ADDITIVE DIVERSITY PARTITIONING IN PALAEOBIOLOGY .
[Palaeontology, Vol. 53, Part 6, 2010, pp. 1237–1254]ADDITIVE DIVERSITY PARTITIONING INPALAEOBIOLOGY: REVISITING SEPKOSKI’SQUESTIONby STEVEN M. HOLLANDDepartment of Geology, University of Georgia, Athens, GA 30602-2501, USA; e-mail stratum@uga.eduTypescript received 17 October 2009; accepted in revised form 24 March 2010ecosystems, regions, and taxa, suggesting that large changesin beta diversity on evolutionary timescales are likely. Butthe question is which scales are the most important. Severalrecent palaeontological studies help to constrain beta diversity within sedimentary basins, and the emergence of samplebased databases puts an answer to Sepkoski’s question withinreach. A new method for calculating diversity partitions forrichness is introduced, which allows the calculation of eachspecies’ contribution to alpha and beta diversity, as well asthe contribution of each sampling unit to beta diversity.Abstract: Using Whittaker’s concepts of alpha, beta, andgamma diversity, Sepkoski asked how global diversity wasassembled at scales ranging from the community to theprovince. In the years since, ecologists have recast diversityin terms of additive partitions where total diversity can bedecomposed into sample-level alpha diversity plus the sumof a series of beta diversity terms that reflect progressivelylarger spatial scales. Given that marine alpha diversity represents a tiny fraction of global diversity, Phanerozoic globaldiversity patterns must be dominated by changes in betadiversity at one or more scales. A ballooning ecological literature demonstrates wide variation in beta diversity amongKey words: diversity, palaeoecology, marine.In two formative studies of vegetational changes alongenvironmental gradients in the United States, Whittaker(1956, 1960) distinguished three types of diversity. Alphadiversity, so named because it was measured with Fisher’salpha (Fisher 1943), was the diversity in an individualstand (sampling area) or community. Beta diversity wasproposed as a measure of the variation in taxonomiccomposition along an environmental gradient. Gammadiversity was the combined diversity for a range of environments present in a region. Since then, the concepts ofall three have changed. Alpha is now regarded as thespecies richness at the finest scale of observation, but isgenerally not measured with Fisher’s alpha. Gamma isgenerally the richness at the largest scale of observation.Beta diversity is commonly expressed as the ratio (Whittaker 1960, 1972) or the difference (Lande 1996) betweengamma and alpha diversity. Regardless of how it is measured, beta diversity reflects the increase in diversity atlarger spatial or temporal scales.The Phanerozoic record of alpha diversity has beenused to evaluate the biological significance of globaldiversity trends in the marine fossil record (Bambach1977). For example, the similar trajectories of genus-leveldiversity within collections and global family-level diversity were taken as evidence that global-scale patternswere not an artefact of available rock volume as hadbeen suggested (Raup 1972, 1976) and that they werebiologically meaningful (Sepkoski et al. 1981). As globalscale diversity patterns continue to be re-evaluated(Alroy et al. 2001, 2008; Peters and Foote 2001; Smith2001, 2007; Jablonski et al. 2003; Wagner et al. 2007;McGowan and Smith 2008; Miller and Foote 2009;Kiessling et al. 2010), patterns in alpha diversity are likewise being re-examined, particularly with regard to therole of taphonomy and lithification in perceptions ofdiversity (Powell and Kowalewski 2002; Bush and Bambach 2004; Kowalewski et al. 2006; Wagner et al. 2006;Hendy 2009).Ecological studies of the present consistently show thatbeta is a significant, if not the most significant, contributor to gamma diversity (e.g. Summerville and Crist 2005;Benedick et al. 2006; Gabriel et al. 2006; Kattan et al.2006; Veech and Crist 2007; Navarrete and Halffter 2008;Rodriguez-Zaragoza and Arias-Gonzalez 2008). Likewise,a simple comparison of global marine diversity (1000–2500 genera during most of the post-Cambrian; Sepkoski1997) and average diversity within habitats (20–60 speciesfor the richest habitat; Bambach 1977) demonstrates theenormous discrepancy between the two. Assuming a conservative average of three species per genus (Krug et al.2008), beta diversity must represent over 99 per cent ofthe global diversity signal. Understanding the Phanerozoicª The Palaeontological Associationdoi: 10.1111/j.1475-4983.2010.01017.x1237
1238PALAEONTOLOGY, VOLUME 53history of alpha diversity is important in its own rightbecause it sheds light on the changing structure ofecological communities (Kowalewski et al. 2006; Wagneret al. 2006), but the contribution of alpha diversity isfar too small to play a central role in the interpretationor evaluation of global diversity. Even if these studiesunderestimate alpha diversity (for example, manyNeogene samples in the Paleobiology Database exceed100 species), the gulf between gamma and alpha is stillgigantic. If we want to understand global diversity, wemust understand the sources and magnitudes of betadiversity.Valentine (1970, 1971; Valentine et al. 1978) proposedthat provinciality exerts the dominant control on globalmarine diversity and argued that the observed increase inPhanerozoic diversity reflected an increased number ofprovinces through geological time. Sepkoski (1988) followed with a challenging question about what drives global diversity patterns, namely to what extent do globalchanges in diversity reflect community-level changes(alpha diversity) or some aspect of beta diversity, such asonshore–offshore differentiation, the waxing and waningof specialised habitats like reefs and hardgrounds, or provinciality? What makes Sepkoski’s question particularlyinteresting is how he framed it, in particular, the idea thatdiversity might be decomposed into a series of partitionsand that some partitions might be substantially largerthan others.MEASURING BETA DIVERSITYIn his original conception of beta diversity, Whittaker(1956) proposed measuring the ecological similarity ofsamples with the Jaccard Coefficient (also called the coefficient of community), that is, the ratio of the number ofshared taxa to the total number of taxa in a pair of samples. To measure the beta diversity along a gradient, theecological similarity of an endpoint along the gradientcould be measured relative to successively more distantsamples along the gradient. At some gradient position,the similarity with the endpoint would be 50 per cent,marking a half-change in composition. Using this newpoint as the starting point, the locations of successivehalf-changes could be found farther along the gradient,until the total length of the gradient could be measuredin terms of the number of half-changes. Although Whittaker’s original formulation of beta was explicitly cast interms of position along an environmental gradient,numerous beta measures have since been proposed, manyof which measure the overlap in taxon compositionbetween samples or sets of samples, without regard toposition along a gradient (see Koleff et al. 2003 for athorough review of beta measures).Whittaker (1977) later recognised the multiple scales oftaxonomic variation among sampling units. In doing so, heredefined gamma diversity as diversity at the landscapescale and added provincial-scale differentiation (deltadiversity) and global-scale diversity (epsilon). Although theterms delta and epsilon never became widely accepted,ecologists held onto the concept that diversity could bemeasured at multiple scales and that diversity shouldincrease with the scale of a study. The diversity at any particular scale is an inventory (a) diversity, and the change indiversity between scales is known as turnover (b) diversity.Alpha, beta, and gamma diversity also have a strongrelationship to the shape of individual-based rarefactioncurves and sample-based rarefaction curves (also calledcollectors curves; Gotelli and Colwell 2001; Olszewski2004). Individual-based rarefaction curves describe taxonrichness as the number of individuals censused increases(Hurlbert 1971; Raup 1975). When all individuals from astudy area are pooled, their individual-based rarefactioncurve provides an estimate of gamma, or total, diversityover a spectrum of sampling intensity. If individuals arerandomly distributed among samples, a sample-basedrarefaction curve will be identical to an individual-basedrarefaction curve, indicating a lack of beta diversity.However, as individuals of taxa become increasinglyaggregated within samples, the two curves will diverge asbeta diversity increases.TWO APPROACHES TO DIVERSITYPARTITIONINGWhittaker (1960, 1972) related mean inventory diversityat a local scale (a) to the inventory diversity at a largerspatial scale (c) through a multiplicative factor, b:c¼ ab:Similarly, diversity at an even larger scale (e) would beequal to mean gamma diversity multiplied by a factor, d:e ¼ cd:In this way, diversity could be examined at a series ofprogressively larger spatial scales, although raising thequestion of whether codifying each scale of diversity witha Greek letter really adds to an understanding of diversity(cf. Rosenzweig 1995; Kowalewski et al. 2006).Although Sepkoski (1988) framed his paper in Whittaker’s terms, he measured beta diversity with the Jaccard coefficient to avoid the distorting effects of variablenumbers of samples. Even so, the point remains thatdiversity can be considered at a variety of spatial scales,with beta diversity reflecting the importance of variationat each scale in contributing to the total diversity of asystem.
HOLLAND: ADDITIVE DIVERSITY PARTITIONINGMacArthur et al. (1966), Levins (1968), Lewontin(1972) and Allan (1975) showed that diversity could alsobe partitioned additively rather than multiplicatively, withLande (1996) later making the explicit connection to thealpha, beta, and gamma diversities of Whittaker. In additive diversity partitioning, beta is redefined as the diversity that is added by examining a larger spatial scale, thatis, the difference between gamma and mean alpha diversity.richness, but also to Shannon’s H and Simpson’s D,which can provide a more informative picture of diversitychanges (Lande 1996). Finally, additive diversity partitionscan be recast as percentages of gamma diversity, allowingone to see not only the absolute changes in the size of apartition but also their relative (i.e. multiplicative)changes.An alternative approach to measuring additive partitionsc¼ aþbIn this formulation, beta diversity can be thought of asthe diversity among a set of samples that is missing froman average sample (Veech et al. 2002). Progressively largerspatial scales are easily incorporated. For a study considering local diversity and three successively larger spatialscales, global diversity (c) is þ b 1 þ b 2 þ bc¼ a3where mean a, mean b1, and mean b2 are the means overthe entire region (e.g. Okuda et al. 2004; Text-fig. 1).Additive diversity partitioning has several advantagesover multiplicative diversity partitioning (Veech et al.2002; Heim 2009). Foremost, alpha, beta and gammadiversity share the same units in additive diversity partitioning and can be compared directly, whereas beta anddelta are dimensionless ratios in the multiplicativeapproach (Lande 1996). Additive diversity partitioning isalso ideally suited for multiple hierarchical levels of sampling because the beta diversity at each level is measuredon the same scale (e.g. Wagner et al. 2000; Crist et al.2003; Okuda et al. 2004), whereas the beta and delta ofthe multiplicative approach cannot be compared directly.Additive diversity partitioning can be applied not only toAdditive partitions are typically calculated by first measuring gamma diversity, then finding mean alpha diversityand finally calculating their difference to give beta diversity. Alternatively, additive partitions may be calculatedby measuring the contribution of each taxon to alpha andbeta diversity. The contribution of the jth taxon to alphadiversity isnjaj ¼Nwhere nj is the number of sampling units that contain thejth taxon and N is the total number of sampling units.Similarly, the contribution of the jth taxon to beta diversity isn jbj ¼Nwhere n)j is the number of sampling units that do notcontain the jth taxon. Mean alpha diversity of a group ofsampling units is given by the sum of the alpha contributions from all taxa, with beta diversity among those sampling units being the sum of all beta contributions of thetaxa. The advantage of this approach is that it allows thecomputation of the contribution of each sampling unit tobeta diversity:83 15.323.717.2 208314.523.519 40652525 β2α2α127α34235α1β3α22821faciesα2taxonomic richness (S)T E X T - F I G . 1 . Schematic of samplingdesign of additive diversity partitioning,with four levels of inventory diversity(a1 through a3, and c) and three levelsof turnover diversity (b1 through b3).Values of diversity are shown as aworked example of the calculation ofdiversity partitions.123921.8172118222223212719282024
1240PALAEONTOLOGY, VOLUME 53bi ¼pXbjj ¼1njRichnessxijwhere i is the sampling unit, p is the total number of taxa(that is, gamma diversity), and xij is the presence (one) orabsence (zero) of taxon i in sampling unit j. Thisapproach could be used, for example, to determine whichhabitats contribute most to landscape-scale beta or whichprovinces contribute most to global-scale beta. R codefor implementing these calculations is included inAppendix S1.Using this formulation, a simple example illustrates howincreased diversity is accommodated. Suppose there areten sites, each with ten taxa unique to that site. Each sitehas an alpha diversity of 10, leading to a mean alpha diversity of 10. As the total (c) diversity is 100, the beta diversityis the difference, or 90. If one endemic taxon is added to asingle site, the alpha diversity of that site is now 11, butthe mean alpha diversity is 10.1. The total (c) is now 101,so beta diversity is 90.9 taxa. When that single taxon wasadded, it contributed 0.1 taxa to the alpha diversity and0.9 taxa to the beta diversity. If, instead, a single cosmopolitan taxon had been added, the alpha of all collectionswould be raised by one, increasing mean alpha diversity byone. Gamma diversity would likewise be raised by one,and as the difference of gamma and mean alpha, betawould show no change. In short, as taxa are added, theymay contribute partly to alpha and partly to beta, depending on how they are distributed among the sites. Samplingunits with a large number of endemic or restricted taxacontribute more to beta diversity than sampling unitsdominated by widespread or cosmopolitan taxa.DIVERSITY MEASURES FOR ADDITIVEPARTITIONINGAdditive diversity can be applied to any measure measured that exhibits strict concavity (Lewontin 1972; Lande1996). A diversity metric is strictly concave if the metricin a pooled set of samples is always greater than or equalto the average value of the metric among the samples;equality occurs when the samples are identical. Threediversity metrics exhibit strict concavity: richness, Shannon’s H, and Simpson’s D (Lande 1996). The term ‘diversity’ has a wide range of meanings among ecologists withlittle consensus. Some allow it to be used for any measurethat describes the number and abundance of taxa, butothers restrict it to simply a count of taxa. I use the termin the broader sense, as is typical of the additive diversitypartitioning literature, but I try to be specific about howdiversity is measured in any given study (cf. Spellerbergand Fedor, 2003).Richness (S), the number of taxa within a sample, isthe most intuitive and widely used metric for additivepartitioning. Richness weights rare taxa equally withabundant taxa and is therefore highly sensitive to sampling effort (Lande 1996). Even when sampling effort isheld constant, richness is generally dominated by raretaxa.Shannon’s HShannon’s H (also called Shannon information and Shannon entropy; Box 1) is defined asH ¼ nXpi logðpi Þi¼1BOX 1Shannon’s H was developed originally for a mathematical theory of communication and shared the form ofGibbs’ thermodynamic entropy (Gibbs 1902; equation450, p. 137), which has roots going back to Boltzmann(1866). Although Claude Shannon (1948) was the firstto present the metric in its familiar form, it is frequently also called the Shannon–Weaver index, theShannon–Wiener index, and the Shannon–Weinerindex (Spellerberg and Fedor 2003). Margalef (1957,translated from Spanish in Margalef 1958) introducedShannon’s metric to ecologists and attributed it toShannon alone. Warren Weaver’s name became associated with the metric because he edited the volume inwhich Shannon’s original paper appeared, and becausehe and Shannon subsequently co-authored a frequentlyreprinted book (Shannon and Weaver 1949) that discusses the metric. At least as early as 1964 (Lloyd andGhelardi 1964), Norbert Wiener’s name was linked tothe metric presumably because he authored an influential book on cybernetics (Wiener 1948), whichincluded an integral formulation of entropy (p. 76) inthe form of Gibbs’ (1902) and because Shannonacknowledges the influence of Weiner (Spellerberg andFedor 2003). Complicating matters, Norbert Wiener’sname is frequently misspelled as Weiner. In short,although many authors have published on entropy,Shannon (1948) was the first to present it in the formcommonly used by ecologists today. For this reason,the metric should be called Shannon’s H or ShannonEntropy, but not Shannon–Weaver, Shannon–Wieneror Shannon–Weiner.
HOLLAND: ADDITIVE DIVERSITY PARTITIONINGwhere pi is the proportion of a sample represented bytaxon i. The minimum value for Shannon’s H occurswhen all taxa but one have an abundance of one, andShannon’s H increases with richness and the total number of individuals. Shannon’s H reaches a maximum atln S, when all taxa are equally abundant. Because Shannon’s H is based on proportions rather than a simplecount of taxa, it is less dominated by the effects of raretaxa than is richness. Shannon’s H is an entropy measure, but can be converted to a diversity measure bythe transformation, Shannon diversity exp(H) (Jost2006).Simpson’s DSimpson’s concentration (k) is related to the probability that any two randomly selected individuals froma population belong to the same taxon (Simpson1949):k¼nXpi2 :i¼1The minimum value of lambda is 1 S, and it occurs whenall taxa are equally abundant. Lambda reaches a maximum approaching one when most individuals belong to asingle taxon. As a result, the inverse of Simpson’s concentration is often used as a measure of diversity (Jost 2006).Because common taxa contribute much more to lambdathan do rare taxa, Simpson’s lambda is a measure ofdominance.D ¼1 kSimpson’s D is a measure of evenness equal to and is alsoknown as the Gini Coefficient (Lande 1996). Hurlbert’sPIE is an unbiased estimator of the probability of interspecific encounter and is equal to Simpson’s D multipliedby N N ) 1, where N is the number of individuals (Hurlbert 1971; Olszewski 2004). Of the three measures suitablefor additive diversity partitioning, Simpson’s D is the leastsensitive to rare taxa. Simpson’s D is also the least biasedat small sample sizes and has the smallest standard error(Lande 1996).Richness, Shannon’s H, and Simpson’s D can be usedin concert to understand how all parts of an abundancedistribution contribute to diversity in the broad sense.Richness provides information on the rare tail of thetaxon abundance distribution, whereas Simpson’s Dreflects the abundant end of the distribution and Shannon’s H is an intermediate measure.These and other diversity measures can be performedwith the vegan package for R (Oksanen et al. 2009) and1241with the software package PAST (Hammer and Harper2005).THREE ECOLOGICAL QUESTIONSAmong ecologists, additive diversity partitioning hasraised three classes of questions. At the most basic level isa discussion about pattern: how is landscape-scale diversity assembled, that is, does it primarily reflect high community-level diversity or strong differentiation? Second,do the observed relative sizes of partitions reflect anythingmore than random sampling? Finally, and most importantly, what ecological processes control the sizes of diversity partitions?What are the important sources of diversity?A survey of modern landscape-scale studies of additivediversity partitioning demonstrates consistently high values of beta in trees, herbs, insects, aquatic invertebrates,fishes, and birds (Table 1). Although in some cases, betamay be as low as 25 per cent, total beta is commonly 70 per cent. Time and resources prevent comprehensivesampling at the largest spatial scale in these studies, andas a result, gamma (total) diversity is almost certainlyunderestimated. Increased numbers of sample localitieswould not likely increase mean alpha diversity, but theywould increase beta, making it likely that all of thesebetas are minimum estimates.Many of these studies sampled at four spatial scales,corresponding approximately to a site (quadrat, for example), a habitat, a location composed of several habitatsand a region or landscape containing multiple locations.Patterns in beta diversity across these scales are inconsistent. For example, beta diversity among sites is largerthan beta diversity among habitats in only about half ofthe studies. The same is true when comparing betaamong habitats to beta among locations. This heterogeneity is an important feature, and it may reflect differencesin the precise scaling of measurements in the study areas,differences in the type of ecosystem studied or organismaldifferences. Regardless of its origin, there appears to beno simple rules to the size of beta at progressively largerscales.Many studies have also compared spatial turnover withtemporal turnover (usually seasonal, but sometimesannual) to estimate the magnitude of temporal changeand to scale it against a spatial factor. Again, results aremixed. In some cases, increased spatial coverage yieldsmore diversity than longer temporal coverage (e.g. Tylianakis et al. 2005; Boonzaaier et al. 2007; Sobek et al.2009). Others find that temporal turnover is an important
1242PALAEONTOLOGY, VOLUME 53TABLE 1.Percentage contribution of diversity partitions to total (c) diversity for a range of modern-day organisms and habitats.Studyalphabeta micro-scalebeta meso-scalebeta macro-scaleForest plants, ItalyChiarucci et al. 2008Forest plants, United StatesChandy et al. 2006Agricultural weeds, GermanyGabriel et al. 2006Prairie plants, United StatesWilsey et al. 2005Butterflies on farms, SwedenRundlof et al. 2008Butterflies, Great Basin,United StatesFleishman et al. 2003Agricultural arthropods, EuropeHendrickx et al. 2007Insects, midwestern United StatesGering et al. 2003Aquatic invertebrates, SwedenStendera and Johnson 2005Reef fish, CaribbeanRodriguez-Zaragoza andArias-Gonzalez 2008Riverine fish, HungaryEros 2007Riverine fish, United StatesPegg and Taylor 2007Birds, United StatesVeech and Crist 2007Birds, Great Basin, United StatesFleishman et al. 2003Coastal wetland birds, BrazilGuadagnin et al. 20053 subplot4 among subplots31 among plots62 among regions44 plot24 among plots12 among subregions3–16 plot5–16 among plots20 among researcharea17–37 among fields25 site75 among sites67–73 field27–33 among field40 site21 among canyonsegments25 among canyons14 amongmountain ranges40–51 local49–60 within landscape9 tree29 among sites5–7 site11 21 among trees among stands44–46 among sites10 site20 among sites20 among habitats29 amongecoregions32–33 amongbioregions50 among reefs21 site37 among sites41 between habitat17–36 site16–23 among sites17–29 among reaches13 point31 among points56 within landscape13 site18 among canyonsegments94 38 among site(fragments only lagoons included)36 among canyons6 62 site (fragmentsonly lagoonsincluded)component of beta diversity (Devries et al. 1999; Summerville and Crist 2005; Hirao et al. 2007). The amountof beta diversity varies seasonally in some cases (Murakami et al. 2008), but doesn’t in others (Walla et al. 2004).Such heterogeneity is important because it suggests thatbeta diversity at various scales may change over geologictime as taxonomic composition and ecological structurechange within communities, landscapes and provinces.Several studies have contrasted the role of beta diversityin terms of taxonomic diversity and functional diversity.Two important recent studies reach the same conclusionthat beta diversity may be high in taxonomic terms, butmay be much lower in terms of functional diversity, suggesting a high degree of functional redundancy amongtaxa (Ricotta and Burrascano 2008; de Bello et al. 2009).From an evolutionary viewpoint, this contrast betweentaxonomic and functional diversity raises the questions ofhow and why functional redundancy within ecosystemshas varied over geologic timescales.16 among ecoregions14–25 among region33 among mountainrangesIs the signal statistically different from random?Because sampling at any level within a hierarchy willnever completely capture all of the diversity at that level,it is possible that observed values of beta diversity merelyreflect the chance effects of sampling rather than true ecological structure. Several approaches to this problem havebeen developed, including anova-like analyses (Couteronand Pelissier 2004) and analytical formulas for confidenceintervals (Kiflawi and Spencer 2004). By far the mostcommon approach is randomisation (e.g. Crist et al.2003, Veech 2005; Freestone and Inouye 2006; Cornellet al. 2007; Deblauwe et al. 2008; Ribeiro et al. 2008).Randomisation tests can be either individual-based orsample-based and can be performed with the PARTITION software (Crist et al. 2003).Individual-based randomisation tests whether theobserved diversity partitions could have been generatedby a random distribution of individuals at the lowest level
HOLLAND: ADDITIVE DIVERSITY PARTITIONINGof the sampling hierarchy. To accomplish this, individualsat the lowest level are randomly shuffled among all of thesamples. This shuffling is done while preserving the totalnumber of individuals in each taxon and the total number of individuals in each sample, that is, randomisingwithout replacement. The number of sampling units ateach level of the hierarchy is maintained as in the originalsampling scheme. After randomisation, the sizes of alldiversity partitions are calculated. This process is repeated1000–10 000 times to generate mean values and confidence limits for the size of each diversity partition. Theobserved partitions are compared to the randomised partitions to identify partitions that contain either more orless diversity than would be predicted by a random distribution of individuals among samples. Because thisapproach involves one randomisation for the entire sampling hierarchy, the partitions in any one randomisationare additive, just as they are in the original sampling.Sample-based randomisation tests whether the observedpartitions reflect sampling design. In it, samples at thei)1 level are randomised relative among samples at the ilevel for all that comprise the same sample at the i)1level. For example, suppose insects were sampled atthe levels of a single tree, a stand of trees, a forest, aregion and a continent. To assess beta diversity at thelevel of a stand of trees, trees would be randomly shuffledamong stands found within a single forest. This randomisation and calculation of beta at this level would berepeated 1000–10 000 times to provide null estimates ofbeta and its confidence interval, which would be compared to the observed estimate of beta. This processwould be repeated separately for each level in the hierarchy. Because the randomisation at each level is performedseparately, the diversity partitions among all of the levelsare not additive as they are in the original data or in anindividual-based randomisation.Although richness is strongly controlled by sample size(Hurlbert 1971; Raup 1975), rarefaction is not required inthese randomisation tests or in the original calculation ofthe diversity partitions because alpha diversity is weightedby sample size, such that larger samples contribute morestrongly to the estimate of diversity than do smaller samples (Crist et al. 2003).What are the underlying controls on beta diversity?Beta diversity reflects the differences in taxonomic composition among sampling areas. Identifying the processesresponsible for beta diversity requires comparison tosome external data, such as the environmental characteristics of sites or geographical distances among them.Comparison with ordinations (e.g. Eros 2007) and regression against external variables (e.g. Hofer et al. 2008) are1243useful techniques for establishing the controls on betadiversity. Although chance has an effect on beta diversity,niche differences and environmental heterogeneity, as wellas geographical distance and dispersal limitation, playdominant roles.Most modern ecological studies of additive diversitypartitioning have focussed on habitat patches within landscapes, and they have most commonly attributed betadiversity to niche differences and environmental heterogeneity (e.g. Stendera and Johnson 2005; Gabriel et al. 2006;Eros 2007; Buckley and Jetz 2008). Because habitatpatches are heterogeneous and because taxa differ in allaspects of their niches (optimum, breadth and maxima),patches differ in their taxonomic composition and in therank abundances of taxa (Kattan et al. 2006). In terrestrialstudies, which dominate research into diversity partitioning, physical factors are often implicated and include suchfactors as canopy coverage and soil temperature (Navarrete and Halfft
Beta diversity is commonly expressed as the ratio (Whit-taker 1960, 1972) or the difference (Lande 1996) between gamma and alpha diversity. Regardless of how it is mea-sured, beta diversity reflects the increase in diversity at larger spatial or temporal scales. The
tion diversity. Alpha diversity Dα measures the average per-particle diversity in the population, beta diversity Dβ mea-sures the inter-particle diversity, and gamma diversity Dγ measures the bulk population diversity. The bulk population diversity (Dγ) is the product of diversity on the per-particle
Oracle's approach to "partitioning" LMS's interpretation of "installed and/or running" "Soft partitioning" vs. "hard partitioning", and Oracle's "Partitioning Policy" Oracle in VMware and the Mars case Case background - Oracle initiated an audit of Mars and requested deployment details for the
Oracle 12c R1 Interval-Reference partitioning Partition Maintenance on multiple partitions Asynchronous global index maintenance Online partition MOVE, Cascading TRUNCATE, Partial indexing Oracle 12c R2 Auto-list partitioning Multi-column list [sub]partitioning Online partition maintenance operations Online table conversion to partitioned table
diversity of the other strata. Beta (β) Diversity: β diversity is the inter community diversity expressing the rate of species turnover per unit change in habitat. Gamma (γ) Diversity : Gamma diversity is the overall diversity at landscape level includes both α and β diversities. The relationship is as follows: γ
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Nov 17, 2015 · Additive diversity partitioning means that mean alpha and beta diversities add up to gamma diver- sity, thus beta diversity is measured in the same dimensions as alpha and gamma (Lande 1996). This additive procedure is then extended across multiple scales in a
Gamma diversity is regional diversity – or species diversity in a region of interest. Estimated by pooling observations from a large number of sites in the area and computing an alpha diversity index. Robert Whittaker (1960, 1972). 1. Whittaker’s alpha, beta and gamma diversities
graduation, positioning them for college and careers. The grade level ELA standards begin in the Prekindergarten and Elementary ELA Standards section. Please see the introduction for more about how the anchor standards and grade level standards connect. New York State Education Department ENGLISH LANGUAGE ARTS LEARNING STANDARDS (2017) 4 NEW YORK STATE EDUCATION DEPARTMENT Reading Anchor .
