An Index Of Diversity And The Relation Of Certain Concepts .

394ROBERT P. McINTOSHhas categorized the causes of departure fromhomogeneity, i.e., heterogeneity. Some animalecologists have considered diversity as a criterionof homogeneity (Hairston 1959, MacArthur1960). MacArthur's (1957) model of diversityin which the densities of the species are assumedto be randomly distributed among the species hasbeen asserted to fit some animal communities(Kohn 1959, MacArthur 1960).Others havefound animal groups which do not fit the model(Hairston 1964, King 1964).MacArthur andKohn state that increasing the homogeneity ofthe sample results in a closer fit of the data toMacArthur's model. Hairston (1959) and Turner (1961), however, found that increased heterogeneity resulted in a closer fit to MacArthur'smodel. King (1964) attempts to reconcile thiscontradiction. Whittaker (1964) found that plantspecies in a variety of vegetation types did notfit MacArthur's model. The model therefore issuggested as an indication of homogeneity in anatural community, and departures from it arepresumed to be caused by heterogeneity in thecommunity.Hutchinson (1958) applied the term "homogeneously diverse" to an area where the scale ofenvironmental variation is small relative to themovements of the animal concerned and "heterogeneously diverse" to an area which includesdifferencesin habitat, cf. woodland and pasture.King (1964) notes that MacArthur's model isapplicable only in a homogeneously diverse area.However, there is no indication that any test ofhomogeneity has been applied in the variouspapers cited above other than the fit to MacArthur's model. Although animal ecologists havestudied the distribution of individual species, likethe plant ecologists they rarely apply statisticaltests of homogeneityin communitystudies. Hairston (1964) assesses homogeneityon the ubiquityof distributionof bird species in a number of subjectively recognized vegetational habitats. Thisis reminiscentof the methods of many plant ecologists.There is a widespread view that organisms aredistributed in nature in aggregations or communities which are homogeneous within themselvesand heterogeneous between two or more suchaggregations (Goodall 1954). Some plant ecologists have elevated the recognition of uniform orhomogeneous plant community to an art (Dahl1956, Becking 1957).Some animal ecologists,tacitly at least, agree in recognizing a homogeneous community having statistical propertieswhich change if heterogeneous material is includedfrom another community (Hairston 1959, King1964).Richness and diversity have commonlyEcology, Vol. 48, No. 3been describedas a characteristicpropertyof aputativehomogeneouscommunityand indicativeof its organization. Lambert and Dale (1965),on the other hand, commentthat homogeneityhas littleuse in ecological studies. They preferto regard vegetationas a heterogeneoussystemand to reduce the heterogeneityby arbitrarymethodsto statisticallydefinedlevels acceptablefor the purpose of a particularstudy. A technique fordoingthishas been proposed (Williamsand Lambert 1959) which seeks to identifyhomogeneousgroupsof samplesin eciesare absent.In small areas it has given interestingresults.However, it is a monotheticmethodinvolvingarigid series of successive divisions. Beckner(1959) pointsout that the possibilityof errorisgreatin such methods. He notesalso thatmonotheticmethodsdo not yield natural groups althoughtheymay produceusefuland clear classifications. A furtherproblemof the techniqueisthat the sequence of choices and hence the resultantgroups will be markedlyaffectedby thesize of the sample used (Kershaw 1961, Austin,personal communication).It is essential to keep in mind that the termhomogeneityis commonlyused by ecologistsintwo senses (Greig-Smith1964): (1) It is applied,as notedabove, to the spatial distributionof species in a singleplot of a communityor in a standin the sense of the plant(the concretecommunityecologist). (2) It mayalso applyto a comparisonof data derivedfromseveralseparatecommunitiesor standswithno referenceto spatialpattern. Inthe conceptof stand or communityunit thereiscommonlyan assumptionof a considerabledegreeof homogeneityin the firstor spatialsense. Standsometimesconnotesthe idea that it is a replicateof a largercommunityrepresentedby an assemblage of individualstands which are similar toeach other, i.e., which are homogeneousin thesecond sense. It is difficultto escape fromthehistoricalimplicationsof thetermscommunityandstand. King (1964), for example, attemptstoavoid confusioninherentin the word communityibysubstitutingassociation,an excellentexampleof goingfromfryingpan to fire. Klopfer(1962)applies the termcommunityonlyto sociallyinteractinglpopulationsof animals,a mostpeculiarandrestrictedusage. Lloyd and Ghelardi (1964) referto a "latch" as an area whose scale coincideswiththe scale of movementsof the animals. Thisis similar to Hutchinson's "homogeneouslydiverse" area but seems mostdescriptiveof a population distribution. Hairston (1964) rigorouslyavoids definitionof community,or even the implicationof a definition,consideringonly coexis-

Late Spring 1967INDEXOF DIVERSITY395and suggestingtence of species in place and time. He deduces above, usingan index of similaritycommunityproperties"from a considerationof an index of diversityderivedfromit.groups of species simultaneously." Some plantDISTANCEAS A MEASUREOF SIMILARITYecologists (Lambert and Dale 1965) prefertoOR HOMOGENEITYignorethe stand concept,and its accumulationofassumptions,and deal only with an area whichDuring the past decade therehas been increastheytermthe "site," withoutprejudice as to the ing interestamong plant and animal ecologistsinhabitat or the homogeneityof the vegetation quantitativemethodsof expressingthe similaritywhich is determinedsecondarilyfromthe data. of samplesof organisms. The earlierwidespreadThis use of site is in conflictwith its widelyac- assumptionof mostplantecologiststhata limitedcepted connotationof habitat of forestersand number of distinctcommunitiesor associationsmanyecologists. The foregoingmay appear as a characterizedthe vegetationof a regionhas beendigression,but in factthe core of manyproblems used as the basis for a frameworkfor the studyis foundin the lack of a clear and consistentter- of animal communities(Gause 1936, Park 1941,in thesecond Dice 1952). Recent interestof animal ecologistsminology.In anyevent,homogeneitysense is a statementof similarityof species com- in the animal communityhas, however,coincidedposition of a group of communities,stands, or withthedevelopmentof manyquestionsabout thesamplesof ensemblesof organisms. It is termed nature and definitionof the plant communityby Dahl (1956, 1960). Stands in- (Watt 1964,Poore 1964). Both plantand animalhomotoneityternallyheterogeneousin the firstor spatial sense ecologistshave developed and used quantitativecould comprisea homogeneousgroup in this sec- expressionsof the similaritybetweensamples orond sense. A numberof quantitativeindices of communities(Odum 1950, Whittaker1952, Macof a numberof samplesor standsof fayden1954, Koch 1957, Bray and Curtis 1957,homogeneityhave been proposed (Bray and Cur- Newbould1960,Martin1960,King 1962). Thesea communitytis 1957, Curtis 1959, Dahl 1960). Dahl has are measures of floristicor faunisticsimilarityshown these to be predictable from Williams which may be weightedby the quantitiesof the(1964) index of diversityand suggests a new componentspecies. These measures of a set of"index of uniformity"(homogeneity) which is communitysamples may be combinedin varioustheratioof themean numberof speciesper sample ways to assess the similaritywithinthe set (Mac(richness) to the Williams (1964) index of fayden1957,Greig-Smith1964).diversity.Austinand Orloci (1966) have arguedon theoSimilarityhas been used by plant ecologiststo reticalgrounds that the distance measure is thesamplesor standsof vegetation. soundestmeasureof ecologicalsimilaritycomparedifferentof standsEssentiallyit is a statementof the identityof spe- or samples. Distance has been discussed as acies, or quantitiesthereof,in two or more stands. measure of taxonomicdistanceby Sokal (1961)A numberof indiceshave been devisedto measure and Sokal and Sneath (1963). The termdistancethe similarityof stands (Whittaker 1952, Bray does not referto a spatial relationin nature. Itand Curtis 1957, Austin and Orloci 1966). In is a measure of the ecological relationshipsugthese,all possible stand pairs of a set are com- gested by the resemblanceor similarityof twopared witheach otherand the resultantmatrixof communitiesor samples thereof. The distancevalues used to assess the similarityof the stands. betweentwo communitiesis the square rootof thebetweenthe meaGoodall (1963) describestheresultin a geometric sum of the squared differencesmodelas a dispersionof points (stands) in multi- sures of each species. The distancebetweentwodimensionalspace. The more similarthe stands standsj and h is calculatedby the formulathe tighterthe cluster, the more differenttheJ(Xij -Xih)2.standsthelooserthe cluster. Further,if theclusDijhi lindicatesalackofinteritter is hypersphericalspecificcorrelationwhichis also an indicationof X is themeasureof theith species in standsj andhomogeneity.Similarityis, in fact,identicalwith h respectively;S is the numberof species. Twohomogeneityof a group of stands or samples in stands in whichthreespecies are representedapspace. Thethe second sense describedabove. In a spatialor pear as pointsin a three-dimensionalis equatedwithprox- formulais equally valid beyondthreedimensionsgeometricalmodelsimilarityis equal to in an n-dimensionalspace (hyperspace). Eachimity,hence its antonym,difference,species is theoreticallyrepresentedby an axis indistance.The similarityof aathesuchistohypotheticalspace.ofthisThe purposeexplorepaperrelations of diversityand similarity,as defined set of stands is representedby the matrixof dis-

396ROBERT P. McINTOSHtance values betweenthe stands. If stands havethe same species in equal amountstheyare identical; distancebetweenthemis zero. Methodsofthecontentof such matricesare consummarizingsidered by Orloci (1965).Ecology, Vol. 48, No. 3one species has the maximumpossiblenumberofindividuals,all theresthavingonlyone. If S1,the index value equals N whichis the maximumpossiblevalue. The minimumpossible value forof N and S is:any combinationAN INDEX OF DIVERSITYS (- )SUSAny sample of a communitycan be identifiedas a pointin space, real or imaginary. This point This representsthe case of completeequitability,is given byall species having equal numbersof individuals.If SN, the index value equals VN, the minimumpossiblevalue. Table 1 gives the maximum-\!-1In this and subsequentformulaen the numberof individualsof a single species and S theTABLE 1. Maximumand minimumindex.\/I n,2 fornumberof species. This value can be regardedasa sample of 100 individualswith given numberof speciesthe distancevalue of the sample froman area ofbare ground with zero individuals. It does notMaximumMinimummeasurethe distancebetweentwo samples but isNumberof speciesindexindexisThepointsimply a value for one sample.1.100.00100.00definedby a factorwhich measuresthe distance 2.99.0070.71fromthe origin of a coordinatesystemwith as4.97.0050.005.96.0244.72manyaxes as thereare species. The n values are10.91.0631.62the observationsof each species encounteredin 20.81.1222.36or the total value for 50.the sample of a community51.4814.1475.27.3911.42each species if a completecensus is made. The90.14.4910.52index as used is dependentupon the numberof 100 S N.10.0010.00individualsin the sample and their distributionamong the species. Hence it is a measure ofand minimumvalues forvarious numbersof spediversity.As the sample size (N total numberof indi- cies in a sample of 100 individuals.vidualsofall species) increases,therateof increaseEFFECTSOF SAMPLINGof the index is dependentupon the addition ofof the indinew species and upon the distributionIn any series of samples the index value invidualsamongthespecies. In thecase of a simple creases as (N), the numberof individualssamsuch as a dune coveredby pure dune pled, if theyare all of one species (Scommunity,1), or asgrass, theremay be no new species added and all VN if eachindividualis a new species (SN).individualsadded will be of the same species. AAny intermediate distribution of individualssample of a communitysuch as a tropicalforestamong species resultsin a rate of increasefallingmay approach the theoreticalextremein whichbetweenthesetwo extremes. Fig. 1 illustratestheeach individualadded to the sample is a new speincrease in index value with increasingnumberscies. For any numberof species (S) in a givenof individualssampledforeach of fourplantcomtotal numberof individuals(N) the species may,munities,and the theoreticalmaximumand miniat one extreme,have equal numbersof individualsmum rates of increase. It is apparent fromrepeatedsamplingof artificialand natural commaximumconor theremaybe a theoreticalmunitiesof varyingdegrees of homogeneitythatcentrationin one species,the rest havinga mini- increasingthe numberof individualssampled ismumof one individualeach. Given N 100 and reflectedin a nearlylinear increaseof the index,S4, each species may have 25 individuals(the and therateof increase(i.e., the slope of theline)mean number) or one species may have 97 indi- is governedby theadditionof new speciesand theviduals and three have 1 individualeach. The way in which the individuals are apportionedmaximumindex value for any combinationof N amongthe species.and S is:In artificialcommunitiesconstructedso thateach species had equal numbersof randomlydisV[N- (S- 1)]2 (S- 1)tributedindividuals,the increase of index valuesThis formulasimply representsthe case where for samples closely approximatedthe expected-(f)

Late Spring1967100INDEX-L, 80-D-j60Ui2340z20-4LI QUALITATIVE20NUMBERFIG. 1.397point,quarter,or quadratsamplingmethods.Thiswas trueforrandomartificialpopulationsand on.It is to be expectedthatclumpingof individualswill resultin fewerspecies beingencounteredandin a higherindexvalue at any samplesize. Quadrat samples (100 individuals) of an artificialcommunityof 712 individualsand 25 species,allrandomlydistributed,resultedin an index valueof 30.9. If the same species were all closelyclumped,the index value was 39.4. This effectismorepronouncedif a quadrat samplingtechniqueis used thanif a pointsamplingtechniqueis used.OF DIVERSITY804060OFINDIVIDUALSIncrease of the index iE100n2 with increaseAND QUANTITATIVEDATASamples of communities may be compared bysimply noting the presence or absence of species(qualitative data), or the species may be weightedby a quantitative measure such as density. Ifpresenceand absencedata are used, onlythe rich-ness of the two communities is involved, i.e., thespecies lists. If a community is maximally diverse (S - N), there is no distinction betweenqualitative and quantitative data. This suggeststhat the value of quantitative data in comparingcommunities diminishes as diversity increases andmore species are represented by one or a fewindividuals. Qualitative data are simply a statement of richness with an assumption of equaldensity (i.e., all 1) and always imply maximumcontainingindexvalues. In artificialcommunitiesdiversity. The index value of any sample whena given numberof individuals,if the numberofN (i.e., for qualitative data) is VS whichspecies was increased,the rate of increaseof the Sthe portion attributable to the qualitativeisthusindex value (slope of the line) was lowered. IfIf quantitative data are used, thecomponent.the sample series went from an area with oneportion of the total index attributableto the quanin sample size (number of individuals).Line 1. Theoretical maximum (N 1 species).Line 2. Quarter method sample of temperatehardwoodforest trees (8 species).Line 3. Quadrat sample of a stabilized dune area (15species).Line 4. Quarter method sample of tropical rainforesttrees (82 species).Line 5. Theoretical minimum. (S N)set of dominantsto one withanotherset of dominants,the rate of increasedroppedabruptlyproducing a sharp break in the line. If the dominants remained the same but lesser specieschanged,the break was less conspicuous.Several samplingtechniqueswere used to examine the relationof numbersof individualstospecies and its effecton this index. The mostdirectmethodis simplyto select a series of random pointsand notethe nearestindividualto eachpoint. With this methoda sample of 100 indigaveviduals (trees only) in a forestcommunityan index value of 55.3. A quartermethod(Cottamand Curtis1956) on thesame populationgavea value of 57.8, whilea quadratsamplegave 58.8.The value calculated froma total census of theranpopulationwas 57.2. Five separatestratifieddom samplesof 100 individualsusing the quartermethod on another foresttree populationgavevalues rangingfrom 52.5 to 56.8 (mean 54.5).In general,the rates of increasewere similarbytitative data of the sample isEi2 -\VS.As diversity increases, the fraction of the indexattributable to the quantitative component decreases until S N when it is 0. Conversely, thecomponentattributableto the qualitative com-ponent increases until it is 100%. Fig. 2 showsthe relation of the qualitative component to themaximum and minimum diversity values respectively. The ratio of minimum to maximum diversity diminishes as the number of species increasesand becomes 1 as S - N (Table 2). This simplyindicates that the quantitative component of thedata becomes less significantas S approaches N.Qualitative data make diverse samples appearmore similar than less diverse samples. If twoqualitative samples differby two species, the distance is V2 ( 1.414); if the stands differ by(14.14). The200 species, the distance is V\200

398comparisonof diversityof the two lists. Theobserved diversityof any sample can thus becomparedwith some other observeddiversityortheoreticaltypessuch as MacArthur's(1960) orthe maximumor minimumpossible.The index describedabove is in fact the complementof diversity. An index whichis directlyrelatedto diversityis obtainedby subtractingfrom1 (Table 3). The diversityofany sampleis givenby:H 100zujz0 80 I00 60 U]H 40 -TABLE j 201 - -40NUMBER60OF80100SPECIESQualitative component of diversity as a percentage of minimum diversity (line 1) and maximumdiversity (line 2) related to number of species in a sample of 100 individuals.FIG. 2.Ratio of maximum to minim

King 1964, Menhinick 1964, Tagawa 1964). The property of number, called abundance by some animal ecologists or density by others and by most plant ecologists, I will call density. The distribu- tion of numbers of individuals (or other quantities such as biomass or productivity) among the spe- cies of a community has long been regarded as of