DISCUSSION
Caughley and Gunn (1993), Gunn (2003), Archer and Tieszen (1980), and Payette et al. (2001) suggested that barren-ground caribou fluctuations were mainly caused by the over-grazing of caribou when subpopulation abundance was at high levels, followed by declines to densities sufficiently low that the arctic/subarctic range could recover. Bongelli et al ., (2020) provided a quantitative demonstration that most barren-ground caribou subpopulations do not just fluctuate, they cycle in a regular manner (sine cycles). Bongelliet al. (2020) found that approximately 96% of the variation in the population cycles of nine sine cyclic North American barren-ground subpopulations could be explained by a relatively simple environmental regression model that considered only the subpopulation range size and range productivity. Recognizing that a sine cycle provides an unambiguous estimate of the Nt for the entire cycle, the Verhulst (1838) equation for logistic growth was employed to show that carrying capacity (Kt) also cycles regularly (Figure 1; Figure 2; Figure 3). Barren-ground caribou logistic carrying capacity (K) is best understood as a dynamic proxy variable (the units for K are subpopulation numbers) for subpopulation range condition.
Subpopulation numbers of Qamanirjuaq, Bathurst, and George River caribou closely tracked their respective carrying capacities (2-3 years) throughout most of their respective cycles (Table 1; Figures 1; Figure 2; Figure 3; Figure 4). However, during the eruption phase of the cycle all three subpopulations required about double the time to reach Kt from intrinsic growth than during the rest of the cycle (Table 1; Figure 4). In these three case studies, barren-ground caribou declined when the range declined due to over-grazing, then increased rapidly once their range had recovered. If some external (perhaps decadal, progressive, catastrophic, or circumstantial) factor were to impact subpopulation numbers, their range, or both there would be a response to the trajectories of both habitat (K) and subpopulation numbers (N). However, Bongelli et al., (2020) showed that the best (Bayesian Information Criteria) mathematical description for barren-ground caribou population trajectories is a sine cycle.The increase phase is symmetrical to the decline phase. Exactly half of the cycle is spent with numbers > ½ the amplitude and exactly half the cycle is spent with numbers < ½ the amplitude. The regularity of barren-ground caribou population dynamics over the history of census data collection for 9 subpopulations suggests a simple and consistent relationship between barren-ground caribou subpopulations and their range. Bongelli et al. (2020) suggests that barren-ground caribou population fluctuations are not merely affected by herbivore/range interactions; they are almost entirely determined by this relationship. Other factors may become relevant if those factors are unusually extreme, persistent, or occur at a sensitive time in the cycle (e.g., immigration during eruption or high harvest rates at low numbers). However, Bongelli et al ., (2020) suggests that anthropogenic factors at historical levels are unlikely to have any lasting or substantive impact on barren-ground caribou range dynamics. In all three of our case studies, carrying capacity cycled similar to subpopulation numbers.
We defined the eruption phase as consecutive years when annual subpopulation growth rate (λt) is greater than 1.0 and increasing each year, and/or λt ≥ 1.2. Sine cyclic barren-ground caribou subpopulations are gradually increasing for half the cycle period, and gradually declining in the other half. The eruption phase for the Qamanirjuaq, Bathurst, and George River subpopulations ranged from 6 to 9 years or 12-19% of the cycle (Table 1). After the eruption phase, barren-ground caribou subpopulations continue to increase in numbers, but λt declines progressively into the low period of the cycle (Figure 1; Figure 2; Figure 3; Figure 4). Post-eruption, the initial decline in λt is due to increasing densities of caribou relative to carrying capacity. Once caribou numbers are sufficient to cause carrying capacity to decline, the decline in λt is mainly due to declining carrying capacity (Figure 1; Figure 2; Figure 3). The decline in carrying capacity continues until declining caribou numbers are insufficient to cause a continued decline in impacted forage communities (through over-grazing and/or trampling) and habitat recovery can begin. Range recovery is slow initially, prolonging the time when caribou are at low numbers and keeping the rate of caribou recovery sufficiently low that increase is difficult to detect given the variance inherent to standard survey methods. The surge of habitat recovery preceded the surge in caribou numbers by 2-4 years in all three subpopulations.
Population growth rates continued to track K throughout the eruption phase for the Qamanirjuaq subpopulation (Figure 1). The Qamanirjuaq subpopulation appeared to be limited by carrying capacity throughout eruption phase (i.e., λt <1.2). The maximum and minimum rates of population growth and decline identified for the Qamanirjuaq subpopulation (λmax = 1.196; λmin = 0.837), align with previously reported (Gunn, 2003) maximum and minimum rates of population growth and decline for barren-ground caribou generally (λmax = 1.17; λmin = 0.830). The Qamanirjuaq subpopulation increased more slowly and declined more slowly than the Bathurst and George River subpopulations. For the Qamanirjuaq subpopulation our estimates of logistic Kt from sine curve Nt indicated that subpopulation recovery cannot occur faster than habitat regeneration occurs. Once range recovery is well under way, Qamanirjuaq caribou have the potential to double in about 4 years with no immigration under natural conditions. Conversely, the maximum observed rate of decline suggests the half-life of the subpopulation during the cycle period of maximum decline was a little over 3 years. Our results suggest that at any given time in the cycle, the number of caribou in the Qamanirjuaq subpopulation is typically only a few years away from the current carrying capacity of its range (Figure 4).
Both the Bathurst and George River subpopulations increased at rates exceeding the maximum possible intrinsic rates of increase (i.e., λt >1.363) during the eruption phase of their cycle (Figure 2; Figure 3; Table 1). Population growth rates greater than the maximum plausible rate of increase (> 1.2) during the eruption phase were observed for 9 of 42 years during the Bathurst cycle, and 9 of 52 years for the George River cycle. Observed population growth rates were greater than the intrinsic (or plausible) maximum for the Bathurst and George River subpopulations during the eruption phase, indicating that immigration was contributing to the eruptive increase from low numbers (Figure 5; Figure 6)
The Qamanirjuaq subpopulation did not experience detectable levels of immigration during any part of its cycle. The George River subpopulation experienced the highest observed subpopulation growth rates (λ = 2.85), the greatest percentage of immigration driven increase (89.2%) and the shortest lag time values (0.765 years) during its eruption period (Figure 5; Figure 6; Figure 7). Immigration during the Bathurst eruption was initially intermediate (~39%), then increased to ~57%, and then gradually declined to about 20% (Figure 6). Net immigration was observed only during the eruption phase for the Bathurst and George River subpopulations. Immigration may be important in initiating and sustaining the eruption phase of barren-ground caribou when immigrants are available from adjacent subpopulations that share the winter range. Our method of calculating immigration assumes net immigration in a given year is a permanent transfer of individuals from an adjacent subpopulation
There were no significant differences between subpopulation cycle mean lag times (p = 0.249) or subpopulation cycle mean annual growth rates (p = 0.249) when the eruption years were excluded from the cycle samples (Table 3; Table 4). When the eruption years were included in the subpopulation cycle samples, the George River subpopulation had a significantly shorter mean lag time than the Qamanirjuaq (p = 0.018), but there were no significant differences between subpopulation annual growth rates (p = 0.958) (Table 3). When only the eruption years were considered, the George River subpopulation mean lag time was shorter than the Qamanirjuaq mean lag time (p = 0.013) and the George River subpopulation mean annual growth rate was larger than the Qamanirjuaq mean annual growth rate (p < 0.001) (Table 3; Table 4). There was no significant difference between the Bathurst mean lag times or the Bathurst mean annual population growth rates when compared to either the George River or the Qamanirjuaq subpopulations (Tables 3; Table 4). The only differences between subpopulation mean lag times and mean annual growth rates were those due to immigration during the eruption phase. Most (89%) of the initial George River eruption increase was due to immigration, while none of the Qamanirjuaq eruption increase was due to immigration (Figure 6). The Bathurst eruption was aided by up to ~50% immigration (Figure 6), but even that contribution was insufficient to cause mean lag times and mean annual subpopulation growth rates during the eruption to be statistically (p ≤ 0.05) different from the no-immigration Qamanirjuaq eruption phase or high immigration George River phase (Table 3; Table 4).
There are no life history limits to the rates at which a subpopulation might decline except those imposed by extirpation. The George River subpopulation had the highest observed λmax values but also the highest observed rates of decline (λmin). Bathurst subpopulation maximum rates of increase and decline were intermediate to the George River and Qamanirjuaq subpopulation λmax and λmin values. Optimal foraging theory suggests that individuals will shift their habitat to maximize energy intake (Stephens and Krebs, 1986). We suggest that as carry capacity continually declines, individuals may emigrate from their home range to an adjacent range if greater availability and quality of forage species are available there. Migration to the best range conditions would increase the rate of subpopulation decline in the contributing subpopulation and increase the rate of growth in the adjacent (receiving) subpopulation.
Bathurst and George River eruption patterns suggest that eruption is predicated on habitat recovery but can be accelerated by immigration from adjacent subpopulations. In both cases immigration peaked early in the eruption phase, and then declined as the subpopulation increased. Although substantial net immigration may occur during the eruption phase, these results are most consistent with the view that barren-ground caribou subpopulations may be demographically closed for management purposes. Although immigration may be a factor in triggering the eruption phase, the role of immigration is quickly subordinated by the intrinsic capacity for Nt to quickly close on Kt. Immigration appears to play only a brief and transitory role in barren-ground caribou cycles relative to intrinsic subpopulation growth rates. The main determinant of barren-ground caribou subpopulation numbers and trend is range carrying capacity for all but a few years of the eruption cycle.
George River caribou have one of the largest total ranges and the largest winter ranges of all barren-ground caribou subpopulations (Bongelli et al., 2020; Schmelzer & Otto, 2003). Schmelzer & Otto (2003) found that George River caribou summer range habitat quickly deteriorated during the mid-1980s which corresponds to a period of significant population growth. During this time, George River caribou experienced what Schmelzer & Otto (2003) termed winter range drift. Schmelzer & Otto (2003) suggest that winter range forage allowed George River caribou to delay the effects of density-dependent population decline due to summer forage limitations by expanding their use of the winter range. The ability of George River caribou to shift winter range to provide a compensatory source of forage, delays the density-dependent feedback of range deterioration allowing them to maintain greater numbers, ultimately increasing over-grazing of the summer range before the inevitable decline occurs. Schmelzer & Otto (2003) suggest that the demographic benefits of winter range drift are limited by the cost of lengthier migration to the traditional calving grounds.
Radio collar telemetry data indicated partial overlap between the George River and Leaf River subpopulations during the 1990s and early 2000s (Taillon et al., 2016) which corresponds to a period of growth and peak abundance in the George River subpopulation. However there has been no overlap identified between the two subpopulations since approximately 2006 (Taillon et al., 2016). We speculate that the eruption period of subpopulation growth throughout the 1960s for George River caribou was augmented by immigrants from the adjacent Leaf River subpopulation which was estimated to be at or near its peak during that time (Bongelli et al., 2020). Leaf River caribou are currently declining (CARMA, 2016), so it is unlikely that immigration from the Leaf River subpopulation will contribute substantially (or at all) to the recovery of the George River subpopulation from its current period of low densities. Asynchrony with the Leaf River subpopulation may delay the recovery of the George River subpopulation.
Another possibility to dependency on immigration from the Leaf River subpopulation for eruptive recovery of the George River subpopulation is that some George River caribou have temporarily foregone the annual migration because the summer range was severely over-grazed. As the George River summer range regenerates, the numbers counted (calving ground census) on the summer range could be augmented by a return to annual migration pattern. Still another possibility is that George River caribou may become more aggregated on the calving ground as it increases, and the appearance of immigration during the eruption phase may be due to disaggregation at low densities (resulting in under-estimation) when they are at low numbers. The data available to us were insufficient to limit or discriminate between these or other possibilities for George River caribou recovery scenarios. We do believe that the eruption phase is imminent for the George River caribou, and suggest researchers monitor not only the numerical increase but also determine how the George River subpopulation recovers in the current cycle to guide future management when subpopulation is once again at the low point of its cycle.
Bathurst caribou are intermediate with respect to total range area and the proportion of the total range that is summer range (Bongelliet al., 2020). Like the George River subpopulation, Bathurst caribou can be found further south in the winter range when cycle numbers are high, possibly extending the period of decline to minimum numbers and thus increasing over-grazing damage to the summer range. Bathurst caribou differ from Qamanirjuaq and George River caribou in that their winter range overlaps with two other barren-ground caribou subpopulations (Bluenose East, and Beverly). The most recent (2021) population abundance estimates for Bathurst and Bluenose-East caribou are 6,240 and 23,200 respectively (Government of Northwest Territories, 2021). The most recent (2018) Beverly population abundance estimate is 103,400 (BQCMB, 2021). The 2021 Bathurst calving ground aerial photo survey and aerial reconnaissance survey found high levels of overlap between Bathurst, Beverly, and Bluenose-East caribou on the calving ground (Government of Northwest Territories, 2021). More recent 2022 telemetry data documents immigration from the Bathurst herd to the Beverly herd. Bathurst spring composition counts appear to contain annually variable fractions of Bluenose-East, and Beverly caribou. Immigration from these adjacent subpopulations could trigger and sustain the eruption phase of the Bathurst subpopulation once its habitat had sufficiently recovered.
Bongelli et al. (2020) found that the sine cycle fit for the Bathurst subpopulation was well-supported, but not as definitive as it was for other more demographically segregated subpopulations. Immigration from adjacent subpopulations sufficient to trigger an eruption would be expected to vary from cycle to cycle depending on the degree of synchrony/asynchrony and the degree of overlap with adjacent subpopulations. Predicting precisely when the Bathurst subpopulation will erupt may be more difficult than predicting the eruption phase for a subpopulation that is more demographically segregated and thus mainly driven by intrinsic subpopulation processes (e.g., Qamanirjuaq).
Some subpopulations (e.g., Qamanirjuaq) appear to cycle entirely due to density dependent intrinsic rates of birth and death. Other subpopulations appear to be demographically distinct yet exhibit a dramatic eruption phase that requires substantial immigration or repatriation (e.g., George River). Some subpopulations have overlapping summer and winter ranges with adjacent subpopulations, and inter-subpopulation exchange between these subpopulations is well documented (e.g., Bathurst). Yet all three subpopulations exhibit stable sine cyclic population dynamics rather than converge on some stable equilibrium density or experience periodic extirpation (Bongelli et al. 2020).
The λt versus lag time phase plane suggests a gradation of barren-ground caribou demographic performance rather than subpopulation specific clusters (Figure, 7). Extremes of eruption periods between mostly immigration (George River) and entirely intrinsic (Qamanirjuaq) were significantly different, but the intermediate immigration eruption subpopulation (Bathurst) was not significantly different from either extreme (Table 2; Figure 7). In all three of our ecologically distinct case studies, Nt closely followed Kt in symmetrical sine cycles. In these three subpopulations, range condition closely moderates caribou numbers except for the eruption phase. Post-eruption phase, range condition is almost immediately reduced by grazing. The Nt→Kt lag time is shortest during the decline portion of the cycle and greatest during the eruption phase that initiates the increase portion of the cycle (Figure 1; Figure 2; Figure 3). The regularity and symmetry of both the increase and decline phases of these cycles suggests that barren-ground caribou cycles are both stable and resilient. Short of extensive range management practices to enhance range productivity and/or availability there is little else managers can do to prevent or mitigate cyclical caribou declines or to speed up recovery.
Our relatively simple view of barren-ground population dynamics has strong quantitative support and is relevant to both wildlife co-management and species status designation. These case studies of the Qamanirjuaq, Bathurst, and George River barren-ground caribou subpopulations identify both similarities and differences in the ecological circumstances of barren-ground caribou subpopulations. The biggest similarity and the central demographic characteristic of these subpopulations is that they are sine cyclic and Ntclosely follows Kt. We suggest that barren-ground caribou subpopulations will cycle indefinitely as a demographic result of herbivore-range dynamics in contiguous tundra/taiga habitat. Barren-ground caribou subpopulations are genetically indistinguishable, but most of them can be regarded as demographically closed for management purposes. Immigration appears to play a role in the initiation and acceleration of the eruption period in some subpopulations, but not all of them. Synchrony and asynchrony with adjacent subpopulations can affect the timing of the eruption phase but cannot independently initiate the eventual recovery of subpopulations until the range has recovered.
Precautionary harvest management at low numbers prior to the eruption phase will likely shorten the recovery time from low numbers. Once habitat has recovered the need for harvest management measures will diminish then disappear because it will become logistically impossible to stop barren-ground caribou from increasing to the point that they over-graze their range. Over-grazing begins and rapidly increases well before subpopulation numbers peak (Figure 1; Figure 2; Figure 3). Given the fundamental importance of the range to barren-ground caribou, wildlife co-managers could consider a “No Net Loss” policy requiring viable habitat replacement and/or critical habitat protection measures (e.g., calving, and calving migrations routes) for commercial developments on caribou range that cause harmful alteration, disruption, or destruction of caribou habitat.  The utility of precautionary up-listing the conservation status of barren-ground caribou during the portion of their cycle when they are declining and at low numbers is unclear and appears inconsistent with their natural history. Species status determination could employ fidelity to an historical cycle as an alternative to generation-based designation criteria for cyclic species.
The underlying assumption of these findings is that the census estimates for these subpopulations of barren-ground caribou were accurate. Our results are empirical, not simulation model results. We are describing the demographic consequences of Bongelli et al. (2020) finding that the Qamanirjuaq, Bathurst, and George River (and most barren-ground caribou subpopulations) are sine cyclic. Our paper is fundamentally descriptive, not theoretical. Population growth rate (λt), carrying capacity (Kt), and lag time (Nt→Kt) are highly correlated (Table 5) because of how these variables are calculated. The axes of Figure 7 are population growth rate (λt) and lag time (Nt→Kt), and obviously we find that shorter lag times are associated with higher population growth rates for all three subpopulations. It is also apparent that there is a geometric relationship between (λt) and (Nt→Kt). We use these axes to define a phase plane useful in determining if there are subpopulation eruption clusters. The apparent geometric relationship between population growth rate and lag time is due to autocorrelation. The linear correlation measures between (λt), (Kt), and lag time (Nt→Kt) (Table 7) are highly significant but imperfect because the relationship between these two variables is geometric not linear; and because immigration rates between these subpopulations during the eruption phase were different. Population growth rate (λt), carrying capacity (Kt), and lag time (Nt→Kt) are descriptive summary statistics that derive from the sine cycles of the three subpopulations considered.