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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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PATERSON ET AL.

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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PATERSON ET AL.

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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PATERSON ET AL.

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

t1Þ(1)

Pl,ae

t∼Binomialðτae,Nae

t1Þ(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

t1), 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,0:5Nl

t1Þ(4)

Nyr

t∼Binomial Sr,0:5Nl

t1

:(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

t1,

Nyr

t1), 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

t1þNae

t1Þ(6)

Nar

t∼Binomial Sr,Nyr

t1þNar

t1

:(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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PATERSON ET AL.

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

1210

. 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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PATERSON ET AL.

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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PATERSON ET AL.

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

pNDVIearlyp,tþβNDVIlate

p

NDVIlatep,tþβPRECearly

pPRECearlyp,tþβPREClate

p

PREClatep,tþβPRECwinter

pPRECwinterp,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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PATERSON ET AL.

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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PATERSON ET AL.

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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PATERSON ET AL.

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.

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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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PATERSON ET AL.

(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 ¼1:63½2:19, 1:09, and the

weakest effect was estimated for Wapiti Ridge

(bβPRECwinter ¼0:14½0:23, 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, 0:79to SlPRECwinter0:95

ðÞ¼0:01½0:002, 0:02,

and lamb survival for Wapiti Ridge decreased

from SlPRECwinter0:05

ðÞ¼0:26½0:24, 0:29to

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, 0:06to 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, 0:70to 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 ¼0:56 ½1:06, 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, 0:64to

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, 0:25to 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.

0:62½1:23, 0:06), which translated into

a predicted decline in lamb survival

from SlNDVIlate0:05

ðÞ¼0:84½0:68, 1:00to

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, 0:08to SlNDVIearly0:95

¼0:27½0:18, 0:35.

The strongest negative effect was for Wapiti

Ridge (bβNDVIearly ¼0:32½0:46, 0:18), which

translated into a predicted decline in lamb sur-

vival from SlNDVIearly0:05

¼0:32½0:28, 0:37to

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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PATERSON ET AL.

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, 0:46to 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 ½0:62, 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

broaden the perspective on the drivers of popula-

tion dynamics for ungulates. We suggest that

any apparent conﬂict between these results and

the expectation for ungulate population dynam-

ics merely reﬂects the fact that populations live

in different portions of the vital rate parameter

space: Where adult survival is high and non-

variable, growth rates are driven by variation in

offspring recruitment, but where adult survival

is variable due to intrinsic/extrinsic drivers, it can

overwhelm the role of offspring recruitment, par-