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.