(4s5g5s6)]TJ/F22 11.955 Tf 11.955 0 Td[(r61CCCCCCAFigure2{2.HeliconiaacuminatatransitionmatrixusedinMonteCarlosamplinganalysis.si=Probindividualinstageisurvivesonetimestepgi=Probindividualinstageigrowsatleastonestageinonetimestep|survivalhx)]TJ/F23 7.97 Tf 6.586 0 Td[(i=Probindividualinstageigrowsatleastxstages|growthofatleastx-1stagesri=Probindividualinstageiregressesatleastonestagepertimestep|survivedanddidnotgrowkx)]TJ/F23 7.97 Tf 6.587 0 Td[(i=Probindividualinstageiregressesatleastxstages|regressionofatleastx-1stagespi=Probplantinstageiowersfi=meannumberoffruitsperoweringplantinstagein=meannumberofseedsperfruite=Probseedgerminatesandestablishes

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15 Figure2{3.Relativebiasinestimatesof,^)]TJ/F22 11.955 Tf 12.67 0 Td[(=100%,atincreasingsamplesizes.Panelsshowthe4combinationsofdistributionofsamplingamongstageclassesrealJ-shapedsamplingdistributionvs.equaleortandsurvivalrealandallsurvivalssetto0.5.Samplesizesonthelistedabscissaarethetotalsamplesizesacrossallsizesclasses.Errorbarsare1standarddeviation.Thedot-dashedlineisatbias=0.

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16 Figure2{4.Histogramsofthe6000estimatesof,combiningalllevelsoffecundityvariance,forsampling25,75,and200individualswithbothrealAand0.5survivalratesB.Solidanddashedverticallinesareexpectedvalueof^andthetrue,respectively. Figure2{5.Samplesizesfrompublishedstudies.Fromeachstudy,weextractedtheaveragenumberofindividualsusedtoparameterizeamatrixmodel.

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17 Table2{1.Resultsfromsamplesizeliteraturereviewof52plantdemographypapers.Weextractedtheaveragenumberofindividualssampledtoparameterizeamatrixmodel.Thesearethesummarystatisticsbrokendownbythe3mostprominentlifehistories. LifehistoryMeanMedianSDRangeN perennialherb747.27214.941223.9230-496328shrub548.52302.62506.233-1275.59tree1311.70575.002040.7291-690516

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CHAPTER3SPATIALSAMPLINGMETHODSFORPLANTDEMOGRAPHY3.1IntroductionThedatausedforparameterizingmatrixmodelsofplants,marineinvertebrates,orothersessileorganismsareoftencollectedinpermanentplotse.g., Bierzychudek 1999 ; Engelenetal. 2005 ; Gotelli 1991 ; Parker 2000 ; Silvertownetal. 1993 .Theprimaryadvantagesofplot-basedmethodsarelogistical:mappedplantsareeasiertorelocateinsubsequentsurveys,surveymethodscanbestandardizedamongsites,andsampledindividualsareclosertogether.Plot-basedmethodsusedindemographicstudiesvarytremendouslyintheplotarrangementandareasampled.Forinstance, Batistaetal. 1998 mappedandtaggedallFagusgran-difolia2cmdbhwithina4.5haplot,while SilvaMatosetal. 1999 studiedthedemographyoftheunderstorypalmEuterpeedulisbysampling100randomlyplacedsubplotswithina1haarea.Incontrast, ValverdeandSilvertown 1998 sampledtheperennialherbPrimulavulgarisatregularintervalsalongatransect.Thesedisparatestudiesillustratethreecommonstylesofplot-basedsamplingusedindemographicstudies:identifyingallindividualswithinasingleplot;randomlysamplingmultiplesubplotswithinadenedarea;andsystematicsamplingatregularintervals.Eventhoughlong-termdemographicstudiesoftenrequirelargeinvestmentsoftimeandcapital,studiesrarelyjustifysamplingdecisions.Interestingly,severalstudieshaveevaluatedtheoptimalplotsizesandshapesformeasuringdiversity Jalonenetal. 1998 ; KenkelandPodani 1991 ; Lauranceetal. 1998 ,individualdensity Gray 2003 ; Picardetal. 2004 ,andintraspeciccompetition HynynenandOjansuu 2003 .However,nosuchcomparisonsofsamplingmethodshavebeenconductedfordemographicstudies. 18

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19 Thechoiceoftheappropriatesamplingdesignisimportantbecauseofthepotentialforspatialvariationofvitalratestoinuencetheoutcomeofmatrixmodelse.g., HorvitzandSchemske 1995 ; Moloney 1988 .Onekeycomponentofthisspatialvariationthatmayaecttheecacyofsamplingdesignsisspatialautocorrelationwithinasamplingsite Legendre 1993 .Inthepresenceofspatiallyautocorrelatedvitalrates,asamplingmethodthatincreasesthedistancebetweensampledindividualsshouldprovidemoreestimatesforagivensamplingeortbecausemorevariabilityiscapturedwiththesamenumberofindividualssampled.Thus,systematicsamplingmightprovidemorepreciseestimatesthanrandomsamplingbecauseitmaximizesthedistancebetweensamplesbysamplingaregulargrid Bellhouse 1977 ; Yates 1948 .Whileecologistshaveconsideredhowspatialautocorrelationinuencesestimatesofindividualdensity,speciesrichnessandotherecologicalparameters Legendreetal. 2004 ,todatenoonehasevaluatedhowspatialautocorrelationofvitalrateswithinasamplingunitmightaecttheecacyofsamplingmethodsfordemographicsutdies.Inthispaper,wefocusonhowalternativesamplingmethodsinuencetheprecisionofestimatesofs.Weusedalong-termandlarge-scaledatasetofplantdemographytocomparethreecommonlyusedsamplingtechniques:asingle,randomlyplacedplot,multiplerandomlyplacedplots,andsystematicallyarrangedplots.Althoughwefocusexclusivelyonplants,ourndingsapplyequallytoothertypesofsessileorganisms.Usingdata-basedsimulationsandareviewoftheecologicalliterature,weaddressedthefollowingtwoquestions:Howdoestheprecisionofestimatesofsvaryamongthethreefocalsamplingmethods?Whatsamplingstrategiesareusedinstudiesofplantdemography,andwhatdoourresultssuggestabouttheecacyofthesesamplingschemes?

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20 3.2MethodsStudysystemWeuseddatacollectedduringalong-termandlarge-scalestudyofplantdemographyconductedatBrazil'sBiologicalDynamicsofForestFragmentsProjectBDFFP;230'S,60W.ThefocalspeciesforthisstudywasHeliconiaacuminata,aperennialherbnativetocentralAmazoniaandtheGuyanas Berry 1991 .Descriptionsofthestudysiteandexperimentaldesigncanbefoundelsewhere Bierregaardetal. 1992 ; Bruna 2003 ; BrunaandKress 2002 .Briey,13siteseachcomposedofapermanent50m100mplotwereestablishedintheBDFFPcontinuousforestreserves.Becauseeachsitewasrelativelylargeandcompletelysurveyed,thesedataareidealfortestingsamplingmethodsviasubsampling.AllH.acuminataineachplotweremarkedandmapped;wealsorecordedthenumberofvegetativeshootseachplanthad;shootnumberwashighlycorrelatedwithdemographicparameterssuchastheprobabilityofoweringandsurvivorship Bruna 2003 .Since1998,theplotshavebeensurveyedannuallytorecordplantgrowth,mortality,andtheemergenceofnewseedlingsi.e.,establishedplantslessthan1yearold.Theplotswerealsosurveyedduringtheoweringseasontorecordtheidentityofreproductiveindividuals.Theanalysespresentedherearebasedondatafromthe1998{2005surveys;duringthistimeperiodwemarked,measured,andrecordedthefatesofN=6591plantsN=3842incontinuousforest,N=1688in10-hafragments,andN=1061in1-hafragments.Heliconiaacuminatadensityin2003rangedfrom256{2248plantsha)]TJ/F21 7.97 Tf 6.586 0 Td[(1.Becauseeach50100mplotwassubdividedinto10m10msubplots,thiswasthenestresolutionatwhichsamplingmethodscouldbesimulated.

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21 Dodierentsamplingmethodsaecttheprecisionofpopulationgrowthestimates?Wesimulatedsamplingfromeachsiteusingthreealternativesamplingmethods:randomlyplacedsinglesubplotsRS,randomlyplacedmultiplesubplotsRM,andasystematicgridofsubplotsSM;Fig 3{1 .AnRSsampleconsistedofasingle,randomlyplacedquadrat,dimensionsrangingfrom20m20mto50m100m.AnRMsampleconsistedof4{50randomlyselected10m10msubplots.AnSMsampleconsistedofagridofregularlyspaced10m10msubplots.Becausethesiteswerenotsquares,onlyanapproximatenumberofsubplotscouldbespeciedtothealgorithmforchoosingaparticularSMsample PebesmaandBivand 2005 .Nevertheless,wespeciedthesamenumbersofsubplotstotheSMmethodastotheRMmethod.Thus,whilethethreemethodsweretestedoverthesamerangeofsamplingintensities,theresolutionofsamplingintensitiesdieredamongthemethods.ThisdierenceinresolutiondidnotaectanyoftheanalysesseeResults.Simulationswereconductedasfollows:foreachsiteandmethod,weselected50samplesforeachofthesamplingintensitiesdescribedabove.Foreachsample,wecalculatedallpossibleannualtransitionmatrices.Becausesomesiteswerenotsurveyedinsomeyears,thenumberofpossibletransitionmatricespersitevaried:7matricesfor10sites,5for2sitesand3fortheremainingsite;see Bruna 2003 fordetails.Wethenestimatedthestochasticpopulationgrowthrate,s,usingtherandomtransitionmatrixmethod Bierzychudek 1982 ; Caswell 2001 ratherthanthemorecomputationallyintensivevitalratesimulationmethodreviewedin MorrisandDoak 2002 .Thus,wecreatedasequenceof30,000transitionmatricesbydrawingfromthesetofannualmatriceswithequalprobabilitywithreplacement.Then,toprojectpopulationgrowth,wemultiplied

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22 aninitialpopulationvectorbythe30,000matrices.Wecalculated^sasN30;000=N0Figure 3{2 .Whenthesamplingareabeingsimulatedwassmall,allmethodsresultedinsamplesthatlackedindividualsinatleastonesizeclass,makingitimpossibletocorrectlycalculatetheannualtransitionmatrix.Eliminatingthesesamplesanddrawingalternativeswouldhavebiasedestimatesofsandarticiallyreducedtheirvariancebecausequadratswithfewerindividualswouldnotbeincludedinsubsequentanalysis.Therefore,wedevelopeda3stepmethodtodetermineifasamplingintensity|denedhereasanarea|wasincapableofestimatingsforaparticularmethodandsite:iftherewerenoindividualsofanysizeclassinasampleforagivenyear,wedidnotestimateadeterministicmatrixforthatyear;ifagivensamplewasinsucienttoestimateatleast50%ofthepossibletransitionmatrices,wedidnotestimateaswiththatsample;nally,iffewerthan90%ofsamplesforagivensamplesizecouldnotestimates,thenwedeterminedthatthatsamplesizewastoosmalltoestimatesanddidnotincludethesamplesizeforthatplotandmethodinfurtheranalyses.Tocomparetheprecisionofthesamplingmethods,wedeterminedtheminimumplotsizeforeachsiteforwhichallestimatesofswerewithin1%ofthemeanwholeplotestimate.Then,weperformedalogisticregressionforeachplottoestimatehowtheprobabilityof^sbeingwithin1%ofthewholeplotvaluechangedasafunctionofareasampledforeachmethod:pi=logisticarea[i]+site[i]+method[i]{1Probj^si)]TJ/F22 11.955 Tf 11.955 0 Td[(ssite[i]j<0:01s=piwherelogisticx=1=+e)]TJ/F23 7.97 Tf 6.587 0 Td[(x.Toevaluatehowthemethodsdecreasedvariation^sasthesampledareaincreased,weusednonlinearleastsquarestotanegativeexponentialtothestandarddeviationof^sasafunctionofthesamplesizeforeach

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23 samplingmethodandsiteaccordingtoSD^sarea=de)]TJ/F23 7.97 Tf 6.586 0 Td[(rarea{2Thenegativeexponentialwasanidealmodeltodescribehowthevariationin^sdecreasedwithareabecauseitsrateandinterceptparametersallowittoexiblymodelmanydecayingrelationships.Inordertocomparetheprecisionofthesamplingmethodsateachsite,weevaluatedthettedcurveat1000m2tostandardizethesamplingintensitiesofthedierentmethods.Finally,wewantedtoensurethatanydierencebetweenmethodswerenotanartifactofdierentnumbersofindividualsbeingsampledwithagivenareaforeachmethod.Todoso,weusedordinaryleastsquareslinearregressiontoestimatethenumberofindividualssampledasafunctionofareaforeachmethod.AllcomputationswereconductedintheRstatisticalcomputingenvironment RDevelopmentCoreTeam 2005 .Whatsamplingstrategiesareusedinstudiesofplantdemography?Toevaluatethesamplingdesignsusedindemographicstudiesofplants,weconductedaWebofSciencesearchonMarch15,2006usingcombinationsofthesearchtermsmatrixmodel,"plant,"demography,"andpopulation."Foreachpaper,weextractedthetotalareaandnumberofplotssampledforeachmatrixmodelparameterized.Wealsoassignedeachstudytooneofthefollowing5samplingmethods:completelysurveyingasingleplotS;samplingrandomlydispersedsubplotsRM;samplingsystematicallydistributedsubplotsSM;completelysurveyingpopulations"P;andnally,somesampledindividualswithoutdelineatingplotsorpopulationsI.Wecategorizedthespeciesstudiedbylifehistoryintothefollowingcategories:shrubs,trees,perennialherbs,andothere.g.,geophytes,grasses,sedges.Ifastudyusedthesamesamplingprocesstostudymultiplespecies,wecountedthesestudiesasasinglestudy.However,when

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24 astudyexaminedasinglespeciesusingmultiplesamplingstrategies,wecountedtheseasseparatestudies.Ifdierentmethodswereusedtosamplefordierentstageclasses,weevaluatedtheprocessusedtosamplethelargestsizeclasses.3.3ResultsDodierentsamplingmethodsaecttheprecisionofpopulationgrowthestimates?Withallsamplingmethods,thevarianceof^sdecreasedastotalareaincreasedseeg 2{3 ,aspredictedbystatisticaltheory Wackerlyetal. 2001 .Theminimumplotsizesrequiredforall^stobewithin1%ofthewholeplotestimatearegiveninTable 3{1 .Systematicsamplingrequiredtheleastareatoguaranteethe1%precisionlevelfor8siteswhilesingleplotswerebestfor2sites,multiplerandomplotswerebestfor1site,and2siteshadtiedbetweensingleandsystematicsamplingFigure 3{3 andTable 3{1 .Accordingtothelogisticregressionseeeq 3{1 ,SMwasthemostpreciseofthethreemethods,withRMbeingstatisticallyindisinguishablefromRS.Specically,theeectofsystematicsamplingonthelog-oddsofbeingwithin1percentofthesitevaluewassignicantlygreaterthantheeectofRSp=0.037,whereastheRMmethodwasnotseeFigure 3{4 .Overall,samplingmethodsignicantlyaectedthelog-oddsthatall^swerewithin1percentofthetruevaluep<0:001.Accordingtothelogisticmodel,systematicsamplingrequireda15.4%smallersamplingareathansingleplotsamplingtoachievea95%probabilitythatasamplewillbewithin1%ofthetruevalue.Thenegativeexponentialmodeleq 3{2 tthedatawellseeg 3{5 .Accordingtoevaluationofthemodelat1000m2,SMsamplingproducedthemostpreciseestimatesin8sites,RSat4sites,andRMat1seeTable 3{2 .Thus,allthreeanalysesindicatedthatsystematicsamplingwastheoverallmostpreciseofthethreemethods.However,whileweexpectedRMtoprovideanintermediatelevelofprecision,itprovedtobetheleastprecise.Thelinear

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25 modeltbetweenareaandnumbersofindividualswasnearlyidenticalamongallmethods,indicatingthatthemethodssampledequalnumbersofindividualsforagivenarea.Whatsamplingstrategiesareusedinstudiesofplantdemography?Ourreviewretrieved63studiesofplantdemographyseeAppendixA.MoststudieswereonperennialherbsN=30studies,followedbytreesN=13studies,shrubsN=7studies,andotherlife-formsN=13studies.ThemostcommonmethodswereRMsubplotsforperennialherbsof20studies,Sfortreesof14studies,RMforothers"0of13,andmethodswereevenlydistributedforshrubsseeFigure 3{6 .Inadditiontothethreemethodsthatweanalyzedabove,wefound8studiesusedthepopulation"method,whichdelineatedpopulationsandcompletelysurveyedthem;and4studiesusednon-plot-basedmethods.Thetotalareastudiedtoparameterizeamatrixmodelrangedfrom0.09{175,000m2mean8150m2,SD26,637.Unsurprisingly,studiesoftreesusedthelargestareasmean31,250m2,SD48,818,whilethoseofperennialherbsusedthesmallestmean575m2,SD1843;g 3{7 ,table 3{3 .3.4DiscussionOurstudysuggeststhattheprecisionofestimatesofscandependonhowthedatawerecollected.Using13sitesofafocalspecies,weshowedthatsystematicsamplingcanestimatevitalratesforparameterizingmatrixmodelsmoreecientlythanthetwomostcommonlyusedmethods:completelysurveyingoneplotorrandomsamplingsubplots.However,wealsofoundevidencethatestimatingsmayrequireplotsizesthataremuchlargerthanthosecommonlyused.WedeterminedthatforH.acuminata,theminimumplotsizeneededtocalculate^swith1%precisionwasabout3000m2.However,thisstatisticvariedamongsitesandsamplingmethods,withsystematicsamplingrequiringthesmallestareamean2400m2andrandomsamplingrequiringthelargestarea

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26 mean3200m2.Thevariationindensitiesofplantsamongsitesmayhavecauseddierencesamongsites.Theabilityofsystematicsamplingtocapturethemostvitalratevariationinasamplethantheothermethodsmayhavecauseditssuperiorprecisionforestimating^s.Wediscussthispointfurtherbelow.HowdothesamplingintensitiesrequiredforHeliconiaacuminatacomparewiththoseforothersystems?Althoughoursimulationssuggestedatleast3000m2werenecessarytopreciselyestimatesinforH.acuminata,only8%ofthestudiesofperennialplantssampledareaslargerthanthis.Furthermore,48%ofthestudiesofperennialspeciessampledanarea<10m2.However,variationinlifehistories,densities,andaggregationamongspeciesmayaccountforatleastpartofthisdierencebetweenoursystem'ssamplingrequirementsandthoseintheliterature.Nevertheless,evenifthesamplingrequirementsofH.acuminataareordersofmagnitudehigherthanothersystems,ourresultsstillsuggestotherstudiesusedimprecisemethodstoestimatevitalratesduetotheirsmallsamplingintensities.Atleastfourfactorsmightexplainthesmallsamplingintensitiesintheliterature.First,thefocalspeciesmaybefoundinalimitedareabecauseofendemismorpopulationdeclinee.g., Garcia 2003 .Second,higherdensitiesofotherplantsmayrequiresmallerareastopreciselyestimatevitalrates.Third,researchersmayhavechosentofocusonsmallareasbutmanysitese.g., HorvitzandSchemske 1995 ; Moloney 1988 ; Oostermeijeretal. 1996 .Finally,thenancialresourcesortimeneededforabroad-scalestudycouldbeunavailable. Gibson 2002 arguedthatallspatialsamplingmethodsareequivalentaslongasthesamenumbersofindividualsaresampledinallmethods.However,ourresultssuggestthatsystematicsamplingcanbeamoreecientmethodforsamplingvitalrates.Interestingly,thiswastheleastcommonmethodusedinthestudiesweanalyzed-onlythreeofthe63studieswereviewedusedasystematicnetworkofsamplingsites.Thispaucityofstudiesusingsystematicsampling

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27 issomewhatsurprising,sincesystematicsampling'seciencyadvantageisawell-knownresultfromsamplingtheory Bellhouse 1977 ; Yates 1948 andstudiesinothereldssuchasweedscience Ambrosioetal. 2004 andplantpopulationgenetics Suzukietal. 2004 .However,whileourstudysuggeststhatsystematicsamplingcouldbethepreferablewaytosampleplantfordemography,thismethodshouldbenotblindlyappliedbecausemisusemayleadtobiasesfortworeasons.First,thesamplinggridmaycoincidewiththeperiodicityofrealspatialstructure Thompson 2002 .However,researcherscanemploynon-alignedsystematicsamplingtoavoidthisproblem Quenouille 1949 ; Thompson 2002 .Second,ifthevarianceofavitalrateiscalculatedasifsamplingweredoneatrandom,thiswillleadtobiasedestimatesofvitalratesamplingvariance AubryandDebouzie 2000 .Thesolutiontothisproblemistousevarianceestimationtechniquesdesignedforsystematicsampling AubryandDebouzie 2000 .Sincesolutionsexistforbothofthesepotentialproblems,systematicsamplingappearsrobust.Becausetheincreasedprecisionofsystematicsamplingresultsfrommaximizingthedistancebetweensamples,theadvantageofsystematicsamplingisexpectedtobedependentonthepresenceofspatialautocorrelationofvitalrates.Despitethefactthatspatialautocorrelationofvitalratescanhaveanimpactontheecacyofsamplingmethods,toourknowledgenostudieshaveinvestigatedifvitalratesarespatiallyautocorrelated.Thisissurprisingbecausetherearemanypossiblemechanismsthatwouldleadtospatialautocorrelationofvitalrates.Forexample,spatialvariationinenvironmentalvariablescancausespatiallyautocorrelatedvitalrates:soilchemistrycanaectplantdefensesandthereforesurvival Koikeetal. 2006 ;lowertemperaturescanreducepollenandfruitproductionandthereforefertility JakobsenandMartens 1994 .Anothermechanismthatcouldleadtospatiallyautocorrelatedvitalratesisshortdistancedispersal.Shortdispersalcanleadtoincreasedrelatedness

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28 amongnearbyindividuals.Thesemorecloselyrelatedindividualssharegeneticmaterialthatmaycontributetowardsvitalrates Chungetal. 2005 ; DelesalleandBlum 1994 .Facilitationfromconspecicneighborsisanotherprocessthatcanincreasespatialautocorrelationofsurvivalandgrowthrates Escuderoetal. 2005 .Withincreasingevidencethatspatialautocorrelationisarealcomponentofmanyecologicalsystems,itisimportanttobeawareofandaccountforspatialautocorrelationwhencollectingdatatoparameterizematrixmodels.Tocomputepreciseestimatesofvitalratesinthepresenceofpositivespatialautocorrelationofvitalrates,largerplotsizesareneededtorepresentthetruevariationinvitalrates Legendre 1993 .Thishasbeenthoroughlydocumentedinagriculturaleldtrialsthatdeterminedthatoptimalsamplingwasdependentuponspatialautocorrelationbyusingvariogramsandothergeostatisticaltools Bhatti 2004 ; FagroudandVanMeirvenne 2002 ; Poultneyetal. 1997 .However,moreresearchisneededontheeectsofspatialautocorrelationonsamplingrequirementsinplantdemography.Aninterestingoutcomeofourliteraturereviewisthatitrevealedalackofconsistencyintheterminologyusedtodescribesamplingmethodsinstudiesofplantdemography.Forexample,cleareruseofthefollowingtermswouldbehelpfultoreaders:subplots,plots,sites,andpopulations.Subplotsshouldbeusedtorefertoareasthatwillbepooledtoestimatepopulationparameters,whereasplotsshouldrefertosingleplotsampling.Multipleplots"impliesamulti-sitestudyratherthanmultiplesamplingareastoestimateasinglevitalrate.Papersusingthepopulation"methodseeResultsoftenstatedthattheysampledapopulationwithoutdierentiatingbetweenastatisticalpopulationandabiologicalpopulation.Iftheysurveyedallindividualsinageneticallyisolatedbiologicalpopulation,thentheyconductedacompletesurveywhichismoreaccurateandprecisethananyofthesamplingmethodswecompared.However,iftheyactually

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29 surveyedastatisticalpopulation,whichmayhavebeenasubsetofabiologicalpopulation,thenthedomainofthemodelresultsaresmallerthanstated.Therigorofsamplingterminologyisimportantanditsclaricationwouldhelpwiththedesignoffuturestudiesandeliminationofguesswork.Wehopethatfuturepapersadoptconsistentsamplingterminologyforplantdemography.Althoughourunderstandingofhowthespatialcomponentofsamplinginuencesmatrixmodelsremainslimited,previousstudieshaveevaluatedothersamplingissuesthatcouldinuenceourresults. Doaketal. 2005 lookedattheeectsofsamplingintensity;theyinvestigatedhowthenumbersofindividualsandyearsstudiedaectdeterministicandstochasticmatrixmodels.Theysuggestedthatwithfewerthan5years'data,deterministicmodelsarebetteratmodelingpopulationsthanstochasticmodels.Todevelopmoreecientmethodstoestimatepopulationgrowthwithmatrixmodels, Gross 2002 and MunzbergovaandEhrlen 2005 lookedathowtheratioofsamplingamongstageclassesaectsprecisionofestimates. Gross 2002 suggestedusingpriorinformationabouttherelativedemographicimportanceofstageclassestoincreasesamplingofthemore`important'stagesandthusminimizethevarianceofestimates.Incontrast, MunzbergovaandEhrlen 2005 advocatedsamplingequalnumbersofindividualsacrossallstageclasses;theycreatedadistinctionbetweentheirmethodandplot-basedmethodswithoutsuggestinganon-plot-basedsamplingprocessforuseintheeld.However,studiesoftensubsampleplotstosampleindividualsformoreabundantstageclassestoachieveasimilarresulttotheirequalnumbersmethod.Thus,plot-basedmethodscanstillbeusedtosampleamoreequaldistributionofclassesbecauseofthelogisticaladvantagesoverindividualornon-plot-basedmethods:plot-basedmethodsmakeiteasiertorelocateindividuals;easiertoplanastudy;andeasiertodescribethesamplingprocess.

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30 Takenasawhole,ourandotherstudiesofsamplinghavesuggestedanumberofguidelinestoconsiderwhendesigningademographicstudy.Thersttaskistoclearlydenethedomainofthestudy,makingsurethatthesamplingextentcoversthisdomainsee LegendreandLegendre 1998 ,Fig 3{1 fordenitionsofsamplingterminology.Next,ifpossible,thesamplingunitssitesshouldbechosenwiththeaimofcomparingamongthesites.Afterall,ecologicallyimportantspatialvariationcanbeevaluatedonlyifamulti-sitestudydesignisused.Basedonourresultsandthoseofstatisticalsamplingtheory Bellhouse 1977 ; Cochran 1946 ; Yates 1948 ,wewouldsuggestasystematicgridofsubplotswithineachsite.Ifasubsequentanalysisrevealshomogeneityinvitalratesacrosssites,thenalldatacanbepooledtoobtainmoreprecisematrixmodelparametersforasingle,largesite.Towardthisend,wealsoadvocatesamplingwithasystematicgridofsites.However,ifvitalratesatsitesareheterogeneous,thensiteshavesucientdatatobemodeledseparatelyforcomparisonsamongsites.Nonetheless,perhapsmoreimportantthanwhichsamplingmethodisemployed,researchersshouldensurethatenoughareaissampled,eitherwithinasite,oracrosssites,toencompasstherangeofspatialvariationinplantsurvival,growth,andreproductionoverthedomainofinterest.

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31 Figure3{1.Illustrationofsamplingterminologyasappliedtoplantdemography.arandomsingleplotmethodRS,brandommultipleplotmethodRM,csystematicmultipleplotmethodSM

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32 Figure3{2.Flowchartofsamplingmethodsimulationsusedtocomparethethreealternativesamplingmethods.

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33 Figure3{3.Minimumsamplingareafor1%precision.Thesearetheplotsizesatwhichallsampleswerewithin1%ofthewholeplotestimateforeachsiteandsamplingmethod

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34 Figure3{4.Logisticmodelofprecisionseeequation 3{1 foreachsiteofthewhetheraparticular^swaswithin1%ofthewholeplotvalueasafunctionofareasampledforall3samplingmethods.

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35 Figure3{5.Negativeexponentialmodelofprecisionequation 3{2 ttedtostandarddeviationof^sateachsampleareaandeverysiteforeachsamplingmethod.Thehorizontallinesshowthevalueofthestandarddeviationpredictedbythemodelforeachsamplingmethodat1000m2.

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36 Figure3{6.Samplingmethodsinpublishedstudies.

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37 Figure3{7.Areassampledinpublishedstudies.Histogramsoftheresultsfromaliteraturereviewof63plantmatrixmodelstudies.Foreachpaper,wecalculatedtheaverageareausedtoestimatethepopulationgrowth.

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38 Table3{1.Minimumsamplingareasfor1%precision.Thisistheminimumareaforwhichall^swerewithin1%ofthewholesiteestimateforeachsiteandsamplingmethod SiteRSRMSM 210730003600240021084500360012002206250032002800575016002000120057512500280021005752250028002100575325004000280057542500160010005756160024001000CaboFrio450050004500DimonaCF450038004000Florestal250024001200PA-CF450045004500 mean3015.383207.692369.23SD1096.09993.71294.47 Table3{2.Negativeexponentialmodelevaluatedat1000m2seeequation2evaluatedat1000m2foreachsiteandmethod. SiteRSRMSM 21070.02180.01220.008921080.03690.01540.005722060.00770.01960.008857500.00530.0070.004657510.00880.01350.01357520.0080.00860.008157530.03260.01570.012557540.00670.00640.00557560.00520.00480.0057CaboFrio0.02680.02550.0249DimonaCF0.02320.03540.0295Florestal0.01060.00720.0048PA-CF0.03980.02520.0195 mean0.0180.01510.0116SD0.01280.00920.0082

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39 Table3{3.Areassampledintheliterature.Theareasm2comefromthe63publishedstudiessampled,classiedinto4lifehistorycategoriesseeMethods PlantformMeanMedianS.D.RangeN perennialherb574.6010.001843.300.6-8018.230tree31250.008100.0048814.301500-17500013shrub775.00612.50496.20375-15007other1673.105.002259.900.1-555013

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CHAPTER4CONCLUSIONChapter1revealedthatalthoughJensen'sInequalityandstatisticaltheorysuggestthatestimatesofpopulationgrowthmaybebiasedwhenfewindividualsaresampled,ourresultsrevealedthatthisbiasisunimportantforcertainorganisms.Theseresultsaremostlikelygeneralizabletootherlong-livedorganismscharacterizedbyhighsurvivalandlowfecundity.However,whenthestudyspeciesischaracterizedbylowersurvival,nearerto0.5,lowsamplingintensitiescancausesevereoverestimationofpopulationgrowth.Therefore,weadvocateconductingasimulationpoweranalysissimilartotheoneweconductedheretohelpdetermineanadequatesamplesize.Becauseresearchersdonothavethematrixmodelfortheirstudyspeciesapriori,wesuggestsamplingsimulationsusingseveralmatricesoforganismswithsimilarlifehistorycharacteristicssimilartothemethodof Gross 2002 .InChapter2,weshowedthatsystematicsamplingcanbeamoreecienteldmethodtocollectdatatoestimatevitalrates.However,systematicsamplingwasbyfar,therarestmethodusedinpublishedstudiestosampleplantsformatrixmodels.Thisdisparitybetweenwhatourresultssuggestisthebestmethodandwhatstudiestypicallyuseissurprisinggiventhatstatisticalsamplingtheoryhasdemonstratedtheeciencyimprovementsofsystematicsamplinge.g., Yates 1948 ; Bellhouse 1977 ; Cochran 1946 andinothereldse.g., Ambrosioetal. 2004 .Therefore,ourresultssuggestthatplantpopulationecologistsmayimprovesamplingdesignswiththeadditionoftwooutcomesfromourstudy.First,anaprioripoweranalysisshouldbeconductedtoreducethepotentialforbiasedpopulationgrowthestimatesfromsmallsamplesizes.Second,whenapopulation 40

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41 isnotcompletelysurveyed,demographersshouldconsiderusingasystematicorstratiedsamplingdesigntomaximizethevitalratevariancecapturedfortheirsamplingeortandobtainthemostprecisevitalrateestimates.Withtechniquesavailabletoimprovetheestimationofvitalrates,demographershavethechoiceofmaintainingtheirsamplingeortandimprovingprecisionorreducingtheireortandmaintainingthesamelevelofprecision.Withthelatterchoice,demographicstudieswillrequirefewerresourcesandperhapsmorespeciescanbestudied.Inthecurrentsituationinwhichtherearemorespeciesthatrequirehelpthanwecanpossiblystudytodesignrecoveryplans,anypotentialaidinreducingthecostofasinglestudyshouldbeappreciated.

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APPENDIXPAPERSREVIEWEDBelowisalistofthepapersthatwereincludedinthesamplingliteraturereviewsinChapters2and3. Allphin,L.,andK.T.Harper.1997.DemographyandlifehistorycharacteristicsoftherareKachinadaisyErigeronkachinensis,Asteraceae.AmericanMidlandNaturalist138:109{120. Alvarez-Buylla,E.R.1994.Density-dependenceandpatchdynamicsintropicalrain-forests-matrixmodelsandapplicationstoatreespecies.AmericanNaturalist143:155{191. Barot,S.,J.Gignoux,R.Vuattoux,andS.Legendre.2000.DemographyofasavannapalmtreeinIvoryCoastLamto:populationpersistenceandlife-history.JournalofTropicalEcology16:637{655. Batista,W.B.,W.J.Platt,andR.E.Macchiavelli.1998.Demographyofashade-toleranttreeFagusgrandifoliainahurricane-disturbedforest.Ecology79:38{53. Berg,H.2002.PopulationdynamicsinOxalisacetosella:thesignicanceofsexualreproductioninaclonal,cleistogamousforestherb.Ecography25:233{243. Bierzychudek,P.1982.ThedemographyofJack-in-the-Pulpit,aforestperennialthatchangessex.EcologicalMonographs52:335{351. Brewer,J.S.2001.Ademographicanalysisofre-stimulatedseedlingestablishmentofSarraceniaalataSarraceniaceae.AmericanJournalofBotany88:1250{1257. Bruna,E.M.2003.Areplantpopulationsinfragmentedhabitatsrecruitmentlimited?TestswithanAmazonianherb.Ecology84:932{947. Brys,R.,H.Jacquemyn,P.Endels,G.DeBlust,andM.Hermy.2004.TheeectsofgrasslandmanagementonplantperformanceanddemographyintheperennialherbPrimulaveris.JournalofAppliedEcology41:1080{1091. Bullock,S.H.1980.DemographyofanundergrowthpalminLittoralCameroon.Biotropica12:247{255. 42

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43 Byers,D.L.,andT.R.Meagher.1997.AcomparisonofdemographiccharacteristicsinarareandacommonspeciesofEupatorium.EcologicalApplications7:519{530. Calvo,R.N.1993.Evolutionarydemographyoforchids-intensityandfrequencyofpollinationandthecostoffruiting.Ecology74:1033{1042. Charron,D.,andD.Gagnon.1991.ThedemographyofnorthernpopulationsofPanaxquinquefoliumAmericanGinseng.JournalofEcology79:431{445. Cipollini,M.L.,D.A.Wallacesenft,andD.F.Whigham.1994.Amodelofpatchdynamics,seeddispersal,andsex-ratiointhedioeciousshrubLinderabenzoinLauraceae.JournalofEcology82:621{633. Cipollini,M.L.,D.F.Whigham,andJ.O'Neill.1993.Populationgrowth,structure,andseeddispersalintheunderstoryherbCynoglossumvirgini-anum:Apopulationandpatchdynamicsmodel.PlantSpeciesBiology8:117{129. Ehrlen,J.1995.DemographyoftheperennialherbLathyrusvernus1.Herbivoryandindividual-performance.JournalofEcology83:287{295. Endress,B.A.,D.L.Gorchov,andR.B.Noble.2004.Non-timberforestproductextraction:Eectsofharvestandbrowsingonanunderstorypalm.EcologicalApplications14:1139{1153. Enright,N.,andJ.Ogden.1979.ApplicationsoftransitionmatrixmodelsinforestdynamicstoAraucariainPapuaNew-GuineaandNothofagusinNew-Zealand.AustralianJournalofEcology4:3{24. Enright,N.J.,andA.D.Watson.1992.PopulationdynamicsoftheNikauPalmRhopalostylissapidawendl.etdrudeinatemperateforestremnantnearAucklandNewZealand.NewZealandJournalofBotany30:29{43. Eriksson,O.1988.RametbehaviorandpopulationgrowthintheclonalherbPotentillaanserina.JournalofEcology76:522{536. Fiedler,P.L.1987.LifeHistoryandPopulationDynamicsofRareandCommonMariposaLiliesCalochortusPurshLiliaceae.JournalofEcology75:977{996. Forbis,T.A.,andD.E.Doak.2004.Seedlingestablishmentandlifehistorytrade-osinalpineplants.AmericanJournalofBotany91:1147{1153. Garcia,M.B.2003.DemographicviabilityofarelictpopulationofthecriticallyendangeredplantBordereachouardii.ConservationBiology17:1672{1680. Guardia,R.,J.Raventos,andH.Caswell.2000.SpatialgrowthandpopulationdynamicsofaperennialtussockgrassAchnatherumcalamagrostisinabadlandarea.JournalofEcology88:950{963.

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46 Svensson,B.M.,B.A.Carlsson,P.S.Karlsson,andK.O.Nordell.1993.Comparativelong-termdemographyof3speciesofPinguicula.JournalofEcology81:635{645. Tolvanen,A.,J.Schroderus,andG.H.R.Henry.2001a.Age-andstage-basedbuddemographyofSalixarcticaundercontrastingmuskoxgrazingpressureintheHighArctic.EvolutionaryEcology15:443{462. Tolvanen,A.,J.Schroderus,andG.H.R.Henry.2001b.DemographyofthreedominantsedgesundercontrastinggrazingregimesintheHighArctic.JournalofVegetationScience12:659{670. Valverde,T.,andJ.Silvertown.1998.VariationinthedemographyofawoodlandunderstoreyherbPrimulavulgarisalongtheforestregenerationcycle:projectionmatrixanalysis.JournalofEcology86:545{562. Vavrek,M.C.,J.B.McGraw,andH.S.Yang.1997.Within-populationvariationindemographyofTaraxacumocinale:Season-andsize-dependentsurvival,growthandreproduction.JournalofEcology85:277{287.

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BIOGRAPHICALSKETCHIanFiskewasborninDenville,NewJersey,onAugust20,1980andwasraisedinJacksonville,Florida.HegraduatedfromStantonCollegePreparatoryHighSchoolwithanInternationalBaccalaureatediplomain1998.Thefollowingfall,Ianbeganabachelor'sdegreeinelectricalengineeringattheUniversityofFlorida.Inthenalyearofhisengineeringeducation,hedecidedtoearnanadditionalbachelor'sdegreeinmathematicstogainskillswouldbeapplicabletoavarietyofeldsbesidesengineering.Soonthereafter,IanbeganaresearchprojectanalyzingHeliconiaacuminatademographywithwildlifeecologyandconservationprofessorEmilioBruna.Aftergraduatingcumlaudein2003inelectricalengineeringandmathematics,Ianspentayearexploringtheworldoutsideofacademics.HeworkedonanorganicfarminHomestead,Florida,andonseveralfarmsinCostaRica.AfterreturningtotheUnitedStatesin2004,hebeganamaster'sdegreeinwildlifeecologyandconservationattheUniversityofFlorida,workingwithDr.Brunaonthisthesisproject.Next,IanwillbeginastatisticsPhDprogramatNorthCarolinaStateUniversity,wherehewillspecializeintheanalysisofecologicalandenvironmentaldata. 53


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DEMOGRAPHIC STUDIES OF PLANTS: DATA LIMITATIONS AND SPATIAL
SAMPLING METHODS
















By

IAN J. FISKE


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2006


































Copyright 2006

by

Ian J. Fiske















ACKNOWLEDGMENTS

There are many individuals who have supported me during this work. First,

I would like to thank my advisor, Emilio Bruna, for giving me the opportunity to

become an ecologist despite my prior training as an engineer and mathematician.

Emilio was endlessly encouraging and humorous during stressful times. I would

also like to thank my committee members Be( i -,iiii Bolker and Madan Oli for

helpful discussions and comments on the manuscript and for all that I learned in

their quantitative ecology courses. In particular, C'! lpter 1 is a direct product

of a comment Ben made at my first committee meeting. I also thank the staff at

the Biological Dynamics of Forest Fragments Project for logistical support and

the many field assistants who helped with data collection. Financial support for

this project came from the University of Florida's College of Agricultural and Life

Sciences and National Science Foundation grants OISE-0437369 and DEB-0309819.

Finally, I am grateful for the never-ending patience and support of both my friends

and family during this project.















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ...........

LIST OF TABLES ..............

LIST OF FIGURES .............

ABSTRACT .................

CHAPTER

1 INTRODUCTION ............

2 SAMPLING INTENSITY AND BIASED
GROW TH .................

2.1 Introduction .............
2.2 M ethods .. .. .. .. ... .. ..
2.3 Results and Discussion .......

3 SPATIAL SAMPLING METHODS FOR

3.1 Introduction .............
3.2 M ethods .. .. .. .. ... .. ..
3.3 R results . . . .
3.4 Discussion ..............

4 CONCLUSION .............

APPENDIX PAPERS REVIEWED .

REFERENCES .................

BIOGRAPHICAL SKETCH ..........


ESTIMATES OF POPULATION






PLANT DEMOGRAPHY ...















LIST OF TABLES
Table page

2-1 Results from sample size literature review ................ .. 17

3-1 Minimum sampling areas for 1. precision ................ .. 38

3-2 Negative exponential model evaluated at 1000 m2 ............. 38

3-3 Areas sampled in the literature .................. .. 39















LIST OF FIGURES
Figure page

2-1 Flowchart of simulations .................. ........ .. 13

2-2 Heliconia acuminata transition matrix .................. 14

2-3 Relative bias in estimates of A .................. .... .. 15

2-4 Histograms of the 6000 estimates of A ................ 16

2-5 Sample sizes from published studies .................. .. 16

3-1 Illustration of sampling terminology .................. .. 31

3-2 Flowchart of sampling method simulations ................ 32

3-3 Minimum sampling area for 1. precision ................. ..33

3-4 Logistic model of precision .................. ....... .. 34

3-5 Negative exponential model of precision ................. 35

3-6 Sampling methods in published studies. .................. 36

3-7 Areas sampled in published studies .................. .. 37















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

DEMOGRAPHIC STUDIES OF PLANTS: DATA LIMITATIONS AND SPATIAL
SAMPLING METHODS

By

Ian J. Fiske

August 2006

C'! In': Emilio Bruna
Major Department: Wildlife Ecology and Conservation

Matrix models are widely used to study the dynamics of structured populations

and have been applied to basic and applied ecological questions. An important but

mostly overlooked issue is how (1) the number of individuals sampled and (2) the

field method used to sample those individuals affect estimates of the .i,-,iii:il '1 ic

population growth rate, A, derived from matrix models. We used data from a

long-term study of the demography of the Amazonian understory herb Heliconia

acuminata (Heliconiaceae) to address both of these questions. To investigate how

the number of individuals sampled affects the accuracy of A estimates, we simulated

the effects of sampling variability when collecting demographic data by drawing

vital rates from appropriate probability distributions. Because sampling variability

of A depends upon the number of individuals sampled, the distribution of sampling

effort among size classes, the variance of fecundity rates, and the survival rates, we

varied these factors to investigate their effects. We found no practically important

bias in A estimates when sampling H. acuminata vital rates, independent of

sample size, variance of fecundity among individuals, and distribution of samples

among stage classes. However, we found significant bias at small sample sizes









when survival was reduced to 0.5 for all stage classes. Furthermore, this bias was

exacerbated when sampling with a realistic J-shaped distribution of individuals

among size classes. Because bias was strongest when survival is equal to 0.5, these

results sl:.--- -1 that the extent of small sample size bias strongly depends upon

the study organism and the particular year that sampling occurs. A review of

literature revealed that many studies have parameterized matrix models using fewer

individuals than sample sizes we found to suffer from small sample size bias.

The second goal of this investigation was to determine how the spatial

method used to sample plants affects the precision of stochastic population growth

estimates. We compared the relative efficiency of three commonly used methods

to sample plants for demographic studies: surveying all plants within a single

plot, multiple randomly placed plots, or multiple systematically placed plots.

To evaluate each method, we simulated sampling from the H. acuminata data.

Systematic sampling provided the most precise estimates of population growth

according to all analyses we used to compare the methods. As with the first

question, we conducted a review of the plant demography literature to determine

how published studies have sampled plants. Although our results -Ii--.; -1 that

systematic sampling was the most efficient method, it was the least common

method found in published studies.















CHAPTER 1
INTRODUCTION

Matrix models (Leslie 1945; Lefkovitch 1965) are an important tool that

ecologists use to study the demography of structured populations. They are

flexible, readily applicable to a diversity of life-history strategies, and there is a

broad body of literature describing their construction, interpretation, assumptions,

and limitations (see Bierzychudek 1999; de Kroon et al. 2000; Wisdom et al.

2000; Caswell 2001). Although matrix models are commonly used to estimate the

.,-vmptotic growth rate of a population (both deterministic, A, and stochastic, As),

increasingly sophisticated computational techniques also allow one to explore the

underlying demographic factors influencing the rate of population growth (Caswell

2001; Morris and Doak 2002; Bruna 2003). For example, elasticity analyses and

other pi"-p. i' I.i, approaches (Horvitz et al. 1997), evaluate how hypothetical

changes in vital rates would alter A, while -, I i'-.' I. r techniques such as Life

Table Response Experiments (Caswell 1989, 2001) decompose observed differences

in A into the actual contributions from individual demographic variables. These

analyses have become increasingly common in applied settings because they can

sir.--- -1 what demographic transitions, and therefore what ecological processes,

might be the focus of conservation and management strategies (Valverde and

Silvertown 1998).

Despite the popularity of matrix models in ecology and conservation, few

studies have investigated how to collect data to parameterize these models for

studying plants (but see Gross 2002; Doak et al. 2005; Miinzbergov4 and Ehrlin

2005). This knowledge gap is surprising because study design in ecology has

gained much attention in over the last few decades (e.g., Green 1979; Scheiner









and Gurevitch 2001). The accuracy and precision of any model projection directly

depend upon the quality of the estimated model parameters and the quality

of these parameters is a result of how the data were collected. Thus, a goal of

this work is to investigate the sampling methods of published studies of plant

populations and to determine if work in plant population biology can be improved

by altering sampling methods or sampling intensity.

In C'!i lpter 1, we address how small sample sizes affect the accuracy of

estimates of population growth. We use long-term demographic data on the

Amazonian understory herb Heliconia acuminata (Bruna 2003) to look at bias of

population growth estimates when vital rates are estimated with few individuals.

In C'!i lpter 2, we compare three commonly used spatial sampling methods for

studying plants with matrix models: single plots, multiple randomly placed plots,

and multiple systematically placed plots. We use the H. acuminata system to

investigate the relative precision of estimates of stochastic population growth that

result from these three sampling methods. In both chapters, we conduct a review of

published demographic papers on plants to compare our findings with how previous

work has sampled.















CHAPTER 2
SAMPLING INTENSITY AND BIASED ESTIMATES OF POPULATION
GROWTH

2.1 Introduction

Matrix models are constructed using demographic vital rates (\.' '! and

Doak 2002; Franco and Silvertown 2004), such as the probability that an individual

survives from one time step to the next or the probability of growing into larger

size classes. Although these rates can be estimated in a number of v--i-, one

common method is to track the fates of individuals of different sizes or ages

through time (Horvitz and Schemske 1995; Meagher 1982). This procedure

requires establishing permanent plots, mark-recapture programs, or other means

of following individuals from a population. As might be expected, the number

of individuals sampled from each size or age class can influence estimates of

vital rates for those stages (Kendall 1998, White 2000). However, how estimates

of population-level statistics such as A are influenced by the total number of

individuals observed in a demographic study remains virtually unexplored (but

see Doak et al. 2005). Despite the central role that sample size estimation p.1',li' in

the design of ecological experiments (e.g., Green 1979; Krebs 1999; Scheiner and

Gurevitch 2001), demographers have had little guidance regarding the appropriate

sample sizes needed to have confidence in estimates of A.

Of fundamental importance to this issue is the difference between the pre-

cision and accur n ;, of estimates of A (Wackerly et al. 2001). Precise estimates

have low variance and will likely be similar after repeated sampling. On the

other hand, accurate estimates are expected to correctly estimate the quantity of

interest, on average, over repeated sampling. Precision does not imply accuracy,









nor does accuracy imply precision. In this study, we investigate how sample size

affects the accuracy of estimates of A. We begin by first reviewing the statistical

framework that guides the estimation of vital rates, with particular emphasis

on the implications of sampling variance and the mathematical theorem known

as Jensen's Inequality (Ruel and Ayres 1999). We then present the results of

simulations used to investigate how the total number of plants sampled in a

demographic study influences the accuracy of projected values of the .,-vmptotic

population growth rate, A. Finally, we review the ecological literature and

describe the range of population sizes used to parameterize matrix models of

plant demography.

Background

Demographic transitions are often modeled as the products of underlying vital

rates, and many of these vital rates are modeled as binomial random variables

(\1!, i i i and Doak 2002). The binomial model assumes the fates of individuals are

independent and that all individuals of a given stage class have equal probabilities

of surviving, growing, and shrinking. For example, the probability that an

individual in stage class i grows to the next stage class from one time step to

the next is the product of the probability that a class i individual survives (si)

and the conditional probability a class i individual grows given that it survived

(gi). Assuming a detection probability of 1, we can estimate these probabilities

with sample proportions (e.g., si = ni(t)/n(t 1)) where ni(t) is the number of

individuals of class i found at time t) (Thompson 2002). If E [si] : si, then the

estimate of this survival probability is biased (Wackerly et al. 2001). When an

estimate is biased and the researcher is unaware of it, the result can be precise

but inaccurate estimates of transition probabilities. These biased estimates can be

eliminated entirely by sampling the entire population of interest; however, complete

population surveys are rarely feasible. Alternatively, one can reduce bias by using






5


techniques developed in statistical sampling and estimation theory, such as the use

of random samples (e.g., Mack 2002).

Nevertheless, unbiased estimates of vital rates do not ensure unbiased

estimates of the population growth rate. Because A is the dominant eigenvalue of

the transition matrix, it is by definition a nonlinear function of the underlying vital

rates (Leslie 1945). According to Jensen's inequality (Jensen 1906), a mathematical

theorem that has been increasingly applied in ecological studies (e.g., Karban et al.

1997; Ruel and Ayres 1999; Inouye 2005), any variance in the vital rates will result

in biased estimates of A. The amount and direction of bias depend on two factors:

first, the strength of the nonlinearity of the relationship between A and the vital

rates and second, the variance of the vital rates themselves.

Variation of vital rates arises from two sources. The first of these is process

variance, which is a result of real variation in the population over space (Stratton

1995) or time (Tuljapurkar 1990). The second is sampling variance, which is a

result of studying a sample rather than the entire population. Methods for dealing

with process variance, especially over time, are well-developed (Lewontin and

Cohen 1969). More recently, methods have been devised that attempt to separate

the sampling variance from the total observed variance in matrix models (Kendall

1998; White 2000). Here, we consider the effect that this sampling variance of

vital rates has on the accuracy of A estimates. The binomial distribution, which

assumes homogeneous vital rates among individuals within a stage class, maximizes

the sampling variance of the vital rate (Kendall and Fox 2002). Therefore, the

binomial distribution is a conservative choice for investigating the effects of vital

rate sampling variance. The sampling variance of these binomial vital rates is:

2 -si(1 si)
o = Var(s) s s= (2-1)
Si i









where ni is the number of individuals in class i in the sample, and si is the true

value of the vital rate (Thompson 2002). For estimates of fecundity, such as the

number of offspring an individual produces given it reproduces, the variance is:


2o = Var(fi) = (2-2)


where fi is the fecundity of individuals in class i and fi is the true variance of

the fecundity among individuals in class i; ni is defined as above. Note that these

sample variances increase as sample size decreases. Therefore, Jensen's Inequality

would lead one to predict that small sample sizes will lead to biased estimates

of A as a result of increased sampling variance in the estimates of vital rates,

irrespective of whether the estimates of the vital rates are themselves accurate.

2.2 Methods

We used simulation models to estimate how the accuracy of projections of

A varied with sample size. These models were based on data collected during

a long-term and large-scale study of plant demography conducted at Brazil's

Biological Dynamics of Forest Fragments Project (BDFFP; 230'S, 60W). The

focal species for this study was Heliconia acuminata, a perennial herb native

to central Amazonia and the Guyanas (Berry 1991). Descriptions of the study

site and experimental design can be found elsewhere (Bierregaard et al. 1992;

Bruna and Kress 2002; Bruna 2003). Briefly, permanent 50 m x 100 m plots were

established in 13 of the BDFFP's reserves in January 1998. All H. acuminata in

each plot were marked and mapped and the number of vegetative shoots each plant

had was recorded (Bruna 2003). Since their establishment the plots have been

surveyed annually to record plant growth, mortality, and the emergence of new

seedlings (i.e., established plants less than 1 year old). The plots were also surv, i, d

during the flowering season to record the identity of reproductive individuals. The

analysis presented here is based on data from the 1998-2003 surve--: during this









time period we marked, measured, and recorded the fates of N = 6591 plants (N

S3842 in continuous forest, N = 1688 in 10-ha fragments, and N = 1061 in 1-ha

fragments).

Using these data, we simulated a sampling process that modeled the sampling

variability of vital rates (see also Figure 2-1). We used the 1998-1999 estimates

of the vital rates as 'true' population vital rates; the results were not sensitive to

this data choice. Then, we used one of two probability distributions to simulate

sampling each vital rate: a beta probability distribution for the binomial vital

rates (e.g., probability of survivorship, probability of growth) and the gamma

distribution for the count-based vital rates (i.e., fecundity). Because the beta

distribution is continuous, bounded by 0 and 1, and can be parameterized to have a

variety of means and variances, it is an appropriate choice for modeling estimates of

the binomial vital rates (\ ..i i i and Doak 2002). We chose the gamma distribution

to model estimates of average fecundity because it is non-negative and can also

be flexibly parameterized. We parameterized both the beta and gamma sampling

distributions according to the method of moments, a technique which parameterizes

a distribution by specifying its expected value and variance (Hilborn and Mangel

1997). To define the sampling process, we set the expected value and variance of

an estimated vital rate equal to the true population's mean vital rate and sampling

variance, respectively. We determined the sampling variance at each sample size

with equations 2-1 and 2-2. Then, we used well-known method of moments

relationships between the parameters of the distributions and their expected value

and variance (e.g., Gelman 2004). For the beta distribution, we used the following

relationship between the parameters and the mean and variance of the distribution:


a Pj(t(l p)/o ) (2-3)

/3 (1 uPJ)(+P(1 P)/J, 1)









where and p~) and au are the mean and variance of the estimate of vital rate si,

respectively. Similarly, to calculate the parameters of the gamma distribution, we

used the relationship:


scale 7/tf (2 4)

shape = p/a-


where irf and ja are the mean and variance of the estimate of fecundity fi,

respectively. According to equation 2-1, the variance of binomial vital rate

estimates is maximized when the rate is 0.5. To test for a difference in bias when

survival estimates vary maximally, we simulated with both the true survival rates

and after replacing the survivorship of all stages with 0.5. Similarly, because the

variance of estimates of fecundity is proportional to the real variance of fecundity

among individuals (equation 2-1), we simulated with 3 levels of fecundity variance,

which we defined in terms of coefficient of variation, a f/pf: 0.5, 2, and 16.

Because the distribution of individuals among size classes may affect

demographic analyses (Gross 2002; Miinzbergov4 and Ehrlin 2005), we simulated

sampling with two different distributions of individuals. First, we used a uniform

distribution of individuals among size classes (i.e., equal numbers of individuals

in all size classes). Next, we used a more realistic distribution, known as the "J

distribution." A "J distribution" contains fewer stage i + 1 individuals than stage i,

such that a histogram of stage classes in a sample is J-shaped. We used the actual

distribution of classes observed in the field averaged over all years and sites to

compute the J distribution.

For each vital rate, we ran 2000 simulations with populations ranging in size

from 25 to 200 individuals for each combination of survival (real and 0.5), fecundity

CV (0.5, 2, and 16), and sampling distribution (real and equal). We chose 25

individuals as the smallest sample size because smaller samples would yield too few









individuals per stage class to resemble what a real study might sample. Because

we used Monte Carlo methods to simulate the vital rates directly, fractional

sample sizes posed no problem. In each run of the simulation, we drew all 31 vital

rates (26 binomial vital rates and 5 fecundities) from their appropriate sampling

distributions, computed a sample transition matrix from these vital rates, and then

estimated A as the dominant eigenvalue of this transition matrix (Figure 2-2).

We then estimated the expected value and standard deviation of the relative

bias of A estimates, (A A)/A x 10A('. at each combination of survival rates,

fecundity CV, distribution of sampling effort, and sample size. All simulations were

conducted using the R statistical computing environment (R Development Core

Team 2005).

To assess the importance of our findings, we conducted a review of sample

sizes used in the plant demography literature. We survn i, .1 this literature using

a Web of Science search on March 15, 2006. Our search terms were combinations

of i,, I,:: model", 1I! .i l", and "demography." We also followed the relevant

citations of these papers to increase the number of papers surveyed. From each

study, we extracted the number of individuals sampled to parameterize one

transition matrix. If a study estimated more than one matrix, as in a multi-site

study, we used the average number of individuals sampled for a matrix. We also

determined the number of studies that did not include a sample size.

2.3 Results and Discussion

We detected no bias when sampling from the real Heliconia vital rates.

However, we found an appreciable positive bias in estimates of A when using

50 or fewer individuals to estimate vital rates from a population with a mean

survival probability of 0.5 (Figure 2-3 and Figure 2-4). In addition, the realistic

J-shaped sampling distribution increased the positive bias compared to sampling

equal numbers of individuals in all stage classes (17.1-2'-. 32.54 SD vs. 11.4 !'.









25.40 SD respectively when sampling 25 individuals). As expected, larger

sample sizes produced more precise estimates of A in all scenarios. Therefore, the

accuracy of estimates of A calculated from small sample sizes strongly depended

upon the true vital rates of the system. Specifically, increasing uncertainty in the

fates of individuals, represented by a 0.5 probability of survival, caused A to be

overestimated.

Our results extend those of previous studies demonstrating projections of

A can be biased due to the effects of Jensen's Inequality. For instance, Boyce

(1977) showed that process variance, in the form of stochasticity over time,

biases estimates of A. Using an analytical derivation with 2 x 2 matrices, he

demonstrated that A is a concave function of the matrix elements and is thus

positively biased. Daley (1979) extended Boyce's wi", 1,-i to Leslie matrices of any

size, and showed that A may be either positively or negatively biased if matrix

elements are stochastic. Although the scope of these studies was limited to process

variance, Houllier et al. (1989) used a similar theoretical approach along with

Monte Carlo simulations to examine bias resulting from sampling variance of the

matrix elements. Using both Leslie matrices and more general Usher models (Usher

1966), they found that bias in estimates of A was small less than 0.5' in all but

one scenario. However, the win, 1,-i- of Houllier et al. (1989) used the transition

elements of the matrix rather than the underlying vital rates. It is important to

recognize that sampling variation affects the vital rates, and this distinction may

affect sampling. Our analyses, which considered a more general Lefkovitch model,

demonstrated that significant overestimation and large sampling variation of A can

result from small sample sizes.

We were able to determine the number of plants that had been used to

parameterize matrix models in 52 of the 68 papers we collected (see Appendix

A). Studies of perennial herbs used fewer individuals than those of trees (herbs:









median = 215, N = 28 studies; trees: median = 575, N = 16 studies; shrubs:

median = 303, N = 9 studies, Figure 2-5, Table 2-1). Over 12"- of the studies on

perennial herbs used fewer than 100 individuals (summed across all stage classes),

and only 25'. of studies were based on 500 or more individuals (range: 30-4963).

Our simulations sl---:- -1 that samples containing as many as 100 individuals have

an expected bias of almost !' This level of bias could lead to severely inflated

estimates of time until extinction and the probability of persistence.

We have shown that the amount of bias in estimates of A that can be

attributed to sampling variation at small sample sizes is a function of the vital

rates of the system. Furthermore, the bias in A estimates also depends upon

the distribution of samples among stage classes. A possible explanation for our

findings is that H. acuminata individuals are characterized by low mortality,

low growth, and low fecundity. Therefore, its population growth is especially

sensitive to changes in the survivorship of individuals in the larger stage classes

while being relatively robust to changes in fecundity (Bruna 2003). This plant's

survival-dominated sensitivity structure could explain both the drastic increase in

bias in response to high variance of survival estimates as well as the independence

of bias to variance of fecundity estimates. In addition, the results of simulations

in which we varied the sampling distribution provide further evidence that the

relative sensitivity structure can determine how the vital rate variance affects bias.

Sampling with the more realistic J-distribution increased the sampling variance

of the more demographically 'important' vital rates of larger stage classes and

decreased the sampling variance of the less 'important' vital rates of the smaller

ones.

Our results si-I-, -1 that sampling proportionally to the elasticity structure of

a population may not only increase the precision of estimates of A (Gross 2002),

but may also increase the accuracy. However, more work is needed to explain the









discrepancy between our theory-backed results and the results of Miinzbergova and

Ehrlin (2005). One potential explanation for the poor performance of this method

presented by Miinzbergova and Ehrl6n (2005) is because the elasticity structure

of a study organism is usually unknown before the data is collected. Sampling

according to the elasticity structures of similar organisms to be problematic

because similar organisms may differ enough that sampling according them would

mistakenly undersample critical stages.

Still, H. acuminata's relative elasticity structure is common to many long-lived

plants (Silvertown et al. 1993; Franco and Silvertown 2004) and sampling with the

real Heliconia vital rates was relatively precise and accurate, even at low sample

sizes. Therefore, our results -,-.. -1 that estimates of A may be robust to low

sample sizes in populations of long-lived organisms. However, these results are

sensitive to the vital rates of the study organism. Therefore, we encourage others to

conduct similar simulations with demographic data from species with different life

histories since these taxa could have demographic matrices with markedly different

characteristics.




























true vital rates
-Sl, S2, s3, .
I


# plants per size class
n1, n2,, 3 (1: -i=N)


define parameters of sampling distributions
a,b for beta distributed vital rates
shape, scale for poisson distributed vital rates


draw sample vital rates compute sample
l1, sl12, ..., sim, s21, 2, ..., an, ... transition matrices
(m simulations)

estimate expected value of
compute asymptotic a-asymptotic growth rates
population growth rates for each sample size


Figure 2-1. Flowchart of simulations used to estimate amount and direction of
bias in estimates of vital rates due to sampling variance. We used
empirical vital rates from Heliconia field studies as the true vital rates
and then simulated sampling from appropriate probability distributions
to approximate real sampling variation.
















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CV 0.5
CV 2
CV 16


25 50 75 100 150 200 25 50 75 100 150 200
Number of individuals sampled


Figure 2-3.


Relative bias in estimates of A, (A A)/A x lI,' at increasing sample
sizes. Panels show the 4 combinations of distribution of sampling
among stage classes (real J-shaped sampling distribution vs. equal
effort) and survival (real and all survivals set to 0.5). Sample sizes on
the listed abscissa are the total sample sizes across all sizes classes.
Error bars are 1 standard deviation. The dot-dashed line is at bias =
0.































I I I I I I I I I
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0


Figure 2-4.


Histograms of the 6000 estimates of A, combining all levels of fecundity
variance, for sampling 25, 75, and 200 individuals with both real (A)
and 0.5 survival rates (B). Solid and dashed vertical lines are expected
value of A and the true A, respectively.


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Number of individuals sampled


Figure 2-5.


Sample sizes from published studies. From each study, we extracted
the average number of individuals used to parameterize a matrix
model.


N= 25 N= 75 N = 200







N= 25 N= 75 N = 200





^^~~~~~~~r ^_i____ ___.^____


40
U)
-


S0
0
60

O- 40

20-

0-









Results from sample size literature review of 52 plant demography
papers. We extracted the average number of individuals sampled to
parameterize a matrix model. These are the summary statistics broken
down by the 3 most prominent life histories.


Life history
perennial herb
shrub
tree


Mean
747.27
548.52
1311.70


Median
214.94
302.62
575.00


SD
1223.92
506.23
2040.72


Range
30 4963
3 1275.5
91 6905


Table 2-1.















CHAPTER 3
SPATIAL SAMPLING METHODS FOR PLANT DEMOGRAPHY

3.1 Introduction

The data used for parameterizing matrix models of plants, marine invertebrates,

or other sessile organisms are often collected in permanent plots (e.g., Bierzychudek

1999; Engelen et al. 2005; Gotelli 1991; Parker 2000; Silvertown et al. 1993).

The primary advantages of plot-based methods are logistical: mapped plants are

easier to relocate in subsequent survy-, survey methods can be standardized

among sites, and sampled individuals are closer together. Plot-based methods

used in demographic studies vary tremendously in the plot arrangement and area

sampled. For instance, Batista et al. (1998) mapped and J--' d all Fagus gran-

1.i: .1.:. > 2 cm dbh within a 4.5 ha plot, while Silva Matos et al. (1999) studied

the demography of the understory palm Euterpe edulis by sampling 100 randomly

placed subplots within a 1 ha area. In contrast, Valverde and Silvertown (1998)

sampled the perennial herb Primula ;; l,/,.i, ;: at regular intervals along a transect.

These disparate studies illustrate three common I- -. of plot-based sampling used

in demographic studies: identifying all individuals within a single plot; randomly

sampling multiple subplots within a defined area; and systematic sampling at

regular intervals. Even though long-term demographic studies often require

large investments of time and capital, studies rarely justify sampling decisions.

Interestingly, several studies have evaluated the optimal plot sizes and shapes

for measuring diversity (Jalonen et al. 1998; Kenkel and Podani 1991; Laurance

et al. 1998), individual density (Gray 2003; Picard et al. 2004), and intraspecific

competition (Hynynen and Ojansuu 2003). However, no such comparisons of

sampling methods have been conducted for demographic studies.









The choice of the appropriate sampling design is important because of the

potential for spatial variation of vital rates to influence the outcome of matrix

models (e.g., Horvitz and Schemske 1995; Moloney 1988). One key component

of this spatial variation that may affect the efficacy of sampling designs is spatial

autocorrelation within a sampling site (Legendre 1993). In the presence of spatially

autocorrelated vital rates, a sampling method that increases the distance between

sampled individuals should provide more estimates for a given sampling effort

because more variability is captured with the same number of individuals sampled.

Thus, systematic sampling might provide more precise estimates than random

sampling because it maximizes the distance between samples by sampling a regular

grid (Bellhouse 1977; Yates 1948). While ecologists have considered how spatial

autocorrelation influences estimates of individual density, species richness and other

ecological parameters (Legendre et al. 2004), to date no one has evaluated how

spatial autocorrelation of vital rates within a sampling unit might affect the efficacy

of sampling methods for demographic sutdies.

In this paper, we focus on how alternative sampling methods influence the

precision of estimates of As. We used a long-term and large-scale dataset of plant

demography to compare three commonly used sampling techniques: a single,

randomly placed plot, multiple randomly placed plots, and systematically arranged

plots. Although we focus exclusively on plants, our findings apply equally to

other types of sessile organisms. Using data-based simulations and a review of the

ecological literature, we addressed the following two questions: (1) How does the

precision of estimates of A8 vary among the three focal sampling methods? (2)

What sampling strategies are used in studies of plant demography, and what do our

results s l---, -1 about the efficacy of these sampling schemes?









3.2 Methods

Study system

We used data collected during a long-term and large-scale study of plant

demography conducted at Brazil's Biological Dynamics of Forest Fragments Project

(BDFFP; 2o30'S, 60W). The focal species for this study was Heliconia acuminata,

a perennial herb native to central Amazonia and the Guyanas (Berry 1991).

Descriptions of the study site and experimental design can be found elsewhere

(Bierregaard et al. 1992; Bruna 2003; Bruna and Kress 2002). Briefly, 13 sites

each composed of a permanent 50 m x 100 m plot were established in the BDFFP

continuous forest reserves. Because each site was relatively large and completely

surveyed, these data are ideal for testing sampling methods via subsampling.

All H. acuminata in each plot were marked and mapped; we also recorded the

number of vegetative shoots each plant had; shoot number was highly correlated

with demographic parameters such as the probability of flowering and survivorship

(Bruna 2003). Since 1998, the plots have been surveyed annually to record plant

growth, mortality, and the emergence of new seedlings (i.e., established plants

less than 1 year old). The plots were also surv, il, .1 during the flowering season to

record the identity of reproductive individuals. The analyses presented here are

based on data from the 1998-2005 surven-: during this time period we marked,

measured, and recorded the fates of N = 6591 plants (N = 3842 in continuous

forest, N = 1688 in 10-ha fragments, and N = 1061 in 1-ha fragments). Heliconia

acuminata density in 2003 ranged from 256-2248 plants ha 1. Because each

50 x 100 m plot was subdivided into 10 m x 10 m subplots, this was the finest

resolution at which sampling methods could be simulated.









Do different sampling methods affect the precision of population growth
estimates?

We simulated sampling from each site using three alternative sampling

methods: randomly placed single subplots (RS), randomly placed multiple subplots

(RM), and a systematic grid of subplots (SM; Fig 3-1). An RS sample consisted

of a single, randomly placed quadrat, dimensions ranging from 20 m x 20 m to

50 m x 100 m. An RM sample consisted of 4-50 randomly selected 10 m x 10 m

subplots. An SM sample consisted of a grid of regularly spaced 10 m x 10 m

subplots. Because the sites were not squares, only an approximate number of

subplots could be specified to the algorithm for choosing a particular SM sample

(Pebesma and Bivand 2005). Nevertheless, we specified the same numbers of

subplots to the SM method as to the RM method. Thus, while the three methods

were tested over the same range of sampling intensities, the resolution of sampling

intensities differed among the methods. This difference in resolution did not affect

any of the analyses (see Results).

Simulations were conducted as follows: for each site and method, we selected

50 samples for each of the sampling intensities described above. For each sample,

we calculated all possible annual transition matrices. Because some sites were

not surveyed in some years, the number of possible transition matrices per site

varied: 7 matrices for 10 sites, 5 for 2 sites and 3 for the remaining site; (see

Bruna 2003 for details). We then estimated the stochastic population growth rate,

As, using the random transition matrix method (Bierzychudek 1982; Caswell

2001) rather than the more computationally intensive vital rate simulation

method (reviewed in Morris and Doak 2002). Thus, we created a sequence of

30,000 transition matrices by drawing from the set of annual matrices with equal

probability with replacement. Then, to project population growth, we multiplied









an initial population vector by the 30,000 matrices. We calculated A8 as N30,ooo/No

(Figure 3-2).

When the sampling area being simulated was small, all methods resulted in

samples that lacked individuals in at least one size class, making it impossible to

correctly calculate the annual transition matrix. Eliminating these samples and

drawing alternatives would have biased estimates of A8 and artificially reduced

their variance because quadrats with fewer individuals would not be included

in subsequent analysis. Therefore, we developed a 3 step method to determine

if a sampling intensity-defined here as an area-was incapable of estimating

A for a particular method and site: (1) if there were no individuals of any size

class in a sample for a given year, we did not estimate a deterministic matrix for

that year; (2) if a given sample was insufficient to estimate at least 5(0' of the

possible transition matrices, we did not estimate a A, with that sample; (3) finally,

if fewer than 911' of samples for a given sample size could not estimate As, then we

determined that that sample size was too small to estimate As and did not include

the sample size for that plot and method in further analyses.

To compare the precision of the sampling methods, we determined the

minimum plot size for each site for which all estimates of A, were within 1 of

the mean whole plot estimate. Then, we performed a logistic regression for each

plot to estimate how the probability of A8 being within 1 of the whole plot value

changed as a function of area sampled for each method:


Pi = logistic(/3are[i] + Osite[i] + '. .. ,[ ) (3-1)

Prob(|, As < 0.01A,) = pi


where logistic(x) = 1/(1 + e-"). To evaluate how the methods decreased variation

As as the sampled area increased, we used nonlinear least squares to fit a negative

exponential to the standard deviation of As as a function of the sample size for each









sampling method and site according to


SD(A,(area)) = de-'xar (3-2)


The negative exponential was an ideal model to describe how the variation in

A, decreased with area because its rate and intercept parameters allow it to

flexibly model many decaying relationships. In order to compare the precision of

the sampling methods at each site, we evaluated the fitted curve at 1000 m2 to

standardize the sampling intensities of the different methods. Finally, we wanted

to ensure that any difference between methods were not an artifact of different

numbers of individuals being sampled with a given area for each method. To do

so, we used ordinary least squares linear regression to estimate the number of

individuals sampled as a function of area for each method. All computations were

conducted in the R statistical computing environment (R Development Core Team

2005).

What sampling strategies are used in studies of plant demography?

To evaluate the sampling designs used in demographic studies of plants, we

conducted a Web of Science search on March 15, 2006 using combinations of the

search terms i,, l 11:: model," pi ,l "demography," and "population." For

each paper, we extracted the total area and number of plots sampled for each

matrix model parameterized. We also assigned each study to one of the following

5 sampling methods: completely surveying a single plot (S); sampling randomly

dispersed subplots (RM); sampling systematically distributed subplots (SM);

completely surveying "populations" (P); and finally, some sampled individuals

without delineating plots or populations (I). We categorized the species studied by

life history into the following categories: shrubs, trees, perennial herbs, and other

(e.g., geophytes, grasses, sedges). If a study used the same sampling process to

study multiple species, we counted these studies as a single study. However, when









a study examined a single species using multiple sampling strategies, we counted

these as separate studies. If different methods were used to sample for different

stage classes, we evaluated the process used to sample the largest size classes.

3.3 Results

Do different sampling methods affect the precision of population growth
estimates?

With all sampling methods, the variance of A, decreased as total area

increased (see fig 2-3), as predicted by statistical theory (Wackerly et al. 2001).

The minimum plot sizes required for all A8 to be within 1 of the whole plot

estimate are given in Table 3-1. Systematic sampling required the least area

to guarantee the 1 precision level for 8 sites while single plots were best for 2

sites, multiple random plots were best for 1 site, and 2 sites had tied between

single and systematic sampling (Figure 3-3 and Table 3-1). According to the

logistic regression (see eq 3-1), SM was the most precise of the three methods,

with RM being statistically indisinguishable from RS. Specifically, the effect of

systematic sampling on the log-odds of being within 1 percent of the site value was

significantly greater than the effect of RS (p = 0.037), whereas the RM method was

not (see Figure 3-4). Overall, sampling method significantly affected the log-odds

that all As were within 1 percent of the true value (p < 0.001). According to the

logistic model, systematic sampling required a 15.!' smaller sampling area than

single plot sampling to achieve a 95'. probability that a sample will be within 1.

of the true value. The negative exponential model (eq 3-2) fit the data well (see

fig 3-5). According to evaluation of the model at 1000 m2, SM sampling produced

the most precise estimates in 8 sites, RS at 4 sites, and RM at 1 (see Table 3-2).

Thus, all three analyses indicated that systematic sampling was the overall most

precise of the three methods. However, while we expected RM to provide an

intermediate level of precision, it proved to be the least precise. The linear









model fit between area and numbers of individuals was nearly identical among

all methods, indicating that the methods sampled equal numbers of individuals for

a given area.

What sampling strategies are used in studies of plant demography?

Our review retrieved 63 studies of plant demography (see Appendix A).

Most studies were on perennial herbs (N = 30 studies), followed by trees (N

= 13 studies), shrubs (N = 7 studies), and other life-forms (N = 13 studies).

The most common methods were RM subplots for perennial herbs (13 of 20

studies), S for trees (9 of 14 studies), RM for .1 !. -;" (10 of 13), and methods

were evenly distributed for shrubs (see Figure 3-6). In addition to the three

methods that we analyzed above, we found 8 studies used the "population"

method, which delineated populations and completely surveyed them; and 4 studies

used non-plot-based methods. The total area studied to parameterize a matrix

model ranged from 0.09 175,000 m2 (mean 8150 m2, SD 26,637). Unsurprisingly,

studies of trees used the largest areas (mean 31,250 m2, SD 48,818), while those of

perennial herbs used the smallest (mean 575 m2, SD 1843; fig 3-7, table 3-3).

3.4 Discussion

Our study -Ii--.: -I that the precision of estimates of A8 can depend on how

the data were collected. Using 13 sites of a focal species, we showed that systematic

sampling can estimate vital rates for parameterizing matrix models more efficiently

than the two most commonly used methods: completely surveying one plot or

random sampling subplots. However, we also found evidence that estimating A8

may require plot sizes that are much larger than those commonly used.

We determined that for H. acuminata, the minimum plot size needed to

calculate A, with 1 precision was about 3000 m2. However, this statistic varied

among sites and sampling methods, with systematic sampling requiring the

smallest area (mean 2400 m2) and random sampling requiring the largest area









(mean 3200 m2). The variation in densities of plants among sites may have caused

differences among sites. The ability of systematic sampling to capture the most

vital rate variation in a sample than the other methods may have caused its

superior precision for estimating A We discuss this point further below.

How do the sampling intensities required for Heliconia acuminata compare

with those for other systems? Although our simulations -ii- .I -I '1, at least 3000 m2

were necessary to precisely estimate A8 in for H. acuminata, only s' of the studies

of perennial plants sampled areas larger than this. Furthermore, !*' of the studies

of perennial species sampled an area < 10 m2. However, variation in life histories,

densities, and ., ::-regation among species may account for at least part of this

difference between our system's sampling requirements and those in the literature.

Nevertheless, even if the sampling requirements of H. acuminata are orders of

magnitude higher than other systems, our results still -ii.:: -1 other studies used

imprecise methods to estimate vital rates due to their small sampling intensities.

At least four factors might explain the small sampling intensities in the literature.

First, the focal species may be found in a limited area because of endemism or

population decline (e.g., Garcia 2003). Second, higher densities of other plants may

require smaller areas to precisely estimate vital rates. Third, researchers may have

chosen to focus on small areas but many sites (e.g., Horvitz and Schemske 1995;

Moloney 1988; 0-. -1i i i i, ier et al. 1996). Finally, the financial resources or time

needed for a broad-scale study could be unavailable.

Gibson (2002) argued that all spatial sampling methods are equivalent as

long as the same numbers of individuals are sampled in all methods. However,

our results -i-i.; -1 that systematic sampling can be a more efficient method for

sampling vital rates. Interestingly, this was the least common method used in the

studies we analyzed only three of the 63 studies we reviewed used a systematic

network of sampling sites. This paucity of studies using systematic sampling









is somewhat surprising, since systematic sampling's efficiency advantage is a

well-known result from sampling theory (Bellhouse 1977; Yates 1948) and studies

in other fields such as weed science (Ambrosio et al. 2004) and plant population

genetics (Suzuki et al. 2004). However, while our study sir.-. -1-' that systematic

sampling could be the preferable way to sample plant for demography, this method

should be not blindly applied because misuse may lead to biases for two reasons.

First, the sampling grid may coincide with the periodicity of real spatial structure

(Thompson 2002). However, researchers can employ non-aligned systematic

sampling to avoid this problem (Quenouille 1949; Thompson 2002). Second, if

the variance of a vital rate is calculated as if sampling were done at random, this

will lead to biased estimates of vital rate sampling variance (Aubry and Debouzie

2000). The solution to this problem is to use variance estimation techniques

designed for systematic sampling (Aubry and Debouzie 2000). Since solutions

exist for both of these potential problems, systematic sampling appears robust.

Because the increased precision of systematic sampling results from maximizing the

distance between samples, the advantage of systematic sampling is expected to be

dependent on the presence of spatial autocorrelation of vital rates.

Despite the fact that spatial autocorrelation of vital rates can have an

impact on the efficacy of sampling methods, to our knowledge no studies have

investigated if vital rates are spatially autocorrelated. This is surprising because

there are many possible mechanisms that would lead to spatial autocorrelation of

vital rates. For example, spatial variation in environmental variables can cause

spatially autocorrelated vital rates: soil chemistry can affect plant defenses and

therefore survival (Koike et al. 2006); lower temperatures can reduce pollen

and fruit production and therefore fertility (Jakobsen and Martens 1994).

Another mechanism that could lead to spatially autocorrelated vital rates

is short distance dispersal. Short dispersal can lead to increased relatedness









among nearby individuals. These more closely related individuals share genetic

material that may contribute towards vital rates (Chuiin et al. 2005; Delesalle

and Blum 1994). Facilitation from conspecific neighbors is another process that

can increase spatial autocorrelation of survival and growth rates (Escudero et al.

2005). With increasing evidence that spatial autocorrelation is a real component

of many ecological systems, it is important to be aware of and account for spatial

autocorrelation when collecting data to parameterize matrix models. To compute

precise estimates of vital rates in the presence of positive spatial autocorrelation

of vital rates, larger plot sizes are needed to represent the true variation in vital

rates (Legendre 1993). This has been thoroughly documented in agricultural

field trials that determined that optimal sampling was dependent upon spatial

autocorrelation by using variograms and other geostatistical tools (Bhatti 2004;

Fagroud and Van Meirvenne 2002; Poultney et al. 1997). However, more research is

needed on the effects of spatial autocorrelation on sampling requirements in plant

demography.

An interesting outcome of our literature review is that it revealed a lack of

consistency in the terminology used to describe sampling methods in studies of

plant demography. For example, clearer use of the following terms would be helpful

to readers: subplots, plots, sites, and populations. Subplots should be used to

refer to areas that will be pooled to estimate population parameters, whereas

plots should refer to single plot sampling. \! !l iip' 1i.1. implies a multi-site

study rather than multiple sampling areas to estimate a single vital rate. Papers

using the "population" method (see Results) often stated that they sampled a

population without differentiating between a statistical population and a ''.:..1... ,,l

population. If they surveyed all individuals in a genetically isolated biological

population, then they conducted a complete survey which is more accurate and

precise than any of the sampling methods we compared. However, if they actually









surveyed a statistical population, which may have been a subset of a biological

population, then the domain of the model results are smaller than stated. The

rigor of sampling terminology is important and its clarification would help with the

design of future studies and elimination of guesswork. We hope that future papers

adopt consistent sampling terminology for plant demography.

Although our understanding of how the spatial component of sampling

influences matrix models remains limited, previous studies have evaluated other

sampling issues that could influence our results. Doak et al. (2005) looked at the

effects of sampling intensity; they investigated how the numbers of individuals and

years studied affect deterministic and stochastic matrix models. They -ii-::. -i.1

that with fewer than 5 years' data, deterministic models are better at modeling

populations than stochastic models. To develop more efficient methods to estimate

population growth with matrix models, Gross (2002) and Miinzbergov4 and

Ehrl6n (2005) looked at how the ratio of sampling among stage classes affects

precision of A estimates. Gross (2002) -, .-.-. -1. 1 using prior information about

the relative demographic importance of stage classes to increase sampling of the

more 'important' stages and thus minimize the variance of A estimates. In contrast,

Miinzbergov4 and Ehrl6n (2005) advocated sampling equal numbers of individuals

across all stage classes; they created a distinction between their method and

plot-based methods without -,.-.i-i iir-; a non-plot-based sampling process for use

in the field. However, studies often subsample plots to sample individuals for more

abundant stage classes to achieve a similar result to their equal numbers method.

Thus, plot-based methods can still be used to sample a more equal distribution

of classes because of the logistical advantages over individual or non-plot-based

methods: plot-based methods make it easier to relocate individuals; easier to plan a

study; and easier to describe the sampling process.









Taken as a whole, our and other studies of sampling have s, l'-- -1 I a number

of guidelines to consider when designing a demographic study. The first task is to

clearly define the domain of the study, making sure that the sampling extent covers

this domain (see Legendre and Legendre (1998), Fig 3-1 for definitions of sampling

terminology). Next, if possible, the sampling units (sites) should be chosen with

the aim of comparing among the sites. After all, ecologically important spatial

variation can be evaluated only if a multi-site study design is used. Based on our

results and those of statistical sampling theory (Bellhouse 1977; Cochran 1946;

Yates 1948), we would -Ii--.- -1 a systematic grid of subplots within each site. If a

subsequent analysis reveals homogeneity in vital rates across sites, then all data

can be pooled to obtain more precise matrix model parameters for a single, large

site. Toward this end, we also advocate sampling with a systematic grid of sites.

However, if vital rates at sites are heterogeneous, then sites have sufficient data to

be modeled separately for comparisons among sites. Nonetheless, perhaps more

important than which sampling method is employ, .1 researchers should ensure that

enough area is sampled, either within a site, or across sites, to encompass the range

of spatial variation in plant survival, growth, and reproduction over the domain of

interest.



































subplots


or or


(a) (b) (c)


Figure 3-1. Illustration of sampling terminology as applied to plant demography.
(a) random single plot method (RS), (b) random multiple plot method
(RM), (c) systematic multiple plot method (SM)


































i-- -- _-"'" 1 choose 50
random
samples
sample 1 sample 2 sample 10
estimate all
Possible annual
year 1 year 2 year 7 transition
transition transition ... transition matrices
matrix matrix matrix

simulate random
30,000 year
sequence of
transition matrices






Figure 3-2. Flowchart of sampling method simulations used to compare the three
alternative sampling methods.







33




















Dimona CF' 'Florestal' 'PA-CF'
5000 -

4000 -

E 3000 -
C 2000 -
0
5 1000 -
S'5752' '5753' '5754' '5756' 'Cabo Frio'
5000

4000

-/ 3000

2000

-_ -_ 1000
E '2107' '2108' '2206' '5750' '5751'
n 5000 -
E
4000 -

3000- ,

2000 -

1000 -
RS RM SM RS RM SM RS RM SM RS RM SM RS RM SM



Figure 3-3. Minimum sampling area for 1 precision. These are the plot sizes at
which all samples were within 1 of the whole plot estimate for each
site and sampling method







34













RS
RM
SM

1000 3000 5000


(,

E



o
C-
(D
a,


0
0-
L-
0










a
a


1000 3000 5000 1000 3000 5000


1000 3000 5000


Area (m2)


Figure 3-4. Logistic model of precision (see equation 3-1) for each site of the
whether a particular A8 was within 1 of the whole plot value as a
function of area sampled for all 3 sampling methods.



















RS
RM
SM
100030005000


100030005000 100030005000

Area (m2)


100030005000


Figure 3-5. Negative exponential model of precision (equation 3-2) fitted to
standard deviation of A~ at each sample area and every site for
each sampling method. The horizontal lines show the value of the
standard deviation predicted by the model for each sampling method
at 1000 m2


0.030
0.025
0.020
0.015
0.010
0.005
0.000


(n
4-
0
0
"O


.0
U0


0.030
0.025
0.020
0.015
0.010
0.005
0.000


0.030
0.025
0.020
0.015
0.010
0.005
0.000


























SSM
M RS
m RM
o Inds
o Pops


II


perennial herb


shrub


Figure 3-6. Sampling methods in published studies.


SO
CN


0 -


tree


other






37




















I I I I I I I I I
perennial herb tree




.. nn nlrl n n n n n
shrub other



fl mrl


I I
0 10000


30000 50000 0 10000
30000 50000 0 10000


30000 50000


Total area sampled (m2)

Figure 3-7. Areas sampled in published studies. Histograms of the results from a
literature review of 63 plant matrix model studies. For each paper, we
calculated the average area used to estimate the population growth.


20 -
15 -
10 -
5-
0-

20 -
15-
10-
5
0-











Minimum sampling areas for 1
for which all As were within 1
and sampling method


. precision. This is the minimum area
Sof the whole site estimate for each site


Site RS
2107 3000
2108 4500
2206 2500
5750 1600
5751 2500
5752 2500
5753 2500
5754 2500
5756 1600
Cabo Frio 4500
Dimona CF 4500
Florestal 2500
PA-CF 4500
mean 3015.38
SD 1096.09


RM
3600
3600
3200
2000
2800
2800
4000
1600
2400
5000
3800
2400
4500
3207.69
993.7


SM
2400
1200
2800
1200
2100
2100
2800
1000
1000
4500
4000
1200
4500
2369.23
1294.47


Table 3-2. Negative exponential model evaluated at 1000 m2 (see equation 2)
evaluated at 1000 m2 for each site and method.

Site RS RM SM
2107 0.0218 0.0122 0.0089
2108 0.0369 0.0154 0.0057
2206 0.0077 0.0196 0.0088
5750 0.0053 0.007 0.0046
5751 0.0088 0.0135 0.013
5752 0.008 0.0086 0.0081
5753 0.0326 0.0157 0.0125
5754 0.0067 0.0064 0.005
5756 0.0052 0.0048 0.0057
Cabo Frio 0.0268 0.0255 0.0249
Dimona CF 0.0232 0.0354 0.0295
Florestal 0.0106 0.0072 0.0048
PA-CF 0.0398 0.0252 0.0195
mean 0.018 0.0151 0.0116
SD 0.0128 0.0092 0.0082


Table 3-1.































Table 3-3. Areas sampled in the literature. The areas (m2) come from the 63
published studies sampled, classified into 4 life history categories (see
Methods)


Plant form
perennial herb
tree
shrub
other


Mean
574.60
31250.00
775.00
1673.10


Median
10.00
8100.00
612.50
5.00


S.D.
1843.30
48814.30
496.20
2259.90


Range
0.6 8018.2
1500 175000
375 1500
0.1 5550















CHAPTER 4
CONCLUSION

C'!i pter 1 revealed that although Jensen's Inequality and statistical theory

si-.-, -i that estimates of population growth may be biased when few individuals

are sampled, our results revealed that this bias is unimportant for certain

organisms. These results are most likely generalizable to other long-lived organisms

characterized by high survival and low fecundity. However, when the study species

is characterized by lower survival, nearer to 0.5, low sampling intensities can cause

severe overestimation of population growth. Therefore, we advocate conducting a

simulation power ',i i,!Ji ; similar to the one we conducted here to help determine

an adequate sample size. Because researchers do not have the matrix model for

their study species a priori, we -Iit-'-. -1 sampling simulations using several matrices

of organisms with similar life history characteristics (similar to the method of Gross

2002).

In C'! lpter 2, we showed that systematic sampling can be a more efficient field

method to collect data to estimate vital rates. However, systematic sampling was

by far, the rarest method used in published studies to sample plants for matrix

models. This disparity between what our results siL-'-- -1 is the best method and

what studies typically use is surprising given that statistical sampling theory has

demonstrated the efficiency improvements of systematic sampling (e.g., Yates 1948;

Bellhouse 1977; Cochran 1946) and in other fields (e.g., Ambrosio et al. 2004).

Therefore, our results -i-i-.; -1 that plant population ecologists may improve

sampling designs with the addition of two outcomes from our study. First, an

a priori power analysis should be conducted to reduce the potential for biased

population growth estimates from small sample sizes. Second, when a population









is not completely surveyed, demographers should consider using a systematic or

stratified sampling design to maximize the vital rate variance captured for their

sampling effort and obtain the most precise vital rate estimates. With techniques

available to improve the estimation of vital rates, demographers have the choice of

maintaining their sampling effort and improving precision or reducing their effort

and maintaining the same level of precision. With the latter choice, demographic

studies will require fewer resources and perhaps more species can be studied. In

the current situation in which there are more species that require help than we can

possibly study to design recovery plans, any potential aid in reducing the cost of a

single study should be appreciated.















APPENDIX
PAPERS REVIEWED

Below is a list of the papers that were included in the sampling literature

reviews in C'! Ilpters 2 and 3.


Allphin, L., and K. T. Harper. 1997. Demography and life history
characteristics of the rare Kachina daisy (Erigeron kachinensis, Asteraceae).
American Midland N .i i .list 138:109-120.

Alvarez-Buylla, E. R. 1994. Density-dependence and patch dynamics in tropical
rain-forests matrix models and applications to a tree species. American
Nl i .1list 143:155-191.

Barot, S., J. Gignoux, R. Vuattoux, and S. Legendre. 2000. Demography of
a savanna palm tree in Ivory Coast (Lamto): population persistence and
life-history. Journal of Tropical Ecology 16:637-655.

Batista, W. B., W. J. Platt, and R. E. Macchiavelli. 1998. Demography of
a shade-tolerant tree (Fagus gr i,',.:.:. J.) in a hurricane-disturbed forest.
Ecology 79:38-53.
Berg, H. 2002. Population dynamics in Oxalis acetosella: the significance
of sexual reproduction in a clonal, cleistogamous forest herb. Ecography
25:233-243.

Bierzychudek, P. 1982. The demography of Jack-in-the-Pulpit, a forest
perennial that changes sex. Ecological Monographs 52:335-351.

Brewer, J. S. 2001. A demographic analysis of fire-stimulated seedling
establishment of Sarracenia alata (Sarraceniaceae). American Journal of
Botany 88:1250-1257.

Bruna, E. M. 2003. Are plant populations in fragmented habitats recruitment
limited? Tests with an Amazonian herb. Ecology 84:932-947.

Brys, R., H. I 1i. .!,, in.n, P. Endels, G. De Blust, and M. Hermy. 2004. The
effects of grassland management on plant performance and demography in
the perennial herb Primula veris. Journal of Applied Ecology 41:1080-1091.

Bullock, S. H. 1980. Demography of an undergrowth palm in Littoral
Cameroon. Biotropica 12:247-255.









Byers, D. L., and T. R. Meagher. 1997. A comparison of demographic
characteristics in a rare and a common species of Eupatorium. Ecological
Applications 7:519-530.

Calvo, R. N. 1993. Evolutionary demography of orchids intensity and
frequency of pollination and the cost of fruiting. Ecology 74:1033-1042.

C'!i ii on, D., and D. Gagnon. 1991. The demography of northern populations of
Panax !;,,':u .! folium (American Ginseng). Journal of Ecology 79:431-445.

Cipollini, M. L., D. A. Wallacesenft, and D. F. Whigham. 1994. A model of
patch dynamics, seed dispersal, and sex-ratio in the dioecious shrub Lindera
benzoin (Lauraceae). Journal of Ecology 82:621-633.

Cipollini, M. L., D. F. Whigham, and J. O'Neill. 1993. Population growth,
structure, and seed dispersal in the understory herb C;,l,..il' .. -',, virgini-
anum: A population and patch dynamics model. Plant Species Biology
8:117-129.

Ehrl6n, J. 1995. Demography of the perennial herb Llil/irus vernus 1.
Herbivory and individual-performance. Journal of Ecology 83:287-295.

Endress, B. A., D. L. Gorchov, and R. B. Noble. 2004. Non-timber forest
product extraction: Effects of harvest and browsing on an understory palm.
Ecological Applications 14:1139-1153.

Enright, N., and J. Ogden. 1979. Applications of transition matrix models
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BIOGRAPHICAL SKETCH

Ian Fiske was born in Denville, New Jersey, on August 20, 1980 and was raised

in Jacksonville, Florida. He graduated from Stanton College Preparatory High

School with an International Baccalaureate diploma in 1998. The following fall,

Ian began a bachelor's degree in electrical engineering at the University of Florida.

In the final year of his engineering education, he decided to earn an additional

bachelor's degree in mathematics to gain skills would be applicable to a variety of

fields besides engineering. Soon thereafter, Ian began a research project analyzing

Heliconia acuminata demography with wildlife ecology and conservation professor

Emilio Bruna. After graduating cum laude in 2003 in electrical engineering and

mathematics, Ian spent a year exploring the world outside of academics. He worked

on an organic farm in Homestead, Florida, and on several farms in Costa Rica.

After returning to the United States in 2004, he began a master's degree in wildlife

ecology and conservation at the University of Florida, working with Dr. Bruna

on this thesis project. Next, Ian will begin a statistics PhD program at North

Carolina State University, where he will specialize in the analysis of ecological and

environmental data.