\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[greek,ngerman,english]{babel}



\usepackage{float}








\begin{document}

\title{North American barren-ground caribou cycles are driven by logistic
trophic dynamics}



\author[1]{Eric Bongelli}%
\author[2]{Martha Dowsley}%
\author[3]{Dean Cluff}%
\author[4]{Mitch Campbell}%
\author[2]{Mitchell Taylor}%
\affil[1]{Lakehead University Faculty of Science and Environmental Studies}%
\affil[2]{Lakehead University}%
\affil[3]{Government of the Northwest Territories}%
\affil[4]{Government of Nunavut}%


\vspace{-1em}



  
  \date{May 04, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
We used the Qamanirjuaq, Bathurst, and George River barren-ground
caribou sine cycles to project numbers (Nt), calculate subpopulation
annual growth rates (\selectlanguage{greek}λ\selectlanguage{english}t) and calculate logistic carrying capacity (Kt).
Maximum annual growth rate was 1.196 and maximum annual rate of decline
was 0.836 for the harvested Qamanirjuaq subpopulation sine cycle.
However, the maximum annual subpopulation growth rates for both the
harvested Bathurst and George River subpopulation sine cycles were
greater than the biologically possible maximum intrinsic rate of
increase during the eruption phase. Subpopulation numbers for
Qamanirjuaq, Bathurst and George River barren-ground caribou
subpopulations all closely tracked carrying capacity for one complete
cycle with lag times between Nt and Kt ranging from \textless{} 1-year
to approximately 5-years. The short lag times observed indicates that
Qamanirjuaq, Bathurst and George River barren-ground caribou
subpopulations closely track their range condition. Range condition
drives barren-ground caribou subpopulation cycles, but range condition
also cycles; presumably because annual barren-ground caribou grazing
rates are proportional to barren-ground caribou numbers and eventually
exceed range annual growth rates. Immigration from adjacent
subpopulations plays a role in the initiation and acceleration of the
eruption period in some subpopulations, but not all of them. Numerical
synchrony and asynchrony with adjacent subpopulation cycles can affect
the timing of the eruption phase through mediation of immigration. Once
subpopulation range has recovered, the rapid recovery of subpopulation
numbers suggest that subpopulations are not restricted by other factors.
The regularity and symmetry of both the increase and decline phases of
these cycles suggests that the barren-ground caribou cycle is both
stable and resilient. Continuation of barren-ground caribou cycles at
historical levels is likely if habitat conservation measures are adopted
so that annual migration patterns are not disrupted, summer and winter
range remain undisturbed and common-sense harvest management policies
are adopted when caribou are at low numbers.%
\end{abstract}%



\sloppy


North American barren-ground caribou cycles are driven by logistic
trophic dynamics

Bongelli, E.\textsuperscript{1}, Dowsley, M.\textsuperscript{2}, Cluff,
H.D \textsuperscript{3}., Campbell, M.\textsuperscript{4}, \& Taylor,
M.\textsuperscript{2}

\emph{Abstract:}

We used the Qamanirjuaq, Bathurst, and George River barren-ground
caribou sine cycles to project numbers (N\textsubscript{t}), calculate
subpopulation annual growth rates (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}) and calculate
logistic carrying capacity (K\textsubscript{t}). Maximum annual growth
rate during the eruption phase was 1.196 and maximum annual rate of
decline was 0.836 for the harvested Qamanirjuaq subpopulation sine
cycle. However, the maximum annual subpopulation growth rates for both
the harvested Bathurst and George River subpopulation sine cycles were
greater than the biologically possible maximum intrinsic rate of
increase during the eruption phase. Subpopulation numbers for
Qamanirjuaq, Bathurst, and George River barren-ground caribou
subpopulations all closely tracked carrying capacity for one complete
cycle with lag times between N\textsubscript{t} and K\textsubscript{t}
ranging from \textless{} 1-year to approximately 5-years (mean lag times
were 2.898, 2.661, and 2.430 years respectively). The short lag times
observed indicates that Qamanirjuaq, Bathurst, and George River
barren-ground caribou subpopulations closely track their range
condition, and that grazing impacts begin to reduce intrinsic
subpopulation growth rates about midway into the increase portion of the
cycle. Range condition drives barren-ground caribou subpopulation
cycles, but range condition also cycles; presumably because annual
barren-ground caribou grazing rates are proportional to barren-ground
caribou numbers and eventually exceed range annual growth rates.
Barren-ground caribou numbers and their carrying capacity
(K\textsubscript{t}) sine cycle because state variables in this
relatively simple two-level trophic relationship are mutually dependent.
Immigration from adjacent subpopulations plays a role in the initiation
and acceleration of the eruption period in some subpopulations, but not
all of them. Numerical synchrony and asynchrony with adjacent
subpopulation cycles can affect the timing of the eruption phase through
mediation of immigration. However once subpopulation range has
recovered, the rapid recovery of subpopulation numbers suggest that
subpopulations are not restricted by other factors. The regularity and
symmetry of both the increase and decline phases of these cycles
suggests that the barren-ground caribou cycle is both stable and
resilient. Continuation of barren-ground caribou cycles at historical
levels is likely if habitat conservation measures are adopted so that
annual migration patterns are not disrupted, summer and winter range
remain undisturbed and common-sense harvest management policies are
adopted when caribou are at low numbers.

Key Words: barren-ground caribou, carrying capacity, cyclic species,
demography, density effects, management, population cycles, resilience,
sine cycle

\textsuperscript{1} Faculty of Science and Environmental Studies.
Department of Geography and Environment, Lakehead University. 955,
Oliver Road, Thunder Bay, ON, Canada P7B 5E1. \textbar{} KBM Resources
Group, 349 Mooney Avenue, Thunder Bay, Canada ON, P7B 5L5.
(esbongel@lakeheadu.ca)

\textsuperscript{2} Faculty of Science and Environmental Studies.
Department of Geography and Environment, Lakehead University. 955,
Oliver Road, Thunder Bay, ON, P7B 5E1

\textsuperscript{3} Regional Biologist North Slave Region. Department of
Environment and Natural Resources, Government of Northwest Territories.
3803 Bretzlaff Drive, Yellowknife, NT. X1A 2P9

\textsuperscript{4} Regional Wildlife Biologist Kivalliq Region.
Wildlife Research Division, Department of Environment, Government of
Nunavut. PO Box 120, Arviat, NU, X0C 0E0.

\textbf{DATA ACCESSIBILITY SATEMENT}

All the data for this paper were taken from public sources. The compiled
dataset used for the analysis described in this manuscript will be
publicly accessible through the Dryad data repository. The dataset will
be submitted to Dryad upon acceptance of this article.

\textbf{INTRODUCTION}

Subpopulations of mainland migratory barren-ground caribou
(\emph{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 \emph{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 (Bongelli\emph{et
al} . 2020, Gunn \& Miller, 1986; Russell \emph{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 \emph{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
(Nagy\emph{et 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-Brock\emph{et al.,} 1987; Messier \emph{et al.,} 1988; Boyce,
1989; McCullough, 1999; Bowyer \emph{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
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max}) (May, 1976; Renshaw, 1991). We examined the
relationship between barren-ground caribou subpopulation growth rate (\selectlanguage{greek}λ\selectlanguage{english})
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 \selectlanguage{greek}λ \selectlanguage{english}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 \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 1.363,
which we confirmed using an independent life table model (Bongelli,
2019). At \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} [?] 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
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} \textgreater{} 2.57 (May, 1976; Renshaw, 1991)
which is greater than the biologically maximum possible \selectlanguage{greek}λ \selectlanguage{english}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 \emph{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 \emph{et al.,} 2020). The George River
subpopulation's sine cycle (Bongelli \emph{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
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}), carrying capacity (K\textsubscript{t}), and lag
time required to reach K\textsubscript{t} from population numbers
(N\textsubscript{t}) for our three case studies. When the observed
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} (N\textsubscript{t+1}/N\textsubscript{t}) 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.

\textbf{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
(Bongelli\emph{et 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 (\selectlanguage{greek}λ\selectlanguage{english}) 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 (N\textsubscript{t}) for the
entire cycle. We estimated annual subpopulation growth rate
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}) as: N\textsubscript{t+1}/N\textsubscript{t}. We
used the discrete formulation of the Verhulst (1838) logistic equation
to calculate carrying capacity (K\textsubscript{t}) 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
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} [?] 1.363). We limited the maximum value for
intrinsic \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} to values that were within the biological
maximum and realistic. We chose \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausible} = 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 (m\textsubscript{x} = 1.0) in a natural
subpopulation. We assumed that any subpopulation increase greater than
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausible} was due to immigration. When
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} values were \textgreater{}1.2
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausible}) we corrected them to 1.2 for our Verhulst
(1838) determinations of K\textsubscript{t} 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
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} 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 N\textsubscript{t} to reach K\textsubscript{t} (lag time =
L\textsubscript{t}) using the uncorrected (intrinsic + immigration)
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} 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*\textsubscript{plausible}) and immigration (I\textsubscript{t}) 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 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}),
carrying capacity (K\textsubscript{t}), and lag time
(N\textsubscript{t}-K\textsubscript{t}) are covariant variables. We used
Descriptive Statistics (Correlations) (IBM Corp, 2021) to determine the
within-subpopulation bivariate correlation coefficients for these
variables.

\textbf{RESULTS}

Sine cycle subpopulation growth rates for the Qamanirjuaq subpopulation
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}) varied from a maximum of 1.196 in the increase
phase to 0.837 in the decline phase (Table 1; Figure 1). All the sine
cycle \selectlanguage{greek}λ \selectlanguage{english}values for the Qamanirjuaq subpopulation were within the maximum
plausible rate of increase (i.e., \textless{} 1.2). However, calculated
sine cyclic estimates of the maximum and minimum rates of annual
subpopulation growth were sometimes greater than the maximum
biologically possible intrinsic rate of increase (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} =
1.363) during the eruption phase for both the Bathurst
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 1.47; \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 0.68) and George
River (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 2.85; \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 0.35)
subpopulations (Table 1; Figure 2; Figure 3).

Barren-ground caribou subpopulation numbers (N\textsubscript{t}) closely
tracked carrying capacity (K\textsubscript{t}) for all three
subpopulations (Table 1; Figure 1; Figure 2; Figure 3; Figure 4). Mean
cycle lag times (years for N\textsubscript{t} -K\textsubscript{t}) at
observed \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} (N\textsubscript{t+1}/N\textsubscript{t})
for the Qamanirjuaq, Bathurst, and George River subpopulations were
2.898 (SE = 0.008), 2.661 (SE = 0.103), and 2.430 (SE = 0.121)
respectively (Table 2; Figure 4). Kruskal-Wallis pairwise comparison of
the observed mean N\textsubscript{t} -K\textsubscript{t} lag times of
each subpopulation over one complete cycle demonstrated that the mean
lag time (intrinsic + immigration) for the George River subpopulation
was significantly different from the Qamanirjuaq (p = 0.018), but not
significantly different from the Bathurst (p = 0.838) subpopulation
(Table 3). The difference in mean lag times between the Qamanirjuaq and
Bathurst subpopulation was also not significant (p = 0.414) (Table 3).
When the period of eruption years was excluded from the comparison of
mean lag time values (i.e., only intrinsic lag times included) there
were no significant differences in the projected lag time values between
any of the three subpopulations (p = 0.249) (Table 3). When pairwise
comparisons were restricted to only the eruption period the mean lag
time for the George River subpopulation was significantly different from
the Qamanirjuaq (p = 0.013) but not significantly different from the
Bathurst (p = 0.690). The difference in mean lag time between the
Qamanirjuaq and Bathurst subpopulations was also not significant (p =
0.221) (Table 3).

Kruskal-Wallis pairwise comparison of mean population growth rates
resulted in no significant difference between subpopulations for one
complete cycle (p = 0.958) or for one complete cycle excluding eruption
years (p = 0.249) (Table 4). When only eruption years were considered
separately, the mean population growth rate for the George River
subpopulation was significantly different than mean population growth
rate for the Qamanirjuaq (p = 0.001) but not significantly different
from the Bathurst mean population growth rate (p = 0.427). The
difference in mean population growth rate between the Qamanirjuaq and
Bathurst subpopulations was also not significant (p = 0.070) (Table 4).

Subpopulation annual growth rates during the eruption phase of the sine
cycle sometimes exceeded the biologically possible and biologically
plausible intrinsic growth rates for barren-ground caribou in both the
Bathurst and George River subpopulations (Figure 5). However, the
Qamanirjuaq subpopulation growth rates were entirely within the
plausible intrinsic value (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausible} = 1.2) (Figure 5).
The additional annual increase observed in the George River and Bathurst
eruption phase was attributed to extrinsic growth (i.e., immigration)
(Figure 6). Immigration rates for the Bathurst subpopulation initially
increased, peaked, and then declined (Figure. 6). Immigration rates for
the George River subpopulation were highest initially then declined
throughout the eruption phase (Figure 6).

K-means cluster analysis identified three distinct clusters of lag time
values and population growth rate values for the eruption years of each
subpopulation cycle. Clusters were characterized as: 1) short N-K time
lags and high lambda values, 2) intermediate N-K time lags and
intermediate lambda values, and 3) long N-K time lags and low lambda
values (Figure 7). Cluster 1 was exclusively associated with the George
River subpopulation (Figure 7). Cluster 2 was mainly comprised of
Bathurst records but also contained George River and Qamanirjuaq
records. Cluster 3 was predominantly associated with the Qamanirjuaq
subpopulation.

Population growth rate, carrying capacity, and lag time were all
correlated (p[?]0.05) for each of the three subpopulations (Table 5).

\textbf{DISCUSSION}

Caughley and Gunn (1993), Gunn (2003), Archer and Tieszen (1980), and
Payette \emph{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 \emph{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).
Bongelli\emph{et 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 N\textsubscript{t} for the
entire cycle, the Verhulst (1838) equation for logistic growth was
employed to show that carrying capacity (K\textsubscript{t}) 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
K\textsubscript{t} 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\textbf{.}The increase phase is symmetrical to the decline phase.
Exactly half of the cycle is spent with numbers \textgreater{} \selectlanguage{ngerman}½ the
amplitude and exactly half the cycle is spent with numbers \textless{} 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 \emph{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 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}) is greater than 1.0 and
increasing each year, and/or \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} [?] 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 \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} declines
progressively into the low period of the cycle (Figure 1; Figure 2;
Figure 3; Figure 4). Post-eruption, the initial decline in
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} 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 \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} 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., \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} \textless{}1.2). The maximum
and minimum rates of population growth and decline identified for the
Qamanirjuaq subpopulation (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 1.196;
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 0.837), align with previously reported (Gunn,
2003) maximum and minimum rates of population growth and decline for
barren-ground caribou generally (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 1.17;
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 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 K\textsubscript{t} from sine curve N\textsubscript{t} 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.,
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} \textgreater{}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 (\textgreater{} 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 (\selectlanguage{greek}λ \selectlanguage{english}= 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 (\textasciitilde{}39\%), then increased to
\textasciitilde{}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 \textless{} 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
\textasciitilde{}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 \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} values but
also the highest observed rates of decline (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min}).
Bathurst subpopulation maximum rates of increase and decline were
intermediate to the George River and Qamanirjuaq subpopulation
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} and \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} 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 N\textsubscript{t} to quickly close on
K\textsubscript{t.} 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 \emph{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 \emph{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 \emph{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 \emph{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 (Bongelli\emph{et
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 \emph{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 \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} 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, N\textsubscript{t} closely followed
K\textsubscript{t} 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
N\textsubscript{t}-K\textsubscript{t} 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
N\textsubscript{t}closely follows K\textsubscript{t}. 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 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}), carrying
capacity (K\textsubscript{t}), and lag time
(N\textsubscript{t}-K\textsubscript{t}) are highly correlated (Table 5)
because of how these variables are calculated. The axes of Figure 7 are
population growth rate (\selectlanguage{greek}λ\selectlanguage{english}t) and lag time
(N\textsubscript{t}-K\textsubscript{t}), 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 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}) and
(N\textsubscript{t}-K\textsubscript{t}). 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 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}), (K\textsubscript{t}), and lag
time (N\textsubscript{t}-K\textsubscript{t}) (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 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}), carrying capacity
(K\textsubscript{t}), and lag time
(N\textsubscript{t}-K\textsubscript{t}) are descriptive summary
statistics that derive from the sine cycles of the three subpopulations
considered.

\textbf{ACKNOWLEDGEMENTS:}

We acknowledge Lakehead University and the Government of Nunavut for
financial and logistic support. Troy Hegel and the
\emph{Rangifer}editorial board and reviewers contributed useful comments
to an earlier version of this paper. The results, interpretations,
opinions, conclusions, and any errors are entirely our own; and do not
necessarily represent the views of any other individual, agency, or
institution. All the data for this paper were taken from public sources
or are acknowledged in the text. We acknowledge the hard work and
professionalism of those who collected these data and made them
open-source available.

\par\null

\textbf{LITERATURE CITED}

\textbf{Archer, S., \& Tieszen, L.L.} 1980. Growth and physiological
responses of tundra plants to defoliation. - \emph{Arctic and Alpine
Research} . 12(4):531-552.

\textbf{Beverly Qamanirjuaq Caribou Management Board (BQCMB).} 2021.
2020/21 Annual Report. 46pp.

\textbf{Birch, L.C.} 1948. The intrinsic rate of natural increase of
insect population. - \emph{Journal of Animal Ecology} 17 (1): 15-26.

\textbf{Bongelli, E., Dowsley, M., Velasco-Herrera, V.M., \& Taylor, M}
. 2020. Do North American migratory barren-ground caribou
subpopulations~cycle?~- \emph{Arctic.~} 73 (3): 326-346.

\textbf{Bongelli., E.} 2019. Barren-ground caribou -- a cyclic species:
The development of a cycle-stratified harvest model and a cycle analysis
of North American barren-ground caribou subpopulations. Masters Thesis.
Lakehead University. Department of Geography. 133pp.

\textbf{Bowyer, T.R., Bleich, V.C., Stewart, K.M., Whiting, J.C. \&
Monteith, K.L.} 2014. Density-dependence in ungulates: A review of
causes and concepts with some clarifications. - \emph{California Fish
and Game.} 100 (3): 550-572.

\textbf{Boyce, M.S.} 1989. The Jackson elk herd: intensive wildlife
management in North America. Cambridge University Press, Cambridge,
United Kingdom.

\textbf{CARMA (CircumArctic Rangifer Monitoring and Assessment
Network).} 2016. Interactive map.
http://carma.caff.is/index.php/carma-interactive-map

\textbf{Caughley, G.} 1977. \emph{Analysis of vertebrate populations} .
John Wiley \& Sons, London. 234pp.

\textbf{Caughley, G. \& A. Gunn.} 1993. Dynamics of large herbivores in
deserts: kangaroos and caribou. Oikos 67:47-55.

\textbf{Clutton-Brock, T.H., Major, M., Albon, S.D. \& Guinness,
F.E.}1987. Early development and population dynamics in Red Deer. I.
density-dependent effects on juvenile survival. - \emph{Journal of
Animal Ecology.} 56 (1): 53-67.

\textbf{Cole, L.C.} 1954. Population consequences of life history
phenomena. - \emph{Quarterly Review of Biology} 29 (2): 103-134.

\textbf{COSEWIC} . 2016. COSEWIC assessment and status report on the
Caribou \emph{Rangifer tarandus} , Barren-ground population, in Canada.
Committee on the Status of Endangered Wildlife in Canada. Ottawa. 123
pp.

\textbf{Demerec, M.,} Organizer. 1957. Population studies: Animal
ecology and demography. Cold Spring Harbor Symposia on Quantitative
Biology 22. Cold Spring Harbor, New York: The Biological Laboratory

\textbf{Fowler, C. W.} 1981. Density-dependence as related to life
history strategy. - \emph{Ecological Society of America.} 62 (3):
602-610.

\textbf{Government of Northwest Territories.} 2021. 2021 Survey Results
for the Bathurst and Bluenose-East Barren-ground Caribou Herds. 9pp.

\textbf{Government of Yukon. 2015.} Barren-ground caribou. --
Environment Yukon. Electronically available at:
http://www.env.gov.yk.ca/animals-habitat/mammals/barrengroundcaribou.php

\textbf{Gunn, A.,} 2003. Voles, lemmings, and caribou - population
cycles revisited? - \emph{Rangifer.} 23 (2003): Special Issue No. 14:
105-112.

\textbf{Gunn, A, and Miller, F. L.} 1986. Traditional behavior and
fidelity to caribou calving grounds by barren-ground caribou. Rangifer.
6 (1986): Special Issues No. 1: 151-158.

\textbf{Heard} , \textbf{D.C.} 1980. The intrinsic rate of increase of
reindeer and caribou populations in arctic environments.
-\emph{Rangifer.} 1990: Special Issue No. 3, 169-173.

\textbf{IBM Corp.} Released 2021. IBM SPSS Statistics for Windows,
Version 28.0. Armonk, NY: IBM Corp.

\textbf{Kie, J.G., \& White, M.} 1985. Population dynamics of
white-tailed deer (\emph{Odocoileus virginianus)} on the Welder Wildlife
Refuge, Texas. - \emph{The Southwestern Naturalist 30 (1): 105-118.}

\textbf{May, R.} 1976. Simple mathematical models with very complicated
dynamics. - \emph{Nature. 261: 459-467.}

\textbf{McCullough, D.R.} 1979. The George Reserve deer herd: Population
ecology of a K-selected species. University of Michigan Press, Ann
Arbor, USA.

\textbf{McCullough, D.R.} 1999. Density-dependence and life-history
strategies of ungulates. - \emph{Journal of Mammalogy.} 80 (4):
1130-1146.

\textbf{Messier, F., Huot, J. Henaff, D., \& Luttich, S.} 1988.
Demography of the George River caribou herd: Evidence of population
regulation by forage exploitation and range expansion. - \emph{Arctic}41
(4): 279-297.

\textbf{Nagy, J.} \textbf{A., Johnson, D. L., Larter, N. C., Campbell,
M. W., Derocher, A. E., Kelly, A., Dumond, M., Allaire, D., \& Croft,
B.} 2011. Subpopulation structure of caribou (\emph{Rangifer
tarandus}L.) in arctic and subarctic Canada. - \emph{Ecological
Applications.} 21 (6): 2334-48.

\textbf{Nagy, J.A., \& Campbell, M.} 2012. Herd structure, movements,
calving grounds, activity periods, home range similarity, and behaviours
of migratory and tundra-wintering barren-ground caribou on mainland
Nunavut and eastern mainland Northwest Territories, Canada. Manuscript
report Prepared for the Beverly and Qamanirjuaq Caribou Management
Board. 155 pp.

\textbf{Payette, S., Fortin, M.J., \& Gamache, I.} 2001. The Subarctic
forest -- Tundra: The structure of a biome in a changing climate: The
shifting of local Subarctic tree lines throughout the forest-tundra
biome, which is linked to ecological processes at different
spatiotemporal scales, will reflect future global changes in climate.
-\emph{BioScience} 51(9):709 -- 718.

\textbf{Pearl, R. and Reed, L. J.} 1920. On the Rate of Growth of the
Population of the United States and its Mathematical Representation.
Proceedings of the National Academy of Sciences. 6: 275-288.

\textbf{Pearl, R., Reed. L. J.} 1922. A Further Note on the Mathematical
Theory of Population Growth. Proceedings of the National Academy of
Sciences, 8: 365-368.

\textbf{Renshaw, E.} 1991. \emph{Modelling biological populations in
space and time} . Cambridge University Press, New York, NY. 350pp.

\textbf{Russell, D.E., Kofinas, G., \& Griffith, B.} 2002. Barren-ground
caribou calving ground workshop: Report of proceedings. Technical report
series no. 390. Canadian Wildlife Service, Ottawa, Ontario.

\textbf{Schmelzer, I., \& Otto, R.} 2003. Winter Ranger drift in the
George River Caribou Herd: a response to summer forage limitation?
--\emph{Rangifer.} Special Issue No. 14: 113-122.

\textbf{Skogland, T.} 1985. The effects of density-dependent resource
limitations on the demography of wild reindeer. -- \emph{Journal of
Animal Ecology} 54 (2): 359-374.

\textbf{Stephens, D.W., and Krebs, J.R. 1986} . Foraging Theory.
Princeton University Press. Princeton, New Jersey.

\textbf{Taillon, J., Brodeur, V., \& Rivard, S.} 2016. Biological status
of migratory caribou, Leaf River herd, minist\selectlanguage{ngerman}ère des Forêts, de la Faune
et des Parcs, Québec, 67 p.

\textbf{Tanner, J.T} . 1966. Effects of population density on growth
rates of animal populations. --  \emph{Ecology} . 47 (5): 733-745.

\textbf{Vandermeer,~J.~} 2010.~How populations grow: The exponential and
logistic equations.~-- \emph{Nature Education Knowledge.} ~3(10):15

\textbf{Verhulst, P.F.} 1838. Notice sur la loi que la population
poursuit dans son accroissement.~\selectlanguage{english} \selectlanguage{english}\selectlanguage{ngerman}\emph{Correspondance Mathématique et
Physique} . 10: 113--121

\textbf{Wilson, D., E. and Reeder, D., M.} (editors)
(2005).\emph{Rangifer tarandus} . Wilson and Reeder's Mammal Species of
the World --- A Taxonomic and Geographic Reference 3rd ed. Baltimore,
Maryland: Johns Hopkins University Press/Bucknell University. 2,142 pp.

Table 1. Summary statistics for the Qamanirjuaq, Bathurst, and George
River barren-ground caribou cycles. The sine cycle maximum and minimum
annual subpopulation growth rates (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}) were updated from
Bongelli \emph{et al.} 2020. Sine cycle estimates of
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}greater than the biologically plausible were limited
to 1.2 (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausible}) with the remainder of population
growth for those years attributed to immigration. Subpopulation growth
that was partially immigration (intrinsic + immigration) occurred only
during the eruption phase.\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
& \textbf{Barren-Ground Caribou Subpopulation} & \textbf{Barren-Ground
Caribou Subpopulation} & \textbf{Barren-Ground Caribou
Subpopulation}\tabularnewline
\midrule
\endhead
\textbf{Cycle Parameter} & Qamanirjuaq & Bathurst & George
River\tabularnewline
Sine Cycle Period & 58 & 42 & 52\tabularnewline
Cycle \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} & 1.196 & 1.469 & 2.848\tabularnewline
Cycle \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} & 0.837 & 0.680 & 0.345\tabularnewline
Cycle \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausible-max} & 1.196 & 1.2 & 1.2\tabularnewline
Max N-K Lag Time & 4.333 & 4.238 & 4.334\tabularnewline
Min N-K Lag Time & 2.086 & 1.640 & 0.765\tabularnewline
Eruption Phase Duration & 6 & 9 & 9\tabularnewline
Years of Partial Immigration & 0 & 8 & 8\tabularnewline
\bottomrule
\end{longtable}

Table 2. Mean lag time (years) values and the mean population growth
rate values associated with three unique phases (complete cycle,
complete cycle excluding eruption years, and eruption years only) of the
Qamanirjuaq, Bathurst and George River subpopulation cycles. Uncorrected
values of annual population growth rate were used to account for both
intrinsic growth and extrinsic growth attributed to immigration.\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\textbf{Subpopulation} & \textbf{Cycle Phase} & \textbf{Mean Lag Time
(SE)} & \selectlanguage{greek}\textbf{Mean λ\textsubscript{t} (SE)}\selectlanguage{english}\tabularnewline
\midrule
\endhead
Qamanirjuaq & Complete cycle & 2.898 (0.008) & 1.006
(0.013)\tabularnewline
& Complete cycle (no eruption years) & 2.805 (0.079) & 0.992
(0.013)\tabularnewline
& Eruption years only & 3.693 (0.2555) & 1.127 (0.031)\tabularnewline
Bathurst & Complete cycle & 2.661 (0.103) & 1.027 (0.032)\tabularnewline
& Complete cycle (no eruption years) & 2.611 (0.118) & 0.942
(0.025)\tabularnewline
& Eruption years only & 2.85 (0.222) & 1.312 (0.047)\tabularnewline
George River & Complete cycle & 2.430 (0.121) & 1.126
(0.072)\tabularnewline
& Complete cycle (no eruption years) & 2.571 (0.116) & 0.921
(0.029)\tabularnewline
& Eruption years only & 2.125 (0.375) & 1.74 (0.192)\tabularnewline
\bottomrule
\end{longtable}

Table 3. Pairwise comparison of mean lag time values associated with
three unique phases (complete cycle, complete cycle excluding eruption
years, and eruption years only) of the Qamanirjuaq, Bathurst and George
River subpopulation cycles. We used a confidence level of p[?]0.05 and
significance values were adjusted by the Bonferroni correction for
multiple tests. Mean lag times for one complete cycle for the
Qamanirjuaq subpopulation were significantly different than the George
River subpopulation, but the difference was not significant between the
Qamanirjuaq and Bathurst or Bathurst and George River subpopulations.
Mean lags times for the entire population cycle, excluding eruption
years (intrinsic growth only) were not significantly different between
any of the subpopulations. Lag time values associated with only the
eruption years for the Qamanirjuaq subpopulation were significantly
different than the George River subpopulation, but the difference was
not significant between the Qamanirjuaq and Bathurst or Bathurst and
George River subpopulations.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\textbf{Subpopulation} & \textbf{Pairwise Comparison} & \textbf{One
Complete Population Cycle: sig. value} & \textbf{One Complete Population
Cycle -- excluding eruption: sig. value} & \textbf{Only Eruption years:
sig. value}\tabularnewline
\midrule
\endhead
Qamanirjuaq & George River & 0.018* & 0.470 & 0.013*\tabularnewline
Bathurst & Qamanirjuaq & 0.414 & 0.498 & 0.221\tabularnewline
George River & Bathurst & 0.838 & 1.00 & 0.690\tabularnewline
Hypothesis Test Summary Statistic & Hypothesis Test Summary Statistic &
0.022* & 0.249 & 0.017*\tabularnewline
\bottomrule
\end{longtable}

*Denotes SPSS Kruskal-Wallis comparison that is significant at p[?]0.05.

Table 4. Mean population growth rate values associated with three unique
phases (complete cycle, complete cycle excluding eruption years, and
eruption years only) of the Qamanirjuaq, Bathurst and George River
subpopulation cycles. We compared the uncorrected values for population
growth rate (i.e., intrinsic + immigration). We used a confidence level
of p[?]0.05 and significance values were adjusted by the Bonferroni
correction for multiple tests. Mean population growth rates for one
complete cycle for were significantly not different between any of the
subpopulations. Mean population growth rates for the entire population
cycle, excluding eruption years (intrinsic growth only) were not
significantly different between any of the subpopulations. Mean
population growth rates associated with only the eruption years for the
Qamanirjuaq versus George River pairwise comparison was significant, but
the Qamanirjuaq and Bathurst pair and the Bathurst and George River pair
were not significantly different.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\textbf{Subpopulation} & \textbf{Pairwise Comparison} & \textbf{One
Complete Population Cycle: sig. value} & \textbf{One complete population
cycle -- excluding eruption years: sig. value} & \textbf{Only eruption
years: sig. value}\tabularnewline
\midrule
\endhead
Qamanirjuaq & George River & 1.000 & 0.470 &
\textless{}0.001*\tabularnewline
Bathurst & Qamanirjuaq & 1.000 & 0.498 & 0.070\tabularnewline
George River & Bathurst & 1.000 & 1.000 & 0.427\tabularnewline
Hypothesis Test Summary Statistic & Hypothesis Test Summary Statistic &
0.958 & 0.249 & 0.002*\tabularnewline
\bottomrule
\end{longtable}

*Denotes SPSS Kruskal-Wallis comparison that is significant at p[?]0.05.

Table 5. Bivariate correlations between population growth rate
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t}), carrying capacity (K\textsubscript{t}), and lag
time (N\textsubscript{t}-K\textsubscript{t}) for the Qamanirjuaq,
Bathurst and George River barren-ground caribou subpopulations during
the eruption phase (IBM Corp, 2021). The within-subpopulation bivariate
correlation coefficients for these variables are significant (p [?] 0.05)
for all three subpopulations.\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\textbf{Subpopulation} & \selectlanguage{greek}\textbf{λ\textsubscript{t} vs
K\textsubscript{t}}\selectlanguage{english} & \selectlanguage{greek}\textbf{λ\textsubscript{t} vs
N\textsubscript{t}-K\textsubscript{t}} & \textbf{K\textsubscript{t} vs
N\textsubscript{t}-K\textsubscript{t}}\tabularnewline
\midrule
\endhead
Qamanirjuaq & 0.435 (p = 0.001) & 0.982 (p \textless{} 0.001) & 0.279 (p
= 0.043)\tabularnewline
Bathurst & 0.885 (p \textless{} 0.001) & 0.973 (p \textless{} 0.001) &
.0778 (p \textless{} 0.001)\tabularnewline
George River & 0.827 (p \textless{} 0.001) & 0.954 (p \textless{} 0.001)
& 0.743 (p \textless{} 0.001)\tabularnewline
\bottomrule
\end{longtable}

Figure 1. The Qamanirjuaq subpopulation numbers (N\textsubscript{t})
tracked carrying capacity (K\textsubscript{t}) throughout its
subpopulation cycle. Maximum and minimum rates of population growth
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 1.19; \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 0.83) closely align
with the maximum reported rates of barren-ground caribou population
growth (\selectlanguage{greek}λ\selectlanguage{english}=1.17) and population decline (\selectlanguage{greek}λ\selectlanguage{english}=0.83) (Gunn, 2003). The
eruption period (i.e., \selectlanguage{greek}λ \selectlanguage{english}\textgreater{} 1.0 and increasing over
consecutive years) lasted for approximately 6-years.
N\textsubscript{t}closely tracked K\textsubscript{t} throughout the
cycle with the maximum lag time over one complete cycle (1971-2030)
being 4.33-years.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Figure 2. The Bathurst subpopulation tracked carrying capacity (K)
throughout its population cycle. Maximum and minimum rates of annual
(intrinsic + immigration) population growth (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} =
1.469; \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 0.68) were greater than the biological
maximum annual population growth rate (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} = 1.363;
Heard, 1980) during the eruption phase. The eruption period (\selectlanguage{greek}λ
\selectlanguage{english}\textgreater{} 1.0 and increasing over consecutive years or \selectlanguage{greek}λ
\selectlanguage{english}\textgreater{}1.20) last for approximately 9-years. When
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} values were [?] 1.2 we limited them to 1.2 for
calculation of the associated Verhulst (1838) carrying capacity
(K\textsubscript{t}) because the Verhulst logistic equation is specific
for intrinsic \selectlanguage{greek}λ \selectlanguage{english}only (no immigration). We choose \selectlanguage{greek}λ \selectlanguage{english}[?] \selectlanguage{english}1.2 as the
threshold because natural populations experience both natural and
harvest mortality and not every reproductive age female would produce
viable offspring each year. N\textsubscript{t} closely tracks
K\textsubscript{t} throughout the cycle with the maximum lag time over
one complete cycle (1978-2020) being 4.238-years.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Figure 3. The George River subpopulation tracked carrying capacity (K)
throughout its population cycle. Maximum and minimum rates of annual
(intrinsic+ immigration) population growth (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} =
2.848; \selectlanguage{greek}λ\selectlanguage{english}\textsubscript{min} = 0.345) were greater than the biological
maximum annual intrinsic population growth rate (\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{max} =
1.363; Heard, 1980) during the eruption phase. The eruption period (\selectlanguage{greek}λ
\selectlanguage{english}\textgreater{} 1.0 and increasing over consecutive years or \selectlanguage{greek}λ
\selectlanguage{english}\textgreater{}1.20) lasted for approximately 9-years. When
\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{t} values were [?]1.2 we limited them to 1.2 for
calculation of the associated Verhulst (1838) determination of
K\textsubscript{t} because the Verhulst logistic equation is specific
for intrinsic \selectlanguage{greek}λ \selectlanguage{english}only (no immigration). We choose \selectlanguage{greek}λ \selectlanguage{english}[?] \selectlanguage{english}1.2 as the
threshold because natural subpopulations experience some natural and
harvest mortality; and because not every reproductive age female
produces viable offspring in every year. N\textsubscript{t} closely
tracked K\textsubscript{t} throughout the cycle with the maximum lag
time over one complete cycle (1965-2017) being

4.334-years.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

Figure 4. Lag time as the number of years for N\textsubscript{t} to
reach K\textsubscript{t} for the Qamanirjuaq, Bathurst and George River
caribou subpopulations considering both intrinsic and extrinsic
population growth over one complete cycle period. All three
subpopulations closely track carrying capacity during the decline phase
of the cycle with the largest deviations occurring the eruption years.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

Figure 5. The Qamanirjuaq, Bathurst and George River subpopulations
experience varying levels of population growth during the eruption
periods of their respective cycles. The Qamanirjuaq subpopulation
eruption period lasts approximately 6-years, and the maximum
subpopulation growth rate was well within plausible intrinsic limits.
The Bathurst and George River subpopulations experience rates of
subpopulation growth during a portion of the eruption period (9-years
respectively) that exceed biologically possible intrinsic rates of
growth (i.e., included immigration.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

Figure 6. Eruption phase relationship with immigration for the
Qamanirjuaq, Bathurst and George River subpopulation cycles.
Biologically plausible maximal intrinsic growth rates
(\selectlanguage{greek}λ\selectlanguage{english}\textsubscript{plausable}) of 1.2 were substituted for the sine-cycle
projected rates providing a measure of maximum intrinsic growth
potential. The remaining increase was ascribed to extrinsic growth
(i.e., immigration) providing a measure of relative contribution to
annual growth. The Qamanirjuaq subpopulation did not experience any
additional population growth due to immigration. Both the Bathurst and
George River subpopulation experienced varying levels of immigration
driven population growth. The maximum contribution of immigration to
total population growth was \textasciitilde{}57\% and
\textasciitilde{}89\% for the Bathurst and George River subpopulations,
respectively.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

Figure 7: Comparison of lag times (years to reach K\textsubscript{t}from
N\textsubscript{t}) and mean population growth rates from the eruption
years only for the Qamanirjuaq, Bathurst, and George River
subpopulations. Three clusters were identified: 1) short lags and high
lambda values, 2) intermediate lag time and intermediate lambda values,
and 3) long lag time and low lambda values. Cluster 1 was associated
with only three George River eruption year records, all of which had
over 70\% immigration. Most (7/9) eruption year records for the Bathurst
subpopulation were associated with intermediate cluster 2. Intermediate
cluster 2 also contained two Qamanirjuaq eruption year records, and four
George River eruption year records. Most (4/6) eruption year records for
the Qamanirjuaq subpopulation were associated with cluster 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\end{document}

