Next steps
We have shown that FEve has critical conceptual and practical drawbacks,
and therefore, we recommend not using this index in studies of
functional variability. However, it is still possible to measure
evenness of functional traits combined with information about abundances
using alternative methods that do not have the limitations of FEve. An
alternative metric based on Hill numbers that combines nearest-neighbor
distances with abundances is the evenness derivative of the diversity
metric of Scheiner (2012):
\({}^{q}{D\left(AT_{N}\right)}=\left(\sum_{i=1}^{S}\left(\frac{n_{i}d_{\text{i\ min}}}{\sum_{j=1}^{S}{n_{j}d_{\text{j\ min}}}}\right)^{q}\right)^{\frac{1}{(1-q)}}\)(4)
\({}^{q}{E\left(AT_{N}\right)}=\frac{{}^{q}{D\left(AT_{N}\right)}}{S}\), (5)
where \({}^{q}{D\left(AT_{N}\right)}\) is effective number of
distinct species that equally contribute to functional interaction and
variability within a community based on nearest-neighbor distances
(\(n_{i}d_{\text{i\ min}}=n_{j}d_{\text{j\ min}}\) for all\(i\neq j\)), S is the number of species,ni is the number of individuals of speciesi , di min is the nearest-neighbor distance
of species i , and q is the exponent of the Hill function.
[The metrics here and below follow the symbol convention of Scheiner
(2019).] This metric measures the evenness of the joint distribution
of abundances and nearest-neighbor distances. Because each species has a
unique nearest-neighbor distance, the resulting metric always has a
single value and small deviations of those values will result in only
small changes in the metric, eliminating the problems that we outlined
above for FEve. It should be used in conjunction with an examination of
the separate evennesses of abundances
[\({}^{q}{E\left(A\right)}\)] and nearest-neighbor distances
[\({}^{q}{E\left(T_{N}\right)}\)]. For example, it is possible
that neither parameter is evenly distributed singly, but that the joint
distribution has an even distribution, which can occur if they are
strongly negatively correlated. Such a combination of values would then
point to the potential importance of processes that jointly affect
traits and abundances (e.g., competitive exclusion).
An alternative approach for combining trait distance and abundance
information is the use of the abundance-weighted distance of speciesi from all of the other \(S-1\) species:
\(d_{i}=\sum_{\par
\begin{matrix}k=1\\
k\neq i\\
\end{matrix}}^{S}{d_{\text{ik}}\left(\frac{n_{k}}{N}\right)}\) , (6)
where \(N=\sum_{j=1}^{S}n_{j}\) is the total number of individuals
in the assemblage. Then functional diversity can be estimated in terms
of Hill numbers as:
\({}^{q}{D\left(AT_{T}\right)}=\left(\sum_{i=1}^{S}\left(\frac{n_{i}d_{i}}{\sum_{j=1}^{S}{n_{j}d_{j}}}\right)^{q}\right)^{\frac{1}{(1-q)}}\),
(7)
which is the effective number of distinct species that equally
contribute to functional interaction and variability within a community
based on abundances and weighted distances of every species from all
other species (\(n_{i}d_{i}=n_{j}d_{j}\) for all \(i\neq j\)). From
this, we can obtain an evenness measure as:
\({}^{q}{E\left(AT_{T}\right)}=\frac{{}^{q}{D\left(AT_{T}\right)}}{S}\). (8)
This measure of evenness would be appropriate if a given species
interacts with all of the other species in a community in a way that
‘averages’ over all of those interactions (e.g., in a system with
diffuse competition).
The evenness metrics given in eqs. 5 and 8 are based on the individual
properties of each species. An alternative approach is to measure
functional variation based on pairs of species:
\({}^{q}{H\left(AT_{P}\right)}=\left(\sum_{i=1}^{S}{\sum_{j=1}^{S}\left(\frac{n_{i}n_{j}d_{\text{ij}}}{\sum_{k=1}^{S}{\sum_{l=1}^{S}{n_{k}n_{l}d_{\text{kl}}}}}\right)^{q}}\right)^{\frac{1}{(1-q)}}\), (9)
which measures the effective number of equally interacting pairs of
species (equal values of \(n_{i}n_{j}d_{\text{ij}}\) for all\(i,j=1,2,\ldots,S\), \(i\neq j\)) (see eq. A23 in Scheiner et al.,
2017), so that the number of equally interacting species is determined
as follows:
\(\frac{{}^{q}{D\left(AT_{P}\right)}=\left(1+\sqrt{1+4^{q}{H\left(AT_{P}\right)}}\right)}{2}\),
(10)
(eqs. 4 and A4, Scheiner et al., 2017). The corresponding metric of
functional evenness is then:
\({}^{q}{E\left(AT_{P}\right)}=^{q}{D\left(AT_{P}\right)}/S\) .
(11).
This measure of evenness would be appropriate if the pairwise
interactions are important and those interactions occur with all of the
other species in the community (e.g., scramble competition for a
spectrum of resources(. The metrics presented here (eqs. 4 – 11), as
well as FEve itself, assume that all individuals within a species are
identical; somewhat different forms are necessary to capture
within-species variation.
More general concepts (Gregorius & Kosman 2017, 2018) and a large
variety of metrics (Scheiner 2019) exist for measuring functional
variation, and can be used as alternative for FEve. We caution, though,
that many of them have not yet been critically evaluated. The metrics
suggested here (eqs. 5, 8, and 11) are all based on a concept of
diversity of the dispersion of an effective number of types. Division of
this effective number by the actual number of types turns these into
metrics of functional evenness. While there is no single best way to
measure functional-trait evenness or its combination with abundance,
there are metrics, such as FEve, that should be avoided.