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REVIEW Quantifying biodiversity procedures and pitfalls in the measurement and comparison REVIEW Quantifying biodiversity procedures and pitfalls in the measurement and comparison

REVIEW Quantifying biodiversity procedures and pitfalls in the measurement and comparison - PDF document

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REVIEW Quantifying biodiversity procedures and pitfalls in the measurement and comparison - PPT Presentation

Gotelli and Robert K Colwell Department of Biology University of Vermont Burlington Vermont 05405 USA Email ngotellizoouvmedu Department of Ecology and Evolutionary Biology U43 University of Connecticut Storrs Connecticut 06269 USA Abstract Species ID: 42757

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Quantifyingbiodiversity:proceduresandpitfallsinthemeasurementandcomparisonofspeciesrichnessNicholasJ.GotelliandRobertK.ColwellDepartmentofBiology,UniversityofVermont,Vermont05405,U.S.A.E-mail:ngotelli@zoo.uvm.eduDepartmentofEcologyandEvolutionaryBiology,U-43,UniversityofConnecticut,Connecticut06269,U.S.A.AbstractSpeciesrichnessisafundamentalmeasurementofcommunityandregionaldiversity,anditunderliesmanyecologicalmodelsandconservationstrategies.Inspiteofitsimportance,ecologistshavenotalwaysappreciatedtheeffectsofabundanceandsamplingeffortonrichnessmeasuresandcomparisons.Wesurveyaseriesofcommonpitfallsinquantifyingandcomparingtaxonrichness.Thesepitfallscanbelargelyavoidedbyusingaccumulationandrarefactioncurves,whichmaybebasedoneitherindividualsorsamples.Thesetaxonsamplingcurvescontainthebasicinformationforvalidrichnesscomparisons,includingcategory±subcategoryratios(species-to-genusandspecies-to-individualratios).Rarefactionmethods±bothsample-basedandindividual-based±allowformeaningfulstandardizationandcomparisonofdatasets.Standardizingdatasetsbyareaorsamplingeffortmayproduceverydifferentresultscomparedtostandardizingbynumberofindividualscollected,anditisnotalwaysclearwhichmeasureofdiversityismoreappropriate.Asymptoticrichnessestimatorsprovidelower-boundestimatesfortaxon-richgroupssuchastropicalarthropods,inwhichobservedrichnessrarelyreachesanasymptote,despiteintensivesampling.Recentexamplesofdiversitystudiesoftropicaltrees,streaminvertebrates,andherbaceousplantsemphasizetheimportanceofcarefullyquantifyingspeciesrichnessusingtaxonsamplingcurves.KeywordsSpeciesrichness,speciesdensity,taxonsampling,taxonomicratios,biodiversity,rarefaction,accumulationcurves,asymptoticrichness,richnessestimation,category±subcategoryratios.EcologyLetters(2001)4:379±391Speciesrichnessisthesimplestwaytodescribecommunityandregionaldiversity(Magurran1988),andthisvariable±numberofspecies±formsthebasisofmanyecologicalmodelsofcommunitystructure(MacArthur&Wilson1967;Connell1978;Stevens1989).Quantifyingspeciesrichnessisimportant,notonlyforbasiccomparisonsamongsites,butalsoforaddressingthesaturationoflocalcommunitiescolonizedfromregionalsourcepools(Cornell1999).Maximizingspeciesrichnessisoftenanexplicitorimplicitgoalofconservationstudies(May1988),andcurrentandbackgroundratesofspeciesextinctionarecalibratedagainstpatternsofspeciesrichness(Simberloff1986).Therefore,itisimportanttoexaminehowecologistshavequanti®edthisfundamentalmeasureofbiodiversityandtohighlightsomerecurrentpitfalls.Eventhemostrecentreviewsofbiodi-versityassessment(Lawtonetal.1998;Gaston2000;Purvis&Hector2000)havenotdiscussedthesamplingissuesweaddressinthisreviewinrelationtothemeasurementandcomparisonofspeciesrichness.Incontrast,theusesandabusesofspeciesdiversityindices,which,bydesign,combinerichnesswithrelativeabundance,enjoyasubstan-tialandvenerableliterature(e.g.Washington1984),andarethusbeyondthescopeofthisreview.Webeginbyplacingseveralconceptsofdiverseorigininacommonconceptualframework.TAXONSAMPLINGCURVESAlthoughspeciesrichnessisanaturalmeasureofbiodiversity,itisanelusivequantitytomeasureproperly(May1988).Theproblemisthat,fordiversetaxa,asmoreindividualsaresampled,morespecieswillberecorded(Bunge&Fitzpatrick1993).Thesame,ofcourse,istrueforhighertaxa,suchasgeneraorfamilies.Thissamplingcurverisesrelativelyrapidlyat®rst,thenmuchmoreslowlyinlatersamplesasincreasinglyraretaxaareadded. EcologyLetters,(2001):379±3912001BlackwellScienceLtd/CNRS Inprinciple,forasurveyofsomewell-de®nedspatialscope,anasymptotewilleventuallybereachedandnofurthertaxawillbeadded.Wedistinguishfourkindsoftaxonsamplingcurves,basedontwodichotomies(Fig.1).Althoughwewillpresentthesecurvesintermsofspeciesrichness,theyapplyjustaswelltorichnessofhighertaxa.The®rstdichotomyconcernsthesamplingprotocolusedtoassessspeciesrichness.Supposeonewishestocomparethenumberoftreespeciesintwocontrasting10-haforestplots.Oneapproachistoexaminesomenumberofindividualtreesatrandomwithineachplot,recordingsequentiallythespeciesidentityofonetreeafteranother.Werefertosuchanassessmentprotocolasindividual-based(Fig.1).Alternatively,onecouldestablishaseriesofquadratsineachplot,recordthenumberandidentityofallthetreeswithineach,andaccumulatethetotalnumberofspeciesasadditionalquadratsarecensused(e.g.Cannonetal.1998;Chazdonetal.1998;Hubbelletal.1999;Vandermeeretal2000).Thisisanexampleofasample-basedassessmentprotocol(Fig.1).Therelativemeritoftheseapproachesforestimatingspeciesrichnessoftreesisnotthepointhere.Rather,weemphasizethatspeciesrichnesscensusescanbevalidlybasedondatasetsconsistingeitherofindividualsorofreplicated,multi-individualsamples.Thekeydistinctionistheunitofreplication:theindividualvs.asampleofindividuals±adistinctionthatturnsouttobefarfromtrivial.Examplesofindividual-basedprotocolsincludebirders'``lifelists''(e.g.Howard&Moore1984),Christmasbirdcounts(e.g.Robbinsetal.1989),time-based``collector'scurves''(e.g.Clench1979;Lamasetal.1991),andtaxon-richnesscounts(oftenfamiliesorgenera)frompalaeonto-logicalsites(e.g.Raup1979).Inaddition,whenanunreplicatedmasssample(suchasadeep-seadredgesample,e.g.Sanders1968)istreatedasifitweresetofrandomlycapturedindividualsfromthesourcehabitat,anindividual-basedtaxon-samplingcurvecanbeproducedforthesample.Examplesofsample-basedprotocolsusingsamplingunitsotherthanquadratsincludereplicatedmist-netsamplesforbirds(Melhop&Lynch1986)andreplicatedtrapdataforarthropods(e.g.Stork1991;Longino&Colwell1997;Gotelli&Arnett2000).A``hybrid''betweenindividual-basedandsample-basedtaxonsamplingcurvesisproducedbythe``-specieslist''method,incurrentusebysomeornithologists(e.g.Poulsenetal.1997).Alistiskeptofthe®rst(usually20)speciesobserved(disregardingabundances)inasamplingarea±anindividual-basedlist.Then,additional``samples'',eachbasedonanewlistofspeciesfromthesamearea,aresuccessivelypooled.Thecumulativenumberofspeciesobservedisplottedasafunctionofthenumberof-specieslistspooledtoproduceacurvethatreachesanasymptotewhenallspecieshavebeenobserved.Sample-andindividual-baseddatasetsaresometimestreatedinterchangeablyinstatisticalanalyses.Forexample,dependinguponthescaleofinterestorthefocusofahypothesis(inthesenseofScheineretal.2000),agroupofindividual-baseddatasetsormasssamplescanbeanalysedasiftheywerereplicatesamplesfromthesamestatisticaluniverse(e.g.Grassle&Maciolek1992).Likewise,asetofreplicatedsamplescanusuallybepooledandtreatedasasingle,individual-baseddataset,forsomepurposes(Engstrom&James1981).(Thisisnotpossiblewith-species-listcurves,sinceabundancesarenotrecorded.)Theseconddichotomydistinguishesaccumulationcurvesrarefactioncurves.Aspecies(orhighertaxon)accumulationrecordsthetotalnumberofspeciesrevealed,duringtheprocessofdatacollection,asadditionalindividualsorsampleunitsareaddedtotheofallpreviouslyobservedorcollectedindividualsorsamples(Fig.1).Accumulationcurvesmaybeeitherindividual-based(e.g.Clench1979;Robbinsetal.1989)orsample-based(e.g.Novotny&Basset Figure1Sample-andindividual-basedrarefactionandaccumula-tioncurves.Accumulationcurves(jaggedcurves)representasingleorderingofindividuals(solid-line,jaggedcurve)orsamples(open-line,jaggedcurve),astheyaresuccessivelypooled.Rarefactioncurves(smoothcurves)representthemeansofrepeatedre-samplingofallpooledindividuals(solid-line,smoothcurve)orallpooledsamples(open-line,smoothcurve).Thesmoothedrarefactioncurvesthusrepresentthestatisticalexpectationforthecorrespondingaccumulationcurves.Thesample-basedcurvesliebelowtheindividual-basedcurvesbecauseofthespatialaggrega-tionofspecies.AllfourcurvesarebasedonthebenchmarkseedbankdatasetofButler&Chazdon(1998),analysedbyColwell&Coddington(1994)andavailableonlinewith2000a).Theindividual-basedaccumulationcurveshowsonepar-ticularrandomorderingofallindividualspooled.Theindividual-basedrarefactioncurvewascomputedbyusingtheColemanmethod(Coleman1981).Thesample-basedaccumulationcurveshowsoneparticularrandomorderingofallsamplesinthedataset.Thesample-basedrarefactioncurvewascomputedbyrepeatedre-sampling,using.Forbothsample-basedcurves,thepatchinessparameterinsetto0.8,toemphasizetheeffectofspatialaggregation. N.J.GotelliandR.K.Colwell2001BlackwellScienceLtd/CNRS Incontrast,ararefactioncurveisproducedbyrepeatedlyre-samplingthepoolofindividualsorsamples,atrandom,plottingtheaveragenumberofspeciesrepresentedby1,2,individualsorsamples(Fig.1).Samplingisgenerallydonewithoutreplacement,withineachre-samp-ling.Thus,rarefactiongeneratestheexpectednumberofspeciesinasmallcollectionofindividuals(orsamples)drawnatrandomfromthelargepoolofindividuals(orsamples;Simberloff1978).Thesetwodichotomiesjointlyde®nefourkindsoftaxonsamplingcurves,asshowninFig.1.Accumulationcurves,ineffect,movefromlefttoright,astheyarefurtherextendedbyadditionalsampling.Incontrast,rarefactioncurvesmovefromrighttoleft,asthefulldatasetisincreasingly``rare®ed''.Becausetheentirerarefactioncurvedependsuponeveryindividualorsampleinthepoolattheaccumulationcurve'sright-handend,eachindividualorsampleisequallylikelytobeincludedinthemeanrichnessvalueforanylevelofre-samplingalongtherarefactioncurve.Thecorrespondingrarefactionandaccumulationcurvesarecloselyrelatedtooneanother.Indeed,ararefactioncurve,whetherbasedonindividualsoronsamples,canbeviewedasthestatisticalexpectationofthecorrespondingaccumulationcurve,overdifferentreorder-ingsoftheindividualsorsamples.InFig.1,notethatthetwosample-basedcurvesliebelowthetwoindividual-basedcurves.Thereasonforthisnearlyuniversalpatternisthatsample-basedprotocolsaggregateindividuals,withineachsample,thatarenearbyinspaceorconsecutiveintime.Anyspatialortemporalautocorrelation(patchinessorheterogeneity)intaxonoccurrencewillcausetaxatooccurnonrandomlyamongsamples.Consequently,whenagroupofsamplesispooled,fewerspecieswillberepresentedbythoseindividualsthanbyanequalnumberofindividualscensusedrandomlyandindependentlyinthesamehabitat.AlthoughthefourkindsoftaxonsamplingcurvesinFig.1provideaunifyingframeworkformeasuringspeciesrichness,theydonotfullyconformtocurrentterminology.Sanders(1968)®rstusedindividual-basedrarefactiontocomparespeciesrichnessbenthicmarinemasscollections.Notingthatcollectionsdifferednotonlyinnumberofspeciesbutalsoinnumberofindividuals,Sanderssuggested``rarefying''eachcollectiontoacommonnumberofindividuals,tomatchthesizeofthesmallestcollection.FollowingSanders,thetermhashistoricallyreferredtoindividual-basedtaxonre-samplingcurves.Althoughsample-basedtaxonre-samplingcurvesarepreciselyanalogous,theyhaveusuallybeenreferredto,instead,as``randomized'',or``smoothed''speciesaccumulationcurves(e.g.Colwell&Coddington1994)±anequallyaccuratecharacterization,whichwedonotoppose.TherandomizedsampleaccumulationcurveofPielou's(1966,1975)``pooledquadratmethod''iseffectivelythesamemethod,althoughoriginallyintendedtobeusedintheestimationofdiversityCOMPARINGASSEMBLAGESUSINGTAXONSAMPLINGCURVESComparingspeciesorhigher-taxonrichnesswithoutrefer-encetoataxonsamplingcurveisproblematicatbest.Communitiesmaydifferinmeasuredspeciesrichnessbecauseofdifferencesinunderlyingspeciesrichness,differencesintheshapeoftherelativeabundancedistribu-tion,orbecauseofdifferencesinthenumberofindividualscountedorcollected(Denslow1995).Differencesinnumbersofindividualscountedmaythemselvesre¯ectbiologicallymeaningfulpatternsofresourceavailabilityorgrowthconditions.However,differencesinabundancemayalsore¯ectdifferencesinsamplingeffortorconditionsforcollectionorobservation.Comparingrawtaxoncountsfortwoormoreassemblageswillquitegenerallyproducemisleadingresults.Rawspeciesrichnesscountsorhighertaxoncountscanbevalidlycomparedonlywhentaxonaccumulationcurveshavereachedaclearasymptote.Forinvertebrateandmicrobialassemblageseverywhereandformanytaxaintropicalhabitats,suchasymptotesmayneverbereached(e.g.Stork1991;Woldaetal.1998;Fisher1999;Anderson&Ashe2000;Novotny&Basset2000).Fortunately,ifoneormoreaccumulationcurvesfailtoreachanasymptote,thecurvesthemselvesmayoftenbecompared,afterappropriateForindividual-baseddatasets,itisnotalwayspossibletoconstructanaccumulationcurveasinFig.1.Theorderofidenti®cationofindividualswithineachsamplemaynothavebeenrecorded,orthecollectionmayconsistofmasscaptures.Insuchcases,rarefactionproducestheonlyappropriatecurvesfordatasetcomparisons.Evenwhentheorderofindividualidenti®cationisknown(asintime-seriesdata),rarefactionproducessmoothcurvesthatfacilitatecomparison.Likewise,inthecaseofsample-baseddatasets,sampleorderisoftenunavailableorarbitrary.Repeated,averagedsample-basedrarefactionproducessmoothcurvesforcomparison,allowingstandardizationofsamplingeffort.Whethertouseindividual-basedorsample-basedrare-factiontocomparerichnessdependsuponthedataavailable.Ifthedataareinherentlyindividual-based,thereisnoalternativetousingindividual-basedrarefactiontocompareassemblages.Ifsample-baseddataareavailable,however,eithersample-basedorindividual-basedrarefac-tioncouldbeused,butitisgenerallypreferabletousethesample-basedapproach,toaccountfornaturallevelsofsampleheterogeneity(patchiness)inthedata.Forpatchy Speciesrichnessmeasurement2001BlackwellScienceLtd/CNRS distributions,individual-basedrarefactioninevitablyoveres-timatesthenumberofspecies(orhighertaxa)thatwouldhavebeenfoundwithlesseffort.Infact,thedifferencebetweenthesample-basedandindividual-basedrarefactioncurvescanbeusedasameasureofpatchiness(Colwell&Coddington1994).Regardlessofwhichapproachisused,itistheindividualthatcarriestaxonomicinformation.Whensample-basedrarefactioncurvesareusedtocomparetaxonrichnessatcomparablelevelsofsamplingeffort,thenumberoftaxashouldbeplottedasafunctionoftheaccumulatednumberindividuals,notaccumulatednumberof,becausedatasetsmaydiffersystematicallyinthemeannumberofindividualspersample.(Here,weareassumingthattaxonrichnessisthequestion,nottaxondensity;seebelow.)Anexamplemakesthispitfallclear.Supposeyouwishtoknowwhethertropicalold-growthforestornearbytropicalsecond-growthforestisricherintreespecies.Youidentifyallindividualstemsin10mrandomlyplacedquadratsineachforesttype.Thesamplerarefactioncurveforsecond-growthforest,plottedasafunctionofliesabovethecorrespondingcurveforold-growthforest,butneitherhasreachedanasymptote(Fig.2a).Themeannumberofstemsperquadratisconsiderablygreaterinthesecond-growthforest,aswouldbeexpected.Aretherereallymorespeciesinthesecond-growthforest?Notevenanapproximateanswercanbegiventothisquestionwithoutre-scalingthe-axistonumberofindividuals(basedontheaveragenumberofindividualspersample).Oncere-scaled,thesecond-growthforestcurvewilldroprelativetotheold-growthforestcurve;itmay(still)lieaboveit,coincide,orfallbelowit(Fig.2b).(Cannonetal.1998demonstratedthispitfallforloggedvs.unloggedforests,whichdifferinstemdensityandinquadrat-basedrichness,buthavesimilarspeciesrichnesswhenre-scaledtoindividuals.)Thisexampleillustratestheimportanceofusingtaxonsamplingcurvestocomparespeciesrichness,evenwhenthecomparisonsarebasedonstandardizedmethodsandidenticalsamplingprotocols.-species-listmethod(Poulsenetal.1997)suffersfromarelatedpitfall.Supposetwocommunitiesaresampledwiththismethod,onemorespecies-richthantheother,using20-specieslists.Inthepoorercommunity,foreach20-speciessample,moreindividualswillneedtobeobservedthanintherichercommunitytoreach20species.Thus,assamplesaccumulate,thepoorercommunitywillbeincreas-inglybettersampledthanthericheronebecausemoreindividualswillhavebeensampled.Infact,thisbiasmaybestrongenoughthatthecumulativenumberofspeciesrevealedinthepoorcommunityequalsorexceedsthatoftherichcommunity,forthesamenumberof20-speciessamples,aslongasbothcurvesareincreasing±aswouldoftenbethecaseforarapid-assessmentsurvey(Fig.3).Ofcourse,eventually,the20-species-sampleaccumulationcurveforbothcommunitieswillreachtheirasymptotes(thespecies-poorcommunity®rst)andthecurveswilldiverge,butthewronginferencecaneasilybemadeifbothcurvesarestillrisingwhensamplingisstopped(Fig.3).Inshort,itisperilousorimpossibletomakeavalidcomparisonbetweentwospeciesaccumulationcurvesthatarebasedonthe-species-listmethod,unlessbothcurveshavereachedanasymptote.Otherpitfallstowatchoutforapplytoindividual-basedrarefactionaswellassample-basedrarefaction.Avalidindividual-basedrarefactionanalysisassumesnotonlythatthespatialdistributionofindividualsintheenvironmentisrandom(Kobayashi1982),asdiscussedabove,butthatsamplesizesaresuf®cient,andthatassemblagesbeingcomparedhavebeensampledinthesameway(Abele&Walters1979).Ifsamplesizesarenotsuf®cient,rarefactionwillnotdistinguishbetweendifferentrichnesspatterns,becauseallrarefactioncurvestendtoconvergeatlowabundances(Tipper1979).Iftheassemblagesare Figure2Theeffectonspeciesrichnessofre-scalingthe-axisofsample-basedrarefactioncurves(randomizedspeciesaccumulationcurves)fromsamplestoindividuals,whenindividualdensitiesvary.Inthishypotheticalexample,speciesrichnessappearstobehigherforasecond-growthforeststandthanforanoldgrowthstand(a,basedoncorrespondingnumbersofaccumulatedsamples.However,stemdensityishigherinthesecond-growthstand(withsmallertrees)thanfortheold-growthstands(withlargertrees).Whenthe-axisisre-scaledtoindividuals,theresultisreversed N.J.GotelliandR.K.Colwell2001BlackwellScienceLtd/CNRS taxonomicallyverydifferent,thesamplingmaynotadequatelycharacterizeeachtaxon(Simberloff1978).Ifthesamplingmethodsarenotidentical,differentkindsofspeciesmaybeover-orunder-representedindifferentsamples,becausenosamplingmethodiscompletelyrandomandunbiased(Boulinieretal.1998).Inaddition,theshapeofindividual-basedrarefactioncurvesdependsuponrelativeabundance±thegreatertheevennessoftherelativeabundancedistribution,thesteepertherarefactioncurve(Gotelli&Graves1996).Forthisreason,rarefactioncurvesfortwocommunitieswithdifferentpatternsofrelativeabundancemaycrossonceoreventwice.Likewise,sample-basedrarefactioncancross,ifbasedoncommunitiesthatdiffersuf®cientlyinpatchiness.Thus,thesamplesizetowhichonerare®escanpotentiallychangetherankorderofestimatedrichnessamongcommunities.COMPUTINGRAREFACTIONCURVESIndividual-basedrarefactionForindividual-basedrarefactioncurves,aprecisemathe-maticalexpressionbasedoncombinatorictheorycanbecomputedforexpectedrichness,givenindividuals,insteadofactuallyre-samplingtorandomize.Sanders(1968)providedwhatwasintendedasanindividual-basedrarefac-tionformulaforcalculatingtheexpectednumberofspeciesinarandomsubsampleofindividualsfromasingle,largecollection.Althoughtheprincipleofrarefactionwassound,Sandersderivedtherarefactionformulaincorrectly(Hurl-bert1971).Thecorrectderivationisbasedonahypergeo-metricsamplingdistribution,inwhichindividualsaresampledrandomlyandwithoutreplacement(HecketalFromthismodel,boththeexpectednumberofspeciesanditsvariancecanbederived.Amathematicallydistinctbutcomputationallymuchfasterwaytoproduceindividual-basedrarefactioncurvesistocomputethecorresponding``randomplacement''curveofColeman(1981;Colemanetal.1982),whichhasbeenshowntoverycloselyapproximatethehypergeometricrarefactioncurve(Brewer&Williamson1994;Colwell&Coddington1994).Sometheoreticalprogresshasbeenmadeinmodifyingtherarefactioncurveforcasesofknownspatialdistribu-tions,suchasthenegativebinomial(Kobayashi1982,1983;etal.1985).However,theseanalysesstillassumethatindividualshavebeensampledrandomly.Inreality,ecolo-gistsrarelysampleindividualsrandomly.Instead,quadratsorsamplingdevicesareimplementedrandomly(orinstrati®edrandomdesign),andalloftheindividualsinasmallcollectionaresorted,yieldingdatasetsappropriateforsample-basedrarefaction.Sample-basedrarefactionBecausethesample-basedrarefactioncurvedependsonthespatialdistributionofindividualsaswellasthesizeandplacementofsamples(Hurlbert1990),itcannotbederivedtheoretically.Thus,computationsrequireMonteCarlore-sampling,inwhichsamplesarerandomlyaccumulatedinmanyiterations.Freesoftwareisavailable(Colwell2000a)tocomputesample-basedrarefactioncurvesaswellasthecorrespondingindividual-basedColemancurves.Mean Figure3Apitfallofthe``-specieslist''methodofcomparingspeciesrichness.Inthismethod(Poulsenetal.1997),listsofthe®rst20(orotherconstant)speciesobservedinrepeatedsamplesareaccumulated,withoutregardtothenumberofindividualsactuallyexaminedtoreach20species.Asthishypotheticalexampleshows,inaspecies-poorcommunity,moreindividualswillinevit-ablyhavetobeexaminedtoreacheachsuccessivesetof20speciesthaninaspecies-richcommunity(a).Nevertheless,assamples1,2,3,4arepooled,inthisexampleanidenticalcumulativenumberofspeciesisreachedasspeciesareplottedagainstnumberoflists(1,2,3,4)onthe-axis(asisstandardforthe-specieslistmethod)(b).Infact,theindividual-basedaccumulationcurvescouldbearrangedtoachieveavarietyofmisleadingresults,whencumulativespeciesareplottedagainstnumberoflists(samples). Speciesrichnessmeasurement2001BlackwellScienceLtd/CNRS numberofaccumulatedindividualsisalsocomputed,toallowre-scalingofsample-basedrarefactioncurves.Freesoftwareisalsoavailablefortheconstructionofindividual-basedrarefactioncurvesandcon®denceintervalsforspeciesrichnessandotherdiversityindices(Gotelli&Entsminger2001).CATEGORY-SUBCATEGORYRATIOSANDTHEIRPITFALLSIndividualsandspeciesTointroducetheconcept,andtheperils,ofwhatwecallcategory±subcategoryratios,letusreturntotheexample(above)ofassessingtreespeciesrichnessinold-growthvs.second-growthforest.Recallthattheproblemwithcomparingsample-basedrarefactioncurvesscaledbynumberofsampleswasthatsecond-growthquadratseachhadmorestemsthanequal-sizedold-growthquadrats,onaverage.Whynotsimplycompareaveragespeciesperstem,amongquadrats,foreachforesttype,toremovetheeffectofstemdensity?Thisindexisthespecies-per-individualratio,aparticularclassofcategory-subcategoryratios.Figure4illustratesthehazardsofusingthespecies-per-individualratiotocomparesamples.EachpanelinFig.4showshypothetical,sample-basedrarefactioncurvesforcontrastingforesthabitats.Eachcurveisbasedonthesamenumberofquadrats,buteachisre-scaledtothenumberofindividualsonthe-axis.Thesoliddotsindicatetotalrichnessforthepooledquadratsineachforesthabitat.Theslopesofthelinesconnectingthesepointstotheoriginequaltheratioofspeciestoindividualsforthedots.InFig.4(a),old-growthandsecond-growthforesthaveiden-ticalspeciesrichness(atleastasfarasthecurvesextend),yetthenumberofspeciesperindividualismuchlowerforthesecond-growthforest.InFig.4(b),speciesrichnessishigherinforestgapsthaninnon-gaps(forestmatrix),yetthenumberofspeciesperindividualisidenticalfortotalrichnessingapsandnon-gaps.Anexamplefromtherecentliteratureillustratestheperilsof``normalizing''richnessbydividingthenumberofspeciesbythenumberofindividuals.Insupportoftheirinferencethattreespeciesrichnessdoesnotdifferbetweengapsandnon-gaps,Hubbelletal.(1999)showedthatnumberofspeciesdividedbynumberofstemsdidnotdifferforsaplingsingapsvs.non-gapsinaPanamanianforest.UsingHubbell'sreportedstemdensitiesandrichnessvaluesforsaplingsin2020-mquadrats,Chazdonetal.(1999)showedthattruesaplingspeciesrichnessmightinfact®tcurvessuchasthoseinFig.4(b)(seealsoKobe1999;Vandermeeretal.2000),withgreatertotalrichnessingaps.Inhisreply,Hubbell(1999)failedtoprovidetheindividual-basedspeciesaccumulationcurvestodisproveChazdon'sconjectureforthesaplingdatasetatissue.Instead,Hubbelletal.(1999)providedindividual-basedaccumulationcurvesforaquitedifferentdataset(nosizeclassspeci®ed)andcitedthefactthat-basedaccumulationcurvesdonotdifferforgapsandnon-gaps,leavingthedebateunresolved.Ourpointhereissimplythat,hadindividual-basedaccumulationcurvesbeenpublishedforthesaplingdatasetatissueinthe®rstplace,theambiguitythatinstigatedthedebatewouldneverhavearisen.Usingthespecies-per-individualratiotocorrectforunequalnumbersofindividualsisinvalidbecauseitassumesthatrichnessincreaseslinearlywithabundance±trueonlyfortheidealizedcaseofextremeunevenness,inwhichonespeciesismaximallydominant(Gotelli&Graves1996).Becauseabundancesarerarelythisextreme,thespecies-per-abundanceratiowilldistortpatternsofspeciesrichness. Figure4Pitfallsofusingspecies/individualratiostocomparedatasets.In(a),anold-growthandasecond-growthforeststandarecompared.The2standshaveidenticalindividual-scaledrarefactioncurves,andthusdonotdifferinspeciesrichness.Thesecondgrowthcurveextendsfarthersimplybecausestemdensityisgreater,sothatmoreindividualshavebeenexaminedforthesamenumberofsamples.However,whentheratioofspecies/individualiscomputedforeach,theratioismuchhigherfortheold-growthstand.In(b),speciesrichnessintreefallgapquadratsiscomparedwithrichnessinnon-gap(forestmatrix)quadrats.Inthiscase,species/individualratiosareidentical,yetthetruespeciesrichnessishigheringaps. N.J.GotelliandR.K.Colwell2001BlackwellScienceLtd/CNRS Thesameproblemaffectstheinverseratio,individualsperspecies(e.g.Irwin1997;histable4.1),whichCodding-ton(Coddingtonetal.1991,1996;Silva&Coddington1996)suggestedasameasureof``samplingintensity.''Forcommunitiesthataremoreorlessequivalentintotalrichnessandrelativeabundancepatterns,thenumberofindividualsperspecies(totalnumberofindividualsdividedbytotalnumberofspecies)isindeedadecentruleofthumbforrelativecompletenessofinventories.However,compu-tingthisratiocanbeamisleadingwayto``standardize''samplingeffortwhencomparingcommunitiesdifferingconsiderablyinrichness,orforwhichcomparativerichnessisunknown.Forexample,inFig.4(b),thenumberofindividualsperspeciesisthesameforbothdatasets,yetsamplingisobviouslymorecompleteforthegaphabitats.SpeciesandgeneraInbiogeography,category±subcategorytaxonomicratioshavebeenrepeatedlyusedandabused.Thebestknownoftheseisthespecies±genusratio,butfamily±orderratiosoranyotherlower-taxon±higher-taxonratiosaresubjecttothesamepitfalls.Figure5depictssample-basedspeciesandgenusrarefactioncurvesforahummingbirddataset(Colwell2000b).Asthenumberofsamplesincreases,thenumberofgenerareachesanasymptotesoonerthanthenumberofspecies.Thispatternisinevitableforanytwotaxonomicranks(exceptintheunrealisticcaseof100%monobasictaxa)sincethehigherrank(thegenus,inthisexample)inevitablyhasfewermembersthanthelowerrank(species).Becauseofthisrelationship,aplotofthenumberofsubtaxapertaxon(speciespergenus,inthisexample±theuppercurveinFig.5)alwayshasapositiveslope,asseeninFig.5.Thespecies-to-genusratiohaslongbeenusedtodescribecommunitypatternsandtoinferlevelsofcompetitiveinteractionsamongspecieswithingenera(reviewsinSimberloff1970;Jarvinen1982).Similarreasoninghasbeenappliedtotheinterpretationofspecies±familyandothertaxonomicratios(e.g.MacArthur&Wilson1967;Cook1969).Alowspecies-to-genusratiowasinterpretedasaproductofstrongintragenericcompetition(Elton1946),whichmightlimitcongenericcoexistence(Darwin1859).Consistentwiththishypothesiswasthewidespreadobser-vationthatspecies-to-genusratioswereusuallysmallerforislandthanmainlandcommunities(Elton1946).However,subtaxon±taxonratiosareanincreasingfunctionofsamplesize,andwouldbeexpectedtodecreaseinsmallcommu-nities,regardlessofthelevelofcompetition(Williams1947,1964;Simberloff1970,1972).Figure6showsthiseffectgraphicallybyre-plottingthedataofFig.5asnumberofspeciesasafunctionofnumberofgenera.Theslopeofthediagonalbrokenlinesshowsthat,inasmallrandomsample(fewspecies),therearefewerspeciespergenusthaninalargerandomsample(morespecies).Sample-sizedependenceintaxonomicratioswas®rstdemonstratedforplantcommunitiesbyMaillefer(1929),whouseddrawsofspeciesfromadeckofshuf¯edcardstocalculatetheexpectedgenericrichnessinsmall Figure5Taxonsamplingcurvesforspeciesandforthegeneratowhichtheybelong,withthespecies±genusratio.Notethatthecurveforgenerareachesitsasymptoteatasmallernumberofsamplesthanthespeciescurve.Forthisreason,theratioofspeciestogeneraisnonlinear.Thispatternsisinevitableforanycaseofcategory-subcategorysamplingcurves.Thecurvesarebasedonasampleofhummingbirdspecimens(Colwell2000b;appendixB,pooled,distributedinto``samples''atrandom,andthenrepeatedlyre-sampledusing(Colwell2000a). Figure6Thespecies-per-genuspitfall.Thesolid-linecurveplotsnumberofspeciesasafunctionofnumberofgeneraforthehummingbirddataofFig.5.Becausetherelationshipisnonlinearwithanincreasingslope,thespecies±genusratio(theslopeofthebrokenlines)isgreaterforalargersampleofspeciesthanforasmallersample. Speciesrichnessmeasurement2001BlackwellScienceLtd/CNRS communities.Foranimalcommunities,Williams(1947,1964)elucidatedthesesamepatternsusingspecies-abun-dancemodelsandcomputersimulations.Althoughtheirworkwasignoredbyecologistsforseveraldecades(Jarvinen1982),re-analysesofspecies-to-genusratiosnowsuggestthatislandcommunitiesharbourslightlyspeciespergenusthanexpectedbychance,inspiteofthelowerabsolutenumberofspeciespergenusexpectedinsmallersamples(Simberloff1970).This®ndingistheoppositeofwhatcompetitiontheorypredicts,perhapsre¯ectinginsteadthesimilardispersalpotentialandecologicalrequirementsofcongenericspecies(theIcarusEffectofColwell&Winkler1984).Despitetheperiodicrediscoveryofthisclassicpitfall,sample-sizedependenceoftaxonomicratioscontinuestotraptheunwary(e.g.Ashton1998).SPECIESRICHNESSVS.SPECIESDENSITYWehaveemphasizedtheimportanceofusingtaxonsamplingcurves(bothindividual-andsample-based)tostandardizedatasetstoacommonnumberofindividualsforthepurposesofcomparingspeciesrichness.Incontrast,mostcommunityecologystudiesstandardizeonthebasisofareaorsamplingeffort.Thus,mostecologicalcomparisonsofbiodiversityareactuallycomparisonsofspeciesdensity:thenumberofspeciesperunitarea(Simpson1964).Suchstudieshingeontheassumptionthatsamplesaredrawnfrompopulationsofindividualsthatareatcomparabledensities.However,speciesdensitydependsonbothspeciesrichnessandonthemeandensityofindividuals(disregard-ingspecies),asdiscussedinrelationtotheexampleofold-growthvs.second-growthforestabove(Fig.2).Conse-quently,theorderingofcommunitiesmaydifferwhenrankedbyspeciesrichnessvs.speciesdensity(James&Wamer1982;McCabe&Gotelli2000).Bothspeciesrichnessandspeciesdensitycanbecomparedusingsample-andindividual-basedrarefactioncurves(Fig.7).Individual-basedrarefactioncurvesstan-dardizeeachoftwoormoresamplesonthebasisofthenumberofindividuals,forthepurposeofcomparingspeciesrichness.Sample-basedrarefactioncurvescanbeusedtocomparerichnessinthesameway,aslongasthe-axisisre-scaledinunitsofindividuals.Incontrast,tocomparespeciesdensitywhensamplesarederivedfromincommen-surateareas,the-axisofindividual-basedrarefactioncurvescanberescaledfromindividualstoarea,basedonaveragedensity.Likewise,ifsample-basedrarefactioncurvesaresimplyleftscaledbynumberofaccumulatedsamples(insteadofre-scalingtoindividuals),thencomparisonsamongdatasetswillbeintermsofspeciesdensity,insteadofspeciesrichness(Fig.7),assumingsamplesarespace-based.Incomparisonsofspeciesdensity,afamiliarpitfallawaitstheunwary,butinanewguise.Foraconstantdensity,areaisaproxyfornumberofindividuals.Thus,``normalizing''speciesdensitydataoftwounequalareasbydividingthenumberofspeciesbytheareameasuredissubjecttotheverysamepitfallsasthespecies-per-individualratio,asshowninFig.4.Althoughitsoundsparadoxical,theratioofrichness Figure7Speciesrichnessvs.speciesdensity.Part(a)showsindividual-basedrarefactioncurvesfortwocontrastingsamples,whereas(b)showssample-basedrarefactioncurvesfortwocontrastingdatasets.A1andA2indicatetheasymptoticrichnessforthetwocurvesineachpanel.In(a),rawspeciestotalsforthetwosamples(blackdots)measurespecies(D1andD2±assumingeachsamplecoversthesamearea),whereasSample2mustberare®edtothesamenumberofindividualsasSample1(theopendot)toallowavalidcomparisonof(R2vs.R1).In(b),rawspeciestotalsforthetwodatasets(blackdots)measuretotalspecies(T1andT2)forthedatasets.Assumingeachsamplecoversthesameamountofspace,Dataset2mustberare®edtothesamenumberofsamplesasDataset1(theopendot)toallowavalidcomparisonofspecies(D2vs.D1).(Foravalidcom-parisonofrichnessbetweenthetwodatasetsinthelowerpanel,the-axiswouldhavetobere-scaledtoindividuals,asinFigs2and4.) N.J.GotelliandR.K.Colwell2001BlackwellScienceLtd/CNRS toareaisnotavalidmeasureofspeciesdensity,becausethenumberofspeciesincreasesnonlinearlywitharea.Instead,speciesdensityisvalidlycomparedonlywiththeappropriatetaxonsamplingcurves(e.g.James&Wamer1982).Whichmeasureismoreappropriate,speciesrichnessorspeciesdensity?Inotherwords,shouldcommunitiesbecomparedonthebasisofastandardizednumberofindividuals(speciesrichness)orastandardizedareaorsamplingunit(speciesdensity)?Forconservationpurposesandappliedproblemsthatfocusonlargeareas,speciesdensityisprobablyofmoreinterestbecauseitmeasuresthenumberofspecieswithinaspeci®edarea.However,rarefactionshouldneverthelessbeusedwhencomparingspeciesdensityindifferentregions,toassessthedegreetowhichdifferencesinspeciesdensitycanbeattributedtopatternsofindividualabundance(whichdeterminesthepositionofthecommunityonthe-axisofthespeciesaccumulationcurve),andhowmuchcanbeattributedtotheshapeandmagnitudeofthespeciesaccumulationcurve(whichdeterminesthespeciesrichnessachievedataparticularlevelofindividualabundance).Ontheotherhand,fortestingmodelsandevaluatingtheoreticalpredictionsinecology,speciesrichnessmaybemoreappropriate.Mosttheoreticalmodelsincommunityecologydonotcontainexplicittermsforareaordensity.Instead,thecurrencyofthesemodelsisabundance()andpopulationgrowthrates(d),whicharemodi®edbycoef®cientsthatdescribeinteractionswithotherspecies(Gotelli2001).Percapitainteractionsmaybeexpressedinsamplesthatarebasedoncommonnumbersofindividuals,whichishowspeciesrichnessismeasuredwithindividual-basedrarefactionorwithsample-basedrarefactionscaledtoindividuals.Westressthatneitherspeciesdensitynorspeciesrichnessisnecessarilythe``correct''waytomeasurediversity,butthatpatternsofdiversitywillbeverysensitivetowhichmeasureisused.Conservationdecisionsmaybecomplicatedwhensomereservesorcandidateareascontainhigherspeciesdensityandotherscontainhigherspeciesrichness.Disturbanceormanagementregimesthataffectabundancemighthavetobeconsideredinchoosingamongsuchareas.Recentstudiesofspeciesrichnessandspeciesdensitypatternshaveledtoare-evaluationofsomefamiliarpatternsofdiversityinnaturalcommunities.Forexample,manyexperimentalandcorrelativestudieshavedocumentedthatdisturbancesreducethediversityofbenthicinvertebrateassemblagesinstreams(Lake1990;Vinson&Hawkins1998).However,mostofthesestudieshavequanti®edspeciesdiversityasspeciesdensity,thenumberofspeciesperunitarea.Becauseecologicaldisturbancesreduceabundance,wewouldexpectdisturbancetodecreasespeciesdensity,simplybecausetherewillbefewerindividualspresenttobesampledafteradisturbance.InanexperimentalstudyofnorthernU.S.streamassemblages,McCabe&Gotelli(2000)manipulatedthearea,intensity,andfrequencyofdisturbanceonarti®cialsubstratesinanorthogonal3-waydesign.Macroinverte-brateswerecollectedfromsubstratesurfacesafter6weeksoftreatmentapplication.Speciesdensity(numberoftaxapersample)wassigni®cantlyreducedinalldisturbancetreatmentscomparedtounmanipulatedcontrols.However,whenthetreatmentswerecomparedbyindividual-basedrarefaction,thepatternswerecompletelyreversed:fora®xednumberofindividuals,taxonrichnesswashigherinalldisturbancetreatmentsthanintheundisturbedcontrols(Fig.8).Thisexampledemonstratestheimportanceofusingthespeciesaccumulationcurvetocarefullyquantifytaxonrichness±eveninexperimentalstudiesinwhichsamplingeffortiscarefullystandardized.Similarly,plantecologistshaverepeatedlymadetheerrorofcomparingrichnessperquadrat(speciesdensity)amongstandsdifferinginoverallplantdensity(Fig.2).Thesecomparisonshaveconfoundedorequateddifferencesindensitywiththedifferencesindisturbance,successional,orproductivityregimesthatarebeingcompared.Asdiscussedabove,attemptingtocorrectforthiserrorbycomputingspeciesperstem(Fig.4)leadstothesamepitfallascomputingspeciespergenusforsamplesdifferinginnumbersofspecies(Fig.6).Forexample,afrequentecologicalpatternisthehump-shapeddiversitycurve,inwhichspeciesrichnesspeaksatintermediateproductivitylevels(DiTomasso&Aarssen1989;Rosenzweig&Abramsky1993).Manymodelsofplantcompetitionassumethatmortalityisnotequalamongspecies,sothatinterspeci®ccompetitionleadstospecieslossesathighlevelsoffertility(Tilman1982,1988;Huston&DeAngelis1994;butseeAbrams1995).However,theassemblage-levelthinninghypothesis1996)accountsforthehump-shapeddiversitycurvebyvariationintotalplantdensity.Asfertilityincreases,individualsgetlarger,crowdingoccurs,anddensity(numberofplants/area)decreases.Therefore,rarespeciesare``lost''athighdensitiesbecausetheyarerepresentedbyfewindividuals,notbecauseofdifferentialmortalityorinter-speci®cdifferencesincompetitiveability.Totesttheassemblage-levelthinninghypothesis,Stevens&Carson(1999)establishedanexperimentalproductivitygradientin1-year-old®eldsinthenorth-easternU.S.Theyfoundthatbothspeciesnumberanddensityofherbaceousplantsdeclinedathighfertility.Asimulationmodelsimilartoindividual-basedrarefactionestablishedthatrandomsurviv-orshipofindividualscouldlargelyaccountforthedeclineindiversityathighproductivity.Althoughmanyplantassem-blagesarecharacterizedbystrongpairwisecompetitiveinteractions(Shipley1993),netcompetitiveeffectsinmultispeciescommunitiesmaybeweak(Miller1994),and Speciesrichnessmeasurement2001BlackwellScienceLtd/CNRS simplechangesindensitymaybetheprimarydeterminantofspeciesrichnessacrossproductivitygradients.ASYMPTOTICESTIMATORSOFSPECIESRICHNESSEstimatesofasymptoticspeciesrichnessmaybeespeciallyimportantinbioticinventoriesandsurveys,whereitisimpracticaltoexhaustivelysamplespeciesrichcommunities,suchastropicalinvertebrate,microbialorplantcommuni-ties(e.g.Cannonetal.1998;Fisher1999;Novotny&Basset2000).Rarefaction(eitherindividual-basedorsample-based)isamethodforinterpolatingtosmallersamplesandestimatingspeciesrichnessintherisingpartofthetaxonsamplingcurve.However,rarefactioncannotbeusedforextrapolation;itdoesnotprovideanestimateofasymptoticrichness(Tipper1979).Statisticalstudieshaveproducedalargenumberofestimatorsoftheasymptoticnumberof``classes''forsamplesofclassi®edobjects(reviewedbyBunge&Fitzpatrick1993),ofwhichspeciesrichnessisoneexample.Themostpromisingofthesearenonparametricestimatorsbasedonmarkandrecapturestatistics(Colwell&Coddington1994;Nichols&Conroy1996;Boulinieretal.1998;Chazdonetal1998;Colwell2000a).Thenonparametricestimatorsuseinformationonthedistributionofrarespeciesintheassemblage±thoserepresentedbyonlyone(singletons),two(doubletons)orafewindividuals.Thegreaterthenumberofrarespeciesinadataset,themorelikelyitisthatotherspeciesarepresentthatwerenotrepresentedinthedataset.Inaddition,asymptotic(andnonasymptotic)richnessmaybeestimatedbycurve-®ttingextrapolationmethods(e.g.Palmer1990;Lamasetal.1991;SoberoÂn&Llorente1993;Mawdsley1996;Keating&Quinn1998;Fisher1999).Althoughextrapolationisinherentlymoreriskythaninterpolation,someoftheseasymptoticestimatorshavesofarperformedwellwhentestedonexhaustivelycensused,benchmarkdatasetsinwhichthespeciessamplingcurvereachesastableasymptote[suchasthetropicalseedbankdatasetofButler&Chazdon1998(analysedbyColwell&Coddington1994)ortheparasitedataofWaltheretal1995].Arichnessestimatoristestedonsuchabenchmarkdatasetbycomputingthesample-basedrarefactioncurve,thencomputingtheestimatorforeachcumulativelevelofsamplepooling,followingPielou(1966,1975;Colwell&Coddington1994).Byrepeatingthecomputationsforalllevelsofsampleaccumulation,acontinuousplotoftheestimatorcanbedisplayedalongwiththesample-basedrarefaction.Resamplingandrecomputingtheestimatorsrepeatedlyandtakingmeansproducessmoothcurves.Anidealestimatorwould(1)reachitsasymptotemuchsoonerthanthesample-basedrarefactioncurvelevelsoff,and(2)approximatetheempiricalasymptoteinanunbiasedway,whentestedovermanybenchmarkdatasets(Anderson&Ashe2000providenumerousexamplesfortropicalbeetles).Ofcourse,asidefromtestingestimators,thereisnoreasontouseanestimatorforadatasetthatreachesasteadyasymptote.Thedatasetsthatneedrichnessestimatorsarethosethat,asyet,arenowherenearanasymptote,suchas 3.006.009.0012.00Disturbance regime 8.0010.0012.0014.0016.00Disturbance regimeAdjusted Species RichnessFigure8Contrastingresultsforspeciesdensityvs.speciesrichnessinassessingpatternsofresponsetodisturbanceamongaquaticinvertebrateassemblages.Eachopenbaristheaveragediversityinoneofeightexperimentaldisturbanceregimes,andthesolidbaristheaveragediversityinunmanipulatedcontrols(C)(7repli-cates/treatment).Theeightregimesarederivedfromafullycrossedthree-factorexperimentwithtwolevelsofdisturbancefrequency(oneortwodisturbances/week),disturbancearea(50%or100%),anddisturbanceintensity(lightvs.heavyscraping)appliedtoarti®cialsubstratesinaVermontstream;(a)showstheconventionalmeasureofspeciesdensity(speciesnumber/sample);(b)showsthesamedata,buttheresponsevariablehasbeencalculatedfromanindividual-basedrarefactioncurveconstructedforeachreplicatethenstandardizedtoacommonnumberofrandomlysubsampledindividuals.Inbothanalyses,treatmentmeansdiffersigni®cantlyby0.01).However,thepatternsofdiversityareoppositeforspeciesdensityvs.speciesrichnessmeasures.Figureadaptedandsimpli®edfromMcCabe&Gotelli(2000). N.J.GotelliandR.K.Colwell2001BlackwellScienceLtd/CNRS mosttropicalarthropoddatasets(e.g.Stork1991;Woldaetal.1998;Fisher1999;Novotny&Basset2000).Thetrickyissueiswhethertheperformanceoftheestimatorsonbenchmarkdatasets±whichusuallyconsistofrelativelysmallnumbersofspecies±accuratelypredictstheperform-anceofthesameestimatorsonnot-yet-asymptoticdatasets,whichusuallyconsistofverylargenumbersofspecies.Oneindicationofthefailureoftheexistingcatalogueofestimatorsforhyperdiversetaxaisthattheyoftenfailtoreachanyasymptoteatall,risingmoreorlessinparallelwiththestill-steepsample-basedrarefactioncurve(e.g.Fisher1999).Inthesecases,theestimatorsmustbeviewedasprovidingonlylower-boundestimatesofspeciesrichness(AnneChao,personalcommunication).Ontheotherhand,restrictingdatasetstoecologicallymorehomogenoussubsetsofsamplessometimesdoesproducewell-behaved,asymptoticrichnessestimates(J.Longinoetal.,inpress).Thisisstillanongoingareaofresearch,andthereismuchneedforcomparativestudiesoftheperformanceofasymptoticspeciesestimatorsondifferentempiricalandtheoreticallyderiveddatasets.Theprinciplesofspeciesaccumulation,rarefaction,speciesrichness,andspeciesdensityhavebeenestablishedformanydecades.However,ecologistshaveonlyrecentlybeguninearnesttoincorporatetheseconceptsintotheirmeasure-mentsofspeciesdiversitypatternsandevaluationoftheoryincommunityecologyandbiogeography.Thesetasksareespeciallyimportantasecologistsattempttoinventoryspecies-richcommunitiesanddocumentthelossofspeciesdiversityfromhabitatdestructionandglobalclimatechange.Ecologistsmayhaveavoidedindividual-basedandsample-basedrarefactioncurvesbecausetheyarecomputationallyintensive,butpublic-domainsoftwareisnowavailableforthesecalculations(Colwell2000a;Gotelli&EntsmingerWethankJ.Groverforinvitingustowritethisreview.EcoSimsoftwaredevelopmentsupportedbyNSFgrantsBIR-9612109andDBI9725930toNJG.EstimateSsoft-waredevelopmentsupportedbyNSFgrantsBSR-9025024,DEB-9401069andDEB-9706976toRKC.PreparationofthispaperwassupportedbyNSFgrantDEB-0072702toAbele,L.G.&Walters,K.(1979).Thestability-timehypothesis:Reevaluationofthedata.Am.Naturalist,114,559±568.Abrams,P.(1995).Monotonicorunimodaldiversity-productivitygradients:whatdoescompetitiontheorypredict?,76,Anderson,R.S.&Ashe,J.S.(2000).LeaflitterinhabitingbeetlesassurrogatesforestablishingprioritiesforconservationofselectedtropicalmontanecloudforestsinHonduras,CentralAmerica(Coleoptera;Staphilidae,Curculionidae).BiodiversityConservation9,617±653.Ashton,P.S.(1998).Nichespeci®cityamongtropicaltrees:aquestionofscales.In:DynamicsofTropicalCommunities,edsNewberyD.M.,BrownN.&PrinsH.H.T.BESSymposium,Vol.37,pp.491±514.BlackwellScience,Oxford,U.K.Boulinier,T.,Nichols,J.D.,Sauer,J.R.,Hines,J.E.&Pollock,K.H.(1998).Estimatingspeciesrichness:theimportanceofhetero-geneityinspeciesdetectability.,79,1018±1028.Brewer,A.&Williamson,M.(1994).Anewrelationshipforrar-BiodiversityConservation,3,373±379.Bunge,J.&Fitzpatrick,M.(1993).Estimatingthenumberofspecies;areview.J.Am.Statist.Assoc.,88,364±373.Butler,B.J.&Chazdon,R.L.(1998).Speciesrichness,spatialvariation,andabundanceofthesoilseedbankofasecondarytropicalrainforest.,30,214±222.Cannon,C.H.,Peart,D.R.&Leighton,M.(1998).TreespeciesdiversityofcommerciallyloggedBorneanrainforest.,281,Chazdon,R.L.,Colwell,R.K.&Denslow,J.S.(1999).Tropicaltreerichnessandresource-basedniches.,285,1459ahttp://Chazdon,R.L.,Colwell,R.K.,Denslow,J.S.&Guariguata,M.R.(1998).StatisticalmethodsforestimatingspeciesrichnessofwoodyregenerationinprimaryandsecondaryrainforestsofNECostaRica.In:ForestBiodiversityResearch,MonitoringandModeling:ConceptualBackgroundandOldWorldCaseStudies,edsDallmeierF.&ComiskeyJ.A.),pp.285±309.ParthenonPublishing,Paris,Clench,H.(1979).Howtomakeregionallistsofbutter¯ies:SomeJournalLepidopterist'sSociety,33,216±231.Coddington,J.A.,Griswold,C.E.,SilvaDaÂvila,D.,Penaranda,E.&Larcher,S.F.(1991).Designingandtestingsamplingprotocolstoestimatebiodiversityintropicalecosystems.In:TheUnityofEvolutionaryBiology:ProceedingsoftheFourthInternationalCongressofSystematicandEvolutionaryBiology,ed.DudleyE.C.,pp.44±60.DioscoridesPress,Portland,Oregon,U.S.A.Coddington,J.A.,Young,L.H.&Coyle,F.A.(1996).EstimatingspiderspeciesrichnessinasouthernAppalachiancovehard-woodforest.J.Arachnol.,24,111±128.Coleman,B.D.(1981).Onrandomplacementandspecies±areaMathematBiosciences,54,191±215.Coleman,B.D.,Mares,M.A.,Willig,M.R.&Hsieh,Y.-H.(1982).Randomness,area,andspeciesrichness.,63,Colwell,R.K.(2000a).EstimateS:StatisticalEstimationofSpeciesRich-nessandSharedSpeciesfromSamples(SoftwareandUser'sGuide),Version6.http://viceroy.eeb.uconn.edu/estimatesColwell,R.K.(2000b).Rensch'sRulecrossestheline:Convergentallometryofsexualsizedimorphisminhummingbirdsand¯owermites.Am.Naturalist,156,495±510.Colwell,R.K.&Coddington,J.A.(1994).Estimatingterrestrialbiodiversitythroughextrapolation.Phil.Trans.RoyalSoc.London,345,101±118. 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