Where \(p_{\text{ij}}\) represents the proportion of resource \(j\) used by the individual \(i\) and \(q_{j}\) represents the proportion of resource \(j\) used by the population. \(\text{PS}_{i}\) is bounded by a theoretical minimum, which is population dependent as described below, and one. The variable population dependent minimum indicates a complete specialist and a \(\text{PS}_{i}\) of one indicates a generalist (Bolnick et al., 2002). Because \(\text{PS}_{i}\) is bounded, we report the overall average value with 95% confidence intervals calculated using Monte-Carlo resampling in the R 3.3.1 package “resample”. Traditionally, prey counts have been used for calculating\(\text{PS}_{i}\), not proportions, as each count is assumed to represent an independent prey capture decision (Araújo et al., 2011; Bolnick et al., 2002). Proportions of prey metabarcoding reads are representations of the prey biomass proportions that were consumed by the predator, and similar proportions can result from consuming a few large or many small prey individuals. Correspondingly, calculating\(\text{PS}_{i}\), using the proportions of prey metabarcoding reads will produce a metric of “biomass specialization” that does not necessarily reflect independent prey capture decisions. Nevertheless, it describes intrapopulation variations in the utilization of different prey species. In addition, we calculated \(\text{PS}_{i}\ \)relative to groups of samples from a certain point in space and time. If individuals in that particular group are encountering the same size distribution of prey, then diet proportions may represent the same relative relationship of prey capture decisions as counts of individual prey items. Despite the potential limitations, there are several benefits to using this type of data. Coupled with scat collection, it allows for large samples sizes, is non-invasive, and gives high taxonomic resolution.
To define our groups for analysis, samples were separated by Location, Sex, Year, and Month of collection, yielding a total of 89 groups (Table S1). \(\text{PS}_{i}\) values for each sample were then calculated for each one of these groups. Within each group, each sample was treated as coming from a different individual due to the low probability of resampling the same individual (Rothstein et al., 2017).
Because different groups for analysis can have different theoretical minima, there is potential bias when comparing specialization metrics across groups. Differences in theoretical minima occur due to differences in sample size (the number of scat in each group) and/or differences in minimum prey densities (the smallest occurring proportion of a prey species in a group’s diet). Due to very low minimum prey densities in our data set, the theoretical minima are determined by sample size (Table S1). We addressed this potential bias in multiple ways. First, we excluded from analysis the smallest groups (those with < 5 samples) as they have the highest theoretical minimum and thus the most potential for bias. We also used Spearman’s rank correlation to estimate how much variance was explained by differences in sample size. This correlation was accomplished by comparing sample size to the average \(\text{PS}_{i}\) for each group we kept. We also calculated the theoretical minima for each group by dividing one by the number of samples in the group and then examined the range, average, and median of those minima. Additionally, sample sizes of each group were included in modeling of the data, which is described below. Lastly, the seasonal changes in sample size were visually compared with the seasonal patterns in \(\text{PS}_{i}\) values.