Analytical framework
Figure 2 illustrates how we use ENS rarefaction to disentangle the
diversity components in practice. For this purpose, imagine a
latitudinal diversity gradient between a temperate (low diversity)
community and a tropical (high diversity) community. We consider three
scenarios of how this diversity gradient can manifest in terms of SAD
and N variation. First, a more-individual effect (panels A,D and G);
second, a change in the regional SAD (panels B, E, H); and third, a
combination of more individual-effect and SAD change (panels C, F and
I). The first row of figure 1 (panels A, B, C) shows the IBR curves
corresponding to the 3 scenarios . Panel A depicts the more-individuals
effect, where the tropical community (green) has twice as many
individuals as the temperate one (yellow) and therefore samples a larger
fraction of its species pool. However, when standardized to a common
number of individuals, both communities are expected to yield the same
diversity (i.e. the IBR curves follow the same trajectory), which
reflects that they are samples from similar regional SADs. Compare that
with panel B, where the number of individuals is the same for both
communities, but their SADs differ (i.e. the IBR curves have different
shapes). In this scenario, the tropical community samples from a larger
species pool with a higher number of relatively common species and many
more relatively rare species, which results in an IBR curve that is
steeper than the temperate one. Finally, panel C represents a scenario
where the diversity gradient is underlain by a combination of
more-individual effects and SAD changes. Not only does the tropical
community sample a more diverse SAD, but also it harbors a larger number
of individuals.
While the IBR curves allow us to
qualitatively and visually distinguish the scenarios, they do not
directly enable a quantitative decomposition of the observed diversity
change into contributions of more-individuals effects and SAD.
Therefore, we apply the ENS transformation to free IBR curves of their
numerical constraints. The resulting ENS curves (second row) are similar
to the IBR curves in that changes in their shape reflect changes in the
SAD, but the start of the curve is no longer constrained. In the
more-individuals scenario (panel D), both communities have the same
diversity for any common number of individuals (up to n=1000). Beyond
that, the tropical community passively samples additional rare species
due its larger sample size (labelled as “N-effect”). In the SAD change
scenario (panel E), the ENS transformation reveals that the tropical
community has a higher number of relatively dominant species to start
with (i.e. E2), and then accumulates relatively rare
species at a higher rate than the temperate community, adding up to the
total SAD effect (labelled “SAD-effect”). The same SAD effect can be
observed in the combined scenario (panel F), but now the tropical
community also has additional rare species due to its higher number of
individuals (labelled “N-effect”). As along the ENS curve all values
are expressed in terms of effective numbers of species, we can directly
compare the magnitudes of the two effects. In this example (panel F),
most of the observed diversity change is attributed to changes in the
regional SAD (ca. 80%), while the contributions of the more-individual
effect are relatively small (ca. 20%).
To apply this approach to any number of communities, we can partition
the total diversity of each community (i.e. EN) into two
components: The SAD-component is simply the ENS for a standard number of
individuals (i.e. En), where n is typically the sample
size of the smallest community in the gradient. Then, the N-component is
the difference between the total diversity and the SAD-component (i.e.
EN - En). It reflects the
more-individuals effect with respect to n individuals (i.e. how much
more diversity does a community have because its sample size exceeds n).
Now, instead of considering the total diversity (EN), we
can analyze these components along the gradient of interest. This is
shown in the last row of Figure 1, where the orange and purple dots
represent the SAD and N components of the two example communities. Note
that adding up the two components yields the total diversity of the
communities. In the first scenario, the diversity change occurs
exclusively in the N component (i.e., a N-effect), while in the second
scenario, the diversity change is driven by the SAD component (i.e., a
SAD-effect). Finally, in the third scenario both components change at
the same time, so that N-effect and SAD-effect add up to the total
diversity gradient. By comparing the slopes of the two components along
the gradient (dashed lines), we can assess the relative contributions of
N-effects and SAD-effects to the observed diversity gradient. The pie
charts in Figure 1 illustrate the contributions of SAD effects and N
effects for each scenario. In the combined scenario (panel I), the SAD
effect contributes 80% toward the total diversity gradient while 20%
of the diversity change occurs because the tropical community has more
than 1000 individuals. In practice, these effect sizes correspond to the
regression coefficients of linear models. However, the components could
also be modelled as non-linear functions of continuous predictors. In
that case, the contributions of N and SAD effects may be variable along
the gradient and cannot be summarized as a simple pie chart.