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.