Partitioning Beta Diversity In Landscape Ecology And Genetics
Partitioning beta diversityin landscape ecology and geneticsPierre LegendreDépartement de sciences biologiquesUniversité de MontréalWorkshop on Mathematics for an evolving biodiversityCentre de recherches mathématiques, Université de Montréal, September 19, 2013
Outline of talk1. Whittaker’s alpha, beta and gamma diversities2. Measuring beta by a single number: different approaches3. BDTotal, SCBD and LCBD4. Landscape ecology example5. Compute BD from a dissimilarity matrix6. Calculation summmary7. Properties of D matrices for beta assessment8. Multiple ways of partitioning BDTotal9. Landscape genetics example10. Conclusion
1. Whittaker’s alpha, beta and gamma diversities Alpha diversity is local diversity –or species diversity at a site. Estimated byspecies richness or by one of the alphadiversity indices. Beta diversity is spatial differentiation –or the variation in species compositionamong sites within a region of interest. Gamma diversity is regional diversity –or species diversity in a region of interest.Estimated by pooling observations from alarge number of sites in the area andcomputing an alpha diversity index.Robert Whittaker (1960, 1972).
123Species. . . !i N0, H1, N2, .123.SitespDiversity levelsY communitycompositiondata# variation in species compositionamong sitesnSumsLegendre & Legendre Numerical ecology (2012, Fig. 6.3). " N0, H1, N2, .
2. Measuring beta by a single number:different approachesStudies of beta diversity may focus on two aspects of communitystructure, distinguishing two types of beta diversity – The first is turnover, or the directional change in communitycomposition from one sampling unit to another along a predefinedspatial, temporal, or environmental gradient. Measure dissimilaritiesbetween neighbouring points along the gradient and relate thechanges to the gradient values (positions in space, time, or other). The second is a non-directional approach to the study ofcommunity variation through space. It does not refer to any explicitgradient but simply focuses on the variation in communitycomposition among the sampling units.Vellend (2001), Legendre et al. (2005), Anderson et al. (2011).
Non-directional beta diversity (Whittaker 1960, 1972)can be summarized by a single number – Computed as " S/# or log( " ) log(S) – log(# )where S number of species in the larger area of interest (γ diversity)and ! is the mean number of species at the sampling sites.β indicates how many more species are present in the regionthan at !an average site!within the region. Or from the Sites Species data table Y: Total sum of squares in thecommunity composition table, SS(Y) Total variance in the data table: BDTotal Var(Y) SS(Y)/(n–1)Many other beta diversity indices are reviewed in Koleff et al.(2003), Anderson et al. (2011), and other papers.
3. BDTotal, SCBD and LCBD1. Centre data table Y by columns, then square the values:2&#S sij % yij " y j ( ')SpeciesSpecies[yij][sij] [yc.ij2] S Sites!Y Sites[ ] (2. Sum all values in matrix S to obtain SS(Y):SSTotal SS(Y) "ni 1p" j 1 sij3. Divide by the degrees of freedom (n – 1) to obtain Var(Y):!BDTotal Var(Y) SSTotal / (n–1)
Note 1 – These equations should not be computed directly on rawspecies abundance or biomass data.Reason: this calculation assumes that the Euclidean distancecorrectly represents the relationships among sites. However, theEuclidean distance is inappropriate and should not be used for betadiversity assessment (Section 7 of the talk).
Note 2 –Community composition data should be transformed in someecologically meaningful way before BDTotal is calculated. The chordand Hellinger transformations1 are appropriate because – chord tranformation:yij′ yijp j 1 yij2chord-tranformed data Euclidean distance chord distanceyij′ Hellinger transformation: yij yi Hellinger-tranformed data Euclidean distance Hellinger distanceSection 7 of the talk will show that the chord and Hellinger distancesare appropriate for beta diversity assessment.1 Legendre& Gallagher (Oecologia 2001)
Note 3 – Are BDTotal statistics comparable?BDTotal statistics computed with the same index are comparable – among taxonomic groups observed at the same sites in a geographicarea of interest, among study areas represented by data sets having the same ordifferent numbers of sampling units (n), for a given taxonomic group,provided that the sampling units are of the same size or represent thesame sampling effort.
An advantage of conceiving beta as the total variation in Y is thatSSTotal can be decomposed into species and site contributions.1. Local Contributions to Beta Diversity (LCBD) are computed as thesum of the values of S in each row i :LCBDi "psj 1 ijSSTotal LCBD values represent the degree of uniqueness of the samplingunits in terms of community composition.!
An advantage of conceiving beta as the total variation in Y is thatSSTotal can be decomposed into species and site contributions.1. Local Contributions to Beta Diversity (LCBD) are computed as thesum of the values of S in each row i :LCBDi "psj 1 ijSSTotal LCBD values represent the degree of uniqueness of the samplingunits in terms of community composition.2. Species !Contributions to Beta Diversity (SCBD) are computed asthe sum of the values of S in each column j :SCBD j "nsi 1 ijSS Total Species with high SCBD values have high abundances at a fewsites, hence high variance.!
Small numerical example – 7 fish species at 11 sites along a riverTRU VAI LOC CAR TAN GAR 253200121455210112455230000012240000025250000013
Small numerical example – 7 fish species at 11 sites along a riverTRU VAI LOC CAR TAN GAR Total SS(Y) 5.301BDTotal Var(Y) 0.5301!!1 Afterchord transformation of the abundance data.
Small numerical example – 7 fish species at 11 sites along a riverTRU VAI LOC CAR TAN GAR BD!!LCBDSSTotal SS(Y) 5.30BDTotal Var(Y) 0.530!
LCBD indices can be tested for significance by random,independent permutations of the columns of Y. Example of apermutation of Y:Matrix Y permutedMatrix YSp.1Site.1 0Site.2 1Site.3 2Site.4 3Site.5 4Site.6 5Site.7 6Site.8 7Site.9 8Site.10 1Site.1 7Site.2 4Site.3 2Site.4 0Site.5 9Site.6 8Site.7 3Site.8 5Site.9 1Site.10 p.433363532343037313938R command to produce a random permutation:Y.perm - apply(Y, 2, sample)Or use a permutation method that preserves spatial correlation.Sp.541404446494845424347
LCBD: squared distance to the centroid in an ordination diagram.The sites near the centre are not exceptional in species combination.
4. Full landscape ecology exampleFish observed at 29 sites along the Doubs river, a tributary of theSaône running near the France-Switzerland border in the JuraMountains, eastern France. Data from Verneaux (1973), available athttp://adn.biol.umontreal.ca/ numericalecology/numecolR/the Web page of Numerical ecology with R (Borcard et al. 2011). Analysis of the chord-transformed fish abundance data:SSTotal SS(Y) 15.243BDTotal Var(Y) 0.544
Entre Laissey et Deluz, peu avant bs (rivière)Les sources du Doubs à Mouthe
Map of LCBD, Doubs River fish150100Downstream50y coordinates (km)200Site 23*0Upstream050Site 1**100150200250LCBD: uniqueness of community composition at each site.
Map of LCBD, Doubs River fishSpecies with high SCBD:150Brown trout/Truite brune (Salmotrutta), Eurasian minnow/Vairon(Phoxinus phoxinus) and stoneloach/Loche franche (Nemacheilusbarbatulus) in oligotrophic sitesupriver100Downstream50y coordinates (km)200Common common bleak/Ablette(Alburnus alburnus) abundant ineutrophic sites mid-river0Upstream050100150200250Two signif. LCBD (sites 1, 23) after correction for multiple testing.Regression of LCBD on environmental variables: LCBD positivelyrelated to slope of the riverbed and BOD; adjusted R2 0.58.
5. Compute BD from a dissimilarity matrixWhittaker (1972) –Beta biversity can be computedfrom a dissimilarity matrix Presence-absence data:Jaccard or Sørensen coefficient, or β computed for pairs of sites. Quantitative community composition data: Odum percentagedifference, Hellinger, chord or chi-square distance, etc. Whittaker (1972): the mean of the dissimilarities is another singlenumber index of beta diversity.
Compute SSTotal from the upper triangular portion of a dissimilaritymatrix1 :SSTotal1 n#1 n2 SS(Y) " "Dhin h 1 i h 12 is Euclidean:For D that are not Euclidean but D(0.5) Dhi !SSTotal1 n 1 n SS(Y) Dhin h 1 i h 1Then, compute BDTotal :BDTotal 1 Proof SS Total Var(Y) n "1in Legendre & Fortin (2010, Appendix 1).!
Compute LCBD from a dissimilarity matrix –Compute Gower-centred matrix G containing the centred dissimilaritiesin principal coordinate analysis (PCoA; classical or metric scaling)1: 11# '11# '2 G &I " ) "0.5Dhi &I " )%%n (n ([] The LCBD values are the diagonal elements of G divided by SSTotal!diag(G)[LCBDi ] SSTotalLCBD indices can be computed and tested for significance. SCBD cannot! be computed from a dissimilarity matrix.1 Legendre& Legendre, Numerical ecology (2012), eq. 9.42.
Range of values of BDTotalAll dissimilarity functions used to analyse beta diversity have amaximum value (Dmax), reached when two sites have completelydifferent community compositions. For example, the Hellinger and chord distances have a minimumvalue of 0 and a maximum of 2 . If all sites in Y have the exact same species composition, alldistances in D are 0 and!n"1n1Var(Y) Dhi2 0##n(n "1) h 1 i h 1 If all sites in Y have entirely different species compositions, alln(n – 1)/2 distances in D are 2 and!1 # n(n "1) 2 &Var(Y) 2 ( 1%'n(n "1) 2!For these distances, BDTotal is in the range [0, 1].!
Dissimilarity indices with Dmax 1 have maximum BD 0.5 whenall sites have different species compositions. Hence the range of theirBDTotal values is [0, 0.5].For these distances, multiply BD by 2 to produce normalized BDvalues in the range [0, 1].
6. Calculation summary
7. Properties of D matrices for beta assessmentBasic necessary propertiesP1 – Minimum of zero, positiveness: D 0.P2 – Symmetry: D(x1,x2) D(x2,x1).P3 – Monotonicity to changes in abundance: D increases whendifferences in abundance increase.P4 – Double-zero asymmetry: D does not change when adding doublezeros but D changes when double-X are added where X 0.P5 – Sites without species in common have the largest D.P6 – D does not decrease in series of nested species assemblages.Comparability between data setsP7 – Species replication invariance.P8 – Invariance to measurement units, e.g. for biomass data.P9 – Existence of a fixed upper bound, Dmax.
Additional properties useful in some studies.Sampling issuesP10 – Invariance to the number of species in each sampling unit.P11 – Invariance to the total abundance in each sampling unit.P12 – Coefficients with corrections for undersampling.Ordination-related propertiesP13 – D or D(0.5) [D0.5] is Euclidean. PCoA ordinations withoutnegative eigenvalues and complex axes.P14 – Dissimilarity function emulated by transformation of the rawfrequency data followed by Euclidean distance.Example: the chord distance can be computed by applying the chord transformationto the community composition data, followed by calculation of the Euclideandistance. The Hellinger, chord, profile and chi-square distances have that property.
#1–9: Necessary properties for beta assessment
The following 11 coefficients are appropriate for BD studiesType II: Hellinger and chord distances. They justify the application ofthe Hellinger and chord transformations to raw abundance data anddirect calculation of BDTotal, followed by partition analysis (next slide).Type III: Canberra, Whittaker, divergence, percentage difference (aliasB-C), Wishart, Kulczyinki.Type IV: Abundance-based quantitative forms of Jaccard, Sørensenand Ochiai coefficients with corrections for undersampling.The following 5 coefficients are inappropriateType I: Euclidean, Manhattan, modified mean character difference,species profiles.Type V: The chi-square distance.
8. Multiple ways of partitioning BDTotal1. Partition BDTotal among species (SCBD) and among sites (LCBD).2. Multivariate analysis of variance (MANOVA) using a single factorpartitions the SSTotal into within- and among-group sums of squares.In MANOVA involving two or several crossed factors, SSTotal ispartitioned SSTotal among the factors and their interaction.3. Partition SSTotal by simple and canonical ordinations, e.g. PCA andredundancy analysis (RDA).4. SSTotal can be partitioned with respect to two or more matrices ofexplanatory variables by variation partitioning (Borcard & Legendre1992, 1994).5. SSTotal can be partitioned as a function of different spatial scales byspatial eigenfunction analysis (MEM, AEM), multivariate variogram,and multiscale ordination analysis (Wagner 2003, 2004).
ReferenceLegendre, P. and M. De Cáceres. 2013. Betadiversity as the variance of community data:dissimilarity coefficients and partitioning.Ecology Letters 16: 951-963.PDF available on:http://adn.biol.umontreal.ca/ numericalecology/Reprints/
9. Landscape genetics exampleFreshwater snail Drepanotrema depressissimumin a fragmented landscape of tropical pondsin Grande-Terre, Guadeloupe. Microsatellite data from Lamy et al. (2012)Photo: Jean-Pierre PointierSee also r Ecology (2012) 21, 1394–1410doi: 10.1111/j.1365-294X.2012.05478.xT esti ng metapop u lation d y nam ics usi ng genetic,demograp h ic an d ecological dataT . L A M Y , * J . P . P O I N T I E R , † P . J A R N E * and P . D A V I D **UMR 5175 CEFE, campus CNRS, 1919 route de Mende, 34293 Montpellier cedex 5, France, †USR 3278 CNRS-EPHE,52 avenue Paul Alduy, 66860 Perpignan cedex, France
Landscape genetics exampleDrepanotrema depressissimum (Gastropoda, Planorbidae)Microsatellite data from Lamy et al. (2012) 25 populations in ponds, rivulets, & swamp grasslands, Guadeloupe 749 individual snails were genotyped (diploids) 10 microsatellite loci, with a mean of 34 alleles per locus LCBD analysis through the genetic chord distance:BDTotal Var(Y) 0.197
LCBD map, freshwater snailsPisPorte lGefBamCouMamL’HenrietteRejRocPointe des ChateauxBloVeeKanPouTitFour sites have significant LCBD indices, indicating the mostgenetically unique populations.
Sites with high LCBD values indicate the most genetically uniquepopulations. Something happened to create exceptional allelecombinations. What was it?Regression tree analysis of LCBD values on environmental variables(pond size, vegetation cover, connectivity, and temporal stability)showed that the four sites with high LCBD are ponds – where temporal stability is the lowest (sites regularly dry up),causing loss of alleles through random sampling (genetic drift), and connectivity is low with neighbouring ponds (no connection atall), preventing migration of snails from adjacent areas.Snails can survive in dessicated ponds by aestivating in the sediment.These mechanisms reduced the gene pool of these four populations toa few alleles per locus.
10. ConclusionBeta diversity (BD) is the spatial variation in community – or geneticcomposition – among sites in a geographic region of interest. BD can be estimated in various ways. The estimator described andused in this talk is the variance of the community composition data,Var(Y). BDTotal Var(Y) is a general, flexible index of beta diversity. BDTotal can be computed either from the [transformed] raw data orfrom a dissimilarity matrix. At least 11 dissimilarity coefficients are appropriate for beta diversitystudies. BDTotal can be decomposed into SCBD and LCBD (– maps). BDTotal Var(Y) links beta diversity to all well-known methods ofmultivariate analysis of community composition data.
Do you have questions?
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
0.9721 0.014. This value of beta is 33% lower than the antarctic beta diversity. The additive beta definition fails to rank these data sets correctly because the beta it produces is confounded with alpha. (When diversity is high, Gini-Simpson alpha and gamma both approach unity. Therefore if beta is defined as gamma minus alpha, beta must .
Beta diversity is a different sort of measure from alpha and gamma diversity because it focuses not only on counting species within a plot but also on comparing the species counts between the different plots in the forest. Beta diversity is a between-plots measure.5 To find the beta diversity between the first and the second plots, we
local diversity (alpha diversity) and the complement of species composition among sites within the region (beta diversity), and how these diversities contribute to regional diversity (gamma diversity) [35, 37]. The influence of alpha and beta diversities on gamma diversity is an essential aspect of local and landscape level conservation plans [38].
Jacksonville, FL ALPHA DELTA Miami, FL ALPHA EPSILON Atlanta, GA ALPHA THETA New Orleans, LA ALPHA OMICRON Tampa, FL ALPHA PHI Pensacola, FL BETA BETA Montgomery, AL BETA ETA Memphis, TN BETA KAPPA West Palm Beach, FL BETA XI Orlando, FL SOUTHEAST REGION CHAPTERS BETA SIGMA Lakeland, FL BETA PSI Knoxville, TN
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
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
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
Accounting for the change in mobility 12 6. Conclusion 13 Notes 15 Tables 16 References 23 Appendices 26. Acknowledgments Jo Blanden is a Lecturer at the Department of Economics, University of Surrey and Research Associate at the Centre for Economic Performance and the Centre for the Economics of Education, London School of Economics. Paul Gregg is a Professor of Economics at the Department of .
