INTRODUCTION
Subpopulations of mainland migratory barren-ground caribou (Rangifer tarandus groenlandicus ) are unique among the deer family (Cervidae) because they make annual migrations between the treeless tundra in the summer and the forested taiga in the winter and are typically sine cyclic (Bongelli et al., 2020). Barren-ground caribou are an age-structured, annual birth pulse species (Caughley 1977) whose natural history and ecology are consistent with the assumptions of life table discrete population modeling (Bongelliet al . 2020, Gunn & Miller, 1986; Russell et al., 2002; Wilson & Reeder, 2005; Government of Yukon, 2015, and COSEWIC 2016). Ecological circumstances (e.g., the ratio of summer range to winter range, range overlap with adjacent subpopulations, subpopulation exchange rates, harvest removal rates, and the degree of industrial development) can vary between subpopulations. However, Bongelli et al., (2020) showed that 96% of the variation in nine subpopulations of barren-ground caribou sine cycle dynamics could be explained by range area and range productivity.
Barren-ground caribou calving period is synchronized for each subpopulation and generally occurs over a 2-week period in June (Nagyet al., 2011; Nagy & Campbell, 2012; COSEWIC, 2016). Females generally produce one calf annually, usually beginning at age 2+. Poor health or nutrition may cause adult females of any age not to conceive or produce viable calves (COSEWIC, 2016). Like all naturally occurring species, barren-ground caribou subpopulation numbers are ultimately regulated by density-dependent reductions in calf production and/or survival rates (Demerec, 1957; Tanner, 1966; Caughley, 1977; McCullough, 1979; Fowler, 1981; Kie & White, 1985; Skogland, 1985; Clutton-Brocket al., 1987; Messier et al., 1988; Boyce, 1989; McCullough, 1999; Bowyer et al., 2014).
Thomas Malthus (1798) is credited with the first formal proposition that there are natural limitations to population growth. A continuous mathematical formulation of this principle termed the logistic equation was initially formulated by Verhulst (1838) and rediscovered by Pearl and Reed (1920) and Pearl and Reed (1922). An alternative formulation of the discrete logistic equation indicates that linear density-effects could cause convergence on K (carrying capacity), converging cycles to K, stable limit cycles, increasing oscillations to extinction and even chaos depending on the population’s maximum intrinsic annual growth rate (λmax) (May, 1976; Renshaw, 1991). We examined the relationship between barren-ground caribou subpopulation growth rate (λ) and K through the cycle by calculating K utilizing the discrete analog of the Verhulst (1838) and Pearl and Reed (1920) formulation of the logistic equation.
The maximum and minimum annual rates of population growth (or decline) reported for barren-ground caribou varies between subpopulations, but typically λ does not exceed 1.17 and is not less than 0.83 (Gunn, 2003). Heard (1980) estimated the maximum stable age population growth rate for barren-ground caribou subpopulations was λmax = 1.363, which we confirmed using an independent life table model (Bongelli, 2019). At λmax ≤ 1.363 the logistic equation suggests that, without lag-times, caribou subpopulations (like other deer species) would converge on carrying capacity (May, 1976; Renshaw, 1991; Vandermeer, 2010). Stable limit cycles are not observed until λmax > 2.57 (May, 1976; Renshaw, 1991) which is greater than the biologically maximum possible λ for barren-ground caribou.
We chose the Qamanirjuaq, Bathurst, and George River subpopulations for our comparison of barren-ground caribou cycles because they are relatively well-known and have been frequently surveyed. We updated the Bongelli et al. (2020) sine model for the Qamanirjuaq subpopulation to include the most recent (2017) abundance estimate. The updated Qamanirjuaq cycle has a period of 58 years (SE = 5.7) and an amplitude of 218,431 (SE = 31,348). The Bathurst subpopulation sine cycle has a period of 42 years (SE = 3.4) and an amplitude of 203,154 (SE = 23,147) (Bongelli et al., 2020). The George River subpopulation’s sine cycle (Bongelli et al., 2020) was updated to include the 2020 subpopulation estimate of 8,100. The updated George River sine cycle has a period of 52 years (SE = 2.6) and an amplitude of 305,000 (SE = 26, 644).
We compared cyclic changes in population growth rates (λt), carrying capacity (Kt), and lag time required to reach Kt from population numbers (Nt) for our three case studies. When the observed λt (Nt+1/Nt) is greater than biologically possible, the fraction of population change that is due to extrinsic growth (i.e., immigration) was estimated, and the role of immigration in triggering subpopulation eruption post habitat recovery was examined. Together these summary statistics and comparisons suggest an alternative explanation for barren-ground caribou subpopulation dynamics from those that are strictly intrinsic and drawn from short-term non-cyclic snapshots of the cycle. We considered the implications of viewing barren-ground caribou as a cyclic species for management and status determination.