diversity from genes to ecosystems: a unifying framework to...

Evolutionary Applications 20181ndash18 emsp|emsp1wileyonlinelibrarycomjournaleva

Received31August2017emsp |emsp Accepted21December2017DOI101111eva12593

S P E C I A L I S S U E O R I G I N A L A R T I C L E

Diversity from genes to ecosystems A unifying framework to study variation across biological metrics and scales

Oscar E Gaggiotti1 emsp|emspAnne Chao2emsp|emspPedro Peres-Neto3emsp|emspChun-Huo Chiu4emsp|emsp Christine Edwards5 emsp|emspMarie-Joseacutee Fortin6emsp|emspLou Jost7emsp|emspChristopher M Richards8emsp|emsp Kimberly A Selkoe910

1SchoolofBiologyScottishOceansInstituteUniversityofStAndrewsStAndrewsUK2InstituteofStatisticsNationalTsingHuaUniversityHsin-ChuTaiwan3DepartmentofBiologyConcordiaUniversityMontrealQCCanada4DepartmentofAgronomyNationalTaiwanUniversityTaipeiTaiwan5CenterforConservationandSustainableDevelopmentMissouriBotanicalGardenSaintLouisMOUSA6DepartmentofEcologyandEvolutionaryBiologyUniversityofTorontoTorontoONCanada7EcomingaFundationBanosTungurahuaEcuador8PlantGermplasmPreservationResearchUnitUSDA-ARSFortCollinsCOUSA9NationalCenterforEcologicalAnalysisandSynthesisUniversityofCaliforniaSantaBarbaraSantaBarbaraCAUSA10HawairsquoiInstituteofMarineBiologyUniversityofHawairsquoiatMānoaKaneoheHIUSA

ThisisanopenaccessarticleunderthetermsoftheCreativeCommonsAttributionLicensewhichpermitsusedistributionandreproductioninanymediumprovidedtheoriginalworkisproperlycitedcopy2018TheAuthorsEvolutionary ApplicationspublishedbyJohnWileyampSonsLtd

CorrespondenceOscarEGaggiottiSchoolofBiologyScottishOceansInstituteUniversityofStAndrewsStAndrewsUKEmailoegst-andrewsacuk

Funding informationUSNationalNaturalScienceFoundation(BioOCEAward)GrantAwardNumber1260169TheMarineAllianceforScienceandTechnologyforScotland(ScottishFundingCouncil)GrantAwardNumberHR09011NationalScienceFoundationGrantAwardNumberDBI-1300426TheUniversityofTennesseeNOAACoralReefConservationProgramtheMinistryofScienceandTechnologyTaiwanCanadaResearchChairinSpatialModellingandBiodiversity

AbstractBiologicaldiversityisakeyconceptinthelifesciencesandplaysafundamentalroleinmanyecologicalandevolutionaryprocessesAlthoughbiodiversityisinherentlyahier-archical conceptcoveringdifferent levelsoforganization (genespopulation speciesecological communities and ecosystems) a diversity index that behaves consistentlyacrossthesedifferentlevelshassofarbeenlackinghinderingthedevelopmentoftrulyintegrativebiodiversitystudiesTofillthisimportantknowledgegapwepresentaunify-ingframeworkforthemeasurementofbiodiversityacrosshierarchicallevelsoforgani-zationOurweightedinformation-baseddecompositionframeworkisbasedonaHillnumber of order q=1whichweightsallelementsinproportiontotheirfrequencyandleadstodiversitymeasuresbasedonShannonrsquosentropyWeinvestigatedthenumericalbehaviourofourapproachwithsimulationsandshowedthatitcanaccuratelydescribecomplex spatial hierarchical structuresTodemonstrate the intuitive and straightfor-wardinterpretationofourdiversitymeasuresintermsofeffectivenumberofcompo-nents(allelesspeciesetc)weappliedtheframeworktoarealdatasetoncoralreefbiodiversityWe expect our frameworkwill havemultiple applications covering thefieldsofconservationbiologycommunitygeneticsandeco-evolutionarydynamics

K E Y W O R D S

biodiversityindicesgeneticdiversityhierarchicalspatialstructureHillnumbersspeciesdiversity

2emsp |emsp emspensp GAGGIOTTI eT Al

1emsp |emspINTRODUCTION

Biologicaldiversityisafoundationalconceptinthelifesciencesandcritical to strategies forecological conservationHowever formanydecades biodiversity has been treated in a piecemealmannerwithecologists focusing on species diversity (but more recently also ontrait andphylogeneticdiversity) andpopulationgeneticists focusingongeneticdiversityThisdichotomyhasledtolargedifferencesinthetypeofdiversityindicesthathavebeenusedtomeasurespeciestraitphylogeneticandgeneticdiversityEcologistswereinitiallyfocusedonempiricaldevelopmentsandgeneratedaverylargenumberofspeciesdiversityindicesthatstronglydifferintheirnumericalbehaviour(Jost2006)andestimationproperties (BungeWillisampWalsh2014)Ontheotherhandpopulationgeneticswasinitiallydominatedbytheo-reticaldevelopmentsandmathematicalmodelsfocusedonaspecificsetofparametersthatdescribedgeneticdiversitywithinandamongpopulationswhichledtothedevelopmentofarestrictedsetofge-neticdiversity indicesThusalthoughbiodiversity is inherentlyahi-erarchical concept coveringdifferent levelsoforganization (geneticpopulationspeciesecologicalcommunitiesandecosystems)thelackof diversity indices that behave consistently across these different levelshasprecludedthedevelopmentoftrulyintegrativebiodiversitystudies

Recentlymotivatedby this lackofcommonmeasures forbiodi-versityatdifferentlevelsofbiologicalorganizationpopulationgenet-icistshavecarriedoutmethodologicaldevelopmentsthatextendtheuseofpopularspeciesdiversity indicestothemeasurementgeneticdiversityatdifferentlevelsofspatialsubdivision[egShannonrsquosandSimpsonrsquos indices (SherwinJabotRushampRossetto2006SmouseWhiteheadampPeakall2015)]However simplyadapting speciesdi-versitymeasuresisnotsufficientfortworeasonsFirstthereismuchcontroversyoverhowtoquantifyabundance-basedspeciesdiversityinacommunity(MendesEvangelistaThomazAgostinhoampGomes2008)Secondtherehasbeenlittleagreementonhowtopartitiondi-versityintoitsspatialcomponents(Ellison2010)ApromisingsolutionforaunifiedmeasureofgeneticdiversitycentresonHillnumbers(Hill1973)IndeedaconsensusisemergingontheuseofHillnumbersasaunifyingconcepttodefinemeasuresofvarioustypesofdiversityin-cludingspeciesphylogeneticandfunctionaldiversities(ChaoChiuampJost2014)ImportantlyHillnumbersfollowthereplicationprincipleensuringthatdiversitymeasuresarelinearinrelationtogrouppool-ingAssuch theycanbeused todevelopproperpartitionschemesacrossspatialscalesorotherhierarchicalstructuressuchaspopula-tionswithinmetapopulationsspecieswithinphylogeniescommuni-tieswithinecosystemsandtopoolinformationacrossdifferentlevelsin a hierarchy

Thepurposeof this studywas topresent a unifying frameworkfor the measurement of biodiversity across hierarchical levels of or-ganizationfromlocalpopulationtoecosystemlevelsWeexpectthatthisnewframeworkwillbeauseful tool forconservationbiologistsandwillalsofacilitatethedevelopmentofthefieldsofcommunityge-netics(Agrawal2003)andeco-evolutionarydynamics(Hendry2013)Thisnewframeworkmayalsofacilitatebridgingcommunityecology

processes(selectionamongspeciesdriftdispersalandspeciation)andthe processes emphasized by population genetics theory (selectionwithinspeciesdriftgeneflowandmutation)asexploredbyVellendetal(2014)Thepaperstartsbyoutlininghistoricaldevelopmentsonthe formulation and use of biodiversity measures in the fields of ecol-ogyandpopulationgenetics(Section2)WethenprovideanoverviewoftheuseofHillnumbersinecologyandtheirrelationshipwithpopu-lationgeneticparameterssuchasNe(Section3)Section4presentsaweightedinformation-baseddecompositionframeworkthatprovidesmeasuresofbothgeneticandspeciesdiversityatallhierarchicallevelsofspatialsubdivisionfrompopulationstoecosystemsThisisfollowedbythedescriptionofsoftwarethatimplementstheapproach(Section5)Section6explorespatternsofspeciesandgeneticdiversityunderdifferentspatialsubdivisionmodelsusingsimulateddatawithknowndiversityhierarchicalstructuresSection7showsanapplicationtoarealdatasetoncoralreefbiodiversity(Selkoeetal2016)Weclosewithadiscussionoftheadvantagesand limitationsofourapproachanditsapplicationsinthefieldsofconservationbiologycommunitygeneticsandeco-evolutionarydynamics

2emsp |emspHISTORICAL DEVELOPMENTS

Arguably the ultimate reason for methodological divergence in diver-sityindicesusedbypopulationgeneticistsandcommunityecologistsresidesintheverydifferentcontextsthat leadtotheemergenceofthesetwodisciplinesEcologistswereinterestedinunderstandingtheprocessesthatdeterminethestructureandcompositionofcommuni-tiesandcoulddirectlymeasurethecommunitytraits(numberofspe-ciesandtheirabundances)neededtocomparedifferentcommunitiesThisrelativelyeasyaccesstorealdataandaninitiallylimitedinterestinmechanisticmodelsfosteredthedevelopmentofalargenumberofdiversitymeasures toexplorespeciesdistributionaldata (Magurran2004) and eventually made the quantification of abundance-basedspecies diversity one of the most controversial issues in ecologyPopulationgeneticsontheotherhandaroseinresponsetoaneedtoreconciletwoopposingviewsofevolutionthathingedonthetypeofdiversityuponwhichnaturalselectionactedDarwinproposedthatitwassmallcontinuousvariationwhileGaltonbelievedthatnaturalselection acted upon large discontinuous variation (Provine 1971)Variation in thiscasewasanabstractconceptandcouldnotbedi-rectlymeasuredwhichmotivatedthedevelopmentofavastbodyoftheory centred around mathematical models describing the behaviour ofarestrictedsetofdiversitymeasures(Provine1971)

Althoughecologistsandpopulationgeneticistsuseverydifferentapproachestomeasurediversitytheyarebothinterestedindescrib-ing spatial patterns by decomposing total diversity intowithin- andamong-communitypopulation components But here again meth-odological developmentsdiffer greatlybetween the twodisciplinesEcologists engaged in intensedebateson the choiceofpartitioningschemes (Jost 2007)while population geneticists remained largelyfaithful to the use of so-called fixation indices proposed byWright(1951) Nevertheless the recently established fields of molecular

emspensp emsp | emsp3GAGGIOTTI eT Al

ecologycommunitygeneticsandeco-evolutionarydynamicsarehelp-ing to foster a convergence between the methods used to measure speciesandgeneticdiversity Indeed in the lastdecadepopulationgeneticistshavebeguntoextendtheuseofpopularspeciesdiversitymetrics to the measurement of genetic diversity by deriving mathe-matical expressions linking themwithevolutionaryparameters suchaseffectivepopulationsizeandmutationandmigrationrates (Chaoetal2015Sherwin2010Sherwinetal2006Smouseetal2015)

Regardless of this very recent methodological convergence ecolo-gistsandpopulationgeneticistsfacethesamechallengeswhentryingtocharacterizehowdiversitycomponents(alphabeta)arestructuredgeographicallyTheseproblemshavebeendescribedingreatdetailinthe literature (eg seeJost 2007 2010) so herewewill only giveaverybrief summaryThe first problem is that the commonlyusedwithin-community andwithin-population abundance diversity mea-sures (eg Shannon-Wiener index and heterozygosity) are in factentropiesmeaningthattheyquantifytheuncertainty inthespeciesor allele identity of randomly sampled individuals or alleles respec-tivelyImportantlytheseindicesdonotscalelinearlywithanincreaseindiversityandsomeofthem(egheterozygosity)reachanasymp-toteforlargevaluesThesecondproblemisthattheldquowithin-rdquo(alpha)andldquobetween-rdquo (beta)componentsofdiversityarenot independentIntuitively ifbetadependsonalpha itwouldbeimpossibletocom-parebetadiversitiesacrossalllevelsatwhichalphadiversitiesdiffer

Partitioning components of diversity is central to progress onthese problems Ecologists have related the traditional alpha betaandgammadiversityusingbothadditiveandmultiplicativeschemesofpartitioningOntheotherhandpopulationgeneticistshavealwaysusedthemultiplicativeschemebasedonthepartitioningoftheprob-abilityofidentitybydescentofpairsofalleles(inbreedingcoefficientsF)Althoughtherehasbeensomeconfusion(cfJost2008Jostetal2010MeirmansampHedrick2011) it iseasytodemonstratethatallestimators of FST a parameter that quantifies genetic structure in-cluding GST (Nei1973) andθ (WeirampCockerham1984) arebasedon thewell-knownmultiplicative decomposition ofWrightrsquos (1951)F-statistics (1minusFIT)= (1minusFIS)(1minusFST) where all terms are entropymeasuresdescribingtheuncertaintyintheidentitybydescentofpairsofalleleswhentheyaresampledfromthewholesetofpopulations(metapopulation)(1minusFIT)fromwithinthesamepopulation(1minusFIS) or fromtwodifferentpopulations(1minusFST)

As mentioned earlier ecologists engaged in intense debates onhow topartition speciesdiversitybut ina recentEcology forum(Ellison 2010) contributors agreed that a first step towards reach-ing a consensus was to adopt Hill numbers to measure diversityDiscussionsamongpopulationgeneticistsarelessadvancedbecauseof their traditional focus on the use of genetic polymorphism datato estimate important evolutionary parameterswhich requires thatgenetic diversity statistics be effective measures of the causes and consequencesofgeneticdifferentiation(egWhitlock2011)MuchtheoreticalworkisstillneededtodemonstratethatdiversitymeasuresbasedoninformationtheorydosatisfythisrequirementHereinsteadwearguethattheadoptionofHillnumbersinpopulationgeneticsisalsoagoodstartingpointtoreachaconsensusonhowtopartition

geneticdiversityInwhatfollowswefirstintroduceHillnumbersandthenpresentaweightedinformation-baseddecompositionframeworkapplicabletobothcommunityandpopulationgeneticsstudies

3emsp |emspOVERVIEW OF HILL NUMBERS

TherearenowmanyarticlesdescribingtheapplicationofHillnum-bers Here we follow Jost (2006) who reintroduced their use inecologyAsJost(2006)notedmostdiversityindicesareinfacten-tropiesthatmeasuretheuncertainty inthe identityofspecies (oralleles) inasampleHowever truediversitymeasuresshouldpro-videestimatesofthenumberofdistinctelements(speciesoralleles)in an aggregate (communityorpopulation) Toderive suchmeas-ureswefirstnotethatdiversityindicescreateequivalenceclassesamong aggregates in the sense that all aggregates with the same diversityindexvaluecanbeconsideredasequivalentForexampleallpopulationswith thesameheterozygosityvalueareequivalentin termsof this indexeven if theyhave radicallydifferentallelesfrequencies (seeAppendixS1 for an example)Moreover for anygivenheterozygosity therewill be an ldquoidealrdquo population inwhichallallelesareequallyfrequentItisthereforepossibletodefineanldquoeffectivenumberofelementsrdquo(allelesinthisexample)asthenum-ber of equally frequent elements in an ldquoideal aggregaterdquo that hasthesamediversityindexvalueastheldquorealaggregaterdquoAnexampleofeffectivenumber inanecologicalcontext istheeffectivenum-berofspeciesintroducedbyMacarthur(1965)whileanequivalentconcept in population genetics is the effective number of alleles(KimuraampCrow1964)

NotethattheconceptofeffectivepopulationsizeNeusedinpop-ulationgeneticsisanalogoustothatofHillnumbersbutisbasedonaratherdifferentconceptMorepreciselyNe is defined as the number ofindividualsinanideal(WrightndashFisher)populationthathasthesamemagnitudeofrandomgeneticdriftastherealpopulationbeingstud-iedTherearedifferentwaysinwhichwecanmeasurethestrengthofgenetic drift the most common being change in average inbreeding coefficientchangeinallelefrequencyvarianceandrateoflossofhet-erozygosityandeachleadtoadifferenttypeofeffectivesizeThustheidealandtherealpopulationsareequivalentintermsoftherateoflossofgeneticdiversityandnotintermsofequalrepresentationofdistinct individuals Probably the only similarity between Ne and the rationaleunderlyingHillnumbersisinthesensethatalltheindividualsintheidealpopulationcontributeequally(onaverage)tothegenepoolofthenextgeneration

Theapplicationoftheabove-statedlogictoanyofthemanydiffer-ententropymeasuresusedinecologyandpopulationgeneticsyieldsasingleexpressionfordiversity

where Sdenotesthenumberofspeciesorallelespi denotes the rel-ativeabundanceorfrequencyofspeciesoralleleiandtheexponent

(1)qDequiv

(

sum

S

i=1pqi

)1∕(1minusq)


andsuperscriptq is the order of the diversity and indicates the sen-sitivity of qD the numbers equivalent of the diversity measure being used to commonand rareelements (Jost 2006)Thediversityoforderzero (q =0) iscompletely insensitivetospeciesorallele fre-quencies and is known respectively as species or allelic richnessdepending onwhether it is applied to species or allele frequencydataThediversityoforderone(q =1)weightsthecontributionofeach speciesor alleleby their frequencywithout favouring eithercommonorrarespeciesallelesAlthoughEquation1isnotdefinedfor q=1itslimitexists(Jost2006)

where H is theShannonentropyAllvaluesofq greater than unity disproportionallyfavourthemostcommonspeciesoralleleForex-ampletheSimpsonconcentrationandtheGinindashSimpsonindexwhicharerespectivelyequivalenttoexpectedhomozygosityandexpectedheterozygositywhenappliedtoallelefrequencydataleadtodiver-sitiesoforder2 andgive the sameeffectivenumberof speciesoralleles

It is worth emphasizing that among all these different numberequivalentsortruediversitymeasuresthediversityoforder1iskeybecauseofitsabilitytoweighelementspreciselybytheirfrequencywithout favouring either rare of common elements (Jost 2006)Thereforewewillusethismeasuretodefineournewframeworkfordiversitydecomposition

4emsp |emspWEIGHTED INFORMATION- BASED DECOMPOSITION FRAMEWORK (Q = 1)

Ourdecomposition framework is focusedon the information-baseddiversitymeasure (Hill number of orderq=1) Inwhat followswefirstdescribetheframeworkintermsofabundance(speciesgenetic)diversitiesandthenweprovideanequivalentformulationintermsofphylogeneticdiversityFor simplicitywewilluse thenotationD to refertoabundancediversitiesandPDtorefertophylogeneticdiversi-ties both of order q=1AppendixS2listsallnotationanddefinitionsoftheparametersandvariablesweused

41emsp|emspFormulation in terms of abundance diversity

Herewedevelopaframeworkapplicabletobothspecies(abundancepresencendashabsencebiomass)andgeneticdatatoestimatealphabetaandgammadiversities(iediversitycomponents)acrossdifferentlev-els of a hierarchical spatial structure In this sectionwe consider averysimpleexampleofanecosystemsubdividedintomultipleregionseach of which in turn are subdivided into a number of communities whenconsideringspeciesdataoranumberofpopulationswhencon-sideringgeneticdataHoweverourformulation isapplicabletoany

number of levelswithin a spatially hierarchical partitioning schemeandtheirassociatednumberofcommunitiesandpopulationsateachlevel(nestedscale)suchastheexampleconsideredinoursimulationstudy below (see Figure1) Indeed the framework described hereallows decomposing species and genetic information on an equalfootingthusallowingcontrastingdiversitycomponentsacrosscom-munitiesandpopulationsInotherwordsifgeneticandspeciesabun-dance(orpresencendashabsence)dataareavailableforeverypopulationandeveryspeciesthengeneticandspeciesdiversitycomponentscanbecontrastedwithinandamongspatialscalesaswellasacrossdiffer-entphylogeneticlevelsNotethatourproposedframeworkisbasedon diversities of order q = 1 which are less sensitive than diversities of higher order to the fact that genetic information is not available for allindividualsinapopulationbutratherbasedonsubsamplesofindi-vidualswithinpopulationsAssuchusingq=1allowsonedecompos-inggeneticvariationconsistentlyacrossdifferentspatialsubdivisionlevels that may vary in abundance

Thefinalobjectivewastodecomposetheglobal(ecosystem)diver-sityintoitsregionalandcommunitypopulation-levelcomponentsWedo thisusing thewell-knownadditivepropertyofShannonentropyacrosshierarchicallevels(andthusmultiplicativepartitioningofdiver-sity)(Batty1976Jost2007)Table1presentsthediversities(numberequivalents)thatneedtobeestimatedateachlevelofthehierarchyForeachleveltherewillbeonevaluecorrespondingtospeciesdiver-sityandanothercorrespondingtoallelic (genetic)diversityofapar-ticularspeciesatagiven locus (oranaverageacross loci)FigureS1providesaschematicrepresentationofthecalculationofdiversities

FromTable1 it isapparentthatweonlyneedtouseEquation2to calculate three diversity indices namely D(1)

α D(2)α andDγThesedi-

versity measures are defined in terms of relative abundances of the distinctelements(speciesoralleles)attherespectivelevelsofthehi-erarchyInwhatfollowswefirstpresenttheframeworkasappliedtoallelecountdataandthenexplainhowasimplechangeinthedefini-tionofasingleparameterallowstheapplicationofthesameframe-worktospeciesabundancedataWeassumethatweareconsideringadiploidspecies(buttheschemecanbeeasilygeneralizedforpolyploidspecies)andfocusonthediversityoforderq = 1 which is based on theShannonentropy(seeEquation1)

Geneticdiversityindicesarecalculatedseparatelyforeachlocusso we focus here on a locus with S alleles Additionally we consider an ecosystem subdivided into K regions each having JklocalpopulationsLetNinjk

bethenumberofdiploidindividualswithn(=012)copiesofallele iinpopulationj and region kThenthetotalnumberofcopiesof allele iinpopulationj and region k is Nijk=

sum2

n=0nNinjk

and from this wecanderivethetotalnumberofallelesinpopulationj and region k as N+jk=

sumS

i=1Nijk the total number of alleles in region k as N++k=

sumJk

j=1N+jk

and the total number of alleles in the ecosystem as N+++ =sumK

k=1N++k

All allele frequencies can be derived from these allele counts Forexample the relative frequency of allele i in any given population j within region k is pi|jk = NijkN+jkInthecaseofregion-andecosystem-levelallelefrequencieswepooloverpopulationswithinregionsandoverallregionsandpopulationswithinanecosystemrespectivelyWedefinetheweightforpopulationjandregionk as wjk = N+jkN+++ the

(2)1D=exp

(

minussumS

i=1pi ln pi

)

=exp (H)

(3)2D=1∕

(

sumS

i=1p2i

)


weight for region k thus becomes w+k=sumJk

j=1wjk=N++k∕N+++Table2

describeshowallelespeciesrelativefrequenciesateachlevelarecal-culated in terms of these weight functions

Using these frequencieswe can calculate the genetic diversi-ties at each level of spatial organizationTable3 presents the for-mulas for D(1)

α D(2)α andDγ all other diversity measures can be derived

fromthem(seeTable1)Inthecaseoftheecosystemdiversitythisamountstosimplyreplacingpi inEquation2bypi|++ the allele fre-quencyattheecosystemlevel(seeTable2)Tocalculatethediver-sityattheregionallevelwefirstcalculatetheentropyH(2)

αk for each

individual region k and then obtain the weighted average over all regions H(2)

α Finallywecalculate theexponentof the region-levelentropytoobtainD(2)

α thealphadiversityat theregional levelWeproceedinasimilarfashiontoobtainD(1)

α thediversityatthepop-ulation level but in this case we need to average over regions and populationswithinregions

The calculation of the equivalent diversities based on speciescount data can be carried out using the exact same procedure de-scribed above but in this case Nijkrepresentsthenumberofindivid-ualsofspeciesiinpopulationj and region k All formulas for gamma

alphaandbetaalongwiththedifferentiationmeasuresateachlevelaregiveninTable3Theformulascanbedirectlygeneralizedtoanyarbitrarynumberoflevels(seeSection5)

42emsp|emspFormulation in terms of phylogenetic diversity

Wefirstpresentanoverviewofphylogeneticdiversitymeasuresap-pliedtoasinglenonhierarchicalcasehenceforthreferredtoassingleaggregateforbrevityandthenextendittoconsiderahierarchicallystructured system

421emsp|emspPhylogenetic diversity measures in a single aggregate

Toformulatephylogeneticdiversityinasingleaggregateweassumethatallspeciesorallelesinanaggregateareconnectedbyarootedul-trametricornonultrametricphylogenetictreewithallspeciesallelesastipnodesAllphylogeneticdiversitymeasuresdiscussedbelowarecomputedfromagivenfixedtreebaseoratimereferencepointthatisancestraltoallspeciesallelesintheaggregateAconvenienttime

F IGURE 1emspThespatialrepresentationof32populationsorganizedintoaspatialhierarchy based on three scale levels subregions(eightpopulationseach)regions(16populationseach)andtheecosystem(all32populations)Thedendrogram(upperpanelmdashhierarchicalrepresentationoflevels)representsthespatialrelationship(iegeographicdistance)inwhicheachtiprepresentsapopulationfoundinaparticularsite(lowerpanel)Thecartographicrepresentation(lowerpanel)representsthespatialdistributionofthesesamepopulationsalongageographiccoordinate system


referencepointistheageoftherootofthephylogenetictreespannedby all elements Assume that there are B branch segments in the tree and thus there are BcorrespondingnodesBgeSThesetofspeciesallelesisexpandedtoincludealsotheinternalnodesaswellastheter-minalnodesrepresentingspeciesalleleswhichwillthenbethefirstS elements(seeFigureS2)

LetLi denote the length of branch i in the tree i = 1 2 hellip BWefirstexpandthesetofrelativeabundancesofelements(p1p2⋯ pS) (seeEquation1) toa largersetaii=12⋯ B by defining ai as the total relative abundance of the elements descended from the ith nodebranch i = 1 2 hellip BInphylogeneticdiversityanimportantpa-rameter is the mean branch length Ttheabundance-weightedmeanofthedistancesfromthetreebasetoeachoftheterminalbranchtipsthat is T=

sumB

i=1LiaiForanultrametrictree themeanbranch length

issimplyreducedtothetree depth TseeFigure1inChaoChiuandJost (2010)foranexampleForsimplicityourfollowingformulationofphylogeneticdiversityisbasedonultrametrictreesTheextensiontononultrametric trees isstraightforward (via replacingT by T in all formulas)

Chaoetal(20102014)generalizedHillnumberstoaclassofphy-logenetic diversity of order q qPDderivedas

This measure quantifies the effective total branch lengthduring the time interval from Tyearsagoto thepresent Ifq = 0 then 0PD=

sumB

i=1Liwhich isthewell-knownFaithrsquosPDthesumof

the branch lengths of a phylogenetic tree connecting all speciesHowever this measure does not consider species abundancesRaorsquos quadratic entropy Q (Rao amp Nayak 1985) is a widely usedmeasure which takes into account both phylogeny and speciesabundancesThismeasureisageneralizationoftheGinindashSimpsonindex and quantifies the average phylogenetic distance between

anytwoindividualsrandomlyselectedfromtheassemblageChaoetal(2010)showedthattheqPDmeasureoforderq = 2 is a sim-ple transformationofquadraticentropy that is2PD=T∕(1minusQ∕T) Again here we focus on qPDmeasureoforderq = 1 which can be expressedasa functionof thephylogenetic entropy (AllenKonampBar-Yam2009)

HereIdenotesthephylogeneticentropy

whichisageneralizationofShannonrsquosentropythatincorporatesphy-logeneticdistancesamongelementsNotethatwhenthereareonlytipnodesandallbrancheshaveunitlengththenwehaveT = 1 and qPDreducestoHillnumberoforderq(inEquation1)

422emsp|emspPhylogenetic diversity decomposition in a multiple- level hierarchically structured system

The single-aggregate formulation can be extended to consider ahierarchical spatially structured system For the sake of simplic-ity we consider three levels (ecosystem region and communitypopulation) aswe did for the speciesallelic diversity decomposi-tion Assume that there are Selements in theecosystemFor therootedphylogenetictreespannedbyallS elements in the ecosys-temwedefineroot(oratimereferencepoint)numberofnodesbranches B and branch length Li in a similar manner as those in a single aggregate

Forthetipnodesasintheframeworkofspeciesandallelicdi-versity(inTable2)definepi|jk pi|+k and pi|++ i = 1 2 hellip S as the ith speciesorallelerelativefrequenciesatthepopulationregionalandecosystemlevelrespectivelyToexpandtheserelativefrequenciesto the branch set we define ai|jk i = 1 2 hellip B as the summed rela-tiveabundanceofthespeciesallelesdescendedfromtheith nodebranchinpopulation j and region k with similar definitions for ai|+k and ai|++ i = 1 2 hellip B seeFigure1ofChaoetal (2015) foran il-lustrativeexampleThedecompositionforphylogeneticdiversityissimilartothatforHillnumberspresentedinTable1exceptthatnowallmeasuresarereplacedbyphylogeneticdiversityThecorrespond-ingphylogeneticgammaalphaandbetadiversitiesateachlevelare

(4)qPD=

sumB

i=1Li

(

ai

T

)q1∕(1minusq)

(5)1PD= lim qrarr1

qPD=exp

[

minussumB

i=1Liai

Tln

(

ai

T

)]

equivT exp (I∕T)

(6)I=minussumB

i=1Liai ln ai

TABLE 1emspVariousdiversitiesinahierarchicallystructuredsystemandtheirdecompositionbasedondiversitymeasureD = 1D(Hillnumberoforder q=1inEquation2)forphylogeneticdiversitydecompositionreplaceDwithPD=1PD(phylogeneticdiversitymeasureoforderq = 1 in Equation5)seeTable3forallformulasforDandPDThesuperscripts(1)and(2)denotethehierarchicalleveloffocus

Hierarchical level

Diversity

DecompositionWithin Between Total

3Ecosystem minus minus Dγ Dγ =D(1)α D

(1)

βD(2)

β

2 Region D(2)α D

(2)

β=D

(2)γ ∕D

(2)α D

(2)γ =Dγ D

γ=D

(2)α D

(2)

β

1Communityorpopulation D(1)α D

(1)β

=D(1)γ ∕D

(1)α D

(1)γ =D

(2)α D

(2)α = D

(1)α D

(1)β

TABLE 2emspCalculationofallelespeciesrelativefrequenciesatthedifferent levels of the hierarchical structure

Hierarchical level Speciesallele relative frequency

Population pijk=Nijk∕N+jk=Nijk∕sumS

i=1Nijk

Region pi+k= Ni+k∕N++k=sumJk

j=1(wjk∕w+k)pijk

Ecosystem pi++ = Ni++∕N+++ =sumK

k=1

sumJk

j=1wjkpijk


giveninTable3alongwiththecorrespondingdifferentiationmea-suresAppendixS3 presents all mathematical derivations and dis-cussesthedesirablemonotonicityandldquotruedissimilarityrdquopropertiesthatourproposeddifferentiationmeasurespossess

5emsp |emspIMPLEMENTATION OF THE FRAMEWORK BY MEANS OF AN R PACKAGE

TheframeworkdescribedabovehasbeenimplementedintheRfunc-tioniDIP(information-basedDiversityPartitioning)whichisprovidedasDataS1Wealsoprovideashortintroductionwithasimpleexam-pledatasettoexplainhowtoobtainnumericalresultsequivalenttothoseprovidedintables4and5belowfortheHawaiianarchipelagoexampledataset

TheRfunctioniDIPrequirestwoinputmatrices

1 Abundancedata specifying speciesalleles (rows) rawor relativeabundances for each populationcommunity (columns)

2 Structure matrix describing the hierarchical structure of spatialsubdivisionseeasimpleexamplegiveninDataS1Thereisnolimittothenumberofspatialsubdivisions

Theoutputincludes(i)gamma(ortotal)diversityalphaandbetadiversityforeachlevel(ii)proportionoftotalbetainformation(among

aggregates)foundateachleveland(iii)meandifferentiation(dissimi-larity)ateachlevel

We also provide the R function iDIPphylo which implementsan information-based decomposition of phylogenetic diversity andthereforecantakeintoaccounttheevolutionaryhistoryofthespe-ciesbeingstudiedThisfunctionrequiresthetwomatricesmentionedaboveplusaphylogenetictreeinNewickformatForinteresteduserswithoutknowledgeofRwealsoprovideanonlineversionavailablefromhttpschaoshinyappsioiDIPThisinteractivewebapplicationwasdevelopedusingShiny (httpsshinyrstudiocom)ThewebpagecontainstabsprovidingashortintroductiondescribinghowtousethetoolalongwithadetailedUserrsquosGuidewhichprovidesproperinter-pretationsoftheoutputthroughnumericalexamples

6emsp |emspSIMULATION STUDY TO SHOW THE CHARACTERISTICS OF THE FRAMEWORK

Here we describe a simple simulation study to demonstrate theutility and numerical behaviour of the proposed framework Weconsidered an ecosystem composed of 32 populations dividedintofourhierarchicallevels(ecosystemregionsubregionpopula-tionFigure1)Thenumberofpopulationsateach levelwaskeptconstant across all simulations (ie ecosystem with 32 popula-tionsregionswith16populationseachandsubregionswitheight

TABLE 3emspFormulasforαβandγalongwithdifferentiationmeasuresateachhierarchicallevelofspatialsubdivisionforspeciesallelicdiversityandphylogeneticdiversityHereD = 1D(Hillnumberoforderq=1inEquation2)PD=1PD(phylogeneticdiversityoforderq = 1 in Equation5)TdenotesthedepthofanultrametrictreeH=Shannonentropy(Equation2)I=phylogeneticentropy(Equation6)

Hierarchical level Diversity Speciesallelic diversity Phylogenetic diversity

Level3Ecosystem gammaDγ =exp

minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)

Level2Region gamma D(2)γ =Dγ PD

(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α

Level1Population or community

gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1

pijk ln pijk I(1)αjk

=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α

Differentiation among aggregates at each level

Level2Amongregions Δ(2)

D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k

Level1Populationcommunitywithinregion

Δ(1)D

=H(2)α minusH

(1)α

minussum

jk wjk ln(wjk∕w+k)Δ(1)PD

=I(2)α minusI

(1)α

minusTsum

jk wjk ln(wjk∕w+k)



populations each) Note that herewe used a hierarchywith fourspatialsubdivisionsinsteadofthreelevelsasusedinthepresenta-tion of the framework This decision was based on the fact thatwewantedtosimplifythepresentationofcalculations(threelevelsused)andinthesimulations(fourlevelsused)wewantedtoverifytheperformanceoftheframeworkinamorein-depthmanner

Weexploredsixscenariosvaryinginthedegreeofgeneticstruc-turingfromverystrong(Figure2topleftpanel)toveryweak(Figure2bottomrightpanel)and foreachwegeneratedspatiallystructuredgeneticdatafor10unlinkedbi-allelic lociusinganalgorithmlooselybased on the genetic model of Coop Witonsky Di Rienzo andPritchard (2010) More explicitly to generate correlated allele fre-quenciesacrosspopulationsforbi-alleliclociwedraw10randomvec-tors of dimension 32 from a multivariate normal distribution with mean zero and a covariancematrix corresponding to the particulargeneticstructurescenariobeingconsideredToconstructthecovari-ancematrixwe firstassumed that thecovariancebetweenpopula-tions decreased with distance so that the off-diagonal elements(covariances)forclosestgeographicneighboursweresetto4forthesecond nearest neighbours were set to 3 and so on as such the main diagonalvalues(variance)weresetto5Bymultiplyingtheoff-diagonalelementsofthisvariancendashcovariancematrixbyaconstant(δ)wema-nipulated the strength of the spatial genetic structure from strong(δ=01Figure2)toweak(δ=6Figure2)Deltavalueswerechosentodemonstrategradualchangesinestimatesacrossdiversitycompo-nentsUsing this procedurewegenerated amatrix of randomnor-mally distributed N(01)deviatesɛilforeachpopulation i and locus l The randomdeviateswere transformed into allele frequencies con-strainedbetween0and1usingthesimpletransform

where pilistherelativefrequencyofalleleA1atthelthlocusinpopu-lation i and therefore qil= (1minuspil)istherelativefrequencyofalleleA2Eachbi-allelic locuswasanalysedseparatelybyour frameworkandestimated values of DγDα andDβ foreachspatial level (seeFigure1)were averaged across the 10 loci

Tosimulatearealisticdistributionofnumberofindividualsacrosspopulationswegeneratedrandomvaluesfromalog-normaldistribu-tion with mean 0 and log of standard deviation 1 these values were thenmultipliedbyrandomlygenerateddeviates fromaPoissondis-tribution with λ=30 toobtainawide rangeofpopulationcommu-nitysizesRoundedvalues(tomimicabundancesofindividuals)werethenmultipliedbypil and qiltogeneratealleleabundancesGiventhat

number of individuals was randomly generated across populationsthereisnospatialcorrelationinabundanceofindividualsacrossthelandscapewhichmeansthatthegeneticspatialpatternsweresolelydeterminedbythevariancendashcovariancematrixusedtogeneratecor-relatedallelefrequenciesacrosspopulationsThisfacilitatesinterpre-tation of the simulation results allowing us to demonstrate that the frameworkcanuncoversubtlespatialeffectsassociatedwithpopula-tionconnectivity(seebelow)

For each spatial structurewe generated 100matrices of allelefrequenciesandeachmatrixwasanalysedseparatelytoobtaindistri-butions for DγDαDβ and ΔDFigure2presentsheatmapsofthecor-relationinallelefrequenciesacrosspopulationsforonesimulateddataset under each δ value and shows that our algorithm can generate a wide rangeofgenetic structurescomparable to thosegeneratedbyothermorecomplexsimulationprotocols(egdeVillemereuilFrichotBazinFrancoisampGaggiotti2014)

Figure3 shows the distribution of DαDβ and ΔDvalues for the threelevelsofgeographicvariationbelowtheecosystemlevel(ieDγ geneticdiversity)TheresultsclearlyshowthatourframeworkdetectsdifferencesingeneticdiversityacrossdifferentlevelsofspatialgeneticstructureAsexpectedtheeffectivenumberofalleles(Dαcomponenttoprow) increasesper regionandsubregionas thespatialstructurebecomesweaker (ie fromsmall to largeδvalues)butremainscon-stant at thepopulation level as there is no spatial structure at thislevel(iepopulationsarepanmictic)sodiversityisindependentofδ

TheDβcomponent(middlerow)quantifiestheeffectivenumberofaggregates(regionssubregionspopulations)ateachhierarchicallevelofspatialsubdivisionThelargerthenumberofaggregatesatagivenlevelthemoreheterogeneousthatlevelisThusitisalsoameasureofcompositionaldissimilarityateachlevelWeusethisinterpretationtodescribetheresultsinamoreintuitivemannerAsexpectedasδ in-creasesdissimilaritybetweenregions(middleleftpanel)decreasesbe-causespatialgeneticstructurebecomesweakerandthecompositionaldissimilarityamongpopulationswithinsubregions(middlerightpanel)increases because the strong spatial correlation among populationswithinsubregionsbreaksdown(Figure3centreleftpanel)Thecom-positional dissimilarity between subregionswithin regions (Figure3middlecentrepanel)firstincreasesandthendecreaseswithincreasingδThisisduetoanldquoedgeeffectrdquoassociatedwiththemarginalstatusofthesubregionsattheextremesofthespeciesrange(extremerightandleftsubregionsinFigure2)Asδincreasesthecompositionofthetwosubregionsat thecentreof the species rangewhichbelong todifferentregionschangesmorerapidlythanthatofthetwomarginalsubregionsThusthecompositionaldissimilaritybetweensubregionswithinregionsincreasesHoweverasδcontinuestoincreasespatialeffectsdisappearanddissimilaritydecreases

pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1

F IGURE 2emspHeatmapsofallelefrequencycorrelationsbetweenpairsofpopulationsfordifferentδvaluesDeltavaluescontrolthestrengthofthespatialgeneticstructureamongpopulationswithlowδshavingthestrongestspatialcorrelationamongpopulationsEachheatmaprepresentstheoutcomeofasinglesimulationandeachdotrepresentstheallelefrequencycorrelationbetweentwopopulationsThusthediagonalrepresentsthecorrelationofapopulationwithitselfandisalways1regardlessoftheδvalueconsideredinthesimulationColoursindicaterangeofcorrelationvaluesAsinFigure1thedendrogramsrepresentthespatialrelationship(iegeographicdistance)betweenpopulations


The differentiation componentsΔD (bottom row) measures themeanproportionofnonsharedalleles ineachaggregateandfollowsthesametrendsacrossthestrengthofthespatialstructure(ieacross

δvalues)asthecompositionaldissimilarityDβThisisexpectedaswekeptthegeneticvariationequalacrossregionssubregionsandpop-ulations If we had used a nonstationary spatial covariance matrix

F IGURE 3emspSamplingvariation(medianlowerandupperquartilesandextremevalues)forthethreediversitycomponentsexaminedinthesimulationstudy(alphabetaanddifferentiationtotaldiversitygammaisreportedinthetextonly)across100simulatedpopulationsasa function of the strength (δvalues)ofthespatialgeneticvariationamongthethreespatiallevelsconsideredinthisstudy(iepopulationssubregionsandregions)


in which different δvalueswouldbeusedamongpopulations sub-regions and regions then the beta and differentiation componentswouldfollowdifferenttrendsinrelationtothestrengthinspatialge-netic variation

Forthesakeofspacewedonotshowhowthetotaleffectivenum-ber of alleles in the ecosystem (γdiversity)changesasafunctionofthestrengthof the spatial genetic structurebutvalues increasemono-tonically with δ minusDγ = 16 on average across simulations for =01 uptoDγ =19 for δ=6Inotherwordstheeffectivetotalnumberofalleles increasesasgeneticstructuredecreases Intermsofanequi-librium islandmodel thismeans thatmigration helps increase totalgeneticvariabilityIntermsofafissionmodelwithoutmigrationthiscouldbeinterpretedasareducedeffectofgeneticdriftasthegenetreeapproachesastarphylogeny(seeSlatkinampHudson1991)Notehowever that these resultsdependon the total numberofpopula-tionswhichisrelativelylargeinourexampleunderascenariowherethetotalnumberofpopulationsissmallwecouldobtainaverydiffer-entresult(egmigrationdecreasingtotalgeneticdiversity)Ourgoalherewastopresentasimplesimulationsothatuserscangainagoodunderstanding of how these components can be used to interpretgenetic variation across different spatial scales (here region subre-gionsandpopulations)Notethatweconcentratedonspatialgeneticstructureamongpopulationsasametricbutwecouldhaveusedthesamesimulationprotocoltosimulateabundancedistributionsortraitvariation among populations across different spatial scales thoughtheresultswouldfollowthesamepatternsasfortheoneswefound

hereMoreoverforsimplicityweonlyconsideredpopulationvariationwithinonespeciesbutmultiplespeciescouldhavebeenequallycon-sideredincludingaphylogeneticstructureamongthem

7emsp |emspAPPLICATION TO A REAL DATABASE BIODIVERSITY OF THE HAWAIIAN CORAL REEF ECOSYSTEM

Alltheabovederivationsarebasedontheassumptionthatweknowthe population abundances and allele frequencies which is nevertrueInsteadestimationsarebasedonallelecountsamplesandspe-ciesabundanceestimationsUsually theseestimationsareobtainedindependentlysuchthatthesamplesizeofindividualsinapopulationdiffers from thesample sizeof individuals forwhichwehaveallelecountsHerewepresentanexampleoftheapplicationofourframe-worktotheHawaiiancoralreefecosystemusingfishspeciesdensityestimates obtained from NOAA cruises (Williams etal 2015) andmicrosatellitedatafortwospeciesadeep-waterfishEtelis coruscans (Andrewsetal2014)andashallow-waterfishZebrasoma flavescens (Ebleetal2011)

TheHawaiianarchipelago(Figure4)consistsoftworegionsTheMainHawaiian Islands (MHI)which are highvolcanic islandswithmany areas subject to heavy anthropogenic perturbations (land-basedpollutionoverfishinghabitatdestructionandalien species)andtheNorthwesternHawaiianIslands(NWHI)whichareastring

F IGURE 4emspStudydomainspanningtheHawaiianArchipelagoandJohnstonAtollContourlinesdelineate1000and2000misobathsGreenindicates large landmass


ofuninhabitedlowislandsatollsshoalsandbanksthatareprimar-ilyonlyaffectedbyglobalanthropogenicstressors (climatechangeoceanacidificationandmarinedebris)(Selkoeetal2009)Inaddi-tionthenortherlylocationoftheNWHIsubjectsthereefstheretoharsher disturbance but higher productivity and these conditionslead to ecological dominance of endemics over nonendemic fishes (FriedlanderBrownJokielSmithampRodgers2003)TheHawaiianarchipelagoisgeographicallyremoteanditsmarinefaunaisconsid-erablylessdiversethanthatofthetropicalWestandSouthPacific(Randall1998)Thenearestcoralreefecosystemis800kmsouth-westof theMHI atJohnstonAtoll and is the third region consid-eredinouranalysisoftheHawaiianreefecosystemJohnstonrsquosreefarea is comparable in size to that ofMaui Island in theMHI andthefishcompositionofJohnstonisregardedasmostcloselyrelatedtotheHawaiianfishcommunitycomparedtootherPacificlocations(Randall1998)

We first present results for species diversity of Hawaiian reeffishesthenforgeneticdiversityoftwoexemplarspeciesofthefishesandfinallyaddressassociationsbetweenspeciesandgeneticdiversi-tiesNotethatwedidnotconsiderphylogeneticdiversityinthisstudybecauseaphylogenyrepresentingtheHawaiianreeffishcommunityis unavailable

71emsp|emspSpecies diversity

Table4 presents the decomposition of fish species diversity oforder q=1TheeffectivenumberofspeciesDγintheHawaiianar-chipelagois49InitselfthisnumberisnotinformativebutitwouldindeedbeveryusefulifwewantedtocomparethespeciesdiversityoftheHawaiianarchipelagowiththatofothershallow-watercoralreefecosystemforexampletheGreatBarrierReefApproximately10speciesequivalentsarelostondescendingtoeachlowerdiver-sity level in the hierarchy (RegionD(2)

α =3777 IslandD(1)α =2775)

GiventhatthereareeightandnineislandsrespectivelyinMHIandNWHIonecaninterpretthisbysayingthatonaverageeachislandcontainsabitmorethanoneendemicspeciesequivalentThebetadiversity D(2)

β=129representsthenumberofregionequivalentsin

theHawaiianarchipelagowhileD(1)

β=1361 is the average number

ofislandequivalentswithinaregionHoweverthesebetadiversi-tiesdependontheactualnumbersof regionspopulationsaswellasonsizes(weights)ofeachregionpopulationThustheyneedto

benormalizedsoastoobtainΔD(seebottomsectionofTable3)toquantifycompositionaldifferentiationBasedonTable4theextentofthiscompositionaldifferentiation intermsofthemeanpropor-tion of nonshared species is 029 among the three regions (MHINWHIandJohnston)and015amongislandswithinaregionThusthere is almost twice as much differentiation among regions than among islands within a region

Wecangainmoreinsightaboutdominanceandotherassemblagecharacteristics by comparing diversity measures of different orders(q=012)attheindividualislandlevel(Figure5a)Thisissobecausethecontributionofrareallelesspeciestodiversitydecreasesasq in-creasesSpeciesrichness(diversityoforderq=0)ismuchlargerthanthose of order q = 1 2 which indicates that all islands contain sev-eralrarespeciesConverselydiversitiesoforderq=1and2forNihoa(andtoalesserextentNecker)areverycloseindicatingthatthelocalcommunityisdominatedbyfewspeciesIndeedinNihoatherelativedensityofonespeciesChromis vanderbiltiis551

FinallyspeciesdiversityislargerinMHIthaninNWHI(Figure6a)Possible explanations for this include better sampling effort in theMHIandhigheraveragephysicalcomplexityofthereefhabitatintheMHI (Friedlander etal 2003) Reef complexity and environmentalconditionsmayalsoleadtomoreevennessintheMHIForinstancethe local adaptationofNWHIendemicsallows themtonumericallydominatethefishcommunityandthisskewsthespeciesabundancedistribution to the leftwhereas in theMHI themore typical tropi-calconditionsmayleadtocompetitiveequivalenceofmanyspeciesAlthoughMHIhavegreaterhumandisturbancethanNWHIeachis-landhassomeareasoflowhumanimpactandthismaypreventhumanimpactfrominfluencingisland-levelspeciesdiversity

72emsp|emspGenetic Diversity

Tables 5 and 6 present the decomposition of genetic diversity forEtelis coruscans and Zebrasoma flavescens respectively They bothmaintain similar amounts of genetic diversity at the ecosystem level abouteightalleleequivalentsandinbothcasesgeneticdiversityatthe regional level is only slightly higher than that maintained at the islandlevel(lessthanonealleleequivalenthigher)apatternthatcon-trastwithwhatisobservedforspeciesdiversity(seeabove)Finallybothspeciesexhibitsimilarpatternsofgeneticstructuringwithdif-ferentiation between regions being less than half that observed

TABLE 4emspDecompositionoffishspeciesdiversityoforderq=1anddifferentiationmeasuresfortheHawaiiancoralreefecosystem

Level Diversity

3HawaiianArchipelago Dγ = 48744

2 Region D(2)γ =Dγ D

(2)α =37773D

(2)

β=1290

1Island(community) D(1)γ =D

(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290

1Island(community) Δ(1)D

=0153


among populationswithin regionsNote that this pattern contrastswiththatobservedforspeciesdiversityinwhichdifferentiationwasgreaterbetween regions thanbetween islandswithin regionsNotealsothatdespitethesimilaritiesinthepartitioningofgeneticdiver-sityacrossspatial scalesgeneticdifferentiation ismuchstronger inE coruscans than Z flavescensadifferencethatmaybeexplainedbythefactthatthedeep-waterhabitatoccupiedbytheformermayhavelower water movement than the shallow waters inhabited by the lat-terandthereforemayleadtolargedifferencesinlarvaldispersalpo-tentialbetweenthetwospecies

Overallallelicdiversityofallorders(q=012)ismuchlessspa-tiallyvariablethanspeciesdiversity(Figure5)Thisisparticularlytruefor Z flavescens(Figure5c)whosehighlarvaldispersalpotentialmayhelpmaintainsimilargeneticdiversitylevels(andlowgeneticdifferen-tiation)acrosspopulations

AsitwasthecaseforspeciesdiversitygeneticdiversityinMHIissomewhathigherthanthatobservedinNWHIdespiteitshigherlevelofanthropogenicperturbations(Figure6bc)

8emsp |emspDISCUSSION

Biodiversityisaninherentlyhierarchicalconceptcoveringseverallev-elsoforganizationandspatialscalesHoweveruntilnowwedidnothaveaframeworkformeasuringallspatialcomponentsofbiodiversityapplicable to both genetic and species diversitiesHerewe use an

information-basedmeasure(Hillnumberoforderq=1)todecomposeglobal genetic and species diversity into their various regional- andcommunitypopulation-levelcomponentsTheframeworkisapplica-bletohierarchicalspatiallystructuredscenarioswithanynumberoflevels(ecosystemregionsubregionhellipcommunitypopulation)Wealsodevelopedasimilarframeworkforthedecompositionofphyloge-neticdiversityacrossmultiple-levelhierarchicallystructuredsystemsToillustratetheusefulnessofourframeworkweusedbothsimulateddatawithknowndiversitystructureandarealdatasetstressingtheimportanceofthedecompositionforvariousapplicationsincludingbi-ologicalconservationInwhatfollowswefirstdiscussseveralaspectsofourformulationintermsofspeciesandgeneticdiversityandthenbrieflyaddresstheformulationintermsofphylogeneticdiversity

Hillnumbersareparameterizedbyorderq which determines the sensitivity of the diversity measure to common and rare elements (al-lelesorspecies)Our framework isbasedonaHillnumberoforderq=1 which weights all elements in proportion to their frequencyand leadstodiversitymeasuresbasedonShannonrsquosentropyThis isafundamentallyimportantpropertyfromapopulationgeneticspointofviewbecauseitcontrastswithmeasuresbasedonheterozygositywhich are of order q=2andthereforegiveadisproportionateweighttocommonalleles Indeed it iswellknownthatheterozygosityandrelatedmeasuresareinsensitivetochangesintheallelefrequenciesofrarealleles(egAllendorfLuikartampAitken2012)sotheyperformpoorly when used on their own to detect important demographicchanges in theevolutionaryhistoryofpopulationsandspecies (eg

F IGURE 5emspDiversitymeasuresatallsampledislands(communitiespopulations)expressedintermsofHillnumbersofordersq = 0 1 and 2(a)FishspeciesdiversityofHawaiiancoralreefcommunities(b)geneticdiversityforEtelis coruscans(c)geneticdiversityforZebrasoma flavescens

(a) species diversity (b) E coruscans

(c) Z flabescens


bottlenecks)Thatsaid it isstillveryuseful tocharacterizediversityof local populations and communities using Hill numbers of orderq=012toobtainacomprehensivedescriptionofbiodiversityatthisscaleForexampleadiversityoforderq = 0 much larger than those of order q=12indicatesthatpopulationscommunitiescontainsev-eralrareallelesspeciessothatallelesspeciesrelativefrequenciesarehighly uneven Also very similar diversities of order q = 1 2 indicate that thepopulationcommunity isdominatedby fewallelesspeciesWeexemplifythisusewiththeanalysisoftheHawaiianarchipelagodata set (Figure5)A continuous diversity profilewhich depictsHill

numberwithrespecttotheorderqge0containsallinformationaboutallelesspeciesabundancedistributions

As proved by Chao etal (2015 appendixS6) and stated inAppendixS3 information-based differentiation measures such asthoseweproposehere(Table3)possesstwoessentialmonotonicitypropertiesthatheterozygosity-baseddifferentiationmeasureslack(i)theyneverdecreasewhenanewunsharedalleleisaddedtoapopula-tionand(ii)theyneverdecreasewhensomecopiesofasharedallelearereplacedbycopiesofanunsharedalleleChaoetal(2015)provideexamplesshowingthatthecommonlyuseddifferentiationmeasuresof order q = 2 such as GSTandJostrsquosDdonotpossessanyofthesetwoproperties

Other uniform analyses of diversity based on Hill numbersfocusona two-levelhierarchy (communityandmeta-community)andprovidemeasuresthatcouldbeappliedtospeciesabundanceand allele count data as well as species distance matrices andfunctionaldata(egChiuampChao2014Kosman2014ScheinerKosman Presley amp Willig 2017ab) However ours is the onlyone that presents a framework that can be applied to hierarchi-cal systems with an arbitrary number of levels and can be used to deriveproperdifferentiationmeasures in the range [01]ateachlevel with desirable monotonicity and ldquotrue dissimilarityrdquo prop-erties (AppendixS3) Therefore our proposed beta diversity oforder q=1ateach level isalways interpretableandrealisticand

F IGURE 6emspDiagrammaticrepresentationofthehierarchicalstructureunderlyingtheHawaiiancoralreefdatabaseshowingobservedspeciesallelicrichness(inparentheses)fortheHawaiiancoralfishspecies(a)Speciesrichness(b)allelicrichnessforEtelis coruscans(c)allelicrichnessfor Zebrasoma flavescens

(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s

TABLE 5emspDecompositionofgeneticdiversityoforderq = 1 and differentiationmeasuresforEteliscoruscansValuescorrespondtoaverage over 10 loci

Level Diversity

3HawaiianArchipelago Dγ=8249


(2)α =8083D

(2)

β=1016

1Island(population) D(1)γ =D

(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062


ourdifferentiationmeasurescanbecomparedamonghierarchicallevels and across different studies Nevertheless other existingframeworksbasedonHillnumbersmaybeextendedtomakethemapplicabletomorecomplexhierarchicalsystemsbyfocusingondi-versities of order q = 1

Recently Karlin and Smouse (2017 AppendixS1) derivedinformation-baseddifferentiationmeasurestodescribethegeneticstructure of a hierarchically structured populationTheirmeasuresarealsobasedonShannonentropydiversitybuttheydifferintwoimportant aspects fromourmeasures Firstly ourproposeddiffer-entiationmeasures possess the ldquotrue dissimilarityrdquo property (Chaoetal 2014Wolda 1981) whereas theirs do not In ecology theproperty of ldquotrue dissimilarityrdquo can be enunciated as follows IfN communities each have S equally common specieswith exactlyA speciessharedbyallofthemandwiththeremainingspeciesineachcommunity not shared with any other community then any sensi-ble differentiationmeasuremust give 1minusAS the true proportionof nonshared species in a community Karlin and Smousersquos (2017)measures are useful in quantifying other aspects of differentiationamongaggregatesbutdonotmeasureldquotruedissimilarityrdquoConsiderasimpleexamplepopulationsIandIIeachhas10equallyfrequental-leles with 4 shared then intuitively any differentiation measure must yield60HoweverKarlinandSmousersquosmeasureinthissimplecaseyields3196ontheotherhandoursgivesthetruenonsharedpro-portionof60Thesecondimportantdifferenceisthatwhenthereare only two levels our information-baseddifferentiationmeasurereduces to thenormalizedmutual information (Shannondifferenti-ation)whereas theirs does not Sherwin (2010) indicated that themutual information is linearly relatedtothechi-squarestatistic fortesting allelic differentiation between populations Thus ourmea-surescanbelinkedtothewidelyusedchi-squarestatisticwhereastheirs cannot

In this paper all diversitymeasures (alpha beta and gamma di-versities) and differentiation measures are derived conditional onknowing true species richness and species abundances In practicespeciesrichnessandabundancesareunknownallmeasuresneedtobe estimated from sampling dataWhen there are undetected spe-ciesoralleles inasample theundersamplingbias for themeasuresof order q = 2 is limited because they are focused on the dominant

speciesoralleleswhichwouldbesurelyobservedinanysampleForinformation-basedmeasures it iswellknownthat theobserveden-tropydiversity (ie by substituting species sample proportions intotheentropydiversityformulas)exhibitsnegativebiastosomeextentNeverthelesstheundersamplingbiascanbelargelyreducedbynovelstatisticalmethodsproposedbyChaoandJost(2015)Inourrealdataanalysis statisticalestimationwasnotappliedbecause thepatternsbased on the observed and estimated values are generally consistent When communities or populations are severely undersampled sta-tisticalestimationshouldbeappliedtoreduceundersamplingbiasAmorethoroughdiscussionofthestatisticalpropertiesofourmeasureswillbepresentedinaseparatestudyHereourobjectivewastoin-troducetheinformation-basedframeworkandexplainhowitcanbeappliedtorealdata

Our simulation study clearly shows that the diversity measuresderivedfromourframeworkcanaccuratelydescribecomplexhierar-chical structuresForexampleourbetadiversityDβ and differentia-tion ΔD measures can uncover the increase in differentiation between marginalandwell-connectedsubregionswithinaregionasspatialcor-relationacrosspopulations(controlledbytheparameterδ in our sim-ulations)diminishes(Figure3)Indeedthestrengthofthehierarchicalstructurevaries inacomplexwaywithδ Structuring within regions declines steadily as δ increases but structuring between subregions within a region first increases and then decreases as δ increases (see Figure2)Nevertheless forvery largevaluesofδ hierarchical struc-turing disappears completely across all levels generating spatial ge-neticpatternssimilartothoseobservedfortheislandmodelAmoredetailedexplanationofthemechanismsinvolvedispresentedintheresults section

TheapplicationofourframeworktotheHawaiiancoralreefdataallowsustodemonstratetheintuitiveandstraightforwardinterpreta-tionofourdiversitymeasuresintermsofeffectivenumberofcompo-nentsThedatasetsconsistof10and13microsatellitelocicoveringonlyasmallfractionofthegenomeofthestudiedspeciesHowevermoreextensivedatasetsconsistingofdenseSNParraysarequicklybeingproducedthankstotheuseofnext-generationsequencingtech-niquesAlthough SNPs are bi-allelic they can be generated inverylargenumberscoveringthewholegenomeofaspeciesandthereforetheyaremorerepresentativeofthediversitymaintainedbyaspeciesAdditionallythesimulationstudyshowsthattheanalysisofbi-allelicdatasetsusingourframeworkcanuncovercomplexspatialstructuresTheRpackageweprovidewillgreatlyfacilitatetheapplicationofourapproachtothesenewdatasets

Our framework provides a consistent anddetailed characteriza-tionofbiodiversityatalllevelsoforganizationwhichcanthenbeusedtouncoverthemechanismsthatexplainobservedspatialandtemporalpatternsAlthoughwestillhavetoundertakeaverythoroughsensi-tivity analysis of our diversity measures under a wide range of eco-logical and evolutionary scenarios the results of our simulation study suggestthatdiversitymeasuresderivedfromourframeworkmaybeused as summary statistics in the context ofApproximateBayesianComputation methods (Beaumont Zhang amp Balding 2002) aimedatmaking inferences about theecology anddemographyof natural

TABLE 6emspDecompositionofgeneticdiversityoforderq = 1 and differentiationmeasuresforZebrasomaflavescensValuescorrespondtoaveragesover13loci

Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033


populations For example our approach provides locus-specific di-versitymeasuresthatcouldbeusedto implementgenomescanap-proachesaimedatdetectinggenomicregionssubjecttoselection

Weexpectour frameworktohave importantapplications in thedomain of community genetics This field is aimed at understand-ingthe interactionsbetweengeneticandspeciesdiversity (Agrawal2003)Afrequentlyusedtooltoachievethisgoal iscentredaroundthe studyof speciesndashgenediversity correlations (SGDCs)There arenowmanystudiesthathaveassessedtherelationshipbetweenspe-ciesandgeneticdiversity(reviewedbyVellendetal2014)buttheyhaveledtocontradictoryresultsInsomecasesthecorrelationispos-itive in others it is negative and in yet other cases there is no correla-tionThesedifferencesmaybe explainedby amultitudeof factorssomeofwhichmayhaveabiologicalunderpinningbutonepossibleexplanationisthatthemeasurementofgeneticandspeciesdiversityis inconsistent across studies andevenwithin studiesForexamplesomestudieshavecorrelatedspecies richnessameasure thatdoesnotconsiderabundancewithgenediversityorheterozygositywhicharebasedonthefrequencyofgeneticvariantsandgivemoreweighttocommonthanrarevariantsInothercasesstudiesusedconsistentmeasuresbutthesewerenotaccuratedescriptorsofdiversityForex-amplespeciesandallelicrichnessareconsistentmeasuresbuttheyignoreanimportantaspectofdiversitynamelytheabundanceofspe-ciesandallelicvariantsOurnewframeworkprovidesldquotruediversityrdquomeasuresthatareconsistentacrosslevelsoforganizationandthere-foretheyshouldhelpimproveourunderstandingoftheinteractionsbetweengenetic and speciesdiversities In this sense it provides amorenuancedassessmentof theassociationbetweenspatial struc-turingofspeciesandgeneticdiversityForexampleafirstbutsome-whatlimitedapplicationofourframeworktotheHawaiianarchipelagodatasetuncoversadiscrepancybetweenspeciesandgeneticdiversityspatialpatternsThedifferenceinspeciesdiversitybetweenregionaland island levels ismuch larger (26)thanthedifference ingeneticdiversitybetweenthesetwo levels (1244forE coruscansand7for Z flavescens)Moreoverinthecaseofspeciesdiversitydifferenti-ationamongregionsismuchstrongerthanamongpopulationswithinregions butweobserved theexactoppositepattern in the caseofgeneticdiversitygeneticdifferentiationisweakeramongregionsthanamong islandswithinregionsThisclearly indicatesthatspeciesandgeneticdiversityspatialpatternsaredrivenbydifferentprocesses

InourhierarchicalframeworkandanalysisbasedonHillnumberof order q=1allspecies(oralleles)areconsideredtobeequallydis-tinctfromeachothersuchthatspecies(allelic)relatednessisnottakenintoaccountonlyspeciesabundancesareconsideredToincorporateevolutionaryinformationamongspecieswehavealsoextendedChaoetal(2010)rsquosphylogeneticdiversityoforderq = 1 to measure hierar-chicaldiversitystructurefromgenestoecosystems(Table3lastcol-umn)Chaoetal(2010)rsquosmeasureoforderq=1reducestoasimpletransformationofthephylogeneticentropywhichisageneralizationofShannonrsquosentropythatincorporatesphylogeneticdistancesamongspecies (Allenetal2009)Wehavealsoderivedthecorrespondingdifferentiation measures at each level of the hierarchy (bottom sec-tion of Table3) Note that a phylogenetic tree encapsulates all the

informationaboutrelationshipsamongallspeciesandindividualsorasubsetofthemOurproposeddendrogram-basedphylogeneticdiver-sitymeasuresmakeuseofallsuchrelatednessinformation

Therearetwoother importanttypesofdiversitythatwedonotdirectlyaddressinourformulationThesearetrait-basedfunctionaldi-versityandmoleculardiversitybasedonDNAsequencedataInbothofthesecasesdataatthepopulationorspecieslevelistransformedinto pairwise distancematrices However information contained inadistancematrixdiffers from thatprovidedbyaphylogenetic treePetcheyandGaston(2002)appliedaclusteringalgorithmtothespe-cies pairwise distancematrix to construct a functional dendrogramand then obtain functional diversity measures An unavoidable issue intheirapproachishowtoselectadistancemetricandaclusteringalgorithm to construct the dendrogram both distance metrics and clustering algorithmmay lead to a loss or distortion of species andDNAsequencepairwisedistanceinformationIndeedMouchetetal(2008) demonstrated that the results obtained using this approacharehighlydependentontheclusteringmethodbeingusedMoreoverMaire Grenouillet Brosse andVilleger (2015) noted that even thebestdendrogramisoftenofverylowqualityThuswedonotneces-sarily suggest theuseofdendrogram-basedapproaches focusedontraitandDNAsequencedatatogenerateabiodiversitydecompositionatdifferenthierarchicalscalesakintotheoneusedhereforphyloge-neticstructureAnalternativeapproachtoachievethisgoalistousedistance-based functionaldiversitymeasuresandseveral suchmea-sureshavebeenproposed (egChiuampChao2014Kosman2014Scheineretal2017ab)Howeverthedevelopmentofahierarchicaldecompositionframeworkfordistance-baseddiversitymeasuresthatsatisfiesallmonotonicityandldquotruedissimilarityrdquopropertiesismathe-maticallyverycomplexNeverthelesswenotethatwearecurrentlyextendingourframeworktoalsocoverthiscase

Theapplicationofourframeworktomoleculardataisperformedunder the assumptionof the infinite allelemutationmodelThus itcannotmakeuseoftheinformationcontainedinmarkerssuchasmi-crosatellitesandDNAsequencesforwhichitispossibletocalculatedistancesbetweendistinctallelesWealsoassumethatgeneticmark-ers are independent (ie theyare in linkageequilibrium)which im-pliesthatwecannotusetheinformationprovidedbytheassociationofallelesatdifferentlociThissituationissimilartothatoffunctionaldiversity(seeprecedingparagraph)andrequirestheconsiderationofadistancematrixMorepreciselyinsteadofconsideringallelefrequen-ciesweneed to focusongenotypicdistancesusingmeasuressuchasthoseproposedbyKosman(1996)andGregoriusetal(GregoriusGilletampZiehe2003)Asmentionedbeforewearecurrentlyextend-ingourapproachtodistance-baseddatasoastoobtainahierarchi-calframeworkapplicabletobothtrait-basedfunctionaldiversityandDNAsequence-basedmoleculardiversity

Anessentialrequirementinbiodiversityresearchistobeabletocharacterizecomplexspatialpatternsusinginformativediversitymea-sures applicable to all levels of organization (fromgenes to ecosys-tems)theframeworkweproposefillsthisknowledgegapandindoingsoprovidesnewtoolstomakeinferencesaboutbiodiversityprocessesfromobservedspatialpatterns


ACKNOWLEDGEMENTS

This work was assisted through participation in ldquoNext GenerationGeneticMonitoringrdquoInvestigativeWorkshopattheNationalInstituteforMathematicalandBiologicalSynthesissponsoredbytheNationalScienceFoundation throughNSFAwardDBI-1300426withaddi-tionalsupportfromTheUniversityofTennesseeKnoxvilleHawaiianfish community data were provided by the NOAA Pacific IslandsFisheries Science Centerrsquos Coral Reef Ecosystem Division (CRED)with funding fromNOAACoralReefConservationProgramOEGwas supported by theMarine Alliance for Science and Technologyfor Scotland (MASTS) A C and C H C were supported by theMinistryofScienceandTechnologyTaiwanPP-Nwassupportedby a Canada Research Chair in Spatial Modelling and BiodiversityKASwassupportedbyNationalScienceFoundation(BioOCEAwardNumber 1260169) and theNational Center for Ecological Analysisand Synthesis

DATA ARCHIVING STATEMENT

AlldatausedinthismanuscriptareavailableinDRYAD(httpsdoiorgdxdoiorg105061dryadqm288)andBCO-DMO(httpwwwbco-dmoorgproject552879)

ORCID

Oscar E Gaggiotti httporcidorg0000-0003-1827-1493

Christine Edwards httporcidorg0000-0001-8837-4872

REFERENCES

AgrawalAA(2003)CommunitygeneticsNewinsightsintocommunityecol-ogybyintegratingpopulationgeneticsEcology 84(3)543ndash544httpsdoiorg1018900012-9658(2003)084[0543CGNIIC]20CO2

Allen B Kon M amp Bar-Yam Y (2009) A new phylogenetic diversitymeasure generalizing the shannon index and its application to phyl-lostomid bats American Naturalist 174(2) 236ndash243 httpsdoiorg101086600101

AllendorfFWLuikartGHampAitkenSN(2012)Conservation and the genetics of populations2ndedHobokenNJWiley-Blackwell

AndrewsKRMoriwakeVNWilcoxCGrauEGKelleyCPyleR L amp Bowen B W (2014) Phylogeographic analyses of sub-mesophotic snappers Etelis coruscans and Etelis ldquomarshirdquo (FamilyLutjanidae)revealconcordantgeneticstructureacrosstheHawaiianArchipelagoPLoS One 9(4) e91665httpsdoiorg101371jour-nalpone0091665

BattyM(1976)EntropyinspatialaggregationGeographical Analysis 8(1)1ndash21

BeaumontMAZhangWYampBaldingDJ(2002)ApproximateBayesiancomputationinpopulationgeneticsGenetics 162(4)2025ndash2035

BungeJWillisAampWalshF(2014)Estimatingthenumberofspeciesinmicrobial diversity studies Annual Review of Statistics and Its Application 1 edited by S E Fienberg 427ndash445 httpsdoiorg101146annurev-statistics-022513-115654

ChaoAampChiuC-H(2016)Bridgingthevarianceanddiversitydecom-positionapproachestobetadiversityviasimilarityanddifferentiationmeasures Methods in Ecology and Evolution 7(8)919ndash928httpsdoiorg1011112041-210X12551

Chao A Chiu C H Hsieh T C Davis T Nipperess D A amp FaithD P (2015) Rarefaction and extrapolation of phylogenetic diver-sity Methods in Ecology and Evolution 6(4) 380ndash388 httpsdoiorg1011112041-210X12247

ChaoAChiuCHampJost L (2010) Phylogenetic diversitymeasuresbasedonHillnumbersPhilosophical Transactions of the Royal Society of London Series B Biological Sciences 365(1558)3599ndash3609httpsdoiorg101098rstb20100272

ChaoANChiuCHampJostL(2014)Unifyingspeciesdiversityphy-logenetic diversity functional diversity and related similarity and dif-ferentiationmeasuresthroughHillnumbersAnnual Review of Ecology Evolution and Systematics 45 edited by D J Futuyma 297ndash324httpsdoiorg101146annurev-ecolsys-120213-091540

ChaoA amp Jost L (2015) Estimating diversity and entropy profilesviadiscoveryratesofnewspeciesMethods in Ecology and Evolution 6(8)873ndash882httpsdoiorg1011112041-210X12349

ChaoAJostLHsiehTCMaKHSherwinWBampRollinsLA(2015) Expected Shannon entropy and shannon differentiationbetween subpopulations for neutral genes under the finite islandmodel PLoS One 10(6) e0125471 httpsdoiorg101371journalpone0125471

ChiuCHampChaoA(2014)Distance-basedfunctionaldiversitymeasuresand their decompositionA framework based onHill numbersPLoS One 9(7)e100014httpsdoiorg101371journalpone0100014

Coop G Witonsky D Di Rienzo A amp Pritchard J K (2010) Usingenvironmental correlations to identify loci underlying local ad-aptation Genetics 185(4) 1411ndash1423 httpsdoiorg101534genetics110114819

EbleJAToonenRJ Sorenson LBasch LV PapastamatiouY PampBowenBW(2011)EscapingparadiseLarvalexportfromHawaiiin an Indo-Pacific reef fish the yellow tang (Zebrasoma flavescens)Marine Ecology Progress Series 428245ndash258httpsdoiorg103354meps09083

Ellison A M (2010) Partitioning diversity Ecology 91(7) 1962ndash1963httpsdoiorg10189009-16921

FriedlanderAMBrown EK Jokiel P L SmithWRampRodgersK S (2003) Effects of habitat wave exposure and marine pro-tected area status on coral reef fish assemblages in theHawaiianarchipelago Coral Reefs 22(3) 291ndash305 httpsdoiorg101007s00338-003-0317-2

GregoriusHRGilletEMampZieheM (2003)Measuringdifferencesof trait distributions between populationsBiometrical Journal 45(8)959ndash973httpsdoiorg101002(ISSN)1521-4036

Hendry A P (2013) Key questions in the genetics and genomics ofeco-evolutionary dynamics Heredity 111(6) 456ndash466 httpsdoiorg101038hdy201375

HillMO(1973)DiversityandevennessAunifyingnotationanditscon-sequencesEcology 54(2)427ndash432httpsdoiorg1023071934352

JostL(2006)EntropyanddiversityOikos 113(2)363ndash375httpsdoiorg101111j20060030-129914714x

Jost L (2007) Partitioning diversity into independent alpha andbeta components Ecology 88(10) 2427ndash2439 httpsdoiorg10189006-17361

Jost L (2008) GST and its relatives do not measure differen-tiation Molecular Ecology 17(18) 4015ndash4026 httpsdoiorg101111j1365-294X200803887x

JostL(2010)IndependenceofalphaandbetadiversitiesEcology 91(7)1969ndash1974httpsdoiorg10189009-03681

JostLDeVriesPWallaTGreeneyHChaoAampRicottaC (2010)Partitioning diversity for conservation analyses Diversity and Distributions 16(1)65ndash76httpsdoiorg101111j1472-4642200900626x

Karlin E F amp Smouse P E (2017) Allo-allo-triploid Sphagnum x fal-catulum Single individuals contain most of the Holantarctic diver-sity for ancestrally indicative markers Annals of Botany 120(2) 221ndash231


KimuraMampCrowJF(1964)NumberofallelesthatcanbemaintainedinfinitepopulationsGenetics 49(4)725ndash738

KosmanE(1996)DifferenceanddiversityofplantpathogenpopulationsAnewapproachformeasuringPhytopathology 86(11)1152ndash1155

KosmanE (2014)MeasuringdiversityFrom individuals topopulationsEuropean Journal of Plant Pathology 138(3) 467ndash486 httpsdoiorg101007s10658-013-0323-3

MacarthurRH (1965)Patternsof speciesdiversityBiological Reviews 40(4) 510ndash533 httpsdoiorg101111j1469-185X1965tb00815x

Magurran A E (2004) Measuring biological diversity Hoboken NJBlackwellScience

Maire E Grenouillet G Brosse S ampVilleger S (2015) Howmanydimensions are needed to accurately assess functional diversity A pragmatic approach for assessing the quality of functional spacesGlobal Ecology and Biogeography 24(6) 728ndash740 httpsdoiorg101111geb12299

MeirmansPGampHedrickPW(2011)AssessingpopulationstructureF-STand relatedmeasuresMolecular Ecology Resources 11(1)5ndash18httpsdoiorg101111j1755-0998201002927x

Mendes R S Evangelista L R Thomaz S M Agostinho A A ampGomes L C (2008) A unified index to measure ecological di-versity and species rarity Ecography 31(4) 450ndash456 httpsdoiorg101111j0906-7590200805469x

MouchetMGuilhaumonFVillegerSMasonNWHTomasiniJAampMouillotD(2008)Towardsaconsensusforcalculatingdendrogram-based functional diversity indices Oikos 117(5)794ndash800httpsdoiorg101111j0030-1299200816594x

Nei M (1973) Analysis of gene diversity in subdivided populationsProceedings of the National Academy of Sciences of the United States of America 70 3321ndash3323

PetcheyO L ampGaston K J (2002) Functional diversity (FD) speciesrichness and community composition Ecology Letters 5 402ndash411 httpsdoiorg101046j1461-0248200200339x

ProvineWB(1971)The origins of theoretical population geneticsChicagoILUniversityofChicagoPress

RandallJE(1998)ZoogeographyofshorefishesoftheIndo-Pacificre-gion Zoological Studies 37(4)227ndash268

RaoCRampNayakTK(1985)CrossentropydissimilaritymeasuresandcharacterizationsofquadraticentropyIeee Transactions on Information Theory 31(5)589ndash593httpsdoiorg101109TIT19851057082

Scheiner S M Kosman E Presley S J ampWillig M R (2017a) Thecomponents of biodiversitywith a particular focus on phylogeneticinformation Ecology and Evolution 7(16) 6444ndash6454 httpsdoiorg101002ece33199

Scheiner S M Kosman E Presley S J amp Willig M R (2017b)Decomposing functional diversityMethods in Ecology and Evolution 8(7)809ndash820httpsdoiorg1011112041-210X12696

SelkoeKAGaggiottiOETremlEAWrenJLKDonovanMKToonenRJampConsortiuHawaiiReefConnectivity(2016)TheDNAofcoralreefbiodiversityPredictingandprotectinggeneticdiversityofreef assemblages Proceedings of the Royal Society B- Biological Sciences 283(1829)

SelkoeKAHalpernBSEbertCMFranklinECSeligERCaseyKShellipToonenRJ(2009)AmapofhumanimpactstoaldquopristinerdquocoralreefecosystemthePapahānaumokuākeaMarineNationalMonumentCoral Reefs 28(3)635ndash650httpsdoiorg101007s00338-009-0490-z

SherwinWB(2010)Entropyandinformationapproachestogeneticdi-versityanditsexpressionGenomicgeographyEntropy 12(7)1765ndash1798httpsdoiorg103390e12071765

SherwinWBJabotFRushRampRossettoM (2006)Measurementof biological information with applications from genes to land-scapes Molecular Ecology 15(10) 2857ndash2869 httpsdoiorg101111j1365-294X200602992x

SlatkinM ampHudson R R (1991) Pairwise comparisons ofmitochon-drialDNAsequencesinstableandexponentiallygrowingpopulationsGenetics 129(2)555ndash562

SmousePEWhiteheadMRampPeakallR(2015)Aninformationaldi-versityframeworkillustratedwithsexuallydeceptiveorchidsinearlystages of speciationMolecular Ecology Resources 15(6) 1375ndash1384httpsdoiorg1011111755-099812422

VellendMLajoieGBourretAMurriaCKembelSWampGarantD(2014)Drawingecologicalinferencesfromcoincidentpatternsofpop-ulation- and community-level biodiversityMolecular Ecology 23(12)2890ndash2901httpsdoiorg101111mec12756

deVillemereuil P Frichot E Bazin E Francois O amp Gaggiotti O E(2014)GenomescanmethodsagainstmorecomplexmodelsWhenand how much should we trust them Molecular Ecology 23(8)2006ndash2019httpsdoiorg101111mec12705

WeirBSampCockerhamCC(1984)EstimatingF-statisticsfortheanaly-sisofpopulationstructureEvolution 38(6)1358ndash1370

WhitlockMC(2011)GrsquoSTandDdonotreplaceF-STMolecular Ecology 20(6)1083ndash1091httpsdoiorg101111j1365-294X201004996x

Williams IDBaumJKHeenanAHansonKMNadonMOampBrainard R E (2015)Human oceanographic and habitat drivers ofcentral and western Pacific coral reef fish assemblages PLoS One 10(4)e0120516ltGotoISIgtWOS000352135600033httpsdoiorg101371journalpone0120516

WoldaH(1981)SimilarityindexessamplesizeanddiversityOecologia 50(3)296ndash302

WrightS(1951)ThegeneticalstructureofpopulationsAnnals of Eugenics 15(4)323ndash354

SUPPORTING INFORMATION

Additional Supporting Information may be found online in thesupportinginformationtabforthisarticle

How to cite this articleGaggiottiOEChaoAPeres-NetoPetalDiversityfromgenestoecosystemsAunifyingframeworkto study variation across biological metrics and scales Evol Appl 2018001ndash18 httpsdoiorg101111eva12593


1emsp |emspINTRODUCTION

Biologicaldiversityisafoundationalconceptinthelifesciencesandcritical to strategies forecological conservationHowever formanydecades biodiversity has been treated in a piecemealmannerwithecologists focusing on species diversity (but more recently also ontrait andphylogeneticdiversity) andpopulationgeneticists focusingongeneticdiversityThisdichotomyhasledtolargedifferencesinthetypeofdiversityindicesthathavebeenusedtomeasurespeciestraitphylogeneticandgeneticdiversityEcologistswereinitiallyfocusedonempiricaldevelopmentsandgeneratedaverylargenumberofspeciesdiversityindicesthatstronglydifferintheirnumericalbehaviour(Jost2006)andestimationproperties (BungeWillisampWalsh2014)Ontheotherhandpopulationgeneticswasinitiallydominatedbytheo-reticaldevelopmentsandmathematicalmodelsfocusedonaspecificsetofparametersthatdescribedgeneticdiversitywithinandamongpopulationswhichledtothedevelopmentofarestrictedsetofge-neticdiversity indicesThusalthoughbiodiversity is inherentlyahi-erarchical concept coveringdifferent levelsoforganization (geneticpopulationspeciesecologicalcommunitiesandecosystems)thelackof diversity indices that behave consistently across these different levelshasprecludedthedevelopmentoftrulyintegrativebiodiversitystudies

Recentlymotivatedby this lackofcommonmeasures forbiodi-versityatdifferentlevelsofbiologicalorganizationpopulationgenet-icistshavecarriedoutmethodologicaldevelopmentsthatextendtheuseofpopularspeciesdiversity indicestothemeasurementgeneticdiversityatdifferentlevelsofspatialsubdivision[egShannonrsquosandSimpsonrsquos indices (SherwinJabotRushampRossetto2006SmouseWhiteheadampPeakall2015)]However simplyadapting speciesdi-versitymeasuresisnotsufficientfortworeasonsFirstthereismuchcontroversyoverhowtoquantifyabundance-basedspeciesdiversityinacommunity(MendesEvangelistaThomazAgostinhoampGomes2008)Secondtherehasbeenlittleagreementonhowtopartitiondi-versityintoitsspatialcomponents(Ellison2010)ApromisingsolutionforaunifiedmeasureofgeneticdiversitycentresonHillnumbers(Hill1973)IndeedaconsensusisemergingontheuseofHillnumbersasaunifyingconcepttodefinemeasuresofvarioustypesofdiversityin-cludingspeciesphylogeneticandfunctionaldiversities(ChaoChiuampJost2014)ImportantlyHillnumbersfollowthereplicationprincipleensuringthatdiversitymeasuresarelinearinrelationtogrouppool-ingAssuch theycanbeused todevelopproperpartitionschemesacrossspatialscalesorotherhierarchicalstructuressuchaspopula-tionswithinmetapopulationsspecieswithinphylogeniescommuni-tieswithinecosystemsandtopoolinformationacrossdifferentlevelsin a hierarchy

Thepurposeof this studywas topresent a unifying frameworkfor the measurement of biodiversity across hierarchical levels of or-ganizationfromlocalpopulationtoecosystemlevelsWeexpectthatthisnewframeworkwillbeauseful tool forconservationbiologistsandwillalsofacilitatethedevelopmentofthefieldsofcommunityge-netics(Agrawal2003)andeco-evolutionarydynamics(Hendry2013)Thisnewframeworkmayalsofacilitatebridgingcommunityecology

processes(selectionamongspeciesdriftdispersalandspeciation)andthe processes emphasized by population genetics theory (selectionwithinspeciesdriftgeneflowandmutation)asexploredbyVellendetal(2014)Thepaperstartsbyoutlininghistoricaldevelopmentsonthe formulation and use of biodiversity measures in the fields of ecol-ogyandpopulationgenetics(Section2)WethenprovideanoverviewoftheuseofHillnumbersinecologyandtheirrelationshipwithpopu-lationgeneticparameterssuchasNe(Section3)Section4presentsaweightedinformation-baseddecompositionframeworkthatprovidesmeasuresofbothgeneticandspeciesdiversityatallhierarchicallevelsofspatialsubdivisionfrompopulationstoecosystemsThisisfollowedbythedescriptionofsoftwarethatimplementstheapproach(Section5)Section6explorespatternsofspeciesandgeneticdiversityunderdifferentspatialsubdivisionmodelsusingsimulateddatawithknowndiversityhierarchicalstructuresSection7showsanapplicationtoarealdatasetoncoralreefbiodiversity(Selkoeetal2016)Weclosewithadiscussionoftheadvantagesand limitationsofourapproachanditsapplicationsinthefieldsofconservationbiologycommunitygeneticsandeco-evolutionarydynamics

2emsp |emspHISTORICAL DEVELOPMENTS

Arguably the ultimate reason for methodological divergence in diver-sityindicesusedbypopulationgeneticistsandcommunityecologistsresidesintheverydifferentcontextsthat leadtotheemergenceofthesetwodisciplinesEcologistswereinterestedinunderstandingtheprocessesthatdeterminethestructureandcompositionofcommuni-tiesandcoulddirectlymeasurethecommunitytraits(numberofspe-ciesandtheirabundances)neededtocomparedifferentcommunitiesThisrelativelyeasyaccesstorealdataandaninitiallylimitedinterestinmechanisticmodelsfosteredthedevelopmentofalargenumberofdiversitymeasures toexplorespeciesdistributionaldata (Magurran2004) and eventually made the quantification of abundance-basedspecies diversity one of the most controversial issues in ecologyPopulationgeneticsontheotherhandaroseinresponsetoaneedtoreconciletwoopposingviewsofevolutionthathingedonthetypeofdiversityuponwhichnaturalselectionactedDarwinproposedthatitwassmallcontinuousvariationwhileGaltonbelievedthatnaturalselection acted upon large discontinuous variation (Provine 1971)Variation in thiscasewasanabstractconceptandcouldnotbedi-rectlymeasuredwhichmotivatedthedevelopmentofavastbodyoftheory centred around mathematical models describing the behaviour ofarestrictedsetofdiversitymeasures(Provine1971)

Althoughecologistsandpopulationgeneticistsuseverydifferentapproachestomeasurediversitytheyarebothinterestedindescrib-ing spatial patterns by decomposing total diversity intowithin- andamong-communitypopulation components But here again meth-odological developmentsdiffer greatlybetween the twodisciplinesEcologists engaged in intensedebateson the choiceofpartitioningschemes (Jost 2007)while population geneticists remained largelyfaithful to the use of so-called fixation indices proposed byWright(1951) Nevertheless the recently established fields of molecular












(1)qDequiv

(

sum

S

i=1pqi

)1∕(1minusq)
















sum2

n=0nNinjk


sumS


sumJk

j=1N+jk


k=1N++k


(2)1D=exp

(

minussumS

i=1pi ln pi

)

=exp (H)

(3)2D=1∕

(

sumS

i=1p2i

)








αk for each















sumB





sumB









(4)qPD=

sumB

i=1Li

(

ai

T

)q1∕(1minusq)

(5)1PD= lim qrarr1

qPD=exp

[

minussumB

i=1Liai

Tln

(

ai

T

)]

equivT exp (I∕T)

(6)I=minussumB

i=1Liai ln ai


Hierarchical level

Diversity



(1)

βD(2)

β

2 Region D(2)α D

(2)

β=D

(2)γ ∕D

(2)α D

(2)γ =Dγ D

γ=D

(2)α D

(2)

β


(1)β

=D(1)γ ∕D

(1)α D

(1)γ =D

(2)α D

(2)α = D

(1)α D

(1)β




i=1Nijk


j=1(wjk∕w+k)pijk


k=1

sumJk

j=1wjkpijk
















minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)


(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α


gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1


=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α



D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k


Δ(1)D

=H(2)α minusH

(1)α

minussum


=I(2)α minusI

(1)α

minusTsum












pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES








































































(1)qDequiv

(

sum

S

i=1pqi

)1∕(1minusq)
















sum2

n=0nNinjk


sumS


sumJk

j=1N+jk


k=1N++k


(2)1D=exp

(

minussumS

i=1pi ln pi

)

=exp (H)

(3)2D=1∕

(

sumS

i=1p2i

)








αk for each















sumB





sumB









(4)qPD=

sumB

i=1Li

(

ai

T

)q1∕(1minusq)

(5)1PD= lim qrarr1

qPD=exp

[

minussumB

i=1Liai

Tln

(

ai

T

)]

equivT exp (I∕T)

(6)I=minussumB

i=1Liai ln ai


Hierarchical level

Diversity



(1)

βD(2)

β

2 Region D(2)α D

(2)

β=D

(2)γ ∕D

(2)α D

(2)γ =Dγ D

γ=D

(2)α D

(2)

β


(1)β

=D(1)γ ∕D

(1)α D

(1)γ =D

(2)α D

(2)α = D

(1)α D

(1)β




i=1Nijk


j=1(wjk∕w+k)pijk


k=1

sumJk

j=1wjkpijk
















minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)


(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α


gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1


=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α



D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k


Δ(1)D

=H(2)α minusH

(1)α

minussum


=I(2)α minusI

(1)α

minusTsum












pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES












































































sum2

n=0nNinjk


sumS


sumJk

j=1N+jk


k=1N++k


(2)1D=exp

(

minussumS

i=1pi ln pi

)

=exp (H)

(3)2D=1∕

(

sumS

i=1p2i

)








αk for each















sumB





sumB









(4)qPD=

sumB

i=1Li

(

ai

T

)q1∕(1minusq)

(5)1PD= lim qrarr1

qPD=exp

[

minussumB

i=1Liai

Tln

(

ai

T

)]

equivT exp (I∕T)

(6)I=minussumB

i=1Liai ln ai


Hierarchical level

Diversity



(1)

βD(2)

β

2 Region D(2)α D

(2)

β=D

(2)γ ∕D

(2)α D

(2)γ =Dγ D

γ=D

(2)α D

(2)

β


(1)β

=D(1)γ ∕D

(1)α D

(1)γ =D

(2)α D

(2)α = D

(1)α D

(1)β




i=1Nijk


j=1(wjk∕w+k)pijk


k=1

sumJk

j=1wjkpijk
















minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)


(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α


gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1


=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α



D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k


Δ(1)D

=H(2)α minusH

(1)α

minussum


=I(2)α minusI

(1)α

minusTsum












pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES




































































αk for each















sumB





sumB









(4)qPD=

sumB

i=1Li

(

ai

T

)q1∕(1minusq)

(5)1PD= lim qrarr1

qPD=exp

[

minussumB

i=1Liai

Tln

(

ai

T

)]

equivT exp (I∕T)

(6)I=minussumB

i=1Liai ln ai


Hierarchical level

Diversity



(1)

βD(2)

β

2 Region D(2)α D

(2)

β=D

(2)γ ∕D

(2)α D

(2)γ =Dγ D

γ=D

(2)α D

(2)

β


(1)β

=D(1)γ ∕D

(1)α D

(1)γ =D

(2)α D

(2)α = D

(1)α D

(1)β




i=1Nijk


j=1(wjk∕w+k)pijk


k=1

sumJk

j=1wjkpijk
















minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)


(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α


gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1


=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α



D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k


Δ(1)D

=H(2)α minusH

(1)α

minussum


=I(2)α minusI

(1)α

minusTsum












pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES
































































sumB





sumB









(4)qPD=

sumB

i=1Li

(

ai

T

)q1∕(1minusq)

(5)1PD= lim qrarr1

qPD=exp

[

minussumB

i=1Liai

Tln

(

ai

T

)]

equivT exp (I∕T)

(6)I=minussumB

i=1Liai ln ai


Hierarchical level

Diversity



(1)

βD(2)

β

2 Region D(2)α D

(2)

β=D

(2)γ ∕D

(2)α D

(2)γ =Dγ D

γ=D

(2)α D

(2)

β


(1)β

=D(1)γ ∕D

(1)α D

(1)γ =D

(2)α D

(2)α = D

(1)α D

(1)β




i=1Nijk


j=1(wjk∕w+k)pijk


k=1

sumJk

j=1wjkpijk
















minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)


(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α


gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1


=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α



D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k


Δ(1)D

=H(2)α minusH

(1)α

minussum


=I(2)α minusI

(1)α

minusTsum












pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES












































































minusSsum

i=1

pi++ lnpi++

equivexp

(

Hγ

)

PDγ =Ttimesexp

minusBsum

i=1

Liai++ lnai++

∕T

equivTtimesexp

(

Iγ∕T)


(2)

γ=PDγ

alpha D(2)α =exp

(

H(2)α

)

PD(2)

α=Ttimesexp

(

I(2)α ∕T

)

where H(2)α =

sum

k

w+kH(2)

αk

where I(2)α =

sum

k

w+kI(2)

αk

H(2)

αk=minus

Ssum

i=1

pi+k ln pi+k I(2)

αk=minus

Bsum

i=1

Liai+k ln ai+k

beta D(2)

β=D

(2)γ ∕D

(2)α PD

(2)

β=PD

(2)

γ∕PD

(2)

α


gamma D(1)γ =D

(2)α PD(

1)γ

=PD(2)

α

alpha D(1)α =exp

(

H(1)α

)

PD(1)α

=Ttimesexp(

I(1)α ∕T

)

where H(1)α =

sum

jk

wjkH(1)αjk

where I(1)α =

sum

jk

wjkI(1)αjk

H(1)αjk

=minusSsum

i=1


=minusBsum

i=1

Liaijk ln aijk

beta D(1)β

=D(1)γ ∕D

(1)α PD

(1)β

=PD(1)γ

∕PD(1)α



D=

HγminusH(2)α

minussum

k w+k lnw+k

Δ(2)

PD=

IγminusI(2)α

minusTsum

k w+k lnw+k


Δ(1)D

=H(2)α minusH

(1)α

minussum


=I(2)α minusI

(1)α

minusTsum












pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES







































































pil=

⎧

⎪

⎨

⎪

⎩

0 if εillt0

εil if 0le εille1

1 if εilgt1



















α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES














































































α =3777 IslandD(1)α =2775)












Level Diversity



(2)α =37773D

(2)

β=1290


(2)α D

(1)α =27752D

(1)β

=1361


2 Region Δ(2)

D=0290


=0153











(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES







































































(c) Z flabescens







(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES



































































(a)

Spe

cies

div

ersi

ty(a

)S

peci

esdi

vers

ity

(b)

Gen

etic

div

ersi

tyE

coru

scan

sG

enet

icdi

vers

ityc

orus

cans

(c)

Gen

etic

div

ersi

tyZ

flab

esce

nsG

enet

icdi

vers

ityyyyyZZZ

flabe

scen

s


Level Diversity



(2)α =8083D

(2)

β=1016


(2)α D

(1)α =7077D

(1)β

=1117


2 Region Δ(2)

D=0023


=0062










Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES






































































Level Diversity



(2)α =8290D

(2)

β=1012


(2)α D

(1)α =7690D

(1)β

=1065


2 Region Δ(2)

D=0014


=0033










ACKNOWLEDGEMENTS




ORCID



REFERENCES






































































ACKNOWLEDGEMENTS




ORCID



REFERENCES





























































diversity from genes to ecosystems: a unifying framework to...

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