Drivers of variation in the population dynamics of bighorn sheep 1, 2 1 3 1 J. TERRILL PATERSON, KELLY PROFFITT , JAY ROTELLA, DOUGLAS MCWHIRTER, AND ROBERT GARROTT

1Department of Ecology, Montana State University, Bozeman, Montana, USA 2Montana Department of Fish, Wildlife and Parks, Bozeman, Montana, USA 3Wyoming 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: [email protected]

INTRODUCTION trajectories (Nichols and Williams 2006). Where management resources are limited, such under- The successful management and conservation standing allows the optimal allocation of of wild populations, particularly those at risk, resources to restoration efforts (Johnson et al. requires information on sources of variation in 2010, Mills 2012). Though the importance of vital rates and the contribution of those vital understanding the relative influence of vital rates rates to variation in demography and population for shaping population trajectories has long been

v www.esajournals.org 1 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. recognized, such assessments are still compara- specialist predators and respiratory disease epi- tively rare due the challenges of estimating the zootics, as well as hunter harvest, can induce vital rates and population structure required variation in this key vital rate and generate very (Heppell et al. 1996, Johnson et al. 2010, Koons different population dynamics and trajectories et al. 2017). Consequently, information is fre- compared to populations not subject to such quently borrowed from species and systems for pressure (Festa-Bianchet 1989, Festa-Bianchet which there is sufficient information. et al. 1997, 2006, Jorgenson et al. 1997, Ross et al. For ungulates, the often-referenced expectation 1997, Cassirer and Sinclair 2007, Brewer et al. for population dynamics is that population 2014, Manlove et al. 2016, Parr et al. 2018). More- growth rates are most sensitive to variation in over, a recent comprehensive estimation of adult survival; however, variation in adult sur- sources of variation in ewe survival suggests vival is small enough to contribute little to actual that, apart from the influence of stochastic events variation in population growth rates (Gaillard and harvest, ewe survival varies among years in et al. 1998). In contrast, substantial variation in response to environmental drivers (Proffitt et al. offspring recruitment is the primary driver of 2021). Lamb survival demonstrates substantial variation in population growth rates, even if among-year variation in response to predation, growth rates are theoretically less sensitive to disease, and environmental variation, which is changes in this vital rate than changes in adult consistent with the empirical and theoretical survival. This paradigm was developed for, and expectation for high variation in this rate for intended to be applied to, specific kinds of ungu- ungulates in general (Gaillard et al. 1998) and late populations, that is, from temperate zones, bighorn sheep in particular (Douglas and Leslie non-harvested and with stable or increasing pop- 1986, Hass 1989, Portier et al. 1998, Cassirer et al. ulation growth rates (Coulson et al. 2005, Nilsen 2001, Smith et al. 2014). However, unlike the et al. 2009). Evidence suggests that the dynamics ephemeral decline of adult survival in response of populations that are small, in decline, subject to respiratory disease epizootics, lamb survival to stochastic and substantial variation in vital can be depressed for years following a disease rates, or in non-temperate ecosystems defy these event (Cassirer and Sinclair 2007, Cassirer et al. expectations (Owen-Smith and Mason 2005, Nil- 2013). Pregnancy rates are typically high (Festa- sen et al. 2009, Johnson et al. 2010, Lee et al. Bianchet 1988, Singer et al. 2000). However, preg- 2016), and the evolving understanding of ungu- nancy is the result of a complex set of metabolic late population dynamics has broadened the per- processes that integrate environmental variation spective to demonstrate how elasticities and and state processes (e.g., previous year’s repro- variances of vital rates integrate to shape trajecto- ductive success) such that pregnancy rates can ries (Hilde et al. 2020). also vary among years (Parker et al. 2009). The population dynamics of bighorn sheep Given the potential for wide variation in vital (Ovis canadensis) may or may not operate accord- rates, it is unsurprising that there is no consensus ing to this paradigm. Populations of bighorn on which rate is most important to the growth of sheep suffered significant declines near the turn bighorn sheep populations. Previous work has of the 20th century in response to a series of pres- suggested that population trajectories are driven sures including disease, competition with live- by recruitment (Bender and Weisenberger 2005, stock, and over-harvest (Buechner 1960, Berger Manlove et al. 2016), adult survival (Rubin et al. 1990), and restoration efforts have demonstrated 2002, Singer and Schoenecker 2004), or a combi- mixed success (Singer et al. 2000, Picton and Lon- nation of the two (Parr et al. 2018). The most ner 2008, Hedrick 2014). Major areas of research comprehensive evaluation to date of the relative required to provide the information for bighorn contributions from vital rates to population sheep restoration have been identified for dec- growth for bighorn sheep strongly suggests that ades (Buechner 1960), and a substantial body of these drivers can differ between populations work has developed characterizing the important over comparatively small spatial scale (Johnson vital rates for bighorn sheep. Although adult sur- et al. 2010), a conclusion supported by recent vival is generally high with limited among-year work assessing variation in population trajecto- variation, stochastic events such as predation by ries for populations of bighorn sheep in the

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Rocky Mountain west (Donovan et al. 2020). METHODS Rather than being contradictory, these disparate conclusions point to the diversity of challenges Study area and populations facing bighorn sheep populations, where the The study was conducted in Montana and aggregated pressure from predation, disease, Wyoming and included data for 17 bighorn small population size, and environmental drivers sheep populations (Fig. 1). In all but two cases, varies spatially and temporally (Buechner 1960, populations were defined based on historic Brewer et al. 2014). A better understanding of the delineations of management units used by the mechanisms through which population trajecto- management agencies in the two states thought ries of bighorn sheep transition from increasing to (generally) represent demographically closed or stable to declining (or vice versa) is important populations. In two cases, we combined data for informed management and restoration efforts from separate management units in Wyoming to (Coulson et al. 2005, Johnson et al. 2010, Koons create single demographically closed populations et al. 2016). Prior work on bighorn sheep and (Whiskey Mountain [Mtn]-West and Whiskey other ungulate species has implicated declines in Mtn-East) that each were the combination of two adult survival as the primary driver of declining underlying management units. Sixteen of the 17 population growth rates, a pattern seen in declin- populations were located in mountainous areas, ing populations in both temperate (Wittmer et al. and one population (Middle Missouri) was 2005, Nilsen et al. 2009, Johnson et al. 2010) and located in a prairie breaks landscape. Although tropical systems (Owen-Smith and Mason 2005, the composition of the communities of pathogens Lee et al. 2016). In contrast, improved offspring that are associated with bighorn sheep respira- recruitment is heavily associated with the recov- tory disease are not known in these populations, ery of populations of ungulates across a similar the most comprehensive assessment to date sug- range of ecosystems such that understanding gests that all of these populations host Pasteurel- limiting factors of offspring recruitment is crucial laceae bacterial pathogens, and all but Paradise, (Beissinger and Peery 2007, Mitchell et al. 2009, Petty Creek, Targhee, and Middle Missouri host Manlik et al. 2016). For bighorn sheep, there are Mycoplasma ovipneumoniae (Butler et al. 2018). All comparatively few studies of how lamb recruit- 17 populations had a management history that ment varies in response to ecological drivers, and included removals from the population due to the extant work suggests a potentially complex translocations of animals to other herds or har- interplay of effects from environmental, preda- vest. Every population had ram removals, with tion, and disease-related factors (Hass 1989, Por- the number of rams taken in any given year tier et al. 1998, Manlove et al. 2016, Butler et al. ranging from 0 to 69 (median = 7 removed). 2018). However, only eight of the 17 populations had Here, we use multiple long-term time series of ewe removals; in contrast to ram removals, the survey data on 17 populations of bighorn sheep number of ewes removed in any given popula- in Montana and Wyoming in combination with a tion varied from year to year ranging from 0 to recent comprehensive estimation of bighorn 79 (median = 0 removed). The management his- sheep pregnancy and survival rates (Proffitt et al. tories indicate that nine of the populations expe- 2021) to estimate the parameters of a population rienced all-age die-off events at least once during model for bighorn sheep to address three objec- the study period. tives: (1) to characterize the variation in key vital rates (lamb survival and adult survival) and pop- Survey data ulation growth rates, (2) to estimate the contribu- Survey data for the 17 populations were com- tions of changes in these vital rates to observed piled based on agency records that contained changes in population trajectories, and (3) to information obtained using multiple methods, identify potentially important sources of varia- including fixed-wing and helicopter- and tion in lamb survival including environmental ground-based survey efforts to count animals sources of variations (e.g., precipitation, primary and classify animals into age and sex classes. The production, and winter severity), predation, and timing of surveys differed among and within disease. bighorn sheep populations (median = March,

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Fig. 1. Study area map showing the approximate locations of the 17 Montana and Wyoming bighorn sheep populations used in this study. range = December–May). Ideally, a survey was nearly every population such that some years conducted annually, and during each survey, a were missing a count (but had classification sample of the total animal inventory was classi- data), some years were missing classification fied as lambs, adult males, and adult females. A data (but had a count), some years were missing count of all observed animals was recorded if the all data, and some years had both count and clas- survey was thought to be representative of the sification data (complete) (Fig. 2, see Appendix population in the consistent area of core seasonal S1 for complete population observation histo- range. However, the nature of the data and the ries). We left-truncated the time series for each practical challenges associated with annual sur- population at the first year for which there was a veys resulted in a discontinuous time series for representative count.

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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 (PRECwinter) (colors repre- sent populations). NDVI, normalized difference vegetation index.

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Modeling approach to avoid bias in estimates of offspring survival Our modeling approach was to characterize (Proffitt et al. 2021). Therefore, our biological the drivers of variation in the population process model used six age–sex classes (lambs, dynamics of bighorn sheep using the time ser- yearlings, and adults by sex). ies of management count and classification Bighorn sheep females typically have a sin- data. Prior work has demonstrated that state- gle offspring, and fecundity in a pre-birth space approaches have desirable characteristics pulse model is therefore the product of preg- for modeling population dynamics (de Valpine nancy rates and lamb survival to the end of and Hastings 2002, Buckland et al. 2004, New- their first year. For each population, we sepa- man et al. 2006, Schaub and Abadi 2011). A rately modeled the number of lambs (Pl) pro- state-space model in this context consists of duced from yearling ewes (ye) and adult ewes two processes: a biological model that connects (ae) in biological year t as a function of the changes in the population structure and size size of these two reproductive classes in the through time via key vital rates such as sur- previous year and age class-specific pregnancy vival and fecundity and an observation model rates (τye and τae), and a binomial process to that handles the stochastic and imperfect obser- incorporate demographic stochasticity (or, vari- vation process (Buckland et al. 2004). One of ation in fates at the population level due to the key features of such state-space models for chance, May 1973, Lande 1993, Kendall and population dynamics is that they separate the Fox 2002): observation process from the biological process l,ye ∼ ðτye ye Þ and, conditional on the model, improve preci- Pt Binomial , Nt1 (1) sion on inference for population dynamics (de l,ae ∼ ðτae ae Þ Valpine and Hastings 2002). Such hierarchical Pt Binomial , Nt1 (2) fi models naturally t within the Bayesian para- such that the total number of lambs produced at fl digm of statistical inference, and the exible the start of biological year t is simply nature of model specification in the popular l ¼ l,ye þ l,ae Pt Pt Pt . The survival of lambs from the and freely available software packages accom- beginning of biological year t to the end of bio- modate such models well (e.g., JAGS, STAN, logical year t (immediately prior to transitioning OpenBUGS) and also have two key additional to the yearling class in year t + 1) was modeled fi bene ts. First, by specifying a biological process using a similar binomial process based on lamb model that connects vital rates to size of age/- survival (Sl): sex classes through time, it is straightforward l ∼ l l : to derive a number of biologically relevant Nt Binomial S , Pt (3) quantities such as population growth rates and ye The number of yearling ewes (Nt ) and year- sex/age class ratios that are the integrated result yr of variation in multiple vital rates. Second, it is ling rams (Nt ) at the end of biological year t was easy to introduce informed priors for vital rates modeled using a binomial process based on the − l that are not directly estimated in the model. number of lambs in year t 1(Nt1), the Biological process model.—We developed a assumption of an equal sex ratio of lambs (0.5), fi e stage-based model for the population dynamics and sex-speci c adult survival (ewes: S , rams: r of bighorn sheep (Caswell 2001). We defined a S ): biological year from June 1 in year t to May 31 in ye e l N ∼ BinomialðS ,0:5 N Þ (4) year t + 1 to account for the pre-birth pulse sur- t t 1 veys. At the time of the surveys in the late spring, yr r l N ∼ Binomial S ,0:5 N : (5) each population was comprised of lambs (<1yr t t 1 > < ae old), yearlings ( 1 and 2 yr old), and adults Finally, the number of adult ewes (Nt ) and ≥ ar ( 2 yr old). Recent work in this system demon- rams (Nt ) at the end of year t were modeled strated that pregnancy rates for yearling ewes using a binomial process based on the number of − ye can be substantially lower than that for older yearling ewes and rams in year t 1(Nt1, yr ewes, suggesting that it is important to treat Nt1), the number of adult ewes and rams in year − e r fi yearlings as a separate class in a biological model t 1(Nt , Nt ), and sex-speci c adult survival:

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ae e ye ae fi l N ∼ BinomialðS , N þ N Þ (6) number of classi ed lambs (Count ), ewes t t 1 t 1 e r (Count ), and rams (Count ) to their proportion ar ∼ r yr þ ar : Nt Binomial S , Nt1 Nt1 (7) in the population: fi Our biological model relied on three simpli - Countl, Counte, Countr cations of bighorn sheep vital rates: (1) The sur- t (9) ∼ Multinomialðπ , Classified Þ vival of male and female lambs are equal, (2) t t fi there was no difference in the sex-speci c sur- l ye þ ae yr þ ar π ¼ N N N N N : vival of yearlings and adults, and (3) there was t total , total , total (10) no age-related variation in either survival or N N N t pregnancy rates (e.g., reproductive or actuarial This observation model connected the survey senescence). Our biological model also assumed data to the underlying vital rates, a central density independence of vital rates, a reasonable assumption of which is that these populations assumption for the generally smaller, recovering were demographically closed. Where this populations used here for whom density- assumption was violated in the presence of dependent resource limitation was unlikely to be immigration/emigration, it would be expected to limiting. — result in biased estimates of vital rates, that is, Observation process model. To connect the survival biased high in years with immigration. expected number of animals in each sex and age Where animals were removed by harvest or for class generated by the biological process model translocations, it would bias survival rates low if to the observed data, we used a model that could removals were additive to other sources of mor- accommodate survey data where only a fraction tality. Additionally, the presence of incomplete of the total number of animals counted were clas- fi data (years with completely missing data, or si ed into the management age/sex categories years missing a total count or a classification) (lamb, adult female, adult male) (Paterson et al. would be expected to inflate the variance of esti- 2019). Frequently, populations were surveyed for mates of vital rates from our model and/or gener- counts, but only a fraction of the total number of ate bias in sex-specific survival rates. fi animals counted were classi ed into the manage- Accommodating inconsistent survey timing.—A ment age/sex categories (lamb, adult female, common complication in our survey data is that adult male). Therefore, we modeled the total surveys were not always conducted at the end of count for each population in each year t (Countt) the biological year in late spring. Thus, if we using a Poisson process where the expected ignored differences in survey timing and naively value was the total population size (the sum of used the biological and process models above, total ¼ l þ yeþ all age and sex classes, Nt Nt Nt we would have introduced bias into our esti- yrþ ae þ ar Nt Nt Nt ): mates of vital rates by conflating the timing of total ζ surveys with the survival processes, that is, early Countt ∼ PoissonðN e t Þ (8) t surveys that missed late winter mortality would ζ where t was an observation-level random effect tend to cause year-specific survival rates to be used to account for overdispersion in the obser- inflated. We therefore adapted the above model vation process. We then connected the total num- to the timing of the surveys with a model for how ber of animals classified in each survey survival changes within a year and used the tim- fi (Classi edt) to the age and sex classes using a ing of the observed data to help estimate the pop- multinomial distribution. During these manage- ulation size at the end of year t.Specifically, we ment surveys for population trends, animals introduced another latent state for the size of each were classified only as lambs, ewes, and rams, age and sex class at the time of the surveys and for example, yearling ewes and adult ewes are connected it to the model for the size of each age not separately observable. However, inference on and sex class at the end of year t. For example, the an unobserved age class is possible in our model number of lambs in the population in year t at the l due to the structure of the biological process time of the survey (Nt∗) was modeled as model. We incorporated the yearling class into l ∼ ð ð l Þ lÞ the classification process in year t by relating the Nt∗ Binomial f S , d∗ , Pt (11)

v www.esajournals.org 7 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. and the number of lambs at the end of the biolog- and (2) a version that used logistic regression to l ical year (Nt, the ideal timing) was based on this assess potential sources of variation in lamb sur- latent state vival in each population using covariates as proxies for environmental variation (COVmodel). l ∼ ð ð l Þ l Þ Nt Binomial g S , d∗ , Nt∗ (12) All other model structures were the same l where f(S , d*) was a function that connected the between models. relationship between the timing of the observa- Common model structures.—We used informed tion (d*) and the expected survival of lambs to priors for adult survival and yearling and adult l that point, and g(S , d*) was a function that con- ewe pregnancy rates, constructed from a recent nected the difference in timing between the sur- comprehensive assessment of pregnancy and veys and the end of the year to the expected survival rates in bighorn sheep in this same area number of lambs at the end of the year. We and for many of these populations (Proffitt et al. chose to use simple functions that factored the 2021). We took the empirical distribution of the survival rate into monthly survival and then cal- estimated probabilities for yearling pregnancy culated survival to the time of observation as and adult pregnancy and fit a beta distribution monthly survival raised to a power equivalent to to each to construct the priors for the population- the number of months since the beginning of specific probabilities of pregnancy in the state- τye ∼ ð : : Þ year t. For example, if the survey was conducted space model ( p Beta 29 737, 9 731 and τae ∼ ð : : Þ in March, it was in month 10 (d* = 10) of biologi- p Beta 384 49, 33 25 ). Similarly, for ewe and 1 10 ram survival, we used the results from Proffitt et al. l ¼ l 12 cal year t then fS,10 S , and (2021) for ewe survival to construct an informed 1 12 10 prior to use for sex-specific, time-varying survival l l 12 gS,10 ¼ S . We used these same e ∼ ð : : Þ rates for each population (Sp,t Beta 10 452, 3 292 r ∼ ð : : Þ functions to model the latent states of all age/sex and Sp,t Beta 10 452, 3 292 ). Although this pre- classes. Such survival functions are a consider- vious work was specific to ewes, we considered this able simplification of a reality in which the distribution (with an approximate mean of 0.76 shape of the survival curve likely differs and standard deviation of 0.11) to be sufficiently between age and sex classes and among years. vague so as to serve as a general prior for adult sur- However, we considered this approach to be a vival. For the initial sizes of each age and sex class reasonable approximation in the face of sparse in each population, we used a vague uniform prior information for this species and a considerable (Uniform(0, upperp)), where upperp was an upper improvement over naively using survey data limit for each population defined as the maximum without accounting for the timing. total count in the time series, that is, the number of individuals in each age and sex class in the initial Modeling goals population had to be less than the maximum aggre- Our state-space model for the time series of gate count of all classes). Observation-level random counts is a hierarchical model with a large set of effects were drawn from a normal distribution ζ ∼ σ2 latent states required to accommodate the struc- ( t Normal 0, ζ )withavaguepriorforthe ture of the data. However, these models for pop- standard deviation (σζ ~ Uniform(0, 10)). ulations are driven by a small number of vital Distributions of vital rates and their association rates: pregnancy rates of yearling and adult ewes with demographic performance.—The first of our (τye and τae), the survival rate of lambs (Sl), and three goals was to estimate the key vital rates the survival rates of ewes (Se) and rams (Sr). Our underlying our process model, and our second goals were to estimate these vital rates from the was to characterize the associations between vital count data, to characterize their associations with rates and the observed trajectories of bighorn population trajectories (a derived parameter), sheep populations. However, unlike survival and and to understand sources of variation in lamb pregnancy rates, we did not have an informed survival. To meet these three goals, we used two prior to place on lamb survival. To provide a versions of the above model: (1) a version moderate amount of shrinkage and improve esti- wherein Sl was estimated for each population- mation for years with missing data while not year using a random effects structure (REmodel) overly influencing estimates from years with

v www.esajournals.org 8 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. complete data, we modeled all population-year adult survival rates and a single, common distri- l lamb survival rates, Sp,t, using a random effects bution of lamb survival rates could have been structure on the logit scale (REmodel), where too restrictive. The priors for adult survival might have been too informative to capture these l ¼ μ þ ξ logit Sp,t p,t (13) dramatically lower rates, and rarity of these events resulted in the estimation of the distribu- ξ ∼ ð σ2Þ p,t Normal 0, ξ (14) tion of lamb survival rates being dominated by “normal” years and masking the signal of these with a Logistic(0, 1) prior for μ and a vague prior events. To evaluate the sensitivity of our results for the standard deviation σξ ~ Uniform(0, 10). It to our prior specification, we constructed an is important to note that lamb survival and sex- alternative version of the model wherein both specific adult survival were assumed to be inde- adult and lamb survival were modeled using pendent (i.e., no explicit modeling of the poten- mixture distributions that allowed for more vari- tial covariation between rates), after initial model ation in vital rates (Appendix S2). runs suggested estimation problems for this Correlates of lamb survival.—Our third main more complex model using count data and gen- goal was to assess a series of covariates as poten- erally vague priors for survival and pregnancy tial sources of variation in lamb survival. Our a (Koons et al. 2017). priori expectation was that lamb survival should Rather than a prospective analysis such as life- be positively related to favorable conditions dur- stage simulation analysis (Wisdom et al. 2000) ing the growing season (e.g., indexed by the nor- that estimates how hypothetical changes in vital malized difference vegetation index [NDVI] or rates may impact population growth rates, we precipitation) and negatively related to winter wanted to decompose the observed variation in conditions (e.g., indexed by indices of winter population growth rates into contributions from severity). We used a derived metric of NDVI as the vital rates. Life-table response experiments an index of primary production during the grow- (LTREs) provide a powerful method of investigat- ing season (Pettorelli et al. 2011). Using the ing these drivers of observed variation in popula- AVHRR (Advanced Very High Resolution tion growth rates, and recent work has derived a Radiometer) time series of NDVI values (Ver- series of transient LTREs that do not assume a sta- mote and NOAA CDR Program 2019), we per- tionary environment (Cooch et al. 2001, Koons formed a series of smoothing steps on each pixel et al. 2016, 2017). The population growth rate in in our study area. First, we calculated the year t (λ ) can be expressed as a function of under- t monthly median of the daily NDVI values pro- lying vital rates and the population structure in duced by the AVHRR time series and then year t − 1, and sensitivities of population growth applied a smoothing spline on the monthly time rates to demographic parameters derived using series of values. Second, we estimated the start of partial derivatives (Appendix S2). Our study the growing season as the point in the smoothed focused on: (1) estimating the contribution of each signal at which the value of NDVI first attained of the vital rates to the total observed variation in ðλ Þ 50% of the seasonal maximum, and the end of λ var t θ t (contributionθ for vital rate i from Koons i the season as the first point in time past the sea- et al. 2016) and (2) understanding the drivers of λ sonal maximum at which the smoothed signal changes in t between successive time steps Δλ that the NDVI value was below 50% of its sea- t θ (contributionθ for vital rate i from Koons et al. i sonal maximum (Bradley et al. 2007). Third, to 2016) (details in Appendix S2). ameliorate the impact of canopy cover on the Sudden, precipitous declines in vital rates that NDVI signal, we calculated the integrated NDVI are associated with epizootics, shifts in predation from the start of the season to the end of the sea- risk, or severe environmental conditions are a son as the sum of the differences between each persistent source of variation in many popula- NDVI value and the value of NDVI at the start of tions of bighorn sheep (Ross et al. 1997, Portier the growing season (Ruimy et al. 1994, Jonsson¨ et al. 1998, Festa-Bianchet et al. 2006, Cassirer and Eklundh 2004). Prior work has suggested and Sinclair 2007, Manlove et al. 2016). For such that population responses to NDVI might circumstances, our use of informed priors for depend on the timing (Paterson et al. 2019), that

v www.esajournals.org 9 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. is, high NDVI during the spring could indicate There were two additional potential sources of substantial primary production and favorable variation in lamb survival that were not amen- growing conditions conducive to neonates that able to assessment using our full data set. First, could be coupled to a late season characterized predation on bighorn sheep lambs can have a by drought and poor growing conditions such dramatic impact on survival rates (Berger 1991, that a single NDVI metric for an entire growing Rominger et al. 2004, Festa-Bianchet et al. 2006, season could be misleading. Therefore, to investi- Brewer et al. 2014) and be critical to consider gate differential impacts of the seasonal timing of when evaluating potential sources of variation in NDVI (e.g., early-season vs late-season growing population dynamics. Although the larger region conditions), we defined the integrated NDVI has seen the recovery of populations of multiple over two periods: from the start of the growing predator species during our study period, moun- season to June 30 (NDVIearly), and from July 1 to tain lions (Puma concolor) are implicated as the the end of the growing season (NDVIlate). Addi- primary predator affecting bighorn sheep popu- tionally, we derived a second metric of environ- lation dynamics (Rominger 2018). However, mental conditions during the growing season. there is no direct measure of predation pressure Using the PRISM data set (PRISM Climate from mountain lions or an index of it (e.g., Group, Oregon State University, http://prism.ore mountain lion abundances) for a region the size gonstate.edu), we derived precipitation metrics of our study area and over the timespan of the similar to those for NDVI for each pixel in our data collection. Therefore, to assess the potential study area as the sum of precipitation from May impact of mountain lions on lamb survival, we 1 to June 30 (PRECearly) and the sum of precipita- developed a spatial covariate by extrapolating a tion from July 1 to September 30 (PREClate). previously published winter resource selection Moreover, we used the sum of precipitation from function (RSF) for Montana (Robinson et al. December to March as an index of winter sever- 2015) over the extent of our study area and then ity (PRECwinter). To connect the values of each of used the median value of the RSF within each these five metrics to the populations in our study, seasonal range as an index of risk from mountain we relied on estimated seasonal ranges for these lions (details in Appendix S2). Due to the under- populations using either GPS collar data from a lying assumption of stationarity, we limited the regional research project on these populations data set for this analysis to the last 10 yr of obser- (n = 16, details in Proffitt et al. 2021), or seasonal vations for each population. We then evaluated ranges derived from professional opinion (n = 1, the strength of evidence for a relationship Dubois Badlands). Environmental conditions in between lamb survival and the median value of each biological year were approximated by using the extrapolated mountain lion RSF for each pop- the median value of all the pixels contained in ulation by modifying Eq. 15 to include the each seasonal range for that year. population-year value of the derived covariate We evaluated potential sources of environmen- (LION). Second, disease processes are thought to tal variation in lamb survival using logistic play a pivotal role in the population dynamics of regression to connect covariates in population p bighorn sheep. To explore the potential role of and year t (e.g., NDVIearlyp,t) to survival using disease as a limiting factor in lamb survival, we the logit link (COVmodel): assessed the evidence for a relationship between NDVI pathogen communities in these populations and l ¼ μ þ β early þ βNDVIlate logit Sp,t p NDVIearlyp,t p lamb survival. Using the results of the most com- PREC þ β early þβPREClate prehensive and rigorous study to date on the NDVIlate p PRECearly p,t p,t p composition of communities of pathogens associ- PREC þ βPRECwinter PREC (15) latep,t p winterp,t ated with respiratory disease in bighorn sheep where the population-specific mean, μ, was given (Butler et al. 2018), we compared the distribu- a Logistic(0, 1) prior and regression coefficients tions of lamb survival for those populations that were given independent Normal(0, 4) priors that hosted both M. ovipneumoniae and Pasteurel- are vague on the logit scale (Northrup and Ger- laceae bacteria (Bibersteinia trehalosi, Pasteurella ber 2018). multocida, and Mannheimia haemolytica)(n = 12) to those populations that only hosted

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Pasteurellacea (n = 4). Pathogen data for the which resulted in a combined total of 10,000 Spanish Peaks population were not published as samples used for inference. Convergence of these part of this study, and data were censored from chains was graphically assessed using traceplots this population for this portion of our analysis. for the underlying vital rates. For the large num- Similar to the mountain lion covariate, our com- ber of derived parameters (population-, sex-, parison of lamb survival values between the two age-, and time-specific sizes of populations), a groups assumed stationarity, that is, the presence graphical assessment was not practical and con- of the pathogen(s) was considered a fixed charac- vergence was evaluated using the Gelman-Rubin teristic that did not vary through time. Therefore, statistic and convergence was assumed for values we only used the last 10 yr of estimates of lamb <1.05 (Gelman and Rubin 1992). We summarized survival for each population from the first ver- the approximate posterior distributions for all of sion of our model (Eq. 14, REmodel) and used the the model parameters using the median and 90% approximate posterior distribution of estimated highest posterior density interval. survival values to estimate the median lamb sur- There are no standard test statistics for evalu- vival for all years as well as annual values for ating the adequacy of the fit of state-space mod- both groups. els to time series data such as these. Therefore, The covariates used to assess sources of varia- we evaluated the goodness of fit of both versions tion in lamb survival showed substantial varia- of our model to each population using two poste- tion both across and within populations (Fig. 2, rior predictive checks (Gelman et al. 1996) that Appendix S3). Prior to being used in analyses, all compared the fit of the models to the data using covariates were screened for collinearity (with a two biologically relevant characteristics of the threshold of 0.5) and were centered and scaled data: the observed population growth rates and using the mean and one standard deviation. To the observed ratio of lambs to ewes (both year- facilitate interpretation for estimated covariate ling and adult) (details in Appendix S2). For each effects, for each covariate for which the 90% iteration of the MCMC chain for each model, we highest posterior density interval of the esti- derived a replicated data set to compare to the mated regression coefficient did not overlap observed data. We calculated a Bayesian p value zero, we made predictions for the population- as the proportion of iterations where the value of specific relationship between the covariate and the statistic for the observed data exceeded the lamb survival by holding all other covariates to value of the statistic for replicated data. Bayesian their mean. For each covariate, we then selected p values close to 0.5 indicate the model is faith- a set of example populations to demonstrate fully replicating the variation of the observed how the predicted relationship translated into data, whereas values close to 0 or 1 indicate a variation in lamb survival from the 5th percentile severe lack of fit (Gelman et al. 1996). l of the covariate values (S (covariate0.05)) to the 95th percentile of the covariate values (Sl (covari- RESULTS ate0.95)). The survey data were comprised of (typically) Model estimation and evaluation discontinuous time series of counts and age and Models were fit in the R programming envi- sex classifications for each population from 1983 ronment (R Development Core Team 2019) using to 2018 (Fig. 2; for population-specific sum- the runjags package (Denwood 2016) as an inter- maries, see Appendix S1). In total, the data set face to the JAGS program (Plummer 2017) for included 470 population-years of observations Markov chain Monte Carlo (MCMC) estimation (including years missing observations within the of our model. For both versions of our model time series). Across populations, there was con- (REmodel and COVmodel), we ran two parallel siderable variation in the median total count chains for 150,000 iterations and discarded the from Targhee (68 animals) to Wapiti Ridge (716 first 50,000 as the burn-in and adaptation phase. animals) coupled to similar variation in range of Due to the large number of derived parameters counts from Targhee (maximum–minimum = saved from these simulations, we kept every 59) to Whiskey Mtn-East (maximum–mini- 20th iteration (i.e., a thinning interval of 20), mum = 640). The ratio of lambs to ewes across

v www.esajournals.org 11 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. populations demonstrated similar variation with Creek (max = 0.87 [0.77, 0.97], min = 0.41 [0.29, the median ratio ranging from a minimum in 0.54], variance in annual values = 0.021 [0.012, Lost Creek (0.14) to a maximum in Middle Mis- 0.031]). Lamb survival was generally lower than souri (0.47) and the range of ratios from Francs adult survival, with a range of population- Peak (maximum ratio–minimum ratio = 0.24) to specific median survival rates across years from Whiskey Mtn-West (maximum–minimum = a low of 0.19 [0.15, 0.23] for Whiskey Mtn-East to 0.81). a high of 0.48 [0.40, 0.54] for Middle Missouri Convergence was achieved for parameter esti- (Fig. 3). Within-population differences between mates in both models; bR values for all latent the maximum and minimum of point estimates states (i.e., sizes of sex/age classes) were <1.05, of lamb survival ranged from 0.26 for Targhee and bothbR values and visual inspection of trace- (max = 0.51 [0.18, 0.83], min = 0.24 [0.08, 0.45], plots for underlying vital rates of both models variance in annual values = 0.035 [0.022, 0.049]) indicated convergence and adequate mixing. All to 0.60 for Dubois Badlands (max = 0.70 [0.45, of the posterior predictive checks (two posterior 0.92], min = 0.10 [0.03, 0.19], variance in annual predictive checks for the REmodel and COVmodel) values = 0.046 [0.032, 0.060]). suggested adequate fit for the models, given that Variation in estimated vital rates translated the one-sided Bayesian p values were far from 0 into large variation in population growth rates or 1 for both population growth (REmodel: with population-specific median values of λ p = 0.67; COVmodel: p = 0.52) and age ratios across years ranging from 0.85 [0.81, 0.90] for (REmodel: p = 0.26; COVmodel: p = 0.81). Lost Creek to 1.03 [1.00, 1.05] for Middle Mis- souri (Fig. 3). Within-population differences Variation in vital rates between the maximum and minimum point esti- Using the REmodel for inference, we found mates of annual growth rate values demon- substantial variation in the vital rates (Fig. 3). strated high variability, ranging from 0.19 for The population-specific median values of ewe Younts Peak (max = 1.09 [1.03, 1.26], min = 0.90 survival across years varied from a minimum of [0.80, 1.00], variance in annual values = 0.012 0.78 for Lost Creek (90% highest posterior den- [0.007, 0.0174]) to 0.60 in Dubois Badlands sity interval = 0.73–0.83, hereafter all similar (max = 1.34 [1.10, 1.60], min = 0.74 [0.57, 0.90], credible intervals are denoted using brackets) to variance in annual values = 0.029 [0.020, 0.039]). 0.88 [0.86, 0.90] for Wapiti Ridge. The variation We compared the characteristics of population- in ewe survival within and among the popula- years with growth rates <1 to those with growth tions was substantial: the within-population dif- rates >1 and found substantial differences in ference between the maximum and minimum population vital rates: Population-years with point estimates of annual ewe survival values bλ ≥ 1 had overall higher survival rates (ewe sur- varied from 0.08 for Wapiti Ridge (max = 0.90 vival: median = 0.88 [0.87, 0.89]; ram survival: [0.82, 0.98], min = 0.82 [0.70, 0.93], variance in median = 0.80 [0.79, 0.82]; lamb survival: annual values = 0.004 [0.002, 0.006]) to 0.26 for median = 0.48 [0.45, 0.51]) than population- Lost Creek (max = 0.85 [0.72, 0.96], min = 0.59 years with bλ<1 (ewe survival: median = 0.82 [0.44, 0.76], variance in annual values = 0.01 [0.81, 0.83]; ram survival: median = 0.73 [0.71, [0.006, 0.020]). Point estimates for the survival 0.74]; lamb survival: median = 0.24 [0.23, 0.26]). rates for rams tended to be lower than those for Finally, we estimated the geometric means for ewes and displayed more variation. The each population and found that nine populations population-specific median values of ram sur- had estimated geometric means with the upper vival across years varied from a minimum of limit of a 90% highest posterior density interval 0.71 [0.67, 0.75] for Clarks Fork to 0.82 [0.80, 0.85] <1 (indicating long-term population decline; for Paradise, and within-population differences Castle Reef, Clarks Fork, Francs Peak, Lost between the maximum and minimum point esti- Creek, Targhee, Trout Peak, Wapiti Ridge, Whis- mates of annual ram survival values ranging key Mtn-East, and Younts Peak), two popula- from 0.10 in Middle Missouri (max = 0.87 [0.76, tions had estimated geometric means with a 0.97], min = 0.77 [0.59, 0.94], variance in annual lower 90% highest posterior density interval limit values = 0.008 [0.004, 0.012]) to 0.47 in Lost >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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Fig. 4. Relationship between key vital rates and the growth rates of the female component of the population (λf). Panel (a) represents the univariate relationships between lamb and ewe survival and estimated λf. Panel (b) represents the multivariate relationship between these two key underlying vital rates and λf. 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).

Middle Missouri and South Madison), and the incorporated into the annual survival rates in the remainder had 90% highest posterior density model. We calculated the ratio of the number intervals that included 1 (Fig. 3). females removed in year t + 1 to the estimated We found evidence for constraining relation- size of the population from the model in year t ships between lamb survival, ewe survival, and and compared it to λf in year t. We found that the population growth rate for the female com- when the ratio of female removals to female pop- ponent of the population (bλf) (Fig. 4). Using the ulation size exceeded approximately 0.18, all medians of the approximate posterior distribu- point estimates of λf were <1. tion from the REmodel for each annual parameter as a point estimate, we found that lamb survival Transient life-table response experiments rates corresponding to estimates of bλf ≥ 1 had a We decomposed the observed variation and similar distribution (median = 0.46; range =- changes in the population growth rates of the 0.21–0.78) to those corresponding to bλf <1 (me- female portion of each population into contribu- dian = 0.28; range = 0.05–0.61). There was tions from changes in constituent demographic similar ambiguity in the relationship between parameters using the REmodel for inference and ewe survival and λf. The distribution of ewe sur- found substantial among-population differences vival rates corresponding to estimates of bλf ≥ 1 in drivers of population trajectories. Partitioning (median = 0.86; range = 0.77–0.90) was essen- the total variation in population growth rates tially the same as the distribution of values corre- (see Fig. 3 for the distribution of point estimates sponding tobλf <1 (median = 0.83; range = 0.60– for annual growth rates) into the contributions 0.90). We found that the association between from lamb survival, ewe survival, and changes in these vital rates depended on their values, such the age structure revealed that, in general, varia- that the impact of lamb or ewe survival on λf tion in ewe survival accounts for a large amount depended on the value of the other rate (Fig. 4). of the variation in bλf (Fig. 5). For six of the 17 For example, at values of ewe survival ≥0.85, we populations (Clarks Fork, Jackson, Lost Creek, found λf ≥ 1 over a wide distribution of lamb Paradise, Petty Creek, and Targhee) the 90% survival values (median = 0.43; range = 0.21– highest posterior density interval for the esti- 0.68). Once ewe survival dropped to ≤0.80, how- mated proportion of variation inbλf attributed to ever, we found that λf ≥ 1 over only a narrow variation in ewe survival did not overlap the sim- and high distribution of lamb survival values ilar estimate for the proportion of variation inbλf (median = 0.58; range = 0.54–0.68). If ewe sur- due to variation in lamb survival, strongly sug- vival dropped below 0.75, no value of lamb sur- gesting that variation in ewe survival dominated vival was capable of yielding λf ≥ 1. Conversely, variation in bλf for these populations. For two of when lamb survival rates were ≥0.40, we found the populations (Middle Missouri and Trout that λf ≥ 1 over a relatively wide distribution of Peak), overlap between the credible intervals pre- ewe survival rates (median = 0.85; range = 0.77– vented strong inference but suggested a similar 0.90). When lamb survival rates were ≤0.20, no relationship. For seven of the 17 populations value of ewe survival resulted in λf ≥ 1. Finally, (Clarks Fork, Dubois Badlands, Francs Peak, we noted a relationship between the growth rate Spanish Peaks, Wapiti Ridge, Whiskey Mtn-East, of the female population (λf) and the number of and Younts Peak), the largely coincident credible females removed from each population due to intervals suggest roughly equal contributions either harvest or translocation, which was from lamb and ewe survival. In only one case

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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 λf. The dot represents the median of the estimated percentage of total variation in λf attributable to each parameter, and the line the 90% highest posterior density interval.

(South Madison) did we find suggestive evi- We also decomposed the actual changes in dence that lamb survival accounted for more population growth rates between years into the variation in bλf than did ewe survival. Variation contributions from demographic parameters. In in the age class structure never accounted for all cases, the changes in population growth rates more than 3% of the variation inbλf. were driven by variation in lamb survival and

v www.esajournals.org 16 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. ewe survival with only a trivial contribution (41%) of the cases. Notably, in 32 of the 54 cases from changes in the age structure (unsurprising associated with declines in lamb survival (59%), given the simple age structure used, that is, only the population trajectory was reversed despite yearlings and adults), and we focused on varia- increases in ewe survival. In nine of the 31 cases tion in those two vital rates as drivers of popula- associated with declines in ewe survival (29%), tion growth rates. Our results suggest a set of the trajectory was reversed despite increases in potential mechanisms associated with changes to lamb survival. When a population’s trajectory bλf < the trajectory of a population. When a popula- changed from decreasing ( t 1) to increasing or ’ bλf ≥ = tion s trajectory changed from increasing or stable ( tþ1 1) (n 60, lower left quadrant of bλf ≥ bλf < = stable ( t 1) to decreasing ( tþ1 1) (n 63, Fig. 6), increases in lamb survival were involved upper right quadrant of Fig. 6), declines in lamb in 57 (95%) of the cases, and increases in ewe sur- survival were involved in 54 (86%) of the cases vival were involved in 40 (67%) of the cases. A and declines in ewe survival were involved in 31 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 λf. The columns represent λf in year t (<1or>1), the rows represent λf in year t + 1, the x-andy-axes represent the contribution of changes in ewe survival and lamb survival to changes in λf (Δλf), and the color repre- sents Δλf. For example, the upper right quadrant is the case where the population was increasing in year t and declining in year t + 1, and the dots denote how ewe survival and lamb survival contributed to that change in λf.

v www.esajournals.org 17 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. increases in ewe survival and decreases in lamb estimated effects translated into substantial changes survival, whereas 20 (33%) involved increases in in predicted lamb survival over the range of lamb survival and decreases in ewe survival. For observed PRECwinter values (Fig. 7). For Lost Creek, successive years in which population growth predicted lamb survival decreased over the range λf < bλf < = lðÞ¼: rates were below 1 ( t 1, tþ1 1) (n 159 of PRECwinter values from S PRECwinter0:05 0 58 ½ : : lðÞ¼: ½ : : cases, upper left quadrant of Fig. 6), there was 0 41, 0 79 to S PRECwinter0:95 0 01 0 002, 0 02 , considerable variation in the contribution of ewe and lamb survival for Wapiti Ridge decreased λf lðÞ¼: ½ : : survival and lamb survival to t, including posi- from S PRECwinter0:05 0 26 0 24, 0 29 to fi lðÞ¼: ½ : : tive changes that were insuf cient to reverse the S PRECwinter0:95 0 17 0 13, 0 22 . We found no negative trajectory of the population. Similarly, evidence that the strength of the association for successive years in which population growth between winter precipitation and lamb survival λf ≥ bλf ≥ = rates were at or above 1 ( t 1, tþ1 1) (n 154 depended on the range of winter conditions cases, upper left quadrant of Fig. 6), there was no experienced (Fig. 8). clear pattern as the positive trajectory of the pop- In contrast to the strong and consistent sup- ulation absorbed wide variation in cases of nega- port that we found for the relationship between tive contributions from lamb and ewe survival. lamb survival and winter severity, we found con- Together, these patterns suggest that these popu- flicting evidence regarding the relationship lations are subject to a complex interplay of between lamb survival and growing conditions changes in vital rates, and changes in single vital as indexed by summer precipitation. Our results rates are typically poor indicators of population suggested both positive and negative relation- trajectories. ships between lamb survival and PRECearly and Finally, we note that our inference on the dis- PREClate covariates. Where the 90% highest pos- tributions of vital rates and sources of variation terior density interval for the regression coeffi- fi in population growth rates was essentially iden- cient for PREClate did not overlap zero ( ve out tical to results from the mixture model (Appen- of the 18 populations), three populations had dix S4). Given the mixture model was an attempt positive relationships with PREClate (Dubois to capture dramatic changes in vital rates due to Badlands, Francs Peak, and Lost Creek) and two the large-scale mortality events in the data set, populations had negative relationship (South the lack of any clear difference in inference Madison and Whiskey Mtn-East). The strongest bβ ¼ between the two modeling approaches suggests positive effect was for Lost Creek ( PREClate that our population model, which was estimated 1:25½0:74, 1:80, which translated into a predicted lðÞ¼ using management data alone, may not be ade- increase in lamb survival from S PREClate0:05 : ½ : : lðÞ¼: ½ : : quate to capture these die-off events. 0 03 0 01, 0 06 to S PREClate0:95 0 81 0 61, 0 98 (Fig. 7). The strongest negative effect was for the Correlates of lamb survival bβ = − South Madison population ( PREClate 0.60 Our second major goal was to understand cor- [−1.01, −0.17], which translated into a predicted fi lðÞ¼ relates of lamb survival, and our rst analysis decline in lamb survival from S PREClate0:05 : ½ : : lðÞ¼: ½ : : using the COVmodel revealed a diverse set of 0 55 0 40, 0 70 to S PREClate0:95 0 23 0 12, 0 32 relationships between environmental conditions (Fig. 7). Where the 90% highest posterior density fi and lamb survival. We found strong support for interval for the regression coef cient for PRECearly our a priori hypothesis that lamb survival would did not overlap zero (four out of the 18 popula- be negatively associated with winter severity as tions), two populations had positive relationships indexed by cumulative winter precipitation. In with PRECearly (Castle Reef and Lost Creek) and eight out of 17 populations, the association two populations had negative relationship (Dubois between PRECwinter and lamb survival was nega- Badlands and Wapiti Ridge). The strongest positive bβ = tive and the associated 90% highest posterior effect was for Lost Creek ( PRECearly 1.26 [0.81, bβ density intervals for PRECwinter did not overlap 1.70]), which translated into a predicted increase in l = zero: The strongest effect was estimated for Lost lamb survival from S PRECearly : 0.04 [0.01, bβ ¼ : ½ : : lðÞ¼: ½0 05: : Creek ( PRECwinter 1 63 2 19, 1 09 , and the 0.07] to S PREClate0:95 0 66 0 47, 0 84 .The weakest effect was estimated for Wapiti Ridge strongest negative effect was for Dubois Bad- bβ ¼ : ½ : : bβ ¼ : ½ : : ( PRECwinter 0 14 0 23, 0 04 ). These lands ( PRECearly 0 56 1 06, 0 07 ,which

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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). translated into a predicted decline in lamb sur- we estimated both positive and negative relation- l ¼ : ½ : : vival from S PRECearly : 0 46 0 30, 0 64 to ships between lamb survival and NDVIearly and l ¼ : ½ 0 05: : S PRECearly0:95 0 13 0 04, 0 23 .Althoughwe NDVIlate covariates. Where the 90% highest pos- found no strong evidence for a relationship terior density interval for the regression coeffi- between the strength and/or direction of the rela- cients for NDVIlate did not overlap zero (seven tionship between early- or late-summer precipita- out of the 18 populations), two populations had tion and lamb survival (Fig. 8), the patterns for the positive relationships with NDVIlate (Dubois coefficient estimates whose 90% highest posterior Badlands and Francs Peak) and five populations density intervals do not include zero provide some had negative relationship (Jackson, Lost Creek, evidence that populations that experience higher Middle Missouri, Wapiti Ridge and Whiskey late-season precipitation have a negative coefficient, Mtn-East) (Fig. 9). The strongest positive effect bβ ¼ : whereas populations that experience higher early- was for Dubois Badlands ( NDVIlate 0 68 season precipitation have a positive coefficient. ½0:02, 1:38), which translated into a predicted lðÞ¼ We found equivocal evidence for a positive increase in lamb survival from S NDVIlate0:05 : ½ : : lðÞ¼ relationship between lamb survival and growing 0 10 0 01, 0 25 to S NDVIlate0:95 conditions as indexed by NDVI. Similar to 0:53½0:30, 0:79. The strongest negative effect was bβ ¼ results for early and late-summer precipitation, for the Middle Missouri population ( NDVIlate

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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.

0:62½1:23, 0:06), which translated into highest posterior density interval for the regres- fi a predicted decline in lamb survival sion coef cients for NDVIearly did not overlap lðÞ¼: ½ : : from S NDVIlate0:05 0 84 0 68, 1 00 to zero (seven out of the 18 populations), three pop- lðÞ¼: ½ : : S NDVIlate0:95 0 43 0 28, 0 61 . Where the 90% ulations had a positive relationship with

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NDVIearly (Castle Reef, Lost Creek and Spanish results for precipitation, we found no strong evi- Peaks) and four populations had a negative rela- dence for a relationship between the magnitude tionship (Jackson, Paradise, Wapiti Ridge, and and/or direction of these estimated effects and Younts Peak). The strongest positive effect was the range of conditions experienced (Fig. 8), but bβ ¼ : ½ : : for Castle Reef ( NDVIearly 0 73 0 37, 1 11 ), suggestive evidence for a relationship in which which translated into a predicted increase in populations with a higher late-season NDVI had l ¼ : fi lamb survival from S NDVIearly : 0 04 a negative coef cient, and populations with a ½ : : l ¼ : ½ 0:05 : 0 01, 0 08 to S NDVIearly0:95 0 27 0 18, 0 35 . higher early-season NDVI had a positive rela- The strongest negative effect was for Wapiti tionship. bβ ¼ : ½ : : Ridge ( NDVIearly 0 32 0 46, 0 18 ), which We note that the predicted relationships translated into a predicted decline in lamb sur- between growing season conditions, winter l ¼ : ½ : : vival from S NDVIearly : 0 32 0 28, 0 37 to severity, and lamb survival have important l ¼ : ½ :0 05 : S NDVIearly0:95 0 15 0 11, 0 18 . Similar to the 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).

v www.esajournals.org 21 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. demonstrated in the previous section, lamb sur- using population models and information on vival plays a significant role in reversing popula- vital rates to estimate vital rates and evaluate tion trajectories. The magnitudes of these how variation in vital rates shapes the trajectories predicted changes in lamb survival associated of populations. Although we note that popula- with environmental variation are large enough tions of ungulates are successfully managed with to suggest environmental drivers can play a sig- a conventional use of age ratios and count data nificant role in altering population growth rates, (Bender 2006, Harris et al. 2008), our model for example, the predicted decline in lamb sur- allows for a more comprehensive assessment of vival associated with winter precipitation for the the mechanisms by which population trajectories lðÞ¼: Clarks Fork population (S PREClate0:05 0 39 may vary. We demonstrated a variety of mecha- ½ : : lðÞ¼: ½ : : 0 30, 0 46 to S PREClate0:95 0 15 0 10, 0 22 ) nisms by which variation in vital rates affects was sufficient to push lamb survival below the changes in population growth rates, including threshold level where no value of ewe survival combinations of lamb survival and adult survival could have resulted in λf ≥ 1. that result in similar population growth rates, We used the last 10 yr of observations and the and suggest that these mechanisms may differ COVmodel to evaluate evidence for whether among populations. Our assessment of sources of mountain lion densities indexed by the quality of variation in lamb survival rates indicated a poten- mountain lion habitat might serve as a potential tially complex interplay of ecological and disease limiting factor for lamb survival. Our estimate processes affecting lamb survival, including gen- was small and quite imprecise such that we erally strong effects of winter climate, contrasting found essentially no evidence for such a relation- effects of summer growing conditions, and no bβ ¼ : ½ : : ship ( LION 0 09 0 62, 0 79 ). effect of mountain lion predation or relationship Our assessment of the relationship between with resident pathogen community. Together, pathogen communities and lamb survival for the these results offer a rigorous and comprehensive last 10 yr of observations provided strong evi- evaluation of bighorn sheep vital rates. dence that the median value of lamb survival for Vital rates demonstrated substantial variation those populations that hosted both M. ovipneu- across years and populations such that a variety moniae and Pasteurellaceae (0.24 [0.22, 0.26]) was of combinations of vital rates could yield similar lower than the median value of lamb survival for population trajectories. The variation in lamb sur- populations that hosted Pasteurellaceae alone vival that we observed is consistent with results (0.41 [0.37, 0.46]) (Fig. 10a), with an estimated from decades of empirical work for other ungu- difference between the medians of 0.18 [0.13, lates that documents high variability in offspring 0.23]. However, although the distribution of survival rates (Gaillard et al. 1998, Gaillard and lamb survival values for populations that hosted Yoccoz 2003, Raithel et al. 2007). The evidence for both pathogens was lower than the distribution very low lamb survival rates in some years and for populations that hosted Pasteurellaceae populations highlights the concern that this rate alone, we note a considerable amount of overlap could play a limiting role in conservation and (Fig. 10b). Approximately 48% of the point esti- restoration efforts for bighorn sheep in particular mates for annual lamb survival values for popu- (Johnson et al. 2010). In contrast to the expecta- lations that hosted both pathogens were equal to tion of high adult survival with minor variation or exceeded the lowest estimated lamb survival for many ungulate populations, our estimated value for populations that hosted Pasteurellaceae adult ram and ewe survival rates were also highly alone; moreover, approximately 18% were equal variable. This is an unsurprising result for popu- to or greater than 0.40, a threshold of lamb sur- lations of ungulates like bighorn sheep that are vival we previously associated with values of subject to stochastic disease and severe weather λf ≥ 1 despite a wide range in ewe survival. events, harvest, or predation that all result in lower and highly variable adult survival rates DISCUSSION (Owen-Smith and Mason 2005, Nilsen et al. 2009, Eacker et al. 2017). The integrated effects of this Results from our novel state-space model in a variation in vital rates on the population growth, Bayesian framework highlight the benefits of particularly in small populations, is poorly

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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. understood. To our knowledge, our work is the to yield positive growth rates (e.g., predator con- first to empirically define the existence of a demo- trol, treatment for respiratory disease, and sup- graphic “safe space” (de Silva and Leimgruber plemental provisioning) and provide minimum 2019) for bighorn sheep, that is, combinations of thresholds of vital rate values required for vital rates that result in positive growth rates. positive population trajectories, below which These results indicate that management can removals from translocations or harvest may address declines in populations by mitigating need to be reduced and/or augmentation may be either (or both) lamb survival and adult survival necessary for population persistence (Johnson

v www.esajournals.org 23 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. et al. 2010). Although our results suggest that a is variable due to intrinsic/extrinsic drivers, it can constraining relationship may exist between overwhelm the role of offspring recruitment, par- removals and the growth rate for the female pop- ticularly for small and declining populations ulation, this relationship was obscured by two (Nilsen et al. 2009). These results caution against sources of uncertainty that prevented strict infer- a one-size-fits-all approach to conservation and ence: (1) uncertainty in estimates of growth rates restoration efforts for bighorn sheep and high- and (2) uncertainty in the relationship between light how managing for a particular objective the larger population from which animals are requires a careful assessment of the range of vital removed and the portion of this larger population rates in the population so as to determine the that is surveyed (uncertainty that arose from the most efficient strategy. distribution of animals on the landscape and the The diversity of mechanisms that underlie quality of the survey process). changes in population growth rates shown here The expectation for the dynamics of unhar- is at apparent odds with two prevailing ideas vested ungulate populations lacking substantial about population dynamics of bighorn sheep. exposure to predation and/or disease is that low All-age die-offs due to disease events in bighorn variation in adult survival (despite high elastic- sheep can potentially devastate populations, ity) renders the high variation in offspring both during the initial epizootic and for years recruitment the dominant source of variation in afterward as enzootic disease reduces lamb sur- population dynamics (despite lower elasticity) vival (Cassirer and Sinclair 2007). Although our (Gaillard et al. 1998, Raithel et al. 2007). A key results do provide some evidence for years in component of that expectation is the low varia- which survival rates of both lambs and adults tion in adult survival; where this rate is affected precipitously declined (consistent with all-age by harvest, disease, or predation, it is unsurpris- die-offs), we also demonstrated that population ing that variation in adult survival can dominate trajectories can change due to a diversity of vital population dynamics given its higher elasticity. rate combinations, many of which are not consis- Our approach found a diversity of relationships tent with all-age die-offs. We found evidence for between vital rates and population growth rates both positive reversals in trajectory due to among populations similar to results from other improved lamb survival even as adult survival work on bighorn sheep that used asymptotic declines, and negative reversals in trajectory due approaches (Johnson et al. 2010). Although varia- to decreased lamb survival even as adult survival tion in adult survival explained the most overall improves. Although all-age die-offs undoubtedly variation in population growth rates, improved play a dominant role in population dynamics lamb survival was found to be the dominant when they occur, population growth rates are mechanisms by which populations actually also shaped by a complex interplay of variation reversed declines (Manlove et al. 2016). The rela- in other vital rates that are (likely) decoupled tive importance of these mechanisms was not from the disease process as well as from one consistent between populations, and we add to another, for example, declines in survival the small number of studies suggesting the dri- induced by harvest. Second, our work suggests vers of population dynamics for ungulates may that the presence of pathogens associated with vary spatially (Albon et al. 2000, Coulson et al. respiratory disease in bighorn sheep is insuffi- 2005, Garrott et al. 2008a, Nilsen et al. 2009, John- cient on its own to account for the realized level son et al. 2010). These results also help further of variation in population dynamics. Consistent broaden the perspective on the drivers of popula- with prior work on many of these same popula- tion dynamics for ungulates. We suggest that tions (Butler et al. 2018), we found that the pres- any apparent conflict between these results and ence of M. ovipneumoniae was not a sufficient the expectation for ungulate population dynam- factor in limiting population growth. Despite our ics merely reflects the fact that populations live conclusion that populations hosting M. ovipneu- in different portions of the vital rate parameter moniae and Pasteurellacea had an overall lower space: Where adult survival is high and non- median lamb survival than those that hosted variable, growth rates are driven by variation in Pasteurellacea alone, the ranges of lamb survival offspring recruitment, but where adult survival estimates for both types of populations were

v www.esajournals.org 24 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. very similar and included values that resulted in differ depending on the range of conditions to positive population growth rates and reversals of which a population is exposed, a possibility that declining population trajectories, regardless of will require further work to elucidate. We specu- ewe survival rate values. If variation in offspring late that these different responses may arise from survival was not well-explained by the composi- the manner in which populations interact with tion of pathogen communities, then ecological their environment, for example, where late- and environmental factors have to be the key season precipitation is typically low, a high players either independent of resident pathogen annual value may indicate greener conditions communities or interacting with pathogen com- during the later season, whereas where late- munities to impact disease expression. Overall, season precipitation is typically high, a high our work strongly suggests that multiple ecologi- annual value may indicate late-season snow cal factors are important drivers of population events that predisposes lambs to nutritional or dynamics with similar or stronger effects on big- other physiological stresses. There are two possi- horn sheep population trajectories as compared ble explanations for the ambiguity of these to disease epizootics (short of all-age die-offs), results. First, it is possible that there is a relation- and thus, bighorn sheep management and ship between the range of environmental condi- restoration should focus on advancing ecological tions experienced by a population and the understanding of the species, rather than assume response of lamb survival that is obscured by disease dynamics are predominately responsible either our small sample size of populations for population trajectories. (n = 17), or the indirect method we used to esti- Our work supports this supposition by mate lamb survival. Second, the broad scale of demonstrating the significant impact that varia- our covariate derivation (median values for tion in environmental conditions can have on entire ranges within a year) only captures a por- lamb survival. In contrast to results for other tion of the variation actually experienced by pop- sympatric ungulates where correlates of off- ulations. These populations are made up of spring recruitment and survival have been well- individuals that experienced different environ- documented to include intrinsic factors such as mental conditions due to different movement body mass and extrinsic factors such as preda- strategies (Lowrey et al. 2019), and our derived tion and environmental factors (Singer et al. covariates may be a crude average of more com- 1997, Cotˆ e´ and Festa-Bianchet 2001, Garrott et al. plex processes. Future work using individual col- 2003, 2008b, White et al. 2010, Creel et al. 2011, lared animals and an appropriate sampling Lukacs et al. 2018), fewer assessments of similar design that accounts for such a population struc- drivers have been done for bighorn sheep lambs ture is required. (Douglas and Leslie 1986, Hass 1989, Portier There are two further limitations of our et al. 1998, Douglas 2001, Brewer et al. 2014). approach that impact the interpretation of our Results of those assessments suggest that big- results and the broader applicability. First, uncer- horn sheep lamb survival can be affected by a tainty in the observation process can cripple similarly diverse set of processes and that lambs inference, and although our approach yields may be quite susceptible to predation from improved inference on vital rates and population mountain lions (Rominger 2018), and our results trajectories, we suggest that the quality of the are broadly consistent with those findings, with inference is related to the quality of the survey the notable exception that we found no relation- data. One practical possibility for improving the ship between lamb survival and our proxy for observation process would be to keep some com- the risk of predation from mountain lions. We ponent of the population collared to continually attribute this result to the difficulties inherent in reassess the distribution of the animals on the developing a suitable proxy for predation in the landscape to inform survey coverage and esti- absence of relationship between lamb survival mate survey-specific detection probabilities. Sec- and growing season conditions (as measured by ond, our method is not a substitute for spring and summer precipitation and primary simultaneously monitoring vital rates and popu- production) can vary among populations and lation trends, and we likely suffer from low pre- suggest that the nature of the relationship might cision on estimates of vital rates due to

v www.esajournals.org 25 July 2021 v Volume 12(7) v Article e03679 PATERSON ET AL. estimating vital rates based on count data alone. would be improved precision on vital rate esti- We speculate that relying solely on count data mates, the ability to detect stochastic die-off reduced our power to detect precipitous drops in events and to assess the impact of predation, an vital rates using flexible mixture models and to improved ability to assess the impacts of estimate the relationships between lamb survival removals on population trajectories, and, as a and environmental covariates. We acknowledge result, a deeper understanding of population- that some of the relationships identified here specific dynamics. Moreover, the detailed informa- may be spurious and actually represent the con- tionwouldbetterbeabletoestimatetheinfluence flated effects of unmodeled processes (e.g., of density-dependent processes in population stochastic die-off events, resulting multi-year dynamics if/when these populations recover to a lower lamb survival values and environmental point where resources may become limiting. In the processes). Future work that incorporates data at face of significant remaining uncertainty regarding the individual level is needed to more clearly elu- the drivers of populations for bighorn sheep, we cidate the relationships between disease, preda- recommend such an enhanced monitoring pro- tion, environmental variation, and lamb survival. gram as a necessary tool for helping to develop In conclusion, we emphasize our broadscale management plans that will conserve and restore finding that the majority of the bighorn sheep this iconic species. populations in this study had marginal popula- tion performance. Given the life-history charac- ACKNOWLEDGMENTS teristics of this species, their overall low density and assuming that resources are not limiting, a We are indebted to the regional biologists and game na¨ıve expectation for population growth from wardens who assisted in gathering and curating the broadly similar species suggest that they are cap- historical data. We thank the many agency and univer- able of rates >1.2 per year (McCorquodale et al. sity staff, capture crews, and volunteers that partici- 1988, Eberhardt et al. 1996). Although marginal pated in all aspects of this bighorn sheep study and population performance is the intended result of specifically to Kevin Monteith, Sarah Dewey, and Aly- harvest and translocations for some bighorn son Courtemanch for sharing survival data from col- sheep populations in this study (by reducing lared animals from the Whiskey Mountain and Targhee populations. Primary funding for this work population growth [Jorgenson et al. 1993] and was provided by the Federal Aid in Wildlife Restora- rendering populations less prone to epizootics tion Grant W-159-R to Montana Fish Wildlife and [Monello et al. 2001]), we found that that 15 of 17 Parks and the annual auction sale of a Montana big- populations had geometric mean growth rates horn sheep hunting license, as well as the Wyoming ≤1, and no point estimate for an annual value of Governor’s Big Game License Coalition, the Wyoming λ >1.08, indicating a general disjunction between Wild Sheep Foundation, and the Wyoming Game and the species’ potential and reality. Unfortunately, Fish Department. our work demonstrated multiple mechanisms by which variation in vital rates affects population LITERATURE CITED growth rates and that these mechanisms may dif- fer among populations, such that conservation Albon, S. D., T. N. Coulson, D. Brown, F. E. Guinness, and restoration efforts need to be population- J. M. Pemberton, and T. H. Clutton-Brock. 2000. specific. We suggest that the first steps to Temporal changes in key factors and key age improving our ability to understand the drivers groups influencing the population dynamics of of bighorn sheep population dynamics are female red deer. Journal of Animal Ecology – population-specific enhanced monitoring pro- 69:1099 1110. grams. By maintaining a long-term, collared Beissinger, S. R., and M. Z. Peery. 2007. Reconstructing the historic demography of an endangered seabird. sample from each population, the observation Ecology 88:296–305. process can be improved by providing more Bender, L. C. 2006. Uses of herd composition and age information on the locations of animals, and the ratios in ungulate management. Wildlife Society biological model can be improved with direct Bulletin 34:1225–1230. estimates of vital rates and estimation of survey- Bender, L. C., and M. E. Weisenberger. 2005. Precipita- specific detection probabilities. The net effect tion, density, and population dynamics of desert

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