Study system and data collection
Data were collected at the Tutakoke River brant colony (61°159′N, 165°37′W) on the Yukon-Kuskokwim Delta, AK, which historically represented about 20% of the breeding population (Sedinger et al. 1993). Brant are long-distance migrants that winter along the Pacific coast of North America from Alaska to the Pacific coast of Mexico (Reed et al. 1998). Brant from the Tutakoke River colony nest in coastal tundra within 2 km of the Bering Sea coast. We captured individual brant by herding them into corral traps during the adult remigial molt in mid-late July each summer (Sedinger et al. 1997). We assigned individuals to one of three age classes upon initial capture based on plumage characteristics: goslings (HY, approximately 1 month old); 1-year-olds (SY,13 months old) and adults (ASY, ≥25 months old). We placed U.S. Geological Survey steel bands and uniquely engraved plastic bands on captured individuals; plastic bands could be read at a distance of >100 m with spotting scopes, which facilitated detection in subsequent years. We determined gender by cloacal examination. Goslings in nests associated with marked adults received uniquely numbered webtags while still in the nest (Sedinger and Flint 1991), which allowed us to determine ages of goslings (±1 day) when captured in banding drives. We weighed and measured all web-tagged goslings when captured during banding drives. We calculated a cohort mean mass adjusted for gender and age (days).
We restricted the analysis to females because (1) most males disperse to other breeding locations (Lindberg et al. 1998), reducing the number of known-age individuals available for study; and (2) brant maintain long-term pair bonds, so fates of males and females were not independent. We encountered marked individuals during the aforementioned banding drives and during nesting. However, few 1-year-olds were observed and none have ever been confirmed as breeders (BRs), encounters of 1-year-olds were therefore removed and we modeled initial survival over the first 2 years of life. At nesting, we visited 50, randomly located 50-m radius plots every 4 days during egg laying and at least on alternate days during the hatch period. We read plastic bands when females were flushed from their nests. Additionally, we attempted to check all nesting females on the colony for plastic bands by flushing them from their nests, beginning shortly after the end of egg laying. We detected about 79% of breeding females that were present on the colony annually (Sedinger et al. 2001).
Analysis was based on female brant marked with individually coded tarsal tags and legbands at the Tutakoke River colony, AK from 1987 to 2009 (Sedinger et al. 1998). We examined individual heterogeneity of female brant based on CMR recruitment models (see e.g., Lebreton et al. 2003) incorporating individual heterogeneity in the various parameters of these models. The resulting model was a multievent model (Pradel 2005), which is a type of hidden Markov chain model (Choquet et al. 2009). We considered a suite of models both with and without individual heterogeneity and other sources of variation to estimate state (i.e., quality) of an individual and capture, apparent survival, and recruitment probability. We used an information-theoretic approach for model selection and to determine what sources of variation were supported (Burnham and Anderson 2002). When included, individual heterogeneity was modeled using a two-level mixture model with two hidden groups. This structure results in four states: 1 = BR group A, 2 = prebreeder (PB) group A, 3 = BR group B, 4 = PB group B. A shortcoming of this approach is that heterogeneity in capture probability is linked with heterogeneity in survival and recruitment, such that an individual in a specific survival or recruitment group has to be in the same group for detection probability. Separating these two types of heterogeneity would require four hidden groups or eight states. Such a model would be quite unstable and present severe identifiability problems and was not considered. The negative sampling correlations between parameters, however, will tend to exaggerate heterogeneity in the demographic parameters of interest, and we consider that issue in the 'Discussion'.
As we are only sampling at a breeding colony, individuals are observed as a BR (only after second year) or PB (in their hatch year) at initial capture and as a BR in any subsequent recapture. However, we do not know to which heterogeneity state (A or B) an individual belongs because this is a hidden state. Therefore, the proportion of individuals in heterogeneity groups A and B have to be estimated within each breeding category. These initial-state probabilities come as extra parameters in multievent models when compared to usual multistate CMR models, in which the state of an individual at time of marking is known. A model with year-specific variation in initial state and 25 occasions would require 75 parameters because the 4th state is estimated as 1−the sum of the probability of the other three states.
Recapture probability (pi) was the probability that an individual alive and in the Tutakoke River population in year i was reencountered in that year. We considered models with effects of year and heterogeneity on p, with an additive and interactive relationship between these factors. With p and other parameters, we considered models with linear trends (logit scale) in heterogeneity to examine whether heterogeneity changed through years because the size of the brant colony was declining during the study period, which may have changed the amount of heterogeneity through time. Apparent survival probability (фi), the probability that a brant alive and associated with the Tutakoke colony at year i survives and does not permanently emigrate between year i and i + 1, was modeled as a function of age, year, mass at capture (i.e., mean year cohort mass adjusted for age; Sedinger and Nicolai 2011), and heterogeneity. Juvenile and adult survival probabilities were modeled independently. Juvenile survival spans the 2 years from fledging to potential age of first breeding, since second year encountered were removed. We estimated survival for the first and second year of life assuming survival was equal for these two annual periods. Recruitment probability (aj), the probability that animal of age j starts to breed at that age (Pradel and Lebreton 1999), was modeled as a function of age from 2-years old up to 6-years old with and without the effects of heterogeneity.
We assessed fit of the model to the data using tests conducted in program U-CARE following Crespin et al. (2006) for recruitment models and Fletcher et al. (2012) for models with heterogeneity. By definition, heterogeneity in capture probability corresponds to a mix of individuals with low and high capturability. Compared with homogeneity, the data set will present an excess of runs of 0 and of runs of 1. This situation results in the simultaneous presence of various degrees of transience and trap happiness, respectively, detected by goodness-of-fit (GOF) components TEST3.SR and TEST2.CT, as shown by Fletcher et al. (2012). We expect all other components to be somewhat sensitive to heterogeneity in all parameters.
All models were run with random initial values repeated 10 times to protect against local minima. In all cases, the minimal value of the deviance was that obtained for the largest number of repeats, so we are confident that the deviances used in QAIC calculation corresponded to the global minimum. For a few of the most complex models, the deviance was obtained after 200 iterations and full convergence was not yet attained, implying a slight overestimation not bearing any consequence on QAIC-based model selection. This was always due to the slow optimization of boundary estimates (i.e., a probability estimated to be 1.0, has to converge in logit to “infinity,” represented by the value 15 in E-SURGE), and this convergence requires many iterations, without substantial changes to the deviance. Our checks with a few of these models showed that the difference in deviance was always <0.100, far from the difference in QAIC with the preferred models.
Using estimates from the best multievent models and published estimates of fecundity (Sedinger et al. 1998; Nicolai and Sedinger 2012), we parameterized a matrix model with two population segments (A and B) in a gosling, prebreeding, or breeding state (six states total) using a postbirth pulse structure (Caswell 2001). Matrix models were run in ULM (Legendre and Clobert 1995) and MATLAB® (Mathworks Inc., Natick, MA). We analyzed the matrix to determine the reproductive value of various states and changes in growth rate (λ) under harvest scenarios that included different proportions of brant from the six states. We calculated reproductive values in the presence of heterogeneity with different levels of inheritance of the states where differences in “heritability” represented individuals producing different proportion of goslings in their quality state. We use the term inheritance rather than heritability, to acknowledge that both genetic and environmental (e.g., timing of reproduction, brood-rearing area) factors may affect quality of offspring. We also considered how reproductive values and λ changed with differences in clutch size among quality classes.
Under the assumptions that females produced 80% offspring of the same class, and in addition to heterogeneity in survival and recruitment, we evaluated the effects of harvest with different proportions of quality classes in the harvest, different clutch sizes between classes, and different proportions of goslings, PBs, or BRs in the harvest. Scenarios for harvest rates were developed based on estimates from Sedinger et al. (2007). Harvest effects were evaluated by comparing resulting λ to the per capita growth rate expected under a uniform harvest proportion (h) irrespective of reproductive value:
where λ(0) is the growth rate in the absence of harvest and h is the overall proportion of the population in the harvest. When reproductive values and proportion of quality classes in the harvest are considered an approximation of λ is obtained based on sensitivity analysis as:
where b is the ratio of harvest proportion weighted by reproductive value of a quality class (hrv) and harvest proportion irrespective of reproductive value (h). Harvest proportion irrespective of reproductive value is:
and harvest proportion weighted by reproductive value is:
where wi is proportion of population in age/quality class i at stable structure and vi is the reproductive value for the ith age/quality class. The effects of harvest on λ when considering reproductive value will be smaller than the effects of uniform harvest irrespective of reproductive value when b < 1.0 (i.e., compensation occurs because harvest is proportionally higher in the population segments with lower reproductive value) and b can therefore be interpreted as the relative strength of harvest effects with stronger effects and less compensation as b approaches 1.0 (i.e., if b = 0 then harvest would hypothetically have no effect and if b = 1.0 no compensation would occur). We also examined the effects of harvest on overall population-level survival for the harvest scenario with the maximum effects of heterogeneity because survival is the parameter often examined to evaluate compensation.