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**Appendix S1**

Understanding a population's survival and reproductive rates is critical to examining its long‐term dynamics and viability, and foundational to effective management and conservation (Caswell, 2001; Morris & Doak, 2002). For wide‐ranging, cryptic, or otherwise hard‐to‐study species, obtaining the mark‐resight data typically used to estimate key demographic rates can be challenging, especially for reproduction (Etterson et al., 2011). For example, if young‐of‐the‐year (YOY) are not always with their mother, or are hard to detect, definitively identifying which adults have reproduced may not be possible. Modelling techniques, such as multievent models, have been developed to deal with such state uncertainty (Gimenez et al., 2012; Kendall et al., 2003; Pradel, 2005), but if offspring remain with their mothers for multiple years and YOY cannot readily be distinguished from older offspring, existing modelling frameworks are inadequate for estimating reproductive rates or accurately estimating offspring survival.

In mark‐resight studies designed to identify breeding state and estimate reproductive parameters, state uncertainty has traditionally been defined as arising when a breeding adult is seen without its offspring. Typically, observing a female with offspring is considered definitive evidence of her being in a breeding state (i.e. having given birth in the observation year), whereas a female seen without offspring may be in a nonbreeding state or may be a breeder whose offspring was not detected. Multievent models account for this uncertainty, but if a species exhibits extended maternal care, observing offspring with their mothers cannot be taken as definitive evidence of current breeding status unless offspring age is known. In such cases, observations of older or unknown‐aged offspring can be excluded from the analysis (Kendall et al., 2004), but such an approach, at best, discards valuable information that could improve estimation, and, at worst, yields biased reproductive rate estimates if YOY are misidentified as older offspring or vice versa. Recently, researchers have begun incorporating sightings of older offspring into analyses to estimate offspring survival, but these approaches assume that offspring age is determined with certainty (Arso Civil et al., 2019; Couet et al., 2019; Cubaynes et al., 2021; Lunn et al., 2016; Regehr et al., 2018; Verborgh et al., 2021). These methods also rely on either relatively high offspring detection rates or high numbers of within‐year surveys, and often assume a fixed (and known) mother‐offspring association period. For many populations, however, offspring age cannot be determined with certainty, the mother‐offspring association period is unknown and variable (sometimes extending beyond the birth of a sibling), and offspring may not be detected as YOY, only as older dependent young. Reproductive and juvenile survival rates cannot be estimated for such data with existing models.

To address these issues, we developed a new multievent mark‐resight modelling framework that uses all available mother‐offspring sighting data, including sightings of offspring of unknown age, to estimate reproductive rates and adult and juvenile survival rates. In standard multievent mark‐resight studies designed to estimate reproductive rates, when a marked individual is observed, its apparent breeding status is recorded (i.e. presence of offspring); this observation ‘event’ is then used to infer the latent (unobserved) true state of the individual. Our model expands what is recorded during the observation event to include not only whether the marked individual was accompanied by offspring, but also the estimated age (or age category) of the offspring. This extension required us to account for uncertainty in both the latent breeding state and latent offspring age, given the observation event, and, in turn, required a host of modifications to the typical state‐transition and observation matrices used in previously developed multievent models.

Below, we describe the basic structure of our model and use a simple example to illustrate its essential components. We demonstrate its usefulness for empirical data by applying it to photo‐ID data from an endangered population of beluga whales, *Delphinapterus leucas*. Finally, we examine the precision and bias of our model with a simulation analysis under a range of detection probabilities and sample sizes using the matrices developed for our beluga application. We conclude by discussing the significance of our new modelling framework, its possible limitations, and how it can be applied to other data.

Multievent models are a class of mark‐resight models that account for the fact that the field observations of marked individuals (i.e. events) may not fully reveal the true latent state of those individuals, such as their breeding status, disease state, or location (if unobserved). The multievent model likelihood function is comprised of three components: initial state vector, state process model, and observation (or event) process model (Gimenez et al., 2012; Pradel, 2005). The initial state vector defines the probabilities, upon first encounter, of an individual being in any one of the states defined in the model. On subsequent occasions, the individual can transition among states (or die) according to the state process model represented by a transition matrix with departure states in rows and arrival states in columns. The observation process model describes the possible observation events (columns) given the true state of marked individuals (rows). Data are in the form of sighting histories for marked individuals indicating the observation event (i.e. the observed status of the individual) on each sighting occasion.

As in other multievent models, our state process matrix describes the probability of a marked individual transitioning from one state at time *t* to another (or the same) state at *t* + 1. For clarity, we construct the overall transition matrix by multiplying a series of four conditional transition matrices representing distinct portions of the state transition process, an approach described previously (e.g. Couet et al., 2019; Gimenez et al., 2012) but used infrequently. We refer to these as component matrices and they account for intermediate states that represent the transition between the states at time *t* and *t* + 1. In order, the component matrices are (Step 1) adult survival, conditioned on state from the previous time‐step; (Step 2) offspring survival, conditioned on adult survival; (Step 3) deterministic offspring aging conditioned on survival (i.e. offspring must age if they survive); and (Step 4) reproduction, conditioned on survival and offspring age.

Similar to the state process matrix, our observation process matrix results from the product of three conditional component matrices. In order, these are (Step 1) adult detection, conditioned on true state; (Step 2) offspring detection, conditioned on adult detection; and (Step 3) categorization of offspring age, conditioned on detection and offspring true age. This third step is a new addition in the multievent modelling canon.

In our model, states describe a combination of a marked individual's breeding status (e.g. nonbreeder or breeder) and the age(s) of any dependent offspring. The observation events describe detection of marked individuals as well as the estimated age of any accompanying offspring. Offspring age estimates are assigned using a set of age categories, some of which incorporate uncertainty (e.g. 0–1 year old [yo], 1 yo, ≤2 yo). The specific age categories used in the model can be defined to match the particular circumstances of any given study (i.e. the level of visual, or other, information available to determine offspring age), which will then guide parameterization of the offspring age‐categorization matrix (see Section 3).

In addition to the standard assumptions of all multievent models, our model relies on two additional simplifying assumptions. We assume no false‐positive offspring sightings (i.e. no marked adults without offspring are identified as having offspring), but offspring can be missed if present. Uncertainty in offspring ages is captured via the age categories used in the observation matrix and we assume no error in offspring age categorization (e.g. a 3 yo offspring will not be categorized as a ≤2 yo, but could be categorized as a ≤3 yo, or ≤4 yo). To meet this latter assumption, age‐categories must be chosen to adequately represent the study‐specific offspring‐age‐categorization uncertainty.

As in all multievent models, the precise form of our full state and observation matrices (and thus the conditional component matrices) will depend on the number of states and observation events used in a given study. Below, we illustrate the flexibility of our modelling framework using, first, a simple hypothetical example, followed by a more complex application with empirical data.

In this example, we use five live states, eight observation events and 14 parameters with the following characteristics. Adult females are marked. During annual sighting occasions, offspring accompanying marked females are recorded, as are any age‐distinguishing characteristics of the offspring (e.g. relative size, neonate features). Offspring must remain with their mother through their second year (i.e. as 1 yo) to survive, but sometimes remain longer (only rarely, however, remaining dependent as ≥3 yo). Females will not give birth if they have dependent young, but if offspring do not survive the transition from *t* to *t* + 1, the adult can give birth in *t* + 1.

In this example, the five true, live states include nonbreeder (NB), breeder with no offspring (*B*), breeder with a YOY (*B*_{Y}), breeder with a 1‐year old (*B*_{c1}), and a breeder with a 2‐year old or older offspring (*B*_{c2*}). The initial state vector gives the probabilities of being in each of these five live states upon initial capture/sighting: $\prod =\left[{\mathrm{\pi}}_{N}\phantom{\rule{0.5em}{0ex}}{\mathrm{\pi}}_{B}\phantom{\rule{0.5em}{0ex}}{\mathrm{\pi}}_{\mathit{By}}\phantom{\rule{0.5em}{0ex}}{\mathrm{\pi}}_{\mathit{Bc}1}\phantom{\rule{0.5em}{0ex}}{\mathrm{\pi}}_{\mathit{Bc}2*}\right]$, where, ${\mathrm{\pi}}_{\mathit{Bc}2*}=1-\left({\mathrm{\pi}}_{N}+{\mathrm{\pi}}_{B}+{\mathrm{\pi}}_{\mathit{By}}+{\mathrm{\pi}}_{\mathit{Bc}1}\right)$.

The state matrix includes six state parameters related to adult and juvenile survival and reproduction (Figure 1). Step 1: The adult survival component matrix allows marked individuals to survive (with probability *φ*_{N} for nonbreeders and *φ*_{B} for breeders) or die (i.e. with probability 1 − *φ*_{N} or 1 − *φ*_{B}, respectively). In this step, they can only transition to the ‘dead’ state or stay in their same alive state. Step 2: The offspring survival matrix allows surviving marked individuals to transition to a state without offspring if the offspring dies, with probability 1 − *S*_{Y} for obligately dependent offspring (≤1 yo) or 1 − *φ*_{J} for older offspring (≥2 yo). *S*_{Y} represents true offspring survival probability, since we assume that offspring ≤1 yo cannot survive if separated from their mother. All other survival rates (denoted by *φ*) represent apparent survival, as true adult survival is confounded with permanent emigration, and true older offspring survival is confounded with gaining independence. Because the offspring survival matrix is conditioned on adult survival, we need to account post‐hoc for offspring that die or become independent when their mothers die; thus, marginal offspring survival is calculated as $S{\prime}_{y}={S}_{y}\times {\phi}_{B}$ and $\phi {\prime}_{J}={\phi}_{J}\times {\phi}_{B}$. Step 3: The offspring‐aging matrix advances surviving offspring from one age to the next, thus transitioning the marked individual to a state with an offspring that is 1 year older. These transitions are deterministic, that is, entries in the matrix are all probabilities equal to zero or one. Step 4: The reproduction matrix allows marked individuals without offspring to transition to a state with a YOY and is based on two parameters: the probability of breeding for the first time (*ψ*_{N}), and the probability of a breeder giving birth again (*ψ*_{B}).

The observation model is defined by eight parameters and eight possible observation events: not seen, seen without offspring, and seen with offspring in one of six age categories (Figure 2). Step 1: The adult detection matrix is based on a single detection rate for all adults (*p*). Step 2: The offspring detection matrix is based on detection rates for young, obligately dependent offspring (i.e. ≤1 yo; *δ*_{Y}) and older offspring (i.e. ≥2 yo; *δ*_{C}), conditional on the mother being detected. Step 3: The offspring‐age‐categorization matrix includes six age categories: ‘J0’ for offspring that can be definitively identified as having been born in the observation year (i.e. YOY), ‘J1−’ for offspring determined to be a maximum of one year old but possibly younger, ‘J1’ for offspring that can be definitively identified as 1 year old, ‘J1+’ for offspring 1 year old or older, ‘J2+’ for offspring 2 years old or older, and ‘Unk’ for offspring whose age category could not be determined. We use five parameters to characterize the age‐categorization process, where each parameter represents the probability of a binary age‐categorization decision, conditional on any previous decision(s). Specifically, *γ* represents the probability that the offspring can be assigned to an age category (vs. unknown). Parameters *α* and *β* apply to YOY and 1 yo, respectively, and, given they can be categorized, are the probabilities they can be assigned to a definitive age category (i.e. YOY assigned to ‘J0’, or 1 yo assigned to ‘J1’). Parameter *κ* represents the probability that a 1 yo offspring, given it was not assigned to the definitive category ‘J1’, is assigned to the ‘younger than or equal to’ age category (e.g. ‘J1−’) versus the ‘equal to or older than’ category (e.g. ‘J1+’). Parameter ω applies to offspring 2 years old or older and is the probability that the 2 yo will be placed in the ‘J1+’ category versus the ‘J2+’ category, given it could be categorized. The probability of a particular age assignment for any given true age will be equal to the product of all parameters along the binary decision path leading to that assignment. For example, a 1 yo will be assigned to the ‘J1+’ category with probability *γ*(1 − *β*)(1 − *κ*) to account for the fact that it could be put into an age category, it could not be aged definitively, and it was not assigned to the ‘younger than’ category, but to the 1 yo ‘or older’ category (see matrix 3 in Figure 2).

This example illustrates a simple application of our model, but the basic framework is designed to be flexible enough to accommodate a wide range of study designs and data from any population with extended offspring care and uncertainty in offspring age determination. As described above, the key characteristics of our model are, (1) the construction of the state transition matrix using a series of conditional transition matrices that account for adult and juvenile survival, juvenile aging, and adult reproduction, and (2) construction of the observation matrix using a series of matrices that account for adult and juvenile detection and juvenile age categorization. Using this basic framework, we are able to build a substantially more complex example to accommodate the ecology and observation of the Cook Inlet beluga whale.

In this section, we apply our model to empirical data from the endangered Cook Inlet beluga whale population, for which managers currently lack robust estimates of reproductive and juvenile survival rates. This beluga case study demonstrates how the structure of our model matrices can be adapted to account for different kinds of data, life histories and model assumptions by modifying the states, events, matrices and parameters while using the same basic model structure. We also use this empirical example to describe the Bayesian model fitting procedure.

We applied our model to photo‐ID data collected from the endangered beluga whale population that resides in Cook Inlet, Alaska, USA (Laidre et al., 2000; O'Corry‐Crowe et al., 1997; Wade et al., 2019). Detailed data collection protocols can be found in McGuire et al. (2020), but we briefly describe them here. Photo‐ID surveys were conducted under General Authorization, Letter of Confirmation No. 481‐1759, and National Marine Fisheries Service Scientific Research Permits #14210 and #18016. The Cook Inlet Beluga Whale Photo‐ID Project (CIBW‐PIP; *n* = 471 individuals; *n* = 13 sightings occasions), where our sighting records consist of the aggregate sightings of an individual made during each field season (*n* = 1931 aggregate sighting records). Based on previous studies and field observations (Bors et al., 2021; McGuire et al., 2020, 2021; Shelden et al., 2020), we assumed that catalogued individuals are a combination of male and female (which are indistinguishable in the field) adults and subadults. We define subadults as independent whales (i.e. no longer with mother) ≥5 yo but not yet of reproductive age (<13–14 yo).

Calves in close association with adults were also photographed (*n* = 367 adult‐calf sightings), but generally could not be individually identified. Calves are thought to remain with their mothers between 2 and 5 years, and can remain closely associated beyond the birth of a sibling (McGuire et al., 2020). In the extremely turbid waters of Cook Inlet, darker‐coloured calves are frequently undetected, and for those that are detected, their age is difficult to determine.

During post‐survey photo‐processing, calves photographed in close association with an adult (their presumed mother) were assigned to an age category based on size relative to adult, colour, and neonate and YOY features (i.e. fetal folds, ‘peanut‐shaped’ head, and presence of an eye ring; McGuire et al., 2020). We assumed presence of neonate or YOY features was definitive evidence the calf was a YOY. Most calves, however, were assigned to a calf‐age category that included uncertainty. At least two photo‐analysts independently assigned an age category to each calf sighting, with discrepancies resolved by using the most‐inclusive category or by a third analyst. In general, age categories were assigned independently across years (i.e. without consideration of previous offspring age categorizations) to avoid age categorization bias and propagation of errors. However, because a few YOY were individually identifiable from distinct permanent markings, following the initial age‐categorization process, photo‐analysts checked for distinct marks that linked to a YOY sighting in previous years, which enabled assignment of a definitive age category (e.g. ‘J2’; 2.7% of calf sightings). When marked adults were observed in close association with two calves of different sizes on a single sighting occasion or on different occasions within the same year, the adult was associated with both calves, and ages were assigned to both (3.8% of calf sightings).

Our beluga application included more states and observation events than the simple example described above. The initial state vector included 13 parameters (*π*_{i}), one for each possible live state (Table 1). We used a total of 72 observation events, with 70 of those corresponding to observation of a marked adult with one or two calves in any combination of 12 calf‐age observation categories (Table 1). Below, we describe the parameters of biological interest, but we list all parameters and present all matrices in Appendix S1.

The state process model included six parameters that differed only slightly from the analogous parameters in the simple example (Figure 3; Appendix S1). We separated adult survival into breeder and nonbreeder survival (*S*_{B} and *S*_{N}, respectively), where breeders are defined as females that have given birth at least once and nonbreeders include subadults, males, and nonbreeding females. These adult survival rates represent true survival because CIBWs are geographically isolated in Cook Inlet and their summer range is currently limited to the portion of the Inlet where CIBW‐PIP surveys are conducted. Thus, we assumed no immigration from or emigration to locations outside the study area. As in the simple example, we estimated two age‐specific calf survival rates (*S*_{Y} for ≤1 yo, and *φ*_{C} for ≥2 yo). We assumed, based on the literature and our own records (McGuire et al., 2020), that beluga calves do not wean before their second ice‐free season and therefore must remain with their mother for at least that long in order to survive. Thus, our young calf survival parameter (*S*_{y}) represents true survival probability, and the older calf survival parameter (*φ*_{C}) represents apparent survival (the probability that the calf both survives and remains with its mother until the next ice‐free season). Given that older calves that survive must become independent at some point, this will by necessity be a negatively biased estimate of true older calf survival. Marginal calf survival was calculated as in the simple example: $S{\prime}_{y}={S}_{y}\times {S}_{B}$ and $\phi {\prime}_{C}={\phi}_{C}\times {S}_{B}$. We did not assume an upper age for calf dependence on its mother, but we did assume a mother would care for no more than two calves concurrently. Our reproductive parameters included a nonbreeder transition parameter (i.e. the probability a nonbreeder gives birth for the first time; *ψ*_{N}) and an established breeder reproductive rate parameter (*ψ*_{B}). Based on the biology of CIBWs, we assume that births cannot occur in consecutive years, thus our reproductive rate is defined as the probability that an established breeder without a YOY in the previous year gives birth. Relatedly, we assume that a breeder whose YOY dies cannot give birth the next year.

The observation model included 13 parameters—five to account for detection and eight to account for the calf‐age‐categorization processes. Adult detection was defined as the probability that an identified individual was detected on at least one of the surveys conducted each year. To account for any potential differences in sociability, habitat use, or photographer bias toward whales with calves, we estimated three separate adult detection probabilities, one for nonbreeders (*p*_{N}), one for breeders with one or more calves of any age (*p*_{Bc}), and one for established breeders with no calf in a given year (*p*_{Bn}). We estimated two calf‐detection probabilities corresponding to young calves (*δ*_{Yc}) and older calves (*δ*_{C}). Our calf‐age‐categorization matrix used eight parameters to account for calf‐age‐categorization uncertainty that together describe the probability a photo‐analyst assigned an observed calf to 1 of 12 calf‐age categories conditional on its true age and detection (Appendix S1). We accomplished this using the type of binary decision‐path approach described in our simple example (and illustrated in detail in Appendix S2), which allowed us to describe 72 observations events using eight parameters.

We fit the beluga model using a Bayesian statistical framework with vague prior distributions for all parameters, including a Dirichlet [1…1] prior for the initial state parameters (*π*_{i}), and Uniform [0, 1] priors for all others. Models were fit by running four chains of 55,000 Markov chain Monte Carlo (MCMC) iterations with a burn‐in of 10,000 iterations and a thin rate of 10; thus, we retained 18,000 samples from each posterior distribution. Below we report the posterior median and 95% credible intervals (CrIs) for all demographic and detection parameters (see Appendix S1 for the complete set of parameter estimates). We performed statistical analyses in R version 3.6.1 (R Core Team, 2017), using a Gibbs sampler algorithm implemented in JAGS version 4.3 (Plummer, 2022), and used the

Our model produced precise estimates of all demographic rates. Breeding females had an estimated survival rate of 0.962 (95% CrI: 0.945–0.975), and nonbreeders of 0.931 (0.917–0.944; Table 2). We estimated a breeding probability for established breeders without a YOY in the previous year of 0.279 (0.226–0.34), which translates to an inter‐birth interval of 4.6 years (3.9–5.4; calculated as, $\frac{1}{{\Psi}_{B}}+1$). Nonbreeders entered the breeding population at a rate of 0.072 (0.059–0.085) per year. Young calves (≤1 yo) had an estimated survival rate of 0.926 (0.85–0.964), and older calves (≥2 yo) had an estimated apparent survival rate (i.e. probability of surviving and remaining with the mother) of 0.492 (0.398–0.595).

Detection rates for all classes of whales were moderate to high (Table 2). For adults and subadults, detection rates varied among breeding classes, ranging from 0.456 (0.43–0.484) for nonbreeders to 0.715 (0.606–0.83) for established breeders with no calf. Calves were detected about half the time they were present with their detected mothers.

To assess bias and precision in our model, we applied it to simulated mark‐resight data. The simulated data resembled the empirical beluga data, with 72 possible observation events and 13 possible live states, and we used the same parameters and model structure used in the beluga case study. We simulated three levels of sample sizes (250, 475, 1000) corresponding to the total number of marked individuals, three levels of adult and calf detection (0.25, 0.5, 0.75), and fixed demographic rate values similar to our empirically based beluga estimates. We generated *n* = 50 datasets for each fully crossed combination of sample size and detection probability. For each parameter and scenario, we calculated relative bias as $\mathrm{RB}=\left|\frac{\left({\sum}_{i=1}^{n}{f}_{0.5,i}\right)-t}{t}\right|\times 100$, and root median square error as $\text{RMSE}=\sqrt{\frac{\sum _{i=1}^{n}{\left({f}_{0.5,i}-t\right)}^{2}}{n}}$, where ${f}_{0.5,i}$ is the estimated posterior median for the *i*th simulation and true value, $t$. To determine CrI coverage, we calculated the mean proportion of simulations in which the 95% CrIs included *t*, that is, ${f}_{0.025}\le t\le {f}_{0.975}$.

The simulation analysis indicated that our model is asymptotically unbiased and performed well under a broad range of detection rates and sample population sizes. Both RB and RMSE decreased at higher sample sizes and higher levels of adult and calf detection rates (Figure 4; Appendix S3). At the highest sample size and detection rates, all survival and reproductive parameters had RMSE <0.03 and RB <1%. Some offspring‐age‐categorization parameters suffered from high bias at low adult and offspring detection, but the bias did not appear to negatively impact parameters of biological interest (Appendix S3). Credible intervals across all model parameters included the data‐generating value in an average of 95% of the simulations.

We developed a novel multievent mark‐resight model that addresses many of the challenges associated with estimating reproductive and survival rates in long‐lived species with complex social structures. This model is the first to produce reproductive rate and juvenile survival estimates in a scenario with classic reproductive state uncertainty coupled with an extended, variable parental care period and uncertain offspring ages. Simulations indicated the model is asymptotically unbiased and performs well under a broad range of detection rates and samples sizes. Our model is applicable to any population for which mark‐resight‐type data are collected on adults along with information on presence and relative size or age of accompanying offspring.

In our case study, we were able to estimate reproductive and survival rates for the Cook Inlet beluga population that were previously inestimable from mark‐resight data due to a combination of reproductive state uncertainty, offspring age uncertainty, and an unknown and variable offspring‐dependency period that can result in overlapping sibling dependency. The vital rate estimates enabled by our new model yield important new information suggesting that nonbreeder survival and reproductive rate may be low and contributing to the population's decline. This knowledge may further the effort to elucidate the exogenous stressors affecting this population.

In our model, as in all mark‐recapture models, capture heterogeneity can negatively bias survival rates (Abadi et al., 2013; Williams et al., 2002, p. 434). The beluga application of our model accounted for potential differences in detection between breeders with calves, breeders without calves, and nonbreeders, but not for the subgroups within the nonbreeder class (subadults, males, nonbreeding females) because those groups were not distinguishable in our data. This has the potential to misrepresent survival of these subgroups. In applying the model to other systems, separate detection parameters should be used for any classes of individuals with potentially different detection patterns to avoid biased survival estimates.

Our model results rely on the assumption that we have no false‐positive adult‐offspring sightings. In the beluga case study, we excluded ambiguous mother‐calf associations from the data, but assumed an adult whale photographed in unambiguous association with a calf was the mother of that calf. If calves commonly associate closely with individuals that are not their mother (i.e. allocare; Aubin et al., 2021), our reproductive rates could be biased. To assess how erroneous adult‐offspring associations might impact demographic rate estimates, we applied our model to a modified beluga dataset in which we excluded the 60 adult‐calf sightings that occurred on only a single survey for any given adult, under the unrealistic, but instructive, assumption that these observations were all erroneous associations. These 60 sightings represented 16% of all adult‐calf sightings and 3% of all sightings. The results from this exploration indicated that violations of the no‐false‐positive assumption in the beluga case study would result in an overestimate of the transition rate from nonbreeder to breeder and an underestimate of established breeder reproductive rates. Quantitatively, a 16% false positive rate in adult‐offspring associations could result in an 80% overestimate of nonbreeder transition and an 18% underestimate of the reproductive rate. In addition, survival rates in the different breeding classes could change because the pool of individuals in the breeder and nonbreeder classes would be different. With respect to the beluga data, we know that ~1/3 of the excluded single adult‐calf sightings came from whales that were seen with calves in years after the end of the present study (T.M., unpublished data) suggesting that the previous single adult‐calf sightings for those individuals were correct. Notably, however, this assessment yields important information about the potential direction and magnitude of bias if allocare or similar misspecification of adult‐offspring association is common within a dataset.

The structure of the state and observation matrices used in our model are driven by the two related, but distinct, adaptations of the multievent modelling framework that we leveraged: (1) defining adult reproductive status in terms of offspring age, and (2) extending state uncertainty to include offspring‐age‐categorization uncertainty. To estimate reproductive rate, offspring ages must be tracked through time to link observations of older offspring (i.e. >YOY) to their birth year, and thus to a reproductive event for the adult. As a result, the ‘reproductive state’ of an adult must include offspring age, and each offspring age must be explicitly accounted for as a separate adult reproductive state in both the state and observation matrices. Thus, for the full state matrix to account for each offspring age plus a breeder‐with‐no‐offspring state and a dead state, its minimum dimensions must equal the number of potential offspring age classes plus two. In addition, the state matrix must account for offspring survival and their transition to the next age‐category. As illustrated in the beluga application, the transition between offspring ages will be deterministic, conditional on offspring (apparent) survival.

In both the simple example and the beluga case study, we defined the oldest adult‐offspring state to be the specified offspring age ‘or older’. This approach was a practical decision to minimize the size of the state matrix while allowing for occasional sightings of older offspring and for uncertainty in how old the oldest offspring is. We found that rare instance of offspring older than the nominal state will not bias model results. However, the model matrices can be constructed to include all possible ages of offspring if that information is available. We also note that use of an ‘or older’ state will bias model results if sightings of the ‘older’ offspring are frequent.

The observation matrix must account for both offspring detection (conditional on adult detection) and offspring age‐categorization. By defining reproductive state in terms of offspring age, we can incorporate this uncertainty in the same way that reproductive state uncertainty has been incorporated in previous work with conditional transition matrices (e.g. Gimenez et al., 2012), albeit with somewhat more complexity. The number of, and relationship between, the offspring‐age‐classification parameters will depend on the number of reproductive states and the complexity and overlap among the age‐categorization categories. Our simple example demonstrated a parameter reduction method based on binary categorization decisions, and we extended that for our extremely complex beluga data with 72 observation events (Appendix S2). Careful a priori decisions about what offspring‐age‐categorization categories will be allowed during the age‐categorization process could help minimize the complexity of the age‐categorization matrix.

The novel multievent model structure we developed is an important step forward in using mark‐resight models to estimate reproductive rates and juvenile survival for species with extended parental care and uncertain offspring ages. Many cetaceans (i.e. whales and dolphins) exhibit social dynamics amenable to our model, as do bears, elephants, and some great apes, bats, and birds (Stearns, 1976; Uomini et al., 2020; van Noordwijk & van Schaik, 2005). In addition to expanding the number of populations for which reproductive rates may be estimable, our model could yield important ecological insights and improved parameter estimates through application to datasets previously analysed under restrictive assumptions (e.g. fixed parental‐offspring association periods, offspring age certainty) or datasets in which observations of older or unknown‐aged offspring were previously discarded. Our model can be adapted to include time‐varying survival and reproductive parameters as well as covariates that can account for variability in demographic and detection rates (A.W., unpublished data).

With respect to the Cook Inlet beluga population, our model provides critical new demographic rate estimates for this endangered and declining population that suggest nonbreeder survival and established breeder reproductive rates are low relative to other beluga populations and likely contributing to negative growth. We suggest our model may yield similarly elucidating information for other hard‐to‐study species with extended parental care.

Tamara McGuire and Gina Himes Boor conceived of the project; Tamara McGuire developed the field methodologies and obtained the field funding and research permits; Tamara McGuire, Amber Stephens and John McClung collected and processed the photo‐ID data; Gina Himes Boor designed the modelling methodology, analysed the data and obtained the analytical funding; Amanda Warlick developed and performed all model validation simulations; Rebecca Taylor provided statistical consulting and error‐checking of the model structure; Sarah Converse provided feedback on model structure and validation. Gina Himes Boor led the writing of the manuscript, but all authors contributed critically to multiple drafts of the manuscript and gave final approval for publication.

This work was funded primarily by North Pacific Research Board (Project #1718), with additional funding from the Knik Tribe and USFWS Tribal Wildlife Grant (F21AS00354). We thank all previous CIBW‐PIP team members, S. St Savage at Axiom Data Science for ongoing software support for the CIBW‐PIP catalogue, all Native and local communities and individuals who have contributed beluga photographs and information, C. Garner (Conservation Department, Joint Base Elmendorf Richardson) and P. Wade (NMFS) for sharing data that allowed for additional sightings and right–left matches, and N. Kellar (NMFS) for genetic sexing of biopsied whales. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. We thank Dr. Tim Ragen for his thoughtful insights and helpful suggestions that significantly improved the manuscript.

None of the authors have any conflicts of interest related to this research.

The peer review history for this article is available at

Model and simulation code and the beluga data are available via GitHub: