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.