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.