Drivers of variation in the population dynamics of bighorn sheep

Abstract

Understanding how variation in vital rates interact to shape the trajectories of populations has long been understood to be a critical component of informed management and restoration efforts. However, an expanding body of work suggests that the expectations for population dynamics of ungulates may not be applicable to small, declining, or threatened populations. Populations of bighorn sheep (Ovis canadensis) suffered declines at the turn of the 20th century, and restoration efforts have been mixed such that many populations remain small and isolated. Here, we utilized survey data collected from 1983 to 2018 from 17 populations of bighorn sheep in Montana and Wyoming to estimate the parameters of a stage‐specific population model that we used to (1) characterize the spatial and temporal variation in key vital rates including whether populations were stable, increasing, or declining; (2) estimate the contributions of vital rates to variation in population growth rates; and (3) evaluate potential sources of variation in lamb survival. We found substantial variation in all vital rates both among years and populations, strong evidence for an overall decline in nine of the 17 populations, and clear evidence for multiple combinations of vital rates that resulted in positive population trajectories. The contribution of ewe survival and lamb survival to the total variation in population growth rates varied among populations; however, declines in ewe survival dominated transitions of population trajectories from stable or increasing to declining, whereas reversals of declining population trajectories were dominated by improved lamb survival. We found strong evidence for a diverse set of associations between lamb survival and environmental covariates related to growing season and winter severity. The estimated relationships predict that environmental drivers can cause important changes in lamb survival and provide suggestive evidence that the presence of Mycoplasma ovipneumoniae is not sufficient to prevent population growth. Although our work demonstrates that the trajectories of these populations of bighorn sheep are driven by a variety of processes, the diversity of relationships between vital rates and population growth rates also suggests that there are multiple pathways to manage for population recovery.

Full text

Drivers of variation in the population dynamics of bighorn sheep
J. TERRILL PATERSON,
1,

KELLY PROFFITT ,
2
JAY ROTELLA,
1
DOUGLAS MCWHIRTER,
3
AND ROBERT GARROTT
1
1
Department of Ecology, Montana State University, Bozeman, Montana, USA
2
Montana Department of Fish, Wildlife and Parks, Bozeman, Montana, USA
3
Wyoming Game and Fish Department, Jackson, Wyoming, USA
Citation: Paterson, J. T., K. Proffitt, J. Rotella, D. McWhirter, and R. Garrott. 2021. Drivers of variation in the population
dynamics of bighorn sheep. Ecosphere 12(7):e03679. 10.1002/ecs2.3679
Abstract. Understanding how variation in vital rates interact to shape the trajectories of populations has
long been understood to be a critical component of informed management and restoration efforts. How-
ever, an expanding body of work suggests that the expectations for population dynamics of ungulates may
not be applicable to small, declining, or threatened populations. Populations of bighorn sheep (Ovis
canadensis) suffered declines at the turn of the 20th century, and restoration efforts have been mixed such
that many populations remain small and isolated. Here, we utilized survey data collected from 1983 to
2018 from 17 populations of bighorn sheep in Montana and Wyoming to estimate the parameters of a
stage-specific population model that we used to (1) characterize the spatial and temporal variation in key
vital rates including whether populations were stable, increasing, or declining; (2) estimate the contribu-
tions of vital rates to variation in population growth rates; and (3) evaluate potential sources of variation in
lamb survival. We found substantial variation in all vital rates both among years and populations, strong
evidence for an overall decline in nine of the 17 populations, and clear evidence for multiple combinations
of vital rates that resulted in positive population trajectories. The contribution of ewe survival and lamb
survival to the total variation in population growth rates varied among populations; however, declines in
ewe survival dominated transitions of population trajectories from stable or increasing to declining,
whereas reversals of declining population trajectories were dominated by improved lamb survival. We
found strong evidence for a diverse set of associations between lamb survival and environmental covari-
ates related to growing season and winter severity. The estimated relationships predict that environmental
drivers can cause important changes in lamb survival and provide suggestive evidence that the presence of
Mycoplasma ovipneumoniae is not sufficient to prevent population growth. Although our work demonstrates
that the trajectories of these populations of bighorn sheep are driven by a variety of processes, the diversity
of relationships between vital rates and population growth rates also suggests that there are multiple path-
ways to manage for population recovery.
Key words: bighorn sheep; juvenile survival; Ovis canadensis; population model; vital rates.
Received 22 January 2021; accepted 18 March 2021. Corresponding Editor: Joseph D. Holbrook.
Copyright: ©2021 The Authors This is an open access article under the terms of the Creative Commons Attribution
License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
E-mail: terrillpaterson@gmail.com
INTRODUCTION
The successful management and conservation
of wild populations, particularly those at risk,
requires information on sources of variation in
vital rates and the contribution of those vital
rates to variation in demography and population
trajectories (Nichols and Williams 2006). Where
management resources are limited, such under-
standing allows the optimal allocation of
resources to restoration efforts (Johnson et al.
2010, Mills 2012). Though the importance of
understanding the relative influence of vital rates
for shaping population trajectories has long been
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recognized, such assessments are still compara-
tively rare due the challenges of estimating the
vital rates and population structure required
(Heppell et al. 1996, Johnson et al. 2010, Koons
et al. 2017). Consequently, information is fre-
quently borrowed from species and systems for
which there is sufficient information.
For ungulates, the often-referenced expectation
for population dynamics is that population
growth rates are most sensitive to variation in
adult survival; however, variation in adult sur-
vival is small enough to contribute little to actual
variation in population growth rates (Gaillard
et al. 1998). In contrast, substantial variation in
offspring recruitment is the primary driver of
variation in population growth rates, even if
growth rates are theoretically less sensitive to
changes in this vital rate than changes in adult
survival. This paradigm was developed for, and
intended to be applied to, specific kinds of ungu-
late populations, that is, from temperate zones,
non-harvested and with stable or increasing pop-
ulation growth rates (Coulson et al. 2005, Nilsen
et al. 2009). Evidence suggests that the dynamics
of populations that are small, in decline, subject
to stochastic and substantial variation in vital
rates, or in non-temperate ecosystems defy these
expectations (Owen-Smith and Mason 2005, Nil-
sen et al. 2009, Johnson et al. 2010, Lee et al.
2016), and the evolving understanding of ungu-
late population dynamics has broadened the per-
spective to demonstrate how elasticities and
variances of vital rates integrate to shape trajecto-
ries (Hilde et al. 2020).
The population dynamics of bighorn sheep
(Ovis canadensis) may or may not operate accord-
ing to this paradigm. Populations of bighorn
sheep suffered significant declines near the turn
of the 20th century in response to a series of pres-
sures including disease, competition with live-
stock, and over-harvest (Buechner 1960, Berger
1990), and restoration efforts have demonstrated
mixed success (Singer et al. 2000, Picton and Lon-
ner 2008, Hedrick 2014). Major areas of research
required to provide the information for bighorn
sheep restoration have been identified for dec-
ades (Buechner 1960), and a substantial body of
work has developed characterizing the important
vital rates for bighorn sheep. Although adult sur-
vival is generally high with limited among-year
variation, stochastic events such as predation by
specialist predators and respiratory disease epi-
zootics, as well as hunter harvest, can induce
variation in this key vital rate and generate very
different population dynamics and trajectories
compared to populations not subject to such
pressure (Festa-Bianchet 1989, Festa-Bianchet
et al. 1997, 2006, Jorgenson et al. 1997, Ross et al.
1997, Cassirer and Sinclair 2007, Brewer et al.
2014, Manlove et al. 2016, Parr et al. 2018). More-
over, a recent comprehensive estimation of
sources of variation in ewe survival suggests
that, apart from the influence of stochastic events
and harvest, ewe survival varies among years in
response to environmental drivers (Proffitt et al.
2021). Lamb survival demonstrates substantial
among-year variation in response to predation,
disease, and environmental variation, which is
consistent with the empirical and theoretical
expectation for high variation in this rate for
ungulates in general (Gaillard et al. 1998) and
bighorn sheep in particular (Douglas and Leslie
1986, Hass 1989, Portier et al. 1998, Cassirer et al.
2001, Smith et al. 2014). However, unlike the
ephemeral decline of adult survival in response
to respiratory disease epizootics, lamb survival
can be depressed for years following a disease
event (Cassirer and Sinclair 2007, Cassirer et al.
2013). Pregnancy rates are typically high (Festa-
Bianchet 1988, Singer et al. 2000). However, preg-
nancy is the result of a complex set of metabolic
processes that integrate environmental variation
and state processes (e.g., previous year’s repro-
ductive success) such that pregnancy rates can
also vary among years (Parker et al. 2009).
Given the potential for wide variation in vital
rates, it is unsurprising that there is no consensus
on which rate is most important to the growth of
bighorn sheep populations. Previous work has
suggested that population trajectories are driven
by recruitment (Bender and Weisenberger 2005,
Manlove et al. 2016), adult survival (Rubin et al.
2002, Singer and Schoenecker 2004), or a combi-
nation of the two (Parr et al. 2018). The most
comprehensive evaluation to date of the relative
contributions from vital rates to population
growth for bighorn sheep strongly suggests that
these drivers can differ between populations
over comparatively small spatial scale (Johnson
et al. 2010), a conclusion supported by recent
work assessing variation in population trajecto-
ries for populations of bighorn sheep in the
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PATERSON ET AL.

Rocky Mountain west (Donovan et al. 2020).
Rather than being contradictory, these disparate
conclusions point to the diversity of challenges
facing bighorn sheep populations, where the
aggregated pressure from predation, disease,
small population size, and environmental drivers
varies spatially and temporally (Buechner 1960,
Brewer et al. 2014). A better understanding of the
mechanisms through which population trajecto-
ries of bighorn sheep transition from increasing
or stable to declining (or vice versa) is important
for informed management and restoration efforts
(Coulson et al. 2005, Johnson et al. 2010, Koons
et al. 2016). Prior work on bighorn sheep and
other ungulate species has implicated declines in
adult survival as the primary driver of declining
population growth rates, a pattern seen in declin-
ing populations in both temperate (Wittmer et al.
2005, Nilsen et al. 2009, Johnson et al. 2010) and
tropical systems (Owen-Smith and Mason 2005,
Lee et al. 2016). In contrast, improved offspring
recruitment is heavily associated with the recov-
ery of populations of ungulates across a similar
range of ecosystems such that understanding
limiting factors of offspring recruitment is crucial
(Beissinger and Peery 2007, Mitchell et al. 2009,
Manlik et al. 2016). For bighorn sheep, there are
comparatively few studies of how lamb recruit-
ment varies in response to ecological drivers, and
the extant work suggests a potentially complex
interplay of effects from environmental, preda-
tion, and disease-related factors (Hass 1989, Por-
tier et al. 1998, Manlove et al. 2016, Butler et al.
2018).
Here, we use multiple long-term time series of
survey data on 17 populations of bighorn sheep
in Montana and Wyoming in combination with a
recent comprehensive estimation of bighorn
sheep pregnancy and survival rates (Proffitt et al.
2021) to estimate the parameters of a population
model for bighorn sheep to address three objec-
tives: (1) to characterize the variation in key vital
rates (lamb survival and adult survival) and pop-
ulation growth rates, (2) to estimate the contribu-
tions of changes in these vital rates to observed
changes in population trajectories, and (3) to
identify potentially important sources of varia-
tion in lamb survival including environmental
sources of variations (e.g., precipitation, primary
production, and winter severity), predation, and
disease.
METHODS
Study area and populations
The study was conducted in Montana and
Wyoming and included data for 17 bighorn
sheep populations (Fig. 1). In all but two cases,
populations were defined based on historic
delineations of management units used by the
management agencies in the two states thought
to (generally) represent demographically closed
populations. In two cases, we combined data
from separate management units in Wyoming to
create single demographically closed populations
(Whiskey Mountain [Mtn]-West and Whiskey
Mtn-East) that each were the combination of two
underlying management units. Sixteen of the 17
populations were located in mountainous areas,
and one population (Middle Missouri) was
located in a prairie breaks landscape. Although
the composition of the communities of pathogens
that are associated with bighorn sheep respira-
tory disease are not known in these populations,
the most comprehensive assessment to date sug-
gests that all of these populations host Pasteurel-
laceae bacterial pathogens, and all but Paradise,
Petty Creek, Targhee, and Middle Missouri host
Mycoplasma ovipneumoniae (Butler et al. 2018). All
17 populations had a management history that
included removals from the population due to
translocations of animals to other herds or har-
vest. Every population had ram removals, with
the number of rams taken in any given year
ranging from 0 to 69 (median =7 removed).
However, only eight of the 17 populations had
ewe removals; in contrast to ram removals, the
number of ewes removed in any given popula-
tion varied from year to year ranging from 0 to
79 (median =0 removed). The management his-
tories indicate that nine of the populations expe-
rienced all-age die-off events at least once during
the study period.
Survey data
Survey data for the 17 populations were com-
piled based on agency records that contained
information obtained using multiple methods,
including fixed-wing and helicopter- and
ground-based survey efforts to count animals
and classify animals into age and sex classes. The
timing of surveys differed among and within
bighorn sheep populations (median =March,
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range =December–May). Ideally, a survey was
conducted annually, and during each survey, a
sample of the total animal inventory was classi-
fied as lambs, adult males, and adult females. A
count of all observed animals was recorded if the
survey was thought to be representative of the
population in the consistent area of core seasonal
range. However, the nature of the data and the
practical challenges associated with annual sur-
veys resulted in a discontinuous time series for
nearly every population such that some years
were missing a count (but had classification
data), some years were missing classification
data (but had a count), some years were missing
all data, and some years had both count and clas-
sification data (complete) (Fig. 2, see Appendix
S1 for complete population observation histo-
ries). We left-truncated the time series for each
population at the first year for which there was a
representative count.
Fig. 1. Study area map showing the approximate locations of the 17 Montana and Wyoming bighorn sheep
populations used in this study.
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Fig. 2. Structure of the bighorn sheep survey data (a) and variation in covariate values used as potential
sources of variation in lamb survival (b). The survey data are largely discontinuous time series, where data in
any year can be missing: both total count data and classification data (missing all data), classification data, or
total count data. Covariates used for lamb survival index growing conditions (primary production [NDVI] and
precipitation [PREC]) during the spring (early) and summer (late) and winter severity (PREC
winter
) (colors repre-
sent populations). NDVI, normalized difference vegetation index.
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Modeling approach
Our modeling approach was to characterize
the drivers of variation in the population
dynamics of bighorn sheep using the time ser-
ies of management count and classification
data. Prior work has demonstrated that state-
space approaches have desirable characteristics
for modeling population dynamics (de Valpine
and Hastings 2002, Buckland et al. 2004, New-
man et al. 2006, Schaub and Abadi 2011). A
state-space model in this context consists of
two processes: a biological model that connects
changes in the population structure and size
through time via key vital rates such as sur-
vival and fecundity and an observation model
that handles the stochastic and imperfect obser-
vation process (Buckland et al. 2004). One of
the key features of such state-space models for
population dynamics is that they separate the
observation process from the biological process
and, conditional on the model, improve preci-
sion on inference for population dynamics (de
Valpine and Hastings 2002). Such hierarchical
models naturally fit within the Bayesian para-
digm of statistical inference, and the flexible
nature of model specification in the popular
and freely available software packages accom-
modate such models well (e.g., JAGS, STAN,
OpenBUGS) and also have two key additional
benefits. First, by specifying a biological process
model that connects vital rates to size of age/-
sex classes through time, it is straightforward
to derive a number of biologically relevant
quantities such as population growth rates and
sex/age class ratios that are the integrated result
of variation in multiple vital rates. Second, it is
easy to introduce informed priors for vital rates
that are not directly estimated in the model.
Biological process model.—We developed a
stage-based model for the population dynamics
of bighorn sheep (Caswell 2001). We defined a
biological year from June 1 in year tto May 31 in
year t+1 to account for the pre-birth pulse sur-
veys. At the time of the surveys in the late spring,
each population was comprised of lambs (<1yr
old), yearlings (>1 and <2 yr old), and adults
(≥2 yr old). Recent work in this system demon-
strated that pregnancy rates for yearling ewes
can be substantially lower than that for older
ewes, suggesting that it is important to treat
yearlings as a separate class in a biological model
to avoid bias in estimates of offspring survival
(Proffitt et al. 2021). Therefore, our biological
process model used six age–sex classes (lambs,
yearlings, and adults by sex).
Bighorn sheep females typically have a sin-
gle offspring, and fecundity in a pre-birth
pulse model is therefore the product of preg-
nancy rates and lamb survival to the end of
their first year. For each population, we sepa-
rately modeled the number of lambs (P
l
) pro-
duced from yearling ewes (ye) and adult ewes
(ae) in biological year tas a function of the
size of these two reproductive classes in the
previous year and age class-specific pregnancy
rates (τ
ye
and τ
ae
), and a binomial process to
incorporate demographic stochasticity (or, vari-
ation in fates at the population level due to
chance, May 1973, Lande 1993, Kendall and
Fox 2002):
Pl,ye
t∼Binomialðτye,Nye
t1Þ(1)
Pl,ae
t∼Binomialðτae,Nae
t1Þ(2)
such that the total number of lambs produced at
the start of biological year tis simply
Pl
t¼Pl,ye
tþPl,ae
t. The survival of lambs from the
beginning of biological year tto the end of bio-
logical year t(immediately prior to transitioning
to the yearling class in year t+1) was modeled
using a similar binomial process based on lamb
survival (S
l
):
Nl
t∼Binomial Sl,Pl
t

:(3)
The number of yearling ewes (Nye
t) and year-
ling rams (Nyr
t) at the end of biological year twas
modeled using a binomial process based on the
number of lambs in year t−1(Nl
t1), the
assumption of an equal sex ratio of lambs (0.5),
and sex-specific adult survival (ewes: S
e
, rams:
S
r
):
Nye
t∼BinomialðSe,0:5Nl
t1Þ(4)
Nyr
t∼Binomial Sr,0:5Nl
t1

:(5)
Finally, the number of adult ewes (Nae
t) and
rams (Nar
t) at the end of year twere modeled
using a binomial process based on the number of
yearling ewes and rams in year t−1(Nye
t1,
Nyr
t1), the number of adult ewes and rams in year
t−1(Ne
t,Nr
t), and sex-specific adult survival:
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PATERSON ET AL.

Nae
t∼BinomialðSe,Nye
t1þNae
t1Þ(6)
Nar
t∼Binomial Sr,Nyr
t1þNar
t1

:(7)
Our biological model relied on three simplifi-
cations of bighorn sheep vital rates: (1) The sur-
vival of male and female lambs are equal, (2)
there was no difference in the sex-specific sur-
vival of yearlings and adults, and (3) there was
no age-related variation in either survival or
pregnancy rates (e.g., reproductive or actuarial
senescence). Our biological model also assumed
density independence of vital rates, a reasonable
assumption for the generally smaller, recovering
populations used here for whom density-
dependent resource limitation was unlikely to be
limiting.
Observation process model.—To connect the
expected number of animals in each sex and age
class generated by the biological process model
to the observed data, we used a model that could
accommodate survey data where only a fraction
of the total number of animals counted were clas-
sified into the management age/sex categories
(lamb, adult female, adult male) (Paterson et al.
2019). Frequently, populations were surveyed for
counts, but only a fraction of the total number of
animals counted were classified into the manage-
ment age/sex categories (lamb, adult female,
adult male). Therefore, we modeled the total
count for each population in each year t(Count
t
)
using a Poisson process where the expected
value was the total population size (the sum of
all age and sex classes, Ntotal
t¼Nl
tþNye
tþ
Nyr
tþNae
tþNar
t):
Countt∼PoissonðNtotal
teζtÞ(8)
where ζ
t
was an observation-level random effect
used to account for overdispersion in the obser-
vation process. We then connected the total num-
ber of animals classified in each survey
(Classified
t
) to the age and sex classes using a
multinomial distribution. During these manage-
ment surveys for population trends, animals
were classified only as lambs, ewes, and rams,
for example, yearling ewes and adult ewes are
not separately observable. However, inference on
an unobserved age class is possible in our model
due to the structure of the biological process
model. We incorporated the yearling class into
the classification process in year tby relating the
number of classified lambs (Count
l
), ewes
(Count
e
), and rams (Count
r
) to their proportion
in the population:
Countl, Counte, Countr

t
∼Multinomialðπt, ClassifiedtÞ(9)
πt¼Nl
Ntotal ,Nye þNae
Ntotal ,Nyr þNar
Ntotal

t
:(10)
This observation model connected the survey
data to the underlying vital rates, a central
assumption of which is that these populations
were demographically closed. Where this
assumption was violated in the presence of
immigration/emigration, it would be expected to
result in biased estimates of vital rates, that is,
survival biased high in years with immigration.
Where animals were removed by harvest or for
translocations, it would bias survival rates low if
removals were additive to other sources of mor-
tality. Additionally, the presence of incomplete
data (years with completely missing data, or
years missing a total count or a classification)
would be expected to inflate the variance of esti-
mates of vital rates from our model and/or gener-
ate bias in sex-specific survival rates.
Accommodating inconsistent survey timing.—A
common complication in our survey data is that
surveys were not always conducted at the end of
the biological year in late spring. Thus, if we
ignored differences in survey timing and naively
used the biological and process models above,
we would have introduced bias into our esti-
mates of vital rates by conflating the timing of
surveys with the survival processes, that is, early
surveys that missed late winter mortality would
tend to cause year-specific survival rates to be
inflated. We therefore adapted the above model
to the timing of the surveys with a model for how
survival changes within a year and used the tim-
ing of the observed data to help estimate the pop-
ulation size at the end of year t.Specifically, we
introduced another latent state for the size of each
age and sex class at the time of the surveys and
connected it to the model for the size of each age
and sex class at the end of year t. For example, the
number of lambs in the population in year tat the
time of the survey (Nl
t∗) was modeled as
Nl
t∗∼BinomialðfðSl,d∗Þ,Pl
tÞ(11)
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and the number of lambs at the end of the biolog-
ical year (Nl
t, the ideal timing) was based on this
latent state
Nl
t∼BinomialðgðSl,d∗Þ,Nl
t∗Þ(12)
where f(S
l
,d
*
) was a function that connected the
relationship between the timing of the observa-
tion (d
*
) and the expected survival of lambs to
that point, and g(S
l
,d
*
) was a function that con-
nected the difference in timing between the sur-
veys and the end of the year to the expected
number of lambs at the end of the year. We
chose to use simple functions that factored the
survival rate into monthly survival and then cal-
culated survival to the time of observation as
monthly survival raised to a power equivalent to
the number of months since the beginning of
year t. For example, if the survey was conducted
in March, it was in month 10 (d
*
=10) of biologi-
cal year tthen fS
l,10

¼Sl

1
12

10
, and
gS
l,10

¼Sl

1
12

1210
. We used these same
functions to model the latent states of all age/sex
classes. Such survival functions are a consider-
able simplification of a reality in which the
shape of the survival curve likely differs
between age and sex classes and among years.
However, we considered this approach to be a
reasonable approximation in the face of sparse
information for this species and a considerable
improvement over naively using survey data
without accounting for the timing.
Modeling goals
Our state-space model for the time series of
counts is a hierarchical model with a large set of
latent states required to accommodate the struc-
ture of the data. However, these models for pop-
ulations are driven by a small number of vital
rates: pregnancy rates of yearling and adult ewes
(τ
ye
and τ
ae
), the survival rate of lambs (S
l
), and
the survival rates of ewes (S
e
) and rams (S
r
). Our
goals were to estimate these vital rates from the
count data, to characterize their associations with
population trajectories (a derived parameter),
and to understand sources of variation in lamb
survival. To meet these three goals, we used two
versions of the above model: (1) a version
wherein S
l
was estimated for each population-
year using a random effects structure (REmodel)
and (2) a version that used logistic regression to
assess potential sources of variation in lamb sur-
vival in each population using covariates as
proxies for environmental variation (COVmodel).
All other model structures were the same
between models.
Common model structures.—We used informed
priors for adult survival and yearling and adult
ewe pregnancy rates, constructed from a recent
comprehensive assessment of pregnancy and
survival rates in bighorn sheep in this same area
and for many of these populations (Proffitt et al.
2021). We took the empirical distribution of the
estimated probabilities for yearling pregnancy
and adult pregnancy and fit a beta distribution
to each to construct the priors for the population-
specific probabilities of pregnancy in the state-
space model (τye
p∼Betað29:737, 9:731Þand
τae
p∼Betað384:49, 33:25Þ). Similarly, for ewe and
ram survival, we used the results from Proffitt et al.
(2021) for ewe survival to construct an informed
prior to use for sex-specific, time-varying survival
rates for each population (Se
p,t∼Betað10:452, 3:292Þ
and Sr
p,t∼Betað10:452, 3:292Þ). Although this pre-
vious work was specific to ewes, we considered this
distribution (with an approximate mean of 0.76
and standard deviation of 0.11) to be sufficiently
vague so as to serve as a general prior for adult sur-
vival. For the initial sizes of each age and sex class
in each population, we used a vague uniform prior
(Uniform(0, upper
p
)), where upper
p
was an upper
limit for each population defined as the maximum
total count in the time series, that is, the number of
individuals in each age and sex class in the initial
population had to be less than the maximum aggre-
gate count of all classes). Observation-level random
effects were drawn from a normal distribution
(ζt∼Normal 0, σ2
ζ

)withavaguepriorforthe
standard deviation (σ
ζ
~Uniform(0, 10)).
Distributions of vital rates and their association
with demographic performance.—The first of our
three goals was to estimate the key vital rates
underlying our process model, and our second
was to characterize the associations between vital
rates and the observed trajectories of bighorn
sheep populations. However, unlike survival and
pregnancy rates, we did not have an informed
prior to place on lamb survival. To provide a
moderate amount of shrinkage and improve esti-
mation for years with missing data while not
overly influencing estimates from years with
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complete data, we modeled all population-year
lamb survival rates, Sl
p,t, using a random effects
structure on the logit scale (REmodel), where
logit Sl
p,t

¼μþξp,t(13)
ξp,t∼Normalð0, σ2
ξÞ(14)
with a Logistic(0, 1) prior for μand a vague prior
for the standard deviation σ
ξ
~Uniform(0, 10). It
is important to note that lamb survival and sex-
specific adult survival were assumed to be inde-
pendent (i.e., no explicit modeling of the poten-
tial covariation between rates), after initial model
runs suggested estimation problems for this
more complex model using count data and gen-
erally vague priors for survival and pregnancy
(Koons et al. 2017).
Rather than a prospective analysis such as life-
stage simulation analysis (Wisdom et al. 2000)
that estimates how hypothetical changes in vital
rates may impact population growth rates, we
wanted to decompose the observed variation in
population growth rates into contributions from
the vital rates. Life-table response experiments
(LTREs) provide a powerful method of investigat-
ing these drivers of observed variation in popula-
tion growth rates, and recent work has derived a
series of transient LTREs that do not assume a sta-
tionary environment (Cooch et al. 2001, Koons
et al. 2016, 2017). The population growth rate in
year t(λ
t
) can be expressed as a function of under-
lying vital rates and the population structure in
year t−1, and sensitivities of population growth
rates to demographic parameters derived using
partial derivatives (Appendix S2). Our study
focused on: (1) estimating the contribution of each
of the vital rates to the total observed variation in
λ
t
(contributionvarðλtÞ
θifor vital rate θ
i
from Koons
et al. 2016) and (2) understanding the drivers of
changes in λ
t
between successive time steps
(contributionΔλt
θifor vital rate θ
i
from Koons et al.
2016) (details in Appendix S2).
Sudden, precipitous declines in vital rates that
are associated with epizootics, shifts in predation
risk, or severe environmental conditions are a
persistent source of variation in many popula-
tions of bighorn sheep (Ross et al. 1997, Portier
et al. 1998, Festa-Bianchet et al. 2006, Cassirer
and Sinclair 2007, Manlove et al. 2016). For such
circumstances, our use of informed priors for
adult survival rates and a single, common distri-
bution of lamb survival rates could have been
too restrictive. The priors for adult survival
might have been too informative to capture these
dramatically lower rates, and rarity of these
events resulted in the estimation of the distribu-
tion of lamb survival rates being dominated by
“normal”years and masking the signal of these
events. To evaluate the sensitivity of our results
to our prior specification, we constructed an
alternative version of the model wherein both
adult and lamb survival were modeled using
mixture distributions that allowed for more vari-
ation in vital rates (Appendix S2).
Correlates of lamb survival.—Our third main
goal was to assess a series of covariates as poten-
tial sources of variation in lamb survival. Our a
priori expectation was that lamb survival should
be positively related to favorable conditions dur-
ing the growing season (e.g., indexed by the nor-
malized difference vegetation index [NDVI] or
precipitation) and negatively related to winter
conditions (e.g., indexed by indices of winter
severity). We used a derived metric of NDVI as
an index of primary production during the grow-
ing season (Pettorelli et al. 2011). Using the
AVHRR (Advanced Very High Resolution
Radiometer) time series of NDVI values (Ver-
mote and NOAA CDR Program 2019), we per-
formed a series of smoothing steps on each pixel
in our study area. First, we calculated the
monthly median of the daily NDVI values pro-
duced by the AVHRR time series and then
applied a smoothing spline on the monthly time
series of values. Second, we estimated the start of
the growing season as the point in the smoothed
signal at which the value of NDVI first attained
50% of the seasonal maximum, and the end of
the season as the first point in time past the sea-
sonal maximum at which the smoothed signal
that the NDVI value was below 50% of its sea-
sonal maximum (Bradley et al. 2007). Third, to
ameliorate the impact of canopy cover on the
NDVI signal, we calculated the integrated NDVI
from the start of the season to the end of the sea-
son as the sum of the differences between each
NDVI value and the value of NDVI at the start of
the growing season (Ruimy et al. 1994, J ¨onsson
and Eklundh 2004). Prior work has suggested
that population responses to NDVI might
depend on the timing (Paterson et al. 2019), that
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is, high NDVI during the spring could indicate
substantial primary production and favorable
growing conditions conducive to neonates that
could be coupled to a late season characterized
by drought and poor growing conditions such
that a single NDVI metric for an entire growing
season could be misleading. Therefore, to investi-
gate differential impacts of the seasonal timing of
NDVI (e.g., early-season vs late-season growing
conditions), we defined the integrated NDVI
over two periods: from the start of the growing
season to June 30 (NDVI
early
), and from July 1 to
the end of the growing season (NDVI
late
). Addi-
tionally, we derived a second metric of environ-
mental conditions during the growing season.
Using the PRISM data set (PRISM Climate
Group, Oregon State University, http://prism.ore
gonstate.edu), we derived precipitation metrics
similar to those for NDVI for each pixel in our
study area as the sum of precipitation from May
1 to June 30 (PREC
early
) and the sum of precipita-
tion from July 1 to September 30 (PREC
late
).
Moreover, we used the sum of precipitation from
December to March as an index of winter sever-
ity (PREC
winter
). To connect the values of each of
these five metrics to the populations in our study,
we relied on estimated seasonal ranges for these
populations using either GPS collar data from a
regional research project on these populations
(n=16, details in Proffitt et al. 2021), or seasonal
ranges derived from professional opinion (n=1,
Dubois Badlands). Environmental conditions in
each biological year were approximated by using
the median value of all the pixels contained in
each seasonal range for that year.
We evaluated potential sources of environmen-
tal variation in lamb survival using logistic
regression to connect covariates in population p
and year t(e.g., NDVIearlyp,t) to survival using
the logit link (COVmodel):
logit Sl
p,t

¼μþβNDVIearly
pNDVIearlyp,tþβNDVIlate
p
NDVIlatep,tþβPRECearly
pPRECearlyp,tþβPREClate
p
PREClatep,tþβPRECwinter
pPRECwinterp,t(15)
where the population-specific mean, μ, was given
a Logistic(0, 1) prior and regression coefficients
were given independent Normal(0, 4) priors that
are vague on the logit scale (Northrup and Ger-
ber 2018).
There were two additional potential sources of
variation in lamb survival that were not amen-
able to assessment using our full data set. First,
predation on bighorn sheep lambs can have a
dramatic impact on survival rates (Berger 1991,
Rominger et al. 2004, Festa-Bianchet et al. 2006,
Brewer et al. 2014) and be critical to consider
when evaluating potential sources of variation in
population dynamics. Although the larger region
has seen the recovery of populations of multiple
predator species during our study period, moun-
tain lions (Puma concolor) are implicated as the
primary predator affecting bighorn sheep popu-
lation dynamics (Rominger 2018). However,
there is no direct measure of predation pressure
from mountain lions or an index of it (e.g.,
mountain lion abundances) for a region the size
of our study area and over the timespan of the
data collection. Therefore, to assess the potential
impact of mountain lions on lamb survival, we
developed a spatial covariate by extrapolating a
previously published winter resource selection
function (RSF) for Montana (Robinson et al.
2015) over the extent of our study area and then
used the median value of the RSF within each
seasonal range as an index of risk from mountain
lions (details in Appendix S2). Due to the under-
lying assumption of stationarity, we limited the
data set for this analysis to the last 10 yr of obser-
vations for each population. We then evaluated
the strength of evidence for a relationship
between lamb survival and the median value of
the extrapolated mountain lion RSF for each pop-
ulation by modifying Eq. 15 to include the
population-year value of the derived covariate
(LION). Second, disease processes are thought to
play a pivotal role in the population dynamics of
bighorn sheep. To explore the potential role of
disease as a limiting factor in lamb survival, we
assessed the evidence for a relationship between
pathogen communities in these populations and
lamb survival. Using the results of the most com-
prehensive and rigorous study to date on the
composition of communities of pathogens associ-
ated with respiratory disease in bighorn sheep
(Butler et al. 2018), we compared the distribu-
tions of lamb survival for those populations that
hosted both M. ovipneumoniae and Pasteurel-
laceae bacteria (Bibersteinia trehalosi,Pasteurella
multocida, and Mannheimia haemolytica)(n=12)
to those populations that only hosted
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Pasteurellacea (n=4). Pathogen data for the
Spanish Peaks population were not published as
part of this study, and data were censored from
this population for this portion of our analysis.
Similar to the mountain lion covariate, our com-
parison of lamb survival values between the two
groups assumed stationarity, that is, the presence
of the pathogen(s) was considered a fixed charac-
teristic that did not vary through time. Therefore,
we only used the last 10 yr of estimates of lamb
survival for each population from the first ver-
sion of our model (Eq. 14, REmodel) and used the
approximate posterior distribution of estimated
survival values to estimate the median lamb sur-
vival for all years as well as annual values for
both groups.
The covariates used to assess sources of varia-
tion in lamb survival showed substantial varia-
tion both across and within populations (Fig. 2,
Appendix S3). Prior to being used in analyses, all
covariates were screened for collinearity (with a
threshold of 0.5) and were centered and scaled
using the mean and one standard deviation. To
facilitate interpretation for estimated covariate
effects, for each covariate for which the 90%
highest posterior density interval of the esti-
mated regression coefficient did not overlap
zero, we made predictions for the population-
specific relationship between the covariate and
lamb survival by holding all other covariates to
their mean. For each covariate, we then selected
a set of example populations to demonstrate
how the predicted relationship translated into
variation in lamb survival from the 5th percentile
of the covariate values (S
l
(covariate
0.05
)) to the
95th percentile of the covariate values (S
l
(covari-
ate
0.95
)).
Model estimation and evaluation
Models were fit in the R programming envi-
ronment (R Development Core Team 2019) using
the runjags package (Denwood 2016) as an inter-
face to the JAGS program (Plummer 2017) for
Markov chain Monte Carlo (MCMC) estimation
of our model. For both versions of our model
(REmodel and COVmodel), we ran two parallel
chains for 150,000 iterations and discarded the
first 50,000 as the burn-in and adaptation phase.
Due to the large number of derived parameters
saved from these simulations, we kept every
20th iteration (i.e., a thinning interval of 20),
which resulted in a combined total of 10,000
samples used for inference. Convergence of these
chains was graphically assessed using traceplots
for the underlying vital rates. For the large num-
ber of derived parameters (population-, sex-,
age-, and time-specific sizes of populations), a
graphical assessment was not practical and con-
vergence was evaluated using the Gelman-Rubin
statistic and convergence was assumed for values
<1.05 (Gelman and Rubin 1992). We summarized
the approximate posterior distributions for all of
the model parameters using the median and 90%
highest posterior density interval.
There are no standard test statistics for evalu-
ating the adequacy of the fit of state-space mod-
els to time series data such as these. Therefore,
we evaluated the goodness of fit of both versions
of our model to each population using two poste-
rior predictive checks (Gelman et al. 1996) that
compared the fit of the models to the data using
two biologically relevant characteristics of the
data: the observed population growth rates and
the observed ratio of lambs to ewes (both year-
ling and adult) (details in Appendix S2). For each
iteration of the MCMC chain for each model, we
derived a replicated data set to compare to the
observed data. We calculated a Bayesian pvalue
as the proportion of iterations where the value of
the statistic for the observed data exceeded the
value of the statistic for replicated data. Bayesian
pvalues close to 0.5 indicate the model is faith-
fully replicating the variation of the observed
data, whereas values close to 0 or 1 indicate a
severe lack of fit (Gelman et al. 1996).
RESULTS
The survey data were comprised of (typically)
discontinuous time series of counts and age and
sex classifications for each population from 1983
to 2018 (Fig. 2; for population-specific sum-
maries, see Appendix S1). In total, the data set
included 470 population-years of observations
(including years missing observations within the
time series). Across populations, there was con-
siderable variation in the median total count
from Targhee (68 animals) to Wapiti Ridge (716
animals) coupled to similar variation in range of
counts from Targhee (maximum–minimum =
59) to Whiskey Mtn-East (maximum–mini-
mum =640). The ratio of lambs to ewes across
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populations demonstrated similar variation with
the median ratio ranging from a minimum in
Lost Creek (0.14) to a maximum in Middle Mis-
souri (0.47) and the range of ratios from Francs
Peak (maximum ratio–minimum ratio =0.24) to
Whiskey Mtn-West (maximum–minimum =
0.81).
Convergence was achieved for parameter esti-
mates in both models; bRvalues for all latent
states (i.e., sizes of sex/age classes) were <1.05,
and bothbRvalues and visual inspection of trace-
plots for underlying vital rates of both models
indicated convergence and adequate mixing. All
of the posterior predictive checks (two posterior
predictive checks for the REmodel and COVmodel)
suggested adequate fit for the models, given that
the one-sided Bayesian pvalues were far from 0
or 1 for both population growth (REmodel:
p=0.67; COVmodel:p=0.52) and age ratios
(REmodel:p=0.26; COVmodel:p=0.81).
Variation in vital rates
Using the REmodel for inference, we found
substantial variation in the vital rates (Fig. 3).
The population-specific median values of ewe
survival across years varied from a minimum of
0.78 for Lost Creek (90% highest posterior den-
sity interval =0.73–0.83, hereafter all similar
credible intervals are denoted using brackets) to
0.88 [0.86, 0.90] for Wapiti Ridge. The variation
in ewe survival within and among the popula-
tions was substantial: the within-population dif-
ference between the maximum and minimum
point estimates of annual ewe survival values
varied from 0.08 for Wapiti Ridge (max =0.90
[0.82, 0.98], min =0.82 [0.70, 0.93], variance in
annual values =0.004 [0.002, 0.006]) to 0.26 for
Lost Creek (max =0.85 [0.72, 0.96], min =0.59
[0.44, 0.76], variance in annual values =0.01
[0.006, 0.020]). Point estimates for the survival
rates for rams tended to be lower than those for
ewes and displayed more variation. The
population-specific median values of ram sur-
vival across years varied from a minimum of
0.71 [0.67, 0.75] for Clarks Fork to 0.82 [0.80, 0.85]
for Paradise, and within-population differences
between the maximum and minimum point esti-
mates of annual ram survival values ranging
from 0.10 in Middle Missouri (max =0.87 [0.76,
0.97], min =0.77 [0.59, 0.94], variance in annual
values =0.008 [0.004, 0.012]) to 0.47 in Lost
Creek (max =0.87 [0.77, 0.97], min =0.41 [0.29,
0.54], variance in annual values =0.021 [0.012,
0.031]). Lamb survival was generally lower than
adult survival, with a range of population-
specific median survival rates across years from
a low of 0.19 [0.15, 0.23] for Whiskey Mtn-East to
a high of 0.48 [0.40, 0.54] for Middle Missouri
(Fig. 3). Within-population differences between
the maximum and minimum of point estimates
of lamb survival ranged from 0.26 for Targhee
(max =0.51 [0.18, 0.83], min =0.24 [0.08, 0.45],
variance in annual values =0.035 [0.022, 0.049])
to 0.60 for Dubois Badlands (max =0.70 [0.45,
0.92], min =0.10 [0.03, 0.19], variance in annual
values =0.046 [0.032, 0.060]).
Variation in estimated vital rates translated
into large variation in population growth rates
with population-specific median values of λ
across years ranging from 0.85 [0.81, 0.90] for
Lost Creek to 1.03 [1.00, 1.05] for Middle Mis-
souri (Fig. 3). Within-population differences
between the maximum and minimum point esti-
mates of annual growth rate values demon-
strated high variability, ranging from 0.19 for
Younts Peak (max =1.09 [1.03, 1.26], min =0.90
[0.80, 1.00], variance in annual values =0.012
[0.007, 0.0174]) to 0.60 in Dubois Badlands
(max =1.34 [1.10, 1.60], min =0.74 [0.57, 0.90],
variance in annual values =0.029 [0.020, 0.039]).
We compared the characteristics of population-
years with growth rates <1 to those with growth
rates >1 and found substantial differences in
population vital rates: Population-years with
bλ≥1 had overall higher survival rates (ewe sur-
vival: median =0.88 [0.87, 0.89]; ram survival:
median =0.80 [0.79, 0.82]; lamb survival:
median =0.48 [0.45, 0.51]) than population-
years with bλ<1 (ewe survival: median =0.82
[0.81, 0.83]; ram survival: median =0.73 [0.71,
0.74]; lamb survival: median =0.24 [0.23, 0.26]).
Finally, we estimated the geometric means for
each population and found that nine populations
had estimated geometric means with the upper
limit of a 90% highest posterior density interval
<1 (indicating long-term population decline;
Castle Reef, Clarks Fork, Francs Peak, Lost
Creek, Targhee, Trout Peak, Wapiti Ridge, Whis-
key Mtn-East, and Younts Peak), two popula-
tions had estimated geometric means with a
lower 90% highest posterior density interval limit
>1 (indicating long-term population growth;
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Fig. 3. Distribution of point estimates of yearly vital rates for the 17 bighorn sheep populations in the study
(a), and all population-years combined (b). The dot denotes the estimated median for the time series and the
black line the range of estimated values. For estimates of the population growth rate, the open square indicates
the geometric mean. For panel (b), the light red corresponds to the distribution of adult female annual survival
rates (Proffitt et al. 2021).
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PATERSON ET AL.

Middle Missouri and South Madison), and the
remainder had 90% highest posterior density
intervals that included 1 (Fig. 3).
We found evidence for constraining relation-
ships between lamb survival, ewe survival, and
the population growth rate for the female com-
ponent of the population (bλf) (Fig. 4). Using the
medians of the approximate posterior distribu-
tion from the REmodel for each annual parameter
as a point estimate, we found that lamb survival
rates corresponding to estimates of bλf≥1 had a
similar distribution (median =0.46; range =-
0.21–0.78) to those corresponding to bλf<1 (me-
dian =0.28; range =0.05–0.61). There was
similar ambiguity in the relationship between
ewe survival and λ
f
. The distribution of ewe sur-
vival rates corresponding to estimates of bλf≥1
(median =0.86; range =0.77–0.90) was essen-
tially the same as the distribution of values corre-
sponding tobλf<1 (median =0.83; range =0.60–
0.90). We found that the association between
these vital rates depended on their values, such
that the impact of lamb or ewe survival on λ
f
depended on the value of the other rate (Fig. 4).
For example, at values of ewe survival ≥0.85, we
found λ
f
≥1 over a wide distribution of lamb
survival values (median =0.43; range =0.21–
0.68). Once ewe survival dropped to ≤0.80, how-
ever, we found that λ
f
≥1 over only a narrow
and high distribution of lamb survival values
(median =0.58; range =0.54–0.68). If ewe sur-
vival dropped below 0.75, no value of lamb sur-
vival was capable of yielding λ
f
≥1. Conversely,
when lamb survival rates were ≥0.40, we found
that λ
f
≥1 over a relatively wide distribution of
ewe survival rates (median =0.85; range =0.77–
0.90). When lamb survival rates were ≤0.20, no
value of ewe survival resulted in λ
f
≥1. Finally,
we noted a relationship between the growth rate
of the female population (λ
f
) and the number of
females removed from each population due to
either harvest or translocation, which was
incorporated into the annual survival rates in the
model. We calculated the ratio of the number
females removed in year t+1 to the estimated
size of the population from the model in year t
and compared it to λ
f
in year t. We found that
when the ratio of female removals to female pop-
ulation size exceeded approximately 0.18, all
point estimates of λ
f
were <1.
Transient life-table response experiments
We decomposed the observed variation and
changes in the population growth rates of the
female portion of each population into contribu-
tions from changes in constituent demographic
parameters using the REmodel for inference and
found substantial among-population differences
in drivers of population trajectories. Partitioning
the total variation in population growth rates
(see Fig. 3 for the distribution of point estimates
for annual growth rates) into the contributions
from lamb survival, ewe survival, and changes in
the age structure revealed that, in general, varia-
tion in ewe survival accounts for a large amount
of the variation in bλf(Fig. 5). For six of the 17
populations (Clarks Fork, Jackson, Lost Creek,
Paradise, Petty Creek, and Targhee) the 90%
highest posterior density interval for the esti-
mated proportion of variation inbλfattributed to
variation in ewe survival did not overlap the sim-
ilar estimate for the proportion of variation in bλf
due to variation in lamb survival, strongly sug-
gesting that variation in ewe survival dominated
variation in bλffor these populations. For two of
the populations (Middle Missouri and Trout
Peak), overlap between the credible intervals pre-
vented strong inference but suggested a similar
relationship. For seven of the 17 populations
(Clarks Fork, Dubois Badlands, Francs Peak,
Spanish Peaks, Wapiti Ridge, Whiskey Mtn-East,
and Younts Peak), the largely coincident credible
intervals suggest roughly equal contributions
from lamb and ewe survival. In only one case
Fig. 4. Relationship between key vital rates and the growth rates of the female component of the population
(λ
f
). Panel (a) represents the univariate relationships between lamb and ewe survival and estimated λ
f
. Panel (b)
represents the multivariate relationship between these two key underlying vital rates and λ
f
. For panel (a), the
dots are the point estimates of rates. For panel (b), the colored and contoured surface represents a simple linear
interpolation of the underlying estimated values (shown by the rug on the x- and y-axes).
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(South Madison) did we find suggestive evi-
dence that lamb survival accounted for more
variation in bλfthan did ewe survival. Variation
in the age class structure never accounted for
more than 3% of the variation inbλf.
We also decomposed the actual changes in
population growth rates between years into the
contributions from demographic parameters. In
all cases, the changes in population growth rates
were driven by variation in lamb survival and
Fig. 5. Across-population results of the life-table response experiment analysis of the relationship between
demographic parameters and the total variation in the population growth rate of the female component of the
population λ
f
. The dot represents the median of the estimated percentage of total variation in λ
f
attributable to
each parameter, and the line the 90% highest posterior density interval.
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PATERSON ET AL.

ewe survival with only a trivial contribution
from changes in the age structure (unsurprising
given the simple age structure used, that is, only
yearlings and adults), and we focused on varia-
tion in those two vital rates as drivers of popula-
tion growth rates. Our results suggest a set of
potential mechanisms associated with changes to
the trajectory of a population. When a popula-
tion’s trajectory changed from increasing or
stable (bλf
t≥1) to decreasing (bλf
tþ1<1) (n=63,
upper right quadrant of Fig. 6), declines in lamb
survival were involved in 54 (86%) of the cases
and declines in ewe survival were involved in 31
(41%) of the cases. Notably, in 32 of the 54 cases
associated with declines in lamb survival (59%),
the population trajectory was reversed despite
increases in ewe survival. In nine of the 31 cases
associated with declines in ewe survival (29%),
the trajectory was reversed despite increases in
lamb survival. When a population’s trajectory
changed from decreasing (bλf
t<1) to increasing or
stable (bλf
tþ1≥1) (n=60, lower left quadrant of
Fig. 6), increases in lamb survival were involved
in 57 (95%) of the cases, and increases in ewe sur-
vival were involved in 40 (67%) of the cases. A
small number (three, or 5%) of cases involved
Fig. 6. Across-population results of the life-table response experiment analysis of the relationship between
demographic parameters and the changes in the estimated population growth rate of the female component of the
population λ
f
. The columns represent λ
f
in year t(<1or>1), the rows represent λ
f
in year t+1, the x-andy-axes
represent the contribution of changes in ewe survival and lamb survival to changes in λ
f
(Δλ
f
), and the color repre-
sents Δλ
f
. For example, the upper right quadrant is the case where the population was increasing in year tand
declining in year t+1, and the dots denote how ewe survival and lamb survival contributed to that change in λ
f
.
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PATERSON ET AL.

increases in ewe survival and decreases in lamb
survival, whereas 20 (33%) involved increases in
lamb survival and decreases in ewe survival. For
successive years in which population growth
rates were below 1 (λf
t<1, bλf
tþ1<1) (n=159
cases, upper left quadrant of Fig. 6), there was
considerable variation in the contribution of ewe
survival and lamb survival to λf
t, including posi-
tive changes that were insufficient to reverse the
negative trajectory of the population. Similarly,
for successive years in which population growth
rates were at or above 1 (λf
t≥1, bλf
tþ1≥1) (n=154
cases, upper left quadrant of Fig. 6), there was no
clear pattern as the positive trajectory of the pop-
ulation absorbed wide variation in cases of nega-
tive contributions from lamb and ewe survival.
Together, these patterns suggest that these popu-
lations are subject to a complex interplay of
changes in vital rates, and changes in single vital
rates are typically poor indicators of population
trajectories.
Finally, we note that our inference on the dis-
tributions of vital rates and sources of variation
in population growth rates was essentially iden-
tical to results from the mixture model (Appen-
dix S4). Given the mixture model was an attempt
to capture dramatic changes in vital rates due to
the large-scale mortality events in the data set,
the lack of any clear difference in inference
between the two modeling approaches suggests
that our population model, which was estimated
using management data alone, may not be ade-
quate to capture these die-off events.
Correlates of lamb survival
Our second major goal was to understand cor-
relates of lamb survival, and our first analysis
using the COVmodel revealed a diverse set of
relationships between environmental conditions
and lamb survival. We found strong support for
our a priori hypothesis that lamb survival would
be negatively associated with winter severity as
indexed by cumulative winter precipitation. In
eight out of 17 populations, the association
between PREC
winter
and lamb survival was nega-
tive and the associated 90% highest posterior
density intervals for bβPRECwinter did not overlap
zero: The strongest effect was estimated for Lost
Creek (bβPRECwinter ¼1:63½2:19, 1:09, and the
weakest effect was estimated for Wapiti Ridge
(bβPRECwinter ¼0:14½0:23, 0:04). These
estimated effects translated into substantial changes
in predicted lamb survival over the range of
observed PREC
winter
values (Fig. 7). For Lost Creek,
predicted lamb survival decreased over the range
of PREC
winter
values from SlPRECwinter0:05
ðÞ¼0:58
½0:41, 0:79to SlPRECwinter0:95
ðÞ¼0:01½0:002, 0:02,
and lamb survival for Wapiti Ridge decreased
from SlPRECwinter0:05
ðÞ¼0:26½0:24, 0:29to
SlPRECwinter0:95
ðÞ¼0:17½0:13, 0:22. We found no
evidence that the strength of the association
between winter precipitation and lamb survival
depended on the range of winter conditions
experienced (Fig. 8).
In contrast to the strong and consistent sup-
port that we found for the relationship between
lamb survival and winter severity, we found con-
flicting evidence regarding the relationship
between lamb survival and growing conditions
as indexed by summer precipitation. Our results
suggested both positive and negative relation-
ships between lamb survival and PREC
early
and
PREC
late
covariates. Where the 90% highest pos-
terior density interval for the regression coeffi-
cient for PREC
late
did not overlap zero (five out
of the 18 populations), three populations had
positive relationships with PREC
late
(Dubois
Badlands, Francs Peak, and Lost Creek) and two
populations had negative relationship (South
Madison and Whiskey Mtn-East). The strongest
positive effect was for Lost Creek (bβPREClate ¼
1:25½0:74, 1:80, which translated into a predicted
increase in lamb survival from SlPREClate0:05
ðÞ¼
0:03½0:01, 0:06to SlPREClate0:95
ðÞ¼0:81½0:61, 0:98
(Fig. 7). The strongest negative effect was for the
South Madison population (bβPREClate =−0.60
[−1.01, −0.17], which translated into a predicted
decline in lamb survival from SlPREClate0:05
ðÞ¼
0:55½0:40, 0:70to SlPREClate0:95
ðÞ¼0:23½0:12, 0:32
(Fig. 7). Where the 90% highest posterior density
interval for the regression coefficient for PREC
early
did not overlap zero (four out of the 18 popula-
tions), two populations had positive relationships
with PREC
early
(Castle Reef and Lost Creek) and
two populations had negative relationship (Dubois
Badlands and Wapiti Ridge). The strongest positive
effect was for Lost Creek (bβPRECearly =1.26 [0.81,
1.70]), which translated into a predicted increase in
lamb survival from SlPRECearly0:05

=0.04 [0.01,
0.07] to SlPREClate0:95
ðÞ¼0:66½0:47, 0:84.The
strongest negative effect was for Dubois Bad-
lands (bβPRECearly ¼0:56 ½1:06, 0:07,which
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PATERSON ET AL.

translated into a predicted decline in lamb sur-
vival from SlPRECearly0:05

¼0:46 ½0:30, 0:64to
SlPRECearly0:95

¼0:13½0:04, 0:23.Althoughwe
found no strong evidence for a relationship
between the strength and/or direction of the rela-
tionship between early- or late-summer precipita-
tion and lamb survival (Fig. 8), the patterns for the
coefficient estimates whose 90% highest posterior
density intervals do not include zero provide some
evidence that populations that experience higher
late-season precipitation have a negative coefficient,
whereas populations that experience higher early-
season precipitation have a positive coefficient.
We found equivocal evidence for a positive
relationship between lamb survival and growing
conditions as indexed by NDVI. Similar to
results for early and late-summer precipitation,
we estimated both positive and negative relation-
ships between lamb survival and NDVI
early
and
NDVI
late
covariates. Where the 90% highest pos-
terior density interval for the regression coeffi-
cients for NDVI
late
did not overlap zero (seven
out of the 18 populations), two populations had
positive relationships with NDVI
late
(Dubois
Badlands and Francs Peak) and five populations
had negative relationship (Jackson, Lost Creek,
Middle Missouri, Wapiti Ridge and Whiskey
Mtn-East) (Fig. 9). The strongest positive effect
was for Dubois Badlands (bβNDVIlate ¼0:68
½0:02, 1:38), which translated into a predicted
increase in lamb survival from SlNDVIlate0:05
ðÞ¼
0:10½0:01, 0:25to SlNDVIlate0:95
ðÞ¼
0:53½0:30, 0:79. The strongest negative effect was
for the Middle Missouri population (bβNDVIlate ¼
Fig. 7. Population-specific predicted relationships between precipitation and lamb survival. All predictions
were constructed by holding other covariates to their mean value. The black line represents the median of the
estimated response to change in that covariate (the gray ribbon represents the 90% highest posterior density
interval).
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PATERSON ET AL.

0:62½1:23, 0:06), which translated into
a predicted decline in lamb survival
from SlNDVIlate0:05
ðÞ¼0:84½0:68, 1:00to
SlNDVIlate0:95
ðÞ¼0:43½0:28, 0:61. Where the 90%
highest posterior density interval for the regres-
sion coefficients for NDVI
early
did not overlap
zero (seven out of the 18 populations), three pop-
ulations had a positive relationship with
Fig. 8. Relationships between estimated regression coefficients for normalized difference vegetation index
(NDVI) and precipitation (PREC) covariates and the median value of each covariate for each population. For each
of our five covariates, we graphed the median value through time for each population (x-axis) against the esti-
mated regression coefficient (y-axis). Estimated regression coefficients whose 90% credible interval did not over-
lap zero are in black; the remainder are in gray. The dot represents the median and the line the 90% highest
posterior density interval.
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PATERSON ET AL.

NDVI
early
(Castle Reef, Lost Creek and Spanish
Peaks) and four populations had a negative rela-
tionship (Jackson, Paradise, Wapiti Ridge, and
Younts Peak). The strongest positive effect was
for Castle Reef (bβNDVIearly ¼0:73 ½0:37, 1:11),
which translated into a predicted increase in
lamb survival from SlNDVIearly0:05

¼0:04
½0:01, 0:08to SlNDVIearly0:95

¼0:27½0:18, 0:35.
The strongest negative effect was for Wapiti
Ridge (bβNDVIearly ¼0:32½0:46, 0:18), which
translated into a predicted decline in lamb sur-
vival from SlNDVIearly0:05

¼0:32½0:28, 0:37to
SlNDVIearly0:95

¼0:15½0:11, 0:18. Similar to the
results for precipitation, we found no strong evi-
dence for a relationship between the magnitude
and/or direction of these estimated effects and
the range of conditions experienced (Fig. 8), but
suggestive evidence for a relationship in which
populations with a higher late-season NDVI had
a negative coefficient, and populations with a
higher early-season NDVI had a positive rela-
tionship.
We note that the predicted relationships
between growing season conditions, winter
severity, and lamb survival have important
implications for population growth rates. As
Fig. 9. Population-specific predicted relationships between normalized difference vegetation index (NDVI)
and lamb survival. All predictions were constructed by holding other covariates to their mean value. The black
line represents the median of the estimated response to change in that covariate (the gray ribbon represents the
90% highest posterior density interval).
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demonstrated in the previous section, lamb sur-
vival plays a significant role in reversing popula-
tion trajectories. The magnitudes of these
predicted changes in lamb survival associated
with environmental variation are large enough
to suggest environmental drivers can play a sig-
nificant role in altering population growth rates,
for example, the predicted decline in lamb sur-
vival associated with winter precipitation for the
Clarks Fork population (SlPREClate0:05
ðÞ¼0:39
½0:30, 0:46to SlPREClate0:95
ðÞ¼0:15 ½0:10, 0:22)
was sufficient to push lamb survival below the
threshold level where no value of ewe survival
could have resulted in λ
f
≥1.
We used the last 10 yr of observations and the
COVmodel to evaluate evidence for whether
mountain lion densities indexed by the quality of
mountain lion habitat might serve as a potential
limiting factor for lamb survival. Our estimate
was small and quite imprecise such that we
found essentially no evidence for such a relation-
ship (bβLION ¼0:09 ½0:62, 0:79).
Our assessment of the relationship between
pathogen communities and lamb survival for the
last 10 yr of observations provided strong evi-
dence that the median value of lamb survival for
those populations that hosted both M.ovipneu-
moniae and Pasteurellaceae (0.24 [0.22, 0.26]) was
lower than the median value of lamb survival for
populations that hosted Pasteurellaceae alone
(0.41 [0.37, 0.46]) (Fig. 10a), with an estimated
difference between the medians of 0.18 [0.13,
0.23]. However, although the distribution of
lamb survival values for populations that hosted
both pathogens was lower than the distribution
for populations that hosted Pasteurellaceae
alone, we note a considerable amount of overlap
(Fig. 10b). Approximately 48% of the point esti-
mates for annual lamb survival values for popu-
lations that hosted both pathogens were equal to
or exceeded the lowest estimated lamb survival
value for populations that hosted Pasteurellaceae
alone; moreover, approximately 18% were equal
to or greater than 0.40, a threshold of lamb sur-
vival we previously associated with values of
λ
f
≥1 despite a wide range in ewe survival.
DISCUSSION
Results from our novel state-space model in a
Bayesian framework highlight the benefits of
using population models and information on
vital rates to estimate vital rates and evaluate
how variation in vital rates shapes the trajectories
of populations. Although we note that popula-
tions of ungulates are successfully managed with
a conventional use of age ratios and count data
(Bender 2006, Harris et al. 2008), our model
allows for a more comprehensive assessment of
the mechanisms by which population trajectories
may vary. We demonstrated a variety of mecha-
nisms by which variation in vital rates affects
changes in population growth rates, including
combinations of lamb survival and adult survival
that result in similar population growth rates,
and suggest that these mechanisms may differ
among populations. Our assessment of sources of
variation in lamb survival rates indicated a poten-
tially complex interplay of ecological and disease
processes affecting lamb survival, including gen-
erally strong effects of winter climate, contrasting
effects of summer growing conditions, and no
effect of mountain lion predation or relationship
with resident pathogen community. Together,
these results offer a rigorous and comprehensive
evaluation of bighorn sheep vital rates.
Vital rates demonstrated substantial variation
across years and populations such that a variety
of combinations of vital rates could yield similar
population trajectories. The variation in lamb sur-
vival that we observed is consistent with results
from decades of empirical work for other ungu-
lates that documents high variability in offspring
survival rates (Gaillard et al. 1998, Gaillard and
Yoccoz 2003, Raithel et al. 2007). The evidence for
very low lamb survival rates in some years and
populations highlights the concern that this rate
could play a limiting role in conservation and
restoration efforts for bighorn sheep in particular
(Johnson et al. 2010). In contrast to the expecta-
tion of high adult survival with minor variation
for many ungulate populations, our estimated
adult ram and ewe survival rates were also highly
variable. This is an unsurprising result for popu-
lations of ungulates like bighorn sheep that are
subject to stochastic disease and severe weather
events, harvest, or predation that all result in
lower and highly variable adult survival rates
(Owen-Smith and Mason 2005, Nilsen et al. 2009,
Eacker et al. 2017). The integrated effects of this
variation in vital rates on the population growth,
particularly in small populations, is poorly
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PATERSON ET AL.

understood. To our knowledge, our work is the
first to empirically define the existence of a demo-
graphic “safe space”(de Silva and Leimgruber
2019) for bighorn sheep, that is, combinations of
vital rates that result in positive growth rates.
These results indicate that management can
address declines in populations by mitigating
either (or both) lamb survival and adult survival
to yield positive growth rates (e.g., predator con-
trol, treatment for respiratory disease, and sup-
plemental provisioning) and provide minimum
thresholds of vital rate values required for
positive population trajectories, below which
removals from translocations or harvest may
need to be reduced and/or augmentation may be
necessary for population persistence (Johnson
Fig. 10. Comparison of median values of estimated lamb survival across population-years (panel a), and the
distribution of estimated annual values of lamb survival (panel b) for populations that host both Mycoplasma
ovipneumoniae and Pasteurellaceae (M. ovi and Past.) and populations that host Pasteurellaceae alone (Past.). For
panel a, the dot represents the median and the line the 90% highest posterior density interval on the estimate of
the median. For panel b, the dot represents the median and the line the range of estimated lamb survival values.
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PATERSON ET AL.

et al. 2010). Although our results suggest that a
constraining relationship may exist between
removals and the growth rate for the female pop-
ulation, this relationship was obscured by two
sources of uncertainty that prevented strict infer-
ence: (1) uncertainty in estimates of growth rates
and (2) uncertainty in the relationship between
the larger population from which animals are
removed and the portion of this larger population
that is surveyed (uncertainty that arose from the
distribution of animals on the landscape and the
quality of the survey process).
The expectation for the dynamics of unhar-
vested ungulate populations lacking substantial
exposure to predation and/or disease is that low
variation in adult survival (despite high elastic-
ity) renders the high variation in offspring
recruitment the dominant source of variation in
population dynamics (despite lower elasticity)
(Gaillard et al. 1998, Raithel et al. 2007). A key
component of that expectation is the low varia-
tion in adult survival; where this rate is affected
by harvest, disease, or predation, it is unsurpris-
ing that variation in adult survival can dominate
population dynamics given its higher elasticity.
Our approach found a diversity of relationships
between vital rates and population growth rates
among populations similar to results from other
work on bighorn sheep that used asymptotic
approaches (Johnson et al. 2010). Although varia-
tion in adult survival explained the most overall
variation in population growth rates, improved
lamb survival was found to be the dominant
mechanisms by which populations actually
reversed declines (Manlove et al. 2016). The rela-
tive importance of these mechanisms was not
consistent between populations, and we add to
the small number of studies suggesting the dri-
vers of population dynamics for ungulates may
vary spatially (Albon et al. 2000, Coulson et al.
2005, Garrott et al. 2008a, Nilsen et al. 2009, John-
son et al. 2010). These results also help further