Supplemental material2.
Figure.Graphical representation of the 11Isotopic Functional Indices in a two-dimensional isotopic δ-space (δ13C-δ15N). A theoretical food web
of 12 species is considered here where species are plotted according to their isotopic signatures with circle areas of species proportional to their
relative biomasses. The IFR indices (IFRic, SEAc and Hullbiom) correspond to the extent of the isotopic space occupied by the community. The
IFD indices (CD, IFDiv, IFDis, IFSpe and IFEnt) maximise the trophic divergence within the community. The IFE indices (NND, IFEve and
IFOri) are indicators of the regularity in the distribution of species biomasses in the isotopic space. Note that IFRic as well as SEAc, CD and
NND do not take into account for species biomasses in their calculations. A detailed description of each index is provided in the following Table.
Table.Detailed description of the Isotopic Functional Indices investigated in this study. Any species i has T traits, where T refers to δ13C and
δ15N in this study and where xik is the coordinate of species i on trait k [1, T]. These metrics potentially apply to a combination of more than two
stable isotopes (e.g., 34S). S species are considered. Relative biomass of species i is noted 𝑤𝑖 , with∑𝑆𝑖=1 𝑤𝑖 = 1.
Species
Index
Index
General
name
code
description
Details of calculation and references
biomass
?
The convex hull volume is computed with the Quickhull algorithm described by Barber, Dobkin & Huhdanpaa
Convex
Hull
(1996). It first determines the most extreme isotopic values of the community (vertices), links them to build the
Isotopic
Volume (minimum
Functional
volume
convex hull and finally calculates the volume encompassed by vertices (Cornwell, Schwilk & Ackerly 2006).
IFRic is standardized by the ‘global’ IFRic that includes all species (of the two communities investigated) so that
IFRic
Richness
encompassed
by
species
the
No
its values are constrained between 0 and 1.This index corresponds to the Total Area index (also called Convex
in
Hull Area) in Layman et al. (2007) and is similar to the Functional Richness index (FRic) described in Villéger,
Isotopic functional Richness
isotopic space)
Mason & Mouillot (2008), but in a -space.
A measure of the
Corrected
Standard
SEAc
Ellipse Area
mean
core
niche
Calculated from the variance and covariance of the δ13C and δ15N data (Jackson et al., 2011). In this study, SEAc
area
of
the
is standardized by the ‘global’ SEAc that corresponds to the sum of the SEAc of the two communities
community isotopic
No
investigated, and therefore SEAc values are constrained between 0 and 1.
niche
This index is a hybrid metric that corresponds to the amount of the space occupied by the community adjusted
Hull biom
Convex
Hull
by species relative biomasses (Clark et al., 2012), but in a δ-space.The convex hull is calculated on vertex
Volume
weighted
species only.To calculate this metric, isotopic values are first standardized to a mean of zero and a standard
species
deviation of one (values between 0 and 1). Isotopic data of vertex species are then weighted by individual
Hull biom
Yes
by
biomasses
species relative biomass prior to calculating the Convex Hull Volume/Area. This weighting procedure moves
rare species further towards the centre of gravity of the distribution of the isotopic values and therefore de-
emphases their influence on isotopic diversity while leaving very abundant species comparatively unchanged
(preserving their influence on isotopic diversity) (Clark et al., 2012).
Corresponds to the average Euclidean distance of species isotopic signatures to the δ13C and δ15N centroid
(Layman et al., 2007), where the centroid (gk) corresponds to the mean δ13C and δ15N value for all species in the
community. CD is expressed as a percentage of the maximal distance observed in the community and therefore
ranges from 0 to 1.
𝐶𝐷 =
Mean Distance of
Distance
to
Isotopic functional Divergence
species to Centroid
Centroid
CD
∑𝑆𝑖=1 𝑑𝐺𝑖
𝑆
where 𝑑𝐺𝑖 corresponds to the Euclidean distance from a species ito the centre of gravity gk
No
(CD)
𝑇
𝑑𝐺𝑖 = √ ∑(𝑥𝑖𝑘 − 𝑔𝑘 )2
𝑘=1
Where
𝑔𝑘 =
∑ 𝑥𝑖𝑘
𝑆
CD is equivalent to IFSpe (described below), where the distance to centroid is called the centre of gravity while
refering to the same calculation (mean of all δ13C and δ15N isotopic values)
Species
deviance
This index first determines the centre of gravity (Gk) of the convex hull on the basis of the vertex species of the
Isotopic
from the
Functional
IFDiv
̅̅̅̅ ) of species to the centre of gravity.
community. It then computes the average biomass-weighted distance (𝑑𝐺
mean distance to the
Divergence
Note that the position of the centre of gravity is not weighted by specific biomasses and is determined only from
centre of gravity
the most extreme isotopic values (vertices) of the community. This index ranges from 0 to 1.
weighted by relative
Yes
biomass
𝐼𝐹𝐷𝑖𝑣 =
∆𝑑 + ̅̅̅̅
𝑑𝐺
∆|𝑑| + ̅̅̅̅
𝑑𝐺
̅̅̅̅ ),
Where ∆𝑑 = ∑𝑆𝑖=1 𝑤𝑖 (𝑑𝐺𝑖 − 𝑑𝐺
and ∆|𝑑| = ∑𝑆𝑖=1 𝑤𝑖 |𝑑𝐺𝑖 − ̅̅̅̅
𝑑𝐺 |
where 𝑑𝐺𝑖 is the Euclidean distance between the species i and the centre of gravity 𝐺𝑘 of the convex hull
1
(coordinates of the centre of gravity 𝐺𝑘 = ∑𝑉𝑖=1 𝑥𝑖𝑘 of the V species forming the vertices of the convex Hull)
𝑉
̅̅̅̅ is the mean distance of the S species to the centre of gravity 𝐺𝑘
and 𝑑𝐺
IFDiv is similar to Functional Divergence index (FDiv) defined in Villéger et al. (2008), but applied in a δ-space
IFDis accounts for relative biomasses by shifting the position of the centre of gravity towards the most biomass-
Isotopic
Functional
IFDis
Dispersion
FDis is the weighted
dominant species (Laliberté & Legendre, 2010). It then computes a weighted average distance to this weighted
mean distance of
centre of gravity, using again the relative biomasses as weights. Contrary to IFDiv, both the centre of gravity and
species
the average distance to this centre are weighted by individual biomasses. IFDis is expressed as a percentage of
to
the
the maximal distance to the centre of gravity observed in the species pool (ranges from 0 to 1).
community
weighted centre of
gravity
c
of
𝐼𝐹𝐷𝑖𝑠 =
Yes
∑ 𝑤𝑖 (𝑥𝑖𝑘 − 𝑐)
∑ 𝑤𝑖
all
IFDis is the weighted mean distance to the weighted centre of gravity c
species
𝒄=
∑ 𝑤𝑖 × 𝑥𝑖𝑘
∑ 𝑤𝑖
Average
biomass-
This index is close to the IFDiv but slightly differs in the computation as it calculates distances from a
weighted
distance
hypothetical mean species (i.e. average value of isotopic signatures exhibited by all species of the community)
of all species to a
rather than to the centre of gravity of the vertex species (Bellwood et al., 2006). IFSpe does not account for
hypothetical
relative biomass of species in the establishment of the hypothetical mean species. It uses however the relative
Isotopic
Functional
IFSpe
Yes
Specialization
mean
𝑔𝑘
species
(or
centre of gravity)
biomasses of species as weight to compute a weighted average distance to the hypothetical mean species. IFSpe
is expressed as a percentage of the maximal specialization observed in the community and therefore ranges from
0 to 1.
𝑆
𝐼𝐹𝑆𝑝𝑒 = ∑(𝑤𝑖 × 𝑑𝐺𝑖 )
𝑖=1
Where 𝑑𝐺𝑖 corresponds to the Euclidean distance from a species ito the centre of gravity gk :
𝑇
𝑑𝐺𝑖 = √ ∑(𝑥𝑖𝑘 − 𝑔𝑘 )2
𝑘=1
𝑔𝑘 =
∑ 𝑥𝑖𝑘
𝑆
IFSpe is equivalent to CD but integrates biomass.
Sum
of
pairwise
It is the sum of pairwise distances between species in the isotopic space weighted by species relative biomasses.
𝑆
distances
Isotopic Rao’s
𝑆
∑ ∑ 𝑑𝑖𝑗 × 𝑤𝑖 × 𝑤𝑗
between species in
𝑖=1 𝑗=1
quadratic
IFEnt
the isotopic space
Where 𝑑𝑖𝑗 is the Euclidean distance between ith and jhspecies
weighted
The quadratic entropy index (Rao, 1982) can be viewed as a multivariate measure of the functional divergence
by species relative
(Mason et al., 2005; Schleuter et al., 2010) and is conceptually similar and highly correlated to IFDis (Laliberté
biomasses
& Legendre, 2010; Clark et al., 2012).
entropy Q
Yes
It measures how evenly species are spaced to their nearest neighbour in the isotopic space. Whilst NND provides
an average measure of nearest neighbour distance, this can also provide a proxy of evenness for these distances
Nearest
Neighbor
NND
Mean distance of
when expressed as a % of maximal distancein the isotopic space, and therefore ranges from 0 to 1. NND
each species to its
considers only species occurrence (presence/absence data only) and does not account for species’ biomasses
nearest neighbour in
(Layman et al., 2007).
No
the δ13C - δ15N bi-
Distance
𝑁𝑁𝐷 =
plot space
∑ 𝑑𝑖𝑗
𝑆
Where 𝑑𝑖𝑗 is the Euclidean distance between species i and its nearest neighbor j
Isotopic functional Evenness
Apart from giving the same weight to all species, NND is equivalent to IFOri (described below).
It is a proxy of the regularity in biomass distribution in the -space. It is based on the minimum spanning tree
which links all the species in the isotopic space with the minimum of branch lengths. This index takes into
account both the regularity of space between species and the evenness in the distribution of species biomasses. It
Evenness
of
ranges from 0 to 1 (1 = species regularly distributed), indicating a level of fullness of the -space by biomasses
Isotopic
(Villéger et al. 2008).
biomasses
Functional
IFEve
distribution in the
Yes
Evenness
minimum spanning
∑𝑆−1
min [
𝑙
𝐹𝐸𝑣𝑒 =
tree
dist(i,j)/(wi +wj )
1
1
,
]−
∑S−1
𝑆−1
l=1 dist(i,j)/(wi +wj ) 𝑆−1
1
1−
𝑆−1
Where dist(i,j) is the Euclidean distance between species i and j, the species involved in branch l of the MST
(minimum spanning tree)
IFEve is close to the Functional Evenness index in a biological trait functional space (FEve)
Isotopic
IFOri
Mean
biomass-
It measures how evenly species are spaced to their nearest neighbor in the isotopic space. The IFOri is based on
the Euclidean distance to the nearest neighbor weighted by the relative biomass of species. This index takes into
Yes
Functional
weighted
distance
Originality
of species to their
account the regularity of space between nearest neighbor species and the evenness in the distribution of
biomasses of species. IFOri is expressed as a percentage of the maximal originality observed in the species pool
which corresponds to an optimal even distribution of species in the isotopic space (i.e. optimal functional
nearest neighbor
regularity). The index is 1 if all distances between nearest neighbor species correspond to the optimal
distribution with all species having the same biomasses (Mouillot et al., 2013).
𝐼𝐹𝑂𝑟𝑖 =
∑𝑆𝑖=1 𝑑𝑖𝑗 × 𝑤𝑖
𝑆
Where 𝑑𝑖𝑗 is the Euclidean distance between species i and its nearest neighbor j
IFOri is the “biomass-weighted” equivalent of NND.
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