Replacement and richness difference
components of beta diversity: new interpretation methods
Pierre Legendre
Département de sciences biologiques Université de Montréal
[email protected]
Genomes to Biomes Conference
Centre Mont-Royal, Montréal, May 29, 2014
Outline of the talk
1. Whittaker’s alpha, beta and gamma diversities
2. Compute total beta diversity (BDTotal)
3. Partitioning D into Repl and RichDiff components
4. Case study: Doubs River fish data
5. Conclusion
Funding: NSERC discovery grant
1. Whittaker’s alpha, beta and gamma diversities
• Alpha diversity is local diversity –
or species diversity at a site. Estimated by
species richness or by one of the alpha
diversity indices (richness, Shannon, Simpson).
• Beta diversity is spatial differentiation –
or the variation in species composition
among sites within a region of interest. • 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).
123
Species
. . .
p
=> αi = N0, H1, N2, ...
1
2
3
.
.
.
Sites
Diversity levels
Y = community
composition
data
β = variation in
=> species composition
among sites
n
Sums
Legendre & Legendre Numerical ecology (2012, Fig. 6.3).
=> γ = N0, H1, N2, ...
2. Compute total beta (BDTotal) as the total variance in the data • either from the transformed community data table
• or from an appropriate dissimilarity matrix
Transformed
species
composition
Squared
differences from
column means
Ytr. nxp
Snxp
Total
dispersion
Total beta
diversity
SSTotal
BDTotal
Species
composition
Ynxp
Dnxn
Dissimilarities among
sampling units
Talk at CSEE 2013, Kelowna. – Legendre & De Caceres, Ecology Letters 2013.
2. Compute total beta (BDTotal) as the total variance in the data • either from the transformed community data table
• or from an appropriate dissimilarity matrix
Transformed
species
composition
Squared
differences from
column means
Ytr. nxp
Snxp
Species
composition
Ynxp
Total
dispersion
Total beta
diversity
SSTotal
BDTotal
SSTotal =
Dnxn
Dissimilarities among
sampling units
1 n−1 n
∑ ∑ Dhi
n h=1 i=h +1
€
Talk at CSEE 2013, Kelowna. – Legendre & De Caceres, Ecology Letters 2013.
3. A different approach
=> Partition the dissimilarities D into Replacement and Richness
difference components.
Two different methods have been proposed: (1)  by Baselga (in Spain) and (2)  by Podani (in Hungary) and coauthors. Each group proposed ways of partitioning the Jaccard and Sørensen
dissimilarity indices for presence-absence data, as well as their
quantitative forms, the Ružička and percentage difference indices.
=> I will focus on the Podani approach. I will describe how these
new indices can be interpreted through graphical analysis.
The dissimilarity D between two sites contains two components:
• replacement (turnover): environmental gradients, competition … • and richness difference due to difference in diversity of niches
causing species thinning, or other processes (e.g. physical barriers). ;<&0<(/(24
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Podani-family indices, presence-absence data
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Podani-family indice, species abundance data
A = abun. common to sites 1 and 2 = A1+A2+0+A4
B = abundances unique to site 1 = 0+B2+B3+0
C = abundances unique to site 2 = C1+0+0+0
C1
B2
A4 A4
B3
A1 A1
A2 A2
Site1 Site2
Site1 Site2
Site1 Site2
Site1 Site2
Spec.1
Spec.2
Spec.3
Spec.4
A = abun. common to sites 1 and 2 = A1+A2+0+A4
B = abundances unique to site 1 = 0+B2+B3+0
C = abundances unique to site 2 = C1+0+0+0
C1
B2
A4 A4
B3
A1 A1
A2 A2
Site1 Site2
Site1 Site2
Site1 Site2
Site1 Site2
Spec.1
Spec.2
Spec.3
Spec.4
A = abun. common to sites 1 and 2 = A1+A2+0+A4
B = abundances unique to site 1 = 0+B2+B3+0
C = abundances unique to site 2 = C1+0+0+0
C1
B2
A4 A4
B3
A1 A1
A2 A2
Site1 Site2
Site1 Site2
Site1 Site2
Site1 Site2
Spec.1
Spec.2
Spec.3
Spec.4
4. Case study:
Doubs River fish data
Fish caught at 29 sites along the Doubs river, a tributary of the Saône
River running near the France-Switzerland border in the Jura
Mountains in eastern France. • Data from Verneaux (1973), available on the Web page of the book
Numerical ecology with R (Borcard et al. 2011):
http://adn.biol.umontreal.ca/~numericalecology/numecolR/
Entre Laissey et Deluz, peu avant Besançon
Besançon
http://fr.wikipedia.org/wiki/Doubs_(rivière) Les sources du Doubs à Mouthe
Decomposing dissimilarity matrices
D
Dissimilarities among
sampling units
Repl
RichDiff
Replacement
among sampling units
Richness difference
among sampling units
Decomposition of the fish beta diversity
for presence-absence data
Jaccard
Sørensen
0.326
65% of BDmax
0.267
53% of BDmax
ReplTotal / BDTotal
0.28
0.28
RichDiffTotal / BDTotal
0.72
0.72
BDTotal
BDmax = 0.5
=> Beta diversity in the Doubs River is dominated by
richness differences among sites
Indices computed from site 30 (confluence site), presence-absence data
= Jaccard D, = replacement, = richness difference
Reference site
0.0 0.2 0.4 0.6 0.8 1.0
D = Jaccard
!"#
1
2
3
4
5
6
7
9
10
11
12
13
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
19
20
21
22
23
24
25
26
27
28
29
30
5
10 15 20 25
Species richness
0
Species richness
!$#
14 15
1
2
3
Upstream
4
5
6
7
9
10
11
12
13
14 15
16
17
18
Sites (site #8 removed)
Downstream
(a) Jaccard dissimilarity (D, ), replacement ( ) and richness
difference ( ) indices for presence-absence data: sites 1–29 are
compared to site 30. (b) Species richness. Grey: polluted #23 to 25. (a) PCoA ordination of
Sorensen dissimilarity
Map of Sorensen D LCBD
0.6
Site 24
7
2
3
30
26 22,27,28
20 21
19
5
4
29
13
16
14 15
17
18
Site 30
Downstream
Upstream
Site 1
0
-0.4
-0.2
11 12
6
10
150
200
9
50
0.0
0.2
1
100
24
25
y coordinates (km)
0.4
23
Axis 2
Site 23
Site 25
-0.4
-0.2
0.0
Axis 1
0.2
0.4
0.6
0
50
100
150
200
250
x coordinates (km)
Left: Ordination of the sites, Sørensen dissimilarity (DS).
Right: Local contributions to beta (LCBDS) on map of Doubs River.
=> Sites 1-3: pristine, few species. Sites 23-25: agricultural pollution.
Map of Replacement LCBD
(b) PCoA ordination of
Replacement (ReplS)
(c) PCoA ordination of
Richness difference
(RichDiffS)
Site 24
Site 23
-0.2
10
3
4
6
25
26
5
15
11
–
s
Downstream
2
23
29
18,19,21
3
50 0.0
7
2
30
22,27,28
0.0
19
20
9 21
1
Si
te
29
16
7,9
16
10-13
-0.2
Upstream4,24,25
-0.2
0.0
0.2
Axis 1
0.4
0.6
Site 1
-0.4
0
-0.4
0
-0.2
50
6,14
5,15
0.0
26,30
17,20,22,27,28
18
-0.4
Axis 2
17
Site 30
24
1
1000.2
14
11,12 15
150
0.4
23
13
y coordinates
Axis 2 (km)
0.2
0.4
200
0.6
Site 25
0.2
100 Axis 1150
0.4
200
x coordinates (km)
Left: Ordination of the sites, ReplS (28% of beta diversity).
Right: Replacement LCBD (ReplLCBD) on map of Doubs River.
=> Higher replacement at sites 11-15 and 23-25. 250
Map of Richness difference LCBD
(c) PCoA ordination of
Richness difference (RichDiffS)
Site 24
Site 23
10-13
4,24,25
-0.4
-0.2
6,14
5,15
0.0
Axis 1
0.2
Site 30
150
200
16
50
7,9
26,30
17,20,22,27,28
0.0
18,19,21
3
Downstream
100
29
Upstream
0.4
Site 1
0
0.2
23
-0.2
Axis 2
2
y coordinates (km)
1
0.4
0.6
Site 25
0
50
100
150
200
250
x coordinates (km)
Left: Ordination of the sites, RichDiffS (72% of beta diversity).
Right: Richness difference LCBD (RichDiffLCBD) on Doubs River map.
=> Higher richness difference at sites 1-3 and 23. Test a hypothesis about the response
of community data to a factor
The environmental data were used to classify the sites in two groups
(Ward clustering): sites 1-22 (upper course) and 23-30 (lower course).
The DJ, ReplJ and RichDiffJ matrices were tested against this
classification using McArdle & Anderson’s (2001) F-test of
significance in RDA for non-Euclidean matrices. Results –
DJ was significantly explained by the classification (p = 0.005)
ReplJ was significantly explained by the classification (p = 0.001)
RichDifflJ was not significantly explained by the classif. (p = 0.943)
=> This analysis could be carried out with an experimental factor.
Identify the environmental factors
responsible for variation in the matrices
The DJ, ReplJ and RichDiffJ matrices were transformed into
rectangular matrices by principal coordinate analysis (PCoA). Environmental variables that significantly explained the variation in
each matrix were selected (forward selection).
Results –
DJ – was explained by {slope, hardness, nitrate, O2}
ReplJ was explained by {O2}
RichDiffJ was explained by {slope, hardness, nitrate}
=> The two components, ReplJ and RichDiffJ, are influenced by
different environmental processes, which all influence DJ.
Other methods of graphical analysis have been suggested – for
example the triangular plots proposed by Podani & Schmera (2011)
and Podani et al. (2013). The analysis of the Doubs River fish data using that triangular plots
is shown in the paper.
Reference
Legendre, P. 2014. Interpreting the replacement and richness
difference components of beta diversity. Global Ecology and
Biogeography 23: 1324-1334.
5. Conclusion
Replacement and richness difference indices can be interpreted and
related to ecosystem processes. Innovations –
• The index values can be summed across all pairs of sites; these
sums decompose total beta diversity into total replacement and total
richness difference components. • Local contributions of replacement and richness difference to total
beta diversity can be computed and mapped. • Within a region, differences among sites measured by these indices
can be analysed and interpreted using explanatory variables. • Replacement and richness difference matrices can be analysed by
all methods of multivariate data analysis for dissimilarity matrices.
Reference
Legendre, P. 2014. Interpreting the replacement and richness
difference components of beta diversity. Global Ecology and
Biogeography 23: 1324-1334.
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