METHODS
We assumed the published subpopulation estimates varied in quality but
were generally accurate because collectively they were a good fit to the
sine cycle that is characteristic of barren-ground caribou (Bongelliet al., 2020). The discrete life table model assumes a constant
time of census (Birch, 1948; Cole, 1954; Caughley, 1977), which we
identified as the calving ground subpopulation surveys. We did not find
reliable estimates of age-specific annual survival or of age-specific
life table annual recruitment rates that were time-of-census referenced
to the subpopulation calving ground survey for any barren-ground caribou
subpopulation that extended over more than a fraction of its entire
cycle. Consequently, we restricted our consideration of density effects
to population growth rate (λ) rather than birth and survival rates.
The sine cycle fit to the subpopulation estimates over time (year) was
used to estimate the number at time = t (Nt) for the
entire cycle. We estimated annual subpopulation growth rate
(λt) as: Nt+1/Nt.
We used the discrete formulation
of the Verhulst (1838) logistic equation to calculate carrying capacity
(Kt) estimates for each year of the Qamanirjuaq,
Bathurst, and George River sine cycles. However, in some cases during
subpopulation increase, the observed growth rate was more than the
biological maximum rate of increase (λmax ≤ 1.363). We
limited the maximum value for intrinsic λt to values
that were within the biological maximum and realistic. We chose
λplausible = 1.20 as our maximum plausible rate of
intrinsic increase considering that natural mortality from harvest and
predation would never be zero and all cows age 2+ would not calve
(mx = 1.0) in a natural subpopulation. We assumed that
any subpopulation increase greater than λplausible was
due to immigration. When λt values were
>1.2 (λplausible) we corrected them to 1.2
for our Verhulst (1838) determinations of Kt because the
Verhulst logistic equation does not include a term for immigration. We
used the term “eruption period” to describe the time interval when
consecutive λt values were greater than 1.0 and
increasing year over year and/or ≥1.20 (20% increase/year). We
calculated K as:
\({}_{\max}=0.363\ (biological\ maximum\ intrinsic\ rate\ of\ increase)\)
\begin{equation}
N_{t+1}-N_{t}\ =\ {{}_{\max}*N}_{t}*\ \left(1-\frac{N_{t}}{K_{t}}\right)\nonumber \\
\end{equation}\begin{equation}
{\frac{N_{t+1}}{N_{t}}=\lambda}_{t}={1+}_{t}\nonumber \\
\end{equation}\({}_{t}=\left(\frac{N_{t}}{K_{t}}\right)-1\)
\begin{equation}
K_{t}\mathrm{\ =\ }\frac{N_{t}}{\left(\frac{{}_{t}}{{}_{\max}\ }-1\right)}\nonumber \\
\end{equation}We estimated the demographic pressure for increase as the time required
for Nt to reach Kt (lag time =
Lt) using the uncorrected (intrinsic + immigration)
λt values:
\(L_{t}=ln(\frac{K_{t}}{N_{t})*ln(}N_{t+1}/N_{t})\)
We compared mean lag time values and mean population growth rate values
between all three subpopulations using the Kruskal-Wallis non-parametric
test (IBM Corp, 2021) for three distinct cycle phases: 1) one complete
cycle, 2) one complete cycle excluding the eruption years (i.e.,
intrinsic growth only) and 3) only the eruption years of the cycle. For
all assessments we assumed pairwise comparisons were nonsignificant if
the probability of the test statistic was ≥ 0.05 using the Bonferroni
corrected significance values (IBM Corp, 2021). We used K-means
clustering (IBM Corp, 2021) to determine whether subpopulations were
grouped or contiguous with respect to lag time values and population
growth rate values for the eruption years of the subpopulation cycles.
We calculated the relative contributions of intrinsic growth (Nt*plausible) and immigration (It) to
population growth during the eruption period from the following
relationship:
\begin{equation}
I_{t}=\ \left(N_{t+1}\right)-\left(N_{t}*\ \lambda_{\text{plausible}}\right)\nonumber \\
\end{equation}As per equations above, population growth rate (λt),
carrying capacity (Kt), and lag time
(Nt→Kt) are covariant variables. We used
Descriptive Statistics (Correlations) (IBM Corp, 2021) to determine the
within-subpopulation bivariate correlation coefficients for these
variables.