A hierarchical modelling framework for estimating individual‐ and population‐level reproductive success from movement data

Rapidly advancing animal telemetry technologies paired with new statistical models can provide insight into the behaviour of otherwise unobservable free‐living animals. Changes in behaviour apparent from pairing telemetry with statistical models often occur as animals undertake key life‐history activities, such as reproduction. For many species that are secretive or occupy remote areas, these life‐history events are difficult to detect with conventional survey techniques, and consequently, vital rates are difficult to estimate. We present a hierarchical modelling framework, which integrates movement data observed via animal‐borne telemetry and optional, infrequent survey data, to estimate individual‐ and population‐level reproductive success. The approach combines a mechanistic movement model and survival model, and allows for assessing the effects of hypothesized individual and environmental covariates on reproductive success. We first tested our approach with simulated data, and then applied it to movement data from migratory golden eagles (Aquila chrysaetos) breeding in southcentral Alaska across four breeding seasons. We show that results supported our biological hypotheses that changes in movement coincided with the timing of reproductive failures, and that changes in movement could be used to assess breeding success (and failure) at the individual and population levels. The analysis also provided evidence of inter‐annual variation in population‐level nest success and the timing of nesting failures. This new approach is adaptable to many species that care for young and can be tracked with telemetry devices, and can provide not only individual‐level information useful for testing ecological hypotheses, but estimates of demographic parameters that can directly inform conservation and management if tagged animals are representative of the population.

1. Rapidly advancing animal telemetry technologies paired with new statistical models can provide insight into the behaviour of otherwise unobservable freeliving animals. Changes in behaviour apparent from pairing telemetry with statistical models often occur as animals undertake key life-history activities, such as reproduction. For many species that are secretive or occupy remote areas, these life-history events are difficult to detect with conventional survey techniques, and consequently, vital rates are difficult to estimate.
2. We present a hierarchical modelling framework, which integrates movement data observed via animal-borne telemetry and optional, infrequent survey data, to estimate individual-and population-level reproductive success. The approach combines a mechanistic movement model and survival model, and allows for assessing the effects of hypothesized individual and environmental covariates on reproductive success. We first tested our approach with simulated data, and then applied it to movement data from migratory golden eagles (Aquila chrysaetos) breeding in southcentral Alaska across four breeding seasons.
3. We show that results supported our biological hypotheses that changes in movement coincided with the timing of reproductive failures, and that changes in movement could be used to assess breeding success (and failure) at the individual and population levels. The analysis also provided evidence of inter-annual variation in population-level nest success and the timing of nesting failures.
4. This new approach is adaptable to many species that care for young and can be tracked with telemetry devices, and can provide not only individual-level information useful for testing ecological hypotheses, but estimates of demographic

| INTRODUC TI ON
Current biotelemetry and biologging technologies provide unprecedented windows into otherwise unobservable animals' lives.
Applications of these technologies have provided insight into patterns ranging from hibernation physiology and behaviour (Chmura et al., 2018) to global changes in movement patterns across taxa (Tucker et al., 2018). Although sometimes lagging behind advances in telemetry technology, development and application of mechanistic movement modelling techniques can pull from the rich data streams these new tags produce to draw nuanced and precise inference about species' biology, yielding a deeper understanding than descriptive techniques often used to analyse telemetry data (Hooten et al., 2017).
Many recent developments in statistical modelling of animal movement data have been made with the goal of drawing inference about an animal's internal state. For example, hidden Markov and other state-switching models have been developed and applied to reveal changes in behavioural states, as well as show how environmental covariates correlate with transitions between states (Breed et al., 2009;Hooten et al., 2017;Langrock et al., 2012). However, changes in behaviour can arise as animals perform key life-history activities. For example, territorial species exhibit marked changes in their movement when they obtain a territory (Eisaguirre et al., 2022), and some taxa, such as ungulates, have been shown to change their movement with the timing of reproductive events (e.g. parturition and offspring mortality; Cameron et al., 2018;DeMars et al., 2013).
In some cases, including the ungulate parturition example just mentioned, metrics computed from movement data collected with satellite telemetry can be used to identify the timing of those life-history events (Cameron et al., 2018;DeMars et al., 2013).
Descriptive metrics, such as changes in speed, may be useful in some cases; however, identifying more subtle changes in movement coinciding with life-history events may require mechanistic approaches. For example, changes in movement rate (a metric) of large magnitude may be easily determined, whereas slight changes in the consistency of movement rate or directionality, second-order parameters following from simple movement rate, may be difficult to identify with simple metrics. Mechanistic models of animal movement typically account for and estimate such secondary and tertiary parameters associated with more nuanced behavioural differences and the inherently autocorrelated nature of animal movement data (Hooten et al., 2017), and subtle changes in those parameters may be diagnostic of behaviours associated with life-history events. There is thus potential to use mechanistic movement modelling techniques to detect signals of life-history events, such as successful (or failed) reproduction, from movement data.
The initiation, success and failure of reproductive attempts are key life-history events. Accurate estimation of these events is critical to our understanding of a population's dynamics and their effects throughout an ecosystem, and for species of conservation concern, crucial for informing management. An ability to precisely estimate reproductive success also provides a means to help investigate individuals' relative fitness and the drivers of individual and interannual variation in reproduction. Being able to employ remote tracking data to understand movement and space use has been one of the most significant advances in studying animal behaviour and ecology in the past century, but extending that inference to detect and assess reproductive effort, success and fitness would further broaden the utility of biotelemetry.
If such inference were possible, it would provide key information about a species' ecology that can also inform conservation and management.
Finally, inferring this information remotely would avoid expensive, disruptive and often logistically infeasible survey efforts, as well as avoid potential biases in vital rate estimates common with some conventional techniques (Brown et al., 2013;Mayfield, 1961).
In this paper, we develop and implement a Bayesian hierarchical model that estimates the reproductive status of individuals throughout the breeding season from movement data collected with satellite telemetry paired with infrequent breeding surveys of a subset of the tracked individuals. We first apply this new approach to simulated data, and then to satellite telemetry data paired with sparse conventional survey data collected for a sample of migratory golden eagles (Aquila chrysaetos) that nest across southcentral Alaska. We show that the model provides inference about movement behaviour and space use dynamics, probability of successful reproduction and the effects of covariates on the probability of success. We also show that results scale up from individuals to estimate population-level reproductive success, revealing interannual variability in reproductive success. Although we demonstrate the method using a migratory raptor, the approach is generalizable, and we discuss some avenues for application to other taxa.

| Model development
We developed a Bayesian hierarchical model that estimates the probability of an animal producing offspring from movement data and, if available, infrequent surveys of breeding status. We present parameters that can directly inform conservation and management if tagged animals are representative of the population.

K E Y W O R D S
Bayesian, continuous time, data integration, movement ecology, Ornstein-Uhlenbeck, raptors, recursive Bayes, reproductive success our model framework as a hierarchical model with three levels: (1) a data model that links imperfect observations to a biological process, (2) a process model that represents our understanding of the biological process and (3) parameter models (or priors) that represent our prior knowledge about the parameters and process (Berliner, 1996).

| Data model
The data model takes the form: where z i,t = 1 if the ith animal is still engaged in a reproductive attempt at time t, or z i,t = 0 if the attempt has failed. The bracket notation a| b represents a probability density or mass function of the variable a given b (Gelfand & Smith, 1990).

| Process model
The movement process model may take any form (e.g. correlated or biased random walk); however, we present an Ornstein-Uhlenbeck (OU) process (Blackwell, 1997;Dunn & Gipson, 1977). The isotropic OU process in two dimensions is continuous time, mean reverting and matches well to the movement patterns of animals moving around a central place as they tend a nest or den (Dunn & Gipson, 1977;Eisaguirre et al., 2021;Johnson et al., 2008). Additionally, it has two biologically interpretable movement parameters that can capture subtle changes in movement patterns. The position likelihood of an OU process takes the form: where The parameter i,t measures the autocorrelation and centralizing tendency of the ith animal's movements around the central point * i ; i,t measures the diffusiveness or spread of the movements around * i ; and I 2 is the 2 × 2 identity matrix. Note that if equation 2 takes the form of a normally-or t-distributed error process, for example, the joint movement model effectively becomes a state-space model (Hooten et al., 2017). Fundamental to our model framework is the idea that individuals will change how they move if a breeding attempt fails. This change could emerge when offspring care, such as brooding, provisioning and defense, is no longer required, among other plausible biological causes of behavioural changes immediately following a reproductive failure. We incorporate this by letting the movement parameters be a function of the breeding state: The parameters i ≡ 0,i , 1,i ′ and i ≡ 0,i , 1,i ′ describe how the individual's movement varies while still engaged in a breeding attempt versus behaviour following a failure. Using 1 − z i,t instead of just z i,t ensures identifiability in cases where individuals successfully reproduce because z i,t = 1 for all t ∈ (0, T .
We construct the breeding state process under the assumption that individuals can only initiate a single breeding attempt per season-although this could be relaxed in a more complex multistate formulation-which effectively equates to a survival model: where Δt i,t represents the probability of a reproductive attempt remaining viable from t − Δt to t, and consequently, the expected probability of successfully reproducing in a given season is T i,t . Indeed, this is analogous to a nest or offspring survival model (Converse et al., 2013;Dinsmore et al., 2002). Furthermore, if Equation 1 is Bernoulli, the joint survival model effectively takes the form of a Cormack-Jolly-Seber model (Lebreton et al., 1992). In many cases, (1) example. However, to better understand population dynamics, population-level reproductive success is also a useful parameter. We

| Derived quantities
There are two potentially useful derived quantities this OU reproductive success modelling approach offers. First is the time of failure t f . An MCMC sample of t f can be obtained by storing the first t for which z (k) i,t = 0 at each of k = 1, 2, … , K iterations. Second is the probability of a successful reproductive attempt. While T i,t can be interpreted as the expected probability of success, the realized probability can be obtained by recognizing that z i,t=T = 1 for a successful attempt. Therefore, we obtain the realized probability of suc-

| Simulation study
To ensure our individual-level model and MCMC algorithm could recover known parameter values, we simulated data under 18 different scenarios and fit the model to each dataset. For simplicity and due to our motivating case study (see below), we assumed no measurement error in neither movement or survey data. That is, y t = z t and s t = t . So, we model the observations directly with the process model.
The datasets were generated with two sets of parameter values, which were (1)  These 18 unique scenarios were chosen based on the system we planned to apply this method to (described below). That is, based on previous work (e.g. Eisaguirre et al., 2022), we anticipated likely effect sizes for the movement parameters, in terms of both spread and correlation of movements, and tested the ability of our approach to recover those. Furthermore, the simulated survey data were formulated similar to the real aerial survey data collected for our application. We encourage researchers interested in applying this model to other systems to conduct similar simulation studies to ensure the method can recover known, realistic values that may be unique to their system(s).
We coded the MCMC algorithm in R (R Core Team, 2019) and used a single chain of 10,000 iterations, discarding the first 1000 as burn in. Convergence was checked by splitting each chain and checking � R < 1.1 (Gelman et al., 1995). We summarized parameters with 90% credible intervals (CrI) and used CrI coverage to assess the ability of our model to estimate the known parameter values.
Eisaguirre (2023) provides code for the simulation study and secondstage MCMC algorithm.

| Application to a migratory raptor: Golden eagle
Migratory raptors are often counted and marked during migration along migration corridors and/or on the wintering grounds. However, many of those individuals breed in extremely remote areas with limited accessibility for direct observation of breeding status. There is also growing concern about the conservation status of many rap- Golden eagles are a long-lived, territorial raptor that reach sexual maturity entering their third breeding season (Watson, 2010). They most commonly nest on cliffs, or less commonly large trees, and are generally central place foragers (Watson, 2010). In North America, golden eagles are a partial migrant with a portion of the population exhibiting annual migration to northern latitudes to nest (Katzner et al., 2020).
Like many raptors (and other taxa), golden eagle pairs partition activities during reproductive attempts: The male does the majority of the provisioning, while the female tends the nest and does most of the incubating and brooding of eggs and nestlings (Watson, 2010). If a nesting attempt fails-although both members of the pair will typically continue to occupy and defend the territory-we would thus expect a change in movement behaviour and space use as the male no longer needs to provision the female and nestlings, and the female no longer must incubate eggs or brood and provision nestlings.
We followed the recommendation of Steenhof et al. (2017) to define a successful reproductive attempt as one where at least one nestling reaches 80% of fledging age (i.e. 51 days for golden eagles).
This corresponds to ∼ 15 July on average in our study area, according to the survey data (described below).

F I G U R E 1
Example simulated movement trajectory to which the Ornstein-Uhlenbeck reproductive success model was fit. Values used to simulate these data were = (4, 2) � , = ( − 1, 1) � , * = (0, 0) � and t f = 1000. Colour corresponds to the posterior mean of z t , and the red point represents the posterior mean of * .

| Telemetry and survey data
The telemetry data we use are described in detail by Eisaguirre et al. (2022). Briefly, golden eagles were captured and tagged in southcentral Alaska during spring migration 2014-2016, and the solar-powered GPS tags were programmed to take fixes 8-14 times per day, with some consecutive fixes being recorded as frequently as hourly and others as long as 16 h apart. Due to low battery voltage, there were occasionally missed fixes.
In four of the breeding seasons for which we had telemetry data, In every case, we spent sufficient time and effort to conclude with confidence the viability of a nesting attempt made by the pair occupying each territory.
Field procedures were conducted following the Alaska April, which implies T = 2184. We also excluded 'floaters' from our sample, which are sexually mature individuals that lack territories and thus do not initiate breeding attempts (Eisaguirre et al., 2022;Penteriani et al., 2011). Consequently, even for individuals in our sample that were not observed engaged in a viable nesting attempt later in the season, z i,t=1 = 1 because any individuals holding and defending a territory at the beginning of the breeding season are likely attempting to nest (Watson, 2010). We note that this implies our estimates of population-level reproductive success correspond to the proportion of territorial individuals that successfully reproduce.
In demonstrating our approach, we chose to estimate annual nest success for the 4 years for which we had survey data, so X i,t = X i can be interpreted as a design matrix indicating each year; this allowed us to implement TARB to estimate i ( Because we typically had information about the nest location for each reproductive attempt, we set based on the known locations and set 2 = 10 2 or 2 = 10,000 2 depending on how well we knew the location of the nest. Since we used a coordinate ref- erence system with units of metres and home ranges of territorial eagles are relatively small in this area (Eisaguirre et al., 2022), these correspond to informative and weakly informative priors respectively. For the priors on the random effect variances, we set q = 2 and r = 0.1. Lastly, we set a = 2184 and b = 1 for the transient prior in the first stage.
The first stage was fit to each individual reproductive attempt using two chains of 10,000 MCMC iterations, discarding the first 1000 as burn in. Convergence was assessed by visually inspecting mixing and checking � R < 1.1 (Gelman et al., 1995). The second stage was fit using the 18,000 retained samples from each first stage as proposals in the MH updates for the individual-level parameters (McCaslin et al., 2021).
Finally, we assessed fit of our model with a leave-one-out crossvalidation approach, leaving out the survey observations for one reproductive attempt, refitting the model and comparing the heldout observation(s) to the estimated breeding state for the respective individual and time step(s). We sequentially held out the survey observations for each of one-third of the reproductive attempts (randomly chosen) and used the probability integral transform (PIT) test to assess fit (Conn et al., 2018). For individuals that were surveyed twice in either 2017 or 2018, both observations were withheld and contributed proportionally to the u-score.

| Simulation study
When fit to simulated data, the MCMC algorithm took approximately 30 min to complete 10,000 iterations (on an 8-core 2.60 GHz processor workstation with 64 GB of RAM) and converged to the posterior in all cases. The results for each of the 18 simulation scenarios are presented in Table S1, and the marginal posteriors for one of the scenarios are shown in Figure 2. Overall, the algorithm performed well in estimating all parameters, as well as the derived failure times t f , with no appreciable bias. The different survey scenarios did not seem to affect the estimates of (and thus ; Table S1). This indicates much of the information about reproductive success arises from the movement pattern, even though the survey data are a direct observation of reproductive status.
While the uncertainty around estimates of seems high (Table S1, Figure S1), the estimates of realized success were highly precise-zero in each case-which is also implied by the high precision in estimates of t f . Furthermore, as demonstrated in the results presented below from our application to golden eagle data, population-level expected reproductive success can be estimated with reasonable precision, despite low precision at the individual level.

| Application: Golden eagles
We applied our model to N = 85 (50 male and 35 female) individual breeding seasons made by GPS-tagged migratory golden eagles. In addition to the movement data available from these birds, a number of the breeding attempts were surveyed; in an approximately Unif(0, 1) distribution ( Figure S2), suggesting no lack of fit (Conn et al., 2018).
We found that nest failure typically corresponded to an abrupt shift to more diffusive movements and larger home ranges (Figure 3).
Failure may have also corresponded to movements that were slightly less autocorrelated and less tending towards the nest (Figure 3). In some cases, changes in movement were subtle but still captured well by the model (e.g. Figure 4). These findings match our biological expectation for the model: After a failure, individuals were less tied to the nest for activities, such as provisioning, incubation and brooding, and that change in behaviour was reflected in their movement patterns.
The combination of telemetry and the limited survey data ap- Signals and timing of nest failure were apparent in the posterior realizations of z i,t (Figure 6), from which failure date t f and the realized probability of nest success were computed. Nest failures seemed to primarily occur early in the season across years, but there was a notable peak around mid-June in 2016 (Figure 7). With a few exceptions, the model identified nest failures and successes with high certainty (Figure 7).

| DISCUSS ION
We While our approach overcomes logistical challenges with estimating reproductive success, it also likely minimizes certain biases common in survey methods (Brown et al., 2013). For example, surveys often record occupancy and breeding status during an early sampling occasion, followed by productivity or reproductive success at a later occasion. However, if a site is occupied and a failure occurs prior to the first sampling occasion, the site would be marked unoccupied, thereby biasing estimates of reproductive success high (Mayfield, 1961;Steenhof et al., 2017). Indeed, we found many failures occurred early in the season (Figure 7 Figure 6. and reproductive success of individuals that occupy those sites (Korpimaki, 1988;Pärt, 2001). Therefore, in many cases, it is impossible to disentangle the effects of site quality and individual quality in explaining variance in annual reproductive rates.
Because our approach explicitly tracks individuals, estimates can be assigned directly to individuals, and if individuals are tracked across multiple age classes and observed using multiple sites, then parsing the effects of site and individual is possible. Indeed, certain intensive study designs can parse these effects, but they require marking many individuals (to ensure a sufficiently large number of re-sights), and extensive and long-term field efforts (e.g. Reynolds et al., 2019). Our approach could accomplish similar objectives remotely and over several years, requiring only occasional tagging and, where logistically feasible, survey effort.
Our application to golden eagles revealed both amongindividual and interannual variation in nest success ( Figure 5).
Additionally, the early failures that were estimated by the model (Figure 7) are consistent with the general pattern that golden eagle productivity is often driven primarily by the proportion of the population laying eggs, rather than nesting attempts failing post-laying (Katzner et al., 2020;McIntyre & Schmidt, 2012;Steenhof et al., 1997). While we did not model nest success as a function of covariates-except for the indicator for year-our method easily accommodates environmental and individual (e.g. age) covariates that could allow for testing hypotheses regarding drivers of both individual-and population-level reproductive success, which would also likely increase precision in estimates of individual expected reproductive success. For example, the relative roles of age and habitat in determining individual nest success could be investigated, as well as the roles of age structure and synoptic or stochastic weather events in determining populationlevel nest success.
In addition to such ecological questions, our approach can directly inform the management of golden eagles and other wildlife.
The ability to use movement data to estimate breeding success opens the door to leveraging existing telemetry data as additional data streams in (integrated) population models for estimating vital rates, assuming tagged animals are representative of the population.
For example, for golden eagles, our results would directly inform a recent regional population model for Alaska , and a broader application of our approach to other data across North America could inform continental-scale models that are used to make management decisions for golden eagles in the United States (Millsap et al., 2022).
While golden eagles were a motivating model system for our approach, similar movement data now exist for many species (Davidson et al., 2020). For example, large telemetry datasets exist for many migratory raptors (e.g. rough-legged hawk) and other birds, making our approach immediately applicable to a number of species of management concern. Furthermore, the OU movement process also matches well the movements of many central place and denning mammals, such as lynx (Lynx spp.) and wolves (Canis lupus), making our approach easily adaptable to assess individual-and populationlevel reproductive success of those and similar species. While we tested 18 scenarios in our simulation study that were relevant to our model system, researchers studying other systems can test specific scenarios to ensure the method has sufficient power to detect reproductive failures. For example, if individuals are expected to have generally larger home ranges, simulating data with larger values of would be worthwhile, and could be chosen based on how F I G U R E 5 Marginal posterior distributions for population-level annual nest success from the Ornstein-Uhlenbeck nest survival model fit to golden eagle data from southcentral Alaska. The curve represents the prior, and the points are the individual-level posterior means.
individuals are expected to move within those home ranges (e.g. due to parental care). Additionally, survey data could be simulated in ways similar to how it was or is planned to be collected, depending on the phase of the study.
A key to the scalability of our model for relevance to population modelling is recursive Bayesian inference . The individual level (first stage) of our model is computationally intensive on its own, so the ability to implement it separately (and in parallel) for each individual is crucial to being able to scale the model to estimate population-level reproductive success. Although only recently introduced to applied ecological statistics (Hooten et al., 2016;McCaslin et al., 2021), recursive Bayes is starting to gain traction in applied studies (Cameron et al., 2021;Eisaguirre et al., 2022;Feuka et al., 2022). Our approach provides another example of its utility in estimating complex, multi-level ecological models for making both individual-and population-level inference.
Population modelling and individual-based movement modelling have developed largely as separate disciplines in ecological statistics. Indeed, an overarching goal in the field of movement ecology is to formally connect individual (Lagrangian) and population (Eulerian) movement processes (Nathan et al., 2008). While there are instances F I G U R E 7 Summary of the posterior modes of Julian day of nest failure (left) and derived realized probability of nest success (right) for each reproductive attempt estimated with the Ornstein-Uhlenbeck nest survival model fit to golden eagle data from southcentral Alaska. F I G U R E 6 Posterior realizations of breeding status z i,t from the Ornstein-Uhlenbeck nest survival model fit to golden eagle data from southcentral Alaska for (top to bottom) a likely midseason failure with an observation of z i,t = 1 in May, a likely late season failure with an observation of z i,t = 1 in June and a likely successful nesting attempt with an observation of z i,t = 1 in May. Red lines correspond to the individual posterior realizations for each MCMC iteration, and the black line is the mean. of mechanistic movement models informing detection and survival processes in spatial capture-recapture models (Gardner et al., 2022;McClintock et al., 2021), our approach directly connects the timing of individual life-history events to a population-level demographic parameter, demonstrating that estimating reproductive success and combining movement and survey data for estimating productivity and other demographic parameters is perhaps also a fruitful area of research that can directly inform population ecology and management.

CO N FLI C T O F I NTE R E S T S TATE M E NT
The authors declare no conflicts of interest.

PEER R E V I E W
The peer review history for this article is available at https:// w w w.web of scien ce.com/api/g atew ay/wos/p e er-revie w/ 10.1111/2041-210X.14159.

DATA AVA I L A B I L I T Y S TAT E M E N T
All movement data used for this manuscript are archived in the online repository Movebank (https://www.moveb ank.org/). Data are not publicly available because the data contain information