## Abstract

To aid managers in assessing status of Pacific black brant *Branta bernicla nigricans* (hereafter brant), I examined pre-existing long-term data series from summer, fall staging, and wintering areas to infer overall population processes and assessed the utility of the various data sources. Variation in demographic parameters measured in subarctic and Arctic locations suggests some form of metapopulation structure likely exists for brant. I used serial autocorrelation coefficients to assess the ability of various indices to track population processes. Based on this approach, the Lincoln–Petersen estimator and the fall aerial survey estimate partitioned using age ratios of staging brant at Izembek Lagoon, Alaska, appear to be the best indicators. However, these two indexes show different trends for the overall brant population. The Lincoln–Petersen estimates showed biologically implausible changes in size among sequential years, whereas the fall Izembek index did not. Annual estimates of survival and productivity fit the patterns of annual variation in the fall Izembek index better than the Lincoln–Petersen estimates. I conclude that the fall age–partitioned Izembek Lagoon index appears to be the best for tracking population processes in brant.

## Introduction

As a migratory bird species that breeds at high latitudes, Pacific black brant *Branta bernicla nigricans* (hereafter brant) has a unique annual cycle (Lewis et al. 2020). These birds breed in large colonies, small colonies, and as individual dispersed pairs. Significant numbers breed in the subarctic on the Yukon-Kuskokwim Delta in western Alaska, whereas the remainder breed in the Arctic from Siberia through the Canadian Arctic islands (Lewis et al. 2020). First breeding does not occur until 2 y of age, and not all adults nest every year (Sedinger et al. 2001b). Subadults and nonbreeding adults may segregate to specific molting areas in summer, although some molting birds also occur on breeding areas (Sedinger et al. 1993). The entire Pacific population is thought to stage at Izembek Lagoon along the Alaska Peninsula in fall. In winter, brant are found along the west coast from Izembek Lagoon, Alaska, to Baja California, Mexico.

Adult female brant show high fidelity to breeding locations, and juvenile females also tend to recruit to their natal location; these two characteristics imply that a metapopulation structure likely exists at some scale (Lindberg et al. 1998). Sedinger et al. (2019) defined two subpopulations as breeding on the Yukon-Kuskokwim Delta and in the Arctic. Although this geographic partitioning is logical based on existing data, it may not fully describe the metapopulation process for brant (Lindberg et al. 1998). Given the very large range over which brant nest, it is almost certain that demographic parameters vary in some geographic manner. In fact, recent studies show greater adult and juvenile survival for brant nesting in Arctic Alaska than for those nesting on the Yukon-Kuskokwim Delta (Leach et al. 2017). Furthermore, detailed studies on the Yukon-Kuskokwim Delta show localized variation in gosling size at 30 d posthatch among breeding colonies (Nicolai et al. 2008; Fondell et al. 2011). Variation in gosling size at 30 d of age shows a link to demographic effects, such as first-year survival and future reproductive output (Sedinger et al. 1995, 1998, 2001b; Sedinger and Chelgren 2007). Thus, it seems clear that some form of metapopulation structure exists for brant, but the scope and scale of this structure are unknown. However, the current Pacific Flyway Council management plan for brant only has objectives for the overall population and does not discuss potential subpopulations (Pacific Flyway Council 2018).

## Using Counts to Infer Population Status and Trends

Multiple surveys, conducted at various points of the annual cycle of brant, could aid in inferring population dynamics. Annual surveys generating brant population indices include the midwinter survey and fall staging and age ratio surveys in the Izembek Lagoon area (Table 1). Recently, Sedinger et al. (2019) developed an estimate of historic total population size based on band recoveries by using the Lincoln–Petersen model. This model estimates adult population size, and it applies at the time of summer banding operations (i.e., July). There is some uncertainty about the overall status of the population due to disparities in trends among these indices (Pacific Flyway Council 2018). The midwinter and fall Izembek surveys provide annual indices of total population size, and these two surveys show a similar long-term mean across the period of years for which data are available (Stehn et al. 2011). However, these ocular surveys suffer from unknown detection probabilities (i.e., missed or double-counted birds) as well as potential estimation bias. Under the assumption of constant detection probability and bias, ocular survey indices can aid in inference of the status and trend of the population.

It is challenging to assess the validity of indices relative to the true, but unknown, population size. However, the degree to which an actual population varies from year to year is functionally constrained by the biology of the species (Koons et al. 2014). That is, there is constraint in the true annual variation in population size, in the absence of immigration and emigration, by the potential variation in annual mortality and recruitment. Thus, the range of variation in annual change for a given index should reasonably correspond to the potential range based on demographics. Furthermore, because demographic processes result in inherent time lags, survey data without immigration or emigration should be serially autocorrelated. Although there may be trends through time, counts from sequential years should be more similar than counts taken a number of years apart. Logically, random measurement error should tend to reduce serial autocorrelation. All else being equal, the degree of random measurement error relative to true annual variation in population size will be negatively correlated with serial autocorrelation, where larger sampling error leads to lower autocorrelation. Thus, although autocorrelation can cloud the interpretation of statistical tests, serial autocorrelation can serve as an assessment of the validity of a survey index relative to population processes. Finally, annual changes in population size should relate to annual fluctuations in demographic parameters such that annual decreases in numbers correspond with lower values of annual survival and productivity and vice versa.

It was my goal to assess the utility of several indices to track brant population dynamics. The current brant management plan uses the midwinter survey as the key management index, with population objectives and management actions based on this index (Pacific Flyway Council 2018). The management plan establishes a series of four categories of population size and defines the harvest regulations associated with each category. Thus, current estimates from the midwinter survey of brant are of use in determining annual harvest regulations. If the midwinter survey is not accurately measuring the population status, there is a high likelihood for the application of incorrect management decisions. The specific objective was to compare and contrast the various indices relative to which might be the most informative from a management perspective.

## Methods

I compiled data on the distribution and abundance of brant across the known summer range including Alaska, Canada, and Russia (Table 2). I used this information to determine the relative size of the subpopulations that occur in Arctic and subarctic areas. Because data on status and trends as well as select life-history parameters exist for Yukon-Kuskokwim Delta and Arctic Alaska subpopulations, I used this metapopulation structure for analyses (Sedinger et al. 2019). Following the brant management plan, I use the term population to refer to the entire Pacific Flyway; I use subpopulation to refer to the Arctic and subarctic components of the larger group (Pacific Flyway Council 2018).

*i*) and assumed an average fall age ratio (i.e., 23.2% juveniles). Partitioning this total number by using the age ratio, my initial values started as 115,145 adults and 34,855 juveniles. I then estimated the next years (

*i*+ 1) fall adult population size by using the year-specific rates for survival of each age class (Figure 1). I then estimated the total fall count in year

*i*+ 1 as follows: where age ratio is proportion of juveniles. Next, I estimated the number of fall juveniles in year

*i*+ 1 by subtracting adult population size

_{i}_{+ 1}from the total fall count

_{i}_{+ 1}. I repeated this process through all years from 1990 to 2013. I included variation in survival and productivity by using a beta distribution and the standard errors provided with the individual estimates. Given that Leach et al. (2017) found differences in survival between Arctic and subarctic birds, I estimated the overall population level survival rate as the weighted average of the estimates, and I used my estimate of distribution from the literature as the weights. In cases where estimates were not available (i.e., juvenile survival in the Arctic from 2001 to 2010), I used the single available estimate. I regressed these population size estimates against year to remove any linear trend and assessed the autocorrelation in the residuals. Theoretically, this demonstrates the potential annual variation in a brant population based on an actual time series of survival and productivity data, including the uncertainty in the survival and productivity estimates, in the absence of any measurement error on total population size. I then added varying degrees of random, normally distributed error to the predicted population estimates by using a range of variation from 0.01 to 0.25 (coefficient of variation) to demonstrate the potential effect of measurement error on the realized autocorrelation coefficient. This can be thought of as the autocorrelation coefficient one would find using estimates from surveys or mark–recapture with varying degrees of error. I detrended these estimates by using linear regression and calculated the residuals. I used Excel to estimate the 1-y lagged autocorrelation coefficient in these residuals, and I used 1,000 replicates in a Monte Carlo simulation to provide 95% confidence intervals to the autocorrelation estimates. I estimated the annual lambda of this hypothetical population follows:

I estimated the minimum and maximum annual lambdas for each series of simulations, including those with additional error.

I used existing data sets that were publicly available (Table 1; Data S1, *Supplemental Material*). Previous publications, unpublished annual U.S. Fish and Wildlife Service (USFWS) reports, or online data repositories described survey methodology and associated data. Because of my interest in comparing among surveys, I was unable to use all years of data from all surveys and restricted analyses to subsets of years in which data sets overlapped (Table 1). Below, I briefly describe these data with the goal of providing details that were relevant to my analyses. Throughout this article, I refer to the data generated from ocular surveys as counts because these are the data actually recorded by the observers. I then refer to these counts as an index by assuming constant detection and estimation bias and averaging among replicates for multiple counts performed within a year.

### Sources of data on population size

#### Midwinter survey.

The midwinter survey index is currently used as the management indicator, and the prescriptive harvest strategy is based on the annual value of this index (Pacific Flyway Council 2018). The midwinter survey of brant began in 1960 (Olson 2017), with several changes in methodology over time. Before 1986, survey coverage excluded Alaska and British Columbia, Canada, because limited data indicated few birds overwintering in these areas. In most years, the survey was a one replicate ocular count by a single crew flying the entire coastline from Washington to Mexico. No survey of the Mexico portion occurred in 2009. Since 2011, ground and boat crews survey the Mexico portion, whereas the U.S. portion counts occur by using ground crews and with some sections flown by local pilots in each state. This shift in methodology may influence the violation of the assumptions of constant detection probability and no double counting while adding the uncertainly of multiple observers. Since 1986, there has been a substantial increase in the number of brant overwintering in Alaska and a decline in number counted in Mexico (Ward et al. 2005, 2009).

#### Izembek survey.

The fall Izembek Lagoon survey is an ocular count flown at low level by using a semiconsistent flight track and counting flocks both on the water and in the air (Wilson 2017). The timing of the survey and number of replicates are variable among years. A variety of aircraft and observers are in use on this survey, both within and among years. Timing of the survey replicates is variable, with some starting in mid-September and others as late as early November. The majority of surveys occur in October, which corresponds with the time when the vast majority of brant from all subpopulations are likely to be present.

#### Izembek Lagoon age ratios.

An estimate of the proportion of juveniles present at Izembek Lagoon has occured every fall since 1963. Observers visually scan the flocks using spotting scopes and record the numbers of adults and juveniles. Ward et al. (2018) demonstrated that the age ratio varied with date and location within the lagoon and as different components of the overall population arrived at Izembek Lagoon in the fall. Overall, Ward et al. (2018) and Amundson et al. (2020) found evidence of a long-term decline in the fall age ratio.

#### Lincoln–Petersen estimators.

Sedinger et al. (2019) used band recoveries in combination with an estimate of the number of banded birds alive at any given time and several estimators of harvest to estimate the overall number of adult brant. This approach requires several assumptions when applied to brant: the marked and recovered sample is representative, estimates of total harvest are accurate, and appropriate estimates of reporting rates and estimates of annual survival for adults and juveniles represent the entire population. Sedinger et al. (2019) concluded that the brant population declined from an average of 765,863 birds (1999–2002) to an average of 157,940 birds (2012–2015) by using flyway scale Harvest Information Program estimates of total harvest. Using harvest estimates and band recoveries from a single hunter check station in Mexico, Sedinger et al. (2019) concluded that the population declined from 227,448 (1999–2002) to 180,118 (2012–2013) birds. The Lincoln–Petersen model applies only to the adult population size at the time of banding and for brant that would be in July.

### Analyses

The winter survey index comes from a single replicate; thus, there was no estimate of variance for the annual data points. The Izembek survey had some level of replication within most years, allowing estimation of variance for the mean annual index. In years with only a single survey, I used the average variance from all other years. The annual Lincoln–Petersen estimates from Sedinger et al. (2019) are combined estimates of annual population size based on a weighted average of their two approaches, presented as posterior distributions. Sedinger et al. (2019) provided the median from these distributions and the associated 95% coverage, which I used to calculate an approximate standard deviation.

*i*) for population size from a normal distribution as population size

*= (population estimate*

_{i}*+ ε*

_{i}*) by using the annual population index and error and then for each iteration of the estimates I used regression to estimate the residuals and calculated the autocorrelation of these residuals. I report the mean autocorrelation coefficient and confidence intervals from the simulations. To assess trends, I report the mean slope and associated 95% confidence intervals from these Monte Carlo simulations. I also derived a simple correlation between the median Lincoln–Petersen estimates, the midwinter index, and Izembek adults index. I calculated the annual lambda for all indices as follows:*

_{i}This summed difference across all years from 1997 to 2013 is functionally similar to a chi-square statistic in that all differences between observed and predicted are made positive and scaled to predicted population size to make them comparable, even when size differed considerably. I report the average deviance as this sum divided by sample size (*n* = 17 y). This statistic describes the average proportional deviance of each index (i.e., Izembek adults, Lincoln–Petersen) from the model-predicted population size.

Similar to Sedinger et al. (2019), I assumed the proportional representation for the Arctic and subarctic subpopulations was 0.7 and 0.3, respectively (Table 2) and weighted the area-specific annual survival rates from Leach et al. (2017) accordingly. Because Leach et al. (2017) did not provide estimates of first-year survival from the Arctic for 2001–2010, I used unweighted subarctic rate for those years. For comparison, I repeated this approach using the average of annual estimates from Leach et al. (2017) as constant rates of adult and juvenile survival across years. This analysis assumes that no adult mortality occurs between banding and fall staging at Izembek in Lincoln–Petersen estimates. This analysis compared annual fluctuations in population indices with annual variation in estimated survival and productivity rates from localized study areas, with the logic being changes in population size from one year to the next should be associated with survival and recruitment rates. It was not the intent for these analyses to model trends through time, but instead only to compare 1-y changes in population indices with annual patterns in survival and productivity estimates. This model builds on the logic that the best predicter for population size in year *i* + 1 is the size in year *i* adjusted for that year's annual survival and recruitment.

## Results

The model of the hypothetical population demonstrates that under published estimates of basic demographic rates (i.e., survival and recruitment), annual variation in the adult brant numbers would be expected to have an autocorrelation coefficient of 0.7 (Figure 2). Counting or other forms of estimation error would tend to reduce the realized autocorrelation coefficient. Thus, larger autocorrelation coefficients will tend to be found in survey indices that contain less random sampling error (Figure 2). I also used these time-series projections to estimate the annual lambda as the population size in year *i* + 1/population size in year *i*. For this hypothetical population across this sample of years, the minimum annual lambda was 0.82 and maximum lambda was 1.25. As estimation error increased in these simulations, the range of annual lambdas increased (Figure 2).

The midwinter index showed no change in numbers through time (Figure 3). There was little effect of controlling for trend in the autocorrelation coefficients (Table 3). There was little correlation between the fall Izembek index and the subsequent midwinter index taken 3–4 mo later (correlation coefficient = 0.09). The annual lambda from the midwinter index ranged from 0.62 to 1.33.

The Izembek total count index showed an increase through time (Figure 4), and this increase showed no relation to timing of surveys within years (Text S1, *Supplemental Material* ). Because of missing data in 1978, I only assessed autocorrelation from 1979 to 2017. Using replicate counts within years, the average standard deviation of the annual mean count was 20,890. Removal of the linear trend reduced the autocorrelation coefficient by 60% (Table 3). Elimination of the autocorrelation coefficient occurred by including counting error and removing the linear trend (Table 3; 95% confidence interval = −0.30 to 0.28). Applying linear regression to the annual counts showed an increase through time (slope = 737 birds/y; 95% confidence interval = 206–1,306). The annual lambda in the total index ranged from 0.69 to 1.44.

The index of adult brant estimated by partitioning the Izembek index using the age ratio showed an increase through time (Figure 5). Removing the linear trend reduced the autocorrelation coefficient by 25% (Table 3). Incorporating the error in the fall index and age ratio, the average standard deviation of the mean adult count was 22,597. Incorporating the error in fall index resulted in a large decrease in the autocorrelation coefficient (Table 3; 95% confidence interval = −0.1 to 0.43). Applying linear regression to the index showed an increase through time (slope = 940 birds/y; 95% confidence interval = 504–1,368). The lambda from the annual estimates ranged from 0.70 to 1.44.

The index of juvenile brant estimated by partitioning the fall Izembek index by using the age ratio showed a decline, and estimates for sequential years were more dissimilar than expected, with an autocorrelation coefficient of −0.30 (95% confidence interval = −0.45 to −0.14; Figure 6). Incorporating the error in the fall index and age ratio, the average standard deviation of the mean juvenile count was 5,551. Applying linear regression to the index showed a gradual decrease through time (slope = −197 birds/y; 95% confidence interval = −38 to −356).

The median Lincoln–Petersen estimates from Sedinger et al. (2019) showed a strong negative trend. Removing the linear trend reduced the autocorrelation by 52% (Table 3). Incorporating the error in the estimates had little additional effect on reducing the autocorrelation coefficient (Table 3; 95% confidence interval = 0.09–0.31). Conceptually, I would expect a strong positive correlation between the Lincoln–Petersen estimates and the adult fall Izembek index, but the observed correlation was negative and weak (*r* = −0.17; Figure 7). The annual lambda from the Lincoln–Petersen estimates ranged from 0.36 to 3.90.

I used a simple model approach to assess the correspondence between annual survival estimates and annual variation in the Izembek adult index and Lincoln–Petersen estimates. The average proportional deviance [i.e., (|observed – expected|)/expected] derived from the fall Izembek adult index was 19%, compared with 52% for the Lincoln–Petersen estimates. However, fit of the constant survival rates was better than annual survival rates in both data sets, because the average proportional deviance derived from the constant survival estimates was 15% for Izembek adult index and 47% for Lincoln–Petersen estimates. Overall, the age-adjusted aerial survey data aligned better with survival estimates than with Lincoln–Petersen estimates; however, neither set of estimates supported the pattern of variation in annual survival provided by Leach et al. (2017), as a constant survival rate fit best.

## Discussion

### Interpretation of population processes with data

The hypothetical model using published survival and productivity estimates for a sequence of years clearly demonstrates that in the absence of counting error, a brant population would be expected to have high serial autocorrelation in numbers through time. This is primarily a result of that fact that brant are a long-lived species with delayed maturation and relatively low reproductive output. Because reproductive success is highly variable among years and female age dependent, there was not a strong correlation between the number of young produced and adult population size. The theoretical maximum autocorrelation from my model only applies to counts of adults. Thus, for counts that include both adults and juveniles, the juveniles become functional error and reduce the autocorrelation. This source of error applies to both the Izembek total index and midwinter index and likely explains why they tend to have lower autocorrelation coefficients. Using just the point estimates, the Izembek total index and midwinter index have the lowest autocorrelation coefficients (i.e., < 0.25), whereas the Izembek adult index and Lincoln–Petersen estimates performed considerably better based on serial autocorrelation coefficients (i.e., > 0.42). Thus, in terms of tracking and understanding population size of adults, the Izembek adult index and Lincoln–Petersen estimates appear to be most useful.

As noted previously, counting error functionally reduces autocorrelation. Trends in data through time also inflate serial autocorrelation coefficients. When I removed trends and incorporated associated error in estimates, serial autocorrelations for both age-adjusted Izembek counts and Lincoln–Petersen estimates declined considerably and were similar. Therefore, I considered how well each of these indices fit with available demographic data through time. Using published data on survival and productivity for brant, I modeled the expected annual variation in a hypothetical brant population (i.e., lambda range = 0.82–1.25). If I accept this as a reasonable approximation of the potential true annual change in population size for adult brant, the Lincoln–Petersen estimates showed biologically implausible annual fluctuations in size (i.e., lambda range = 0.36–3.90; Sedinger et al. 2019), whereas the Izembek adult index was much closer to the predicted range (i.e., lambda range = 0.70–1.44). The range of annual lambda estimates from the Lincoln–Petersen estimates falls well outside the range possible based on published survival and recruitment data (Leach et al. 2017; Ward et al. 2018). Thus, the Lincoln–Petersen approach appears to be overestimating the annual variation in population size.

Specifically, interannual changes in population size should be related to specific annual survival and productivity values. Leach et al. (2017) and Ward et al. (2018) describe the variation and trends in survival and productivity for brant. Sedinger et al. (2019) argued that the long-term decline in numbers they describe fit the general patterns in these survival and recruitment data. My approach takes this one step further and assesses whether annual variation in population indices reflects annual patterns in these demographic data. The fall Izembek adult index fit the patterns in these demographic data much better than the Lincoln–Petersen estimates. This difference in fit is partially a function of the relative greater variation in the Lincoln–Petersen estimates, as demonstrated by the extreme range of annual lambda estimates. That is, the annual changes in population size from the Lincoln–Petersen approach are more likely to exceed what is possible given available survival and recruitment data. I conclude that the Izembek adult index provides the best index for inferring overall population size and trend because it fits better with available demographic data. Because of the sampling differences not all indices can be compared at all levels, but the Izembek adult index had larger autocorrelation in the point estimates than the midwinter index. The Izembek adult index and Lincoln–Petersen estimates had similar autocorrelation coefficients, but the Izembek index fit better based on the range of variation in the annual lambda estimates and patterns in published survival and productivity data. The age-partitioned Izembek indices allow direct assessment of population size as well as annual recruitment.

### Assumptions and issues with survey counts

Both midwinter and Izembek fall surveys are considered to be a census because they attempt to count the entire population in the survey area as opposed to some form of statistical sampling. Because of this design, individual survey replicates have no measure of uncertainty. This approach assumes that all birds from the population of interest are present in the survey area, all birds are observed and accurately counted, and no birds are double counted. The extent of possible violation of these assumptions is unknown, but if they are violated, there will be bias in the population estimates. If the bias is consistent across surveys (i.e., the proportion missed or double counted is constant), the estimated population size may be biased, but it can still serve as an index to population size and should correctly reflect population trends. If violation of these assumptions is inconsistent, then the bias will vary among replicates and years, which would functionally appear as error among surveys. As my hypothetical model has shown, as this source of error increases, the serial autocorrelation in the counts should decrease and the range of annual lambdas should increase (Figure 1). The fact that the annual point estimates from the Izembek adult index had relatively high autocorrelation and the range of annual lambda estimates was close to the predicted range demonstrates that violation of these assumptions does not vary substantially among survey replicates.

The Izembek total index includes all birds (i.e., adults and juveniles) counted in the lagoon system. This index has high annual variation and essentially no autocorrelation, which would lead to the conclusion that these data have substantial error (Figure 2). This pattern demonstrates that observers are not generating estimates that are spuriously similar to previous years as a result of estimation bias. At the time of the survey, the age ratio for that year has not been estimated (Ward et al. 2018). I found autocorrelation in these data only with partitioning of the total count by age, separating estimates of the adult and juvenile portions. This pattern strongly supports the conclusion that ocular counts are reasonably tracking population processes. If the ocular counts were functionally random numbers due to the combination of estimation error, nonclosure of the population, and inconsistent detection probability, adjusting these random numbers by the age ratio would not result in autocorrelated estimates of the number of adults. Although the timing of this survey varies somewhat among years, this variation would not tend to cause the observed population increase (Text S1, *Supplemental Material*). The relatively high serial autocorrelation of the adult index supports the conclusion that the staging population of brant is not changing substantially within seasons, between survey replicates. Such changes in population size would appear as error, which would reduce the serial autocorrelation.

### Assumptions and issues with Lincoln–Petersen estimates

The range of annual lambda estimates from the Lincoln–Petersen approach was biologically implausible. Even when using the upper and lower limits of the 95% plausible intervals (i.e., using the upper limit of the low estimate and the lower limit of the high estimate) to minimize estimates of annual lambda, several annual changes in population size based on the Lincoln–Petersen estimates remain biologically implausible. As such, the confidence in the individual Lincoln–Petersen estimates would seem to be overestimated and the approach likely includes some additional annual sampling error that is underestimated. Use of the Lincoln–Petersen estimates as the basis for management decisions would have resulted in selection of more restrictive regulation packages between 2009 and 2012 compared with either the midwinter or Izembek adult indices (Pacific Flyway Council 2018). The biologically implausibly low Lincoln–Petersen estimates in 2009 and 2011 would drive this outcome (Figure 7; Sedinger et al. 2019). This situation would force managers to defend management decisions based on estimates that do not fit with our understanding of brant survival and productivity.

The Lincoln–Petersen estimates suggest a declining trend, whereas the fall Izembek index showed an increase. The Lincoln–Petersen estimates were considerably larger than the fall count index before 2004, but estimates have been similar in more recent years (i.e., since 2004; Figure 4). The discrepancy between these indexes could be explained by one of these approaches being biased, but if that is the case, the source of bias appears to have diminished over time. Lincoln–Petersen estimates are based on a small number of annual band recoveries (average of 87 band recoveries/y) and a relatively low level of harvest. This makes the Lincoln–Petersen population estimates highly sensitive to the other parameters in the model, such as bias in harvest estimates and band reporting rates. Sedinger et al. (2019) assumed a constant band reporting rate and constant bias in the harvest estimates from the Harvest Information Program, but if these parameters have changed through time, this would force a trend into the Lincoln–Petersen population estimates. Arnold et al. (2019) concluded that band reporting rate has been gradually increasing through time for multiple species of ducks and geese. If true band reporting rate has been increasing, but modeled as constant, the Lincoln–Petersen model would tend to indicate a negative or reduced population trend. Potential trends in band and harvest reporting for brant are unknown.

The basic assumptions of the Lincoln–Petersen approach as applied to brant include that 1) the marked sample is representative sample of the total population; 2) the marked birds distribute randomly through the entire population; and 3) harvest represents a random, representative sample of the entire population. All of these assumptions may have been violated to some extent in the estimation of population size of brant (Sedinger et al. 2019). Subtracting the breeding and molting brant counted annually in Alaska (Table 2) from either the Izembek index or Sedinger et al. (2019) population estimates (Figure 4) implies that greater than 33% of the brant population occurs in unknown areas of Arctic Russia or Canada during summer where they are unmonitored and unmarked (Sedinger et al. 1993). Assuming these birds are adequately represented by the existing bandings and recoveries in a Lincoln–Petersen model may not be valid. Brant from different summer areas may have different migration schedules and be somewhat segregated at Izembek Lagoon in the fall and during winter, as is the case for brant breeding in the western high Arctic (Boyd et al. 2013). The number of brant remaining in Alaska and wintering at Izembek has increased from less than 10,000 to the point where greater than 20% of the population may now winter in Alaska (Wilson 2017). It is unknown whether the birds that winter in Alaska are from a specific subpopulation, but clearly these birds are not subjected to harvest along the southern Pacific Flyway where approximately 50% of harvest has occurred in recent years (Olson 2017). Thus, it seems clear that the basic assumptions of the Lincoln–Petersen approach may have been violated to some extent and that there may be trends in the degree to which these assumptions have been violated, which would invoke trends in the population index generated from a Lincoln–Petersen model. Further work is likely required to assess potential bias in Lincoln–Petersen estimates for Pacific black brant and determine whether the variance is underestimated.

### Opposing trends in estimates of adults and juveniles

The Izembek adult index shows a long-term increasing trend, whereas the juvenile index and age ratio show a slightly declining trend (Ward et al. 2018). Interestingly, the estimate of fall juveniles shows a negative serial autocorrelation coefficient, which implies a cyclic function in terms of overall productivity. This is likely a function of the delayed age of first breeding such that years of high productivity result in the following year having a high number of immature nonbreeding birds, which lowers the age ratio. Leach et al. (2017) questioned the validity of the Izembek and midwinter indices given that they showed increases when survival estimates and fall age ratios declined. Furthermore, the number of brant counted in the five main nesting colonies on the Yukon-Kuskokwim Delta are in a long-term decline (Wilson 2018). However, brant nesting populations in Arctic Alaska and likely Russia are on the increase (Pacific Flyway Council 2018; Amundson et al. 2019). Hupp et al. (2017) and Sedinger et al. (2001a) demonstrated that brant goslings on the Arctic Coastal Plain were considerably larger at 32 d of age than those from the Yukon-Kuskokwim Delta. Associated with this pattern, Leach et al. (2017) reported that first-year survival was twice as high for juveniles from the Arctic Coastal Plain compared with those of the Yukon-Kuskokwim Delta. Ward et al. (2004) showed that the majority of first-year mortality occurs before arrival at Izembek Lagoon. Given these patterns within the subpopulations, it follows that the proportional representation of birds from the Yukon-Kuskokwim Delta in the Izembek juvenile index is declining, whereas the proportional representation of those from the Arctic is increasing. Therefore, on average, the body mass of fall juveniles should also be increasing as the population becomes increasingly represented by the larger Arctic-derived birds (Hupp et al. 2017). If these larger juveniles have greater post-Izembek survival, this would effectively increase the recruitment rate of fall-counted juveniles to the breeding population. Thus, the fall age ratio may be declining, but this may not equate to a similar decline in recruitment rate. Understanding the proportional representation from each of the subpopulations, seasonal partitioning of juvenile survival, and how these may have changed through time is required to assess the validity of this hypothesis.

### Interpretation of demographic parameters

There are detailed studies of brant demography for a limited number of sites, primarily the Tutakoke River colony on the Yukon-Kuskokwim Delta and Colville River Delta on the Arctic Coastal Plain. In some cases, for some years, very detailed information regarding aspects of life history exists for birds at these locations. It is unknown to what extent these site-specific results can be assumed to fit larger population segments. When I used the estimates of survival to explain the fluctuations in the Izembek adult index and Lincoln–Petersen estimates, the specific annual estimates fit more poorly than a constant estimate of survival. It is not clear whether this indicates that annual indices to population size are not really tracking population processes or whether the annual survival estimates from localized study areas do not apply at the population level. Geographic variation in demographic rates may explain why survival estimates from a specific study site do not necessarily apply to a larger population. Fondell et al. (2011) looked at gosling growth rates from several colonies on the Yukon-Kuskokwim Delta and noted substantial differences among colonies. This would imply associated geographic variation in juvenile survival and recruitment among Yukon-Kuskokwim Delta colonies (Sedinger et al. 1995). Furthermore, juvenile survival varies substantially between the Yukon-Kuskokwim Delta and Arctic Coastal Plain study areas (Leach et al. 2017). Thus, data would imply that juvenile survival rates measured at a single colony would not be applicable to the larger population. Similarly, Flint et al. (2016) reported spectacled eider *Somateria fischeri* demographics at two sites on the Yukon-Kuskokwim Delta and found substantial differences in survival and productivity estimates among sites and data from neither site seemed to fit the larger Yukon-Kuskokwim Delta population. Population models derived from site-specific data on the Yukon-Kuskokwim Delta also failed to fit population indices for northern pintail *Anas acuta*, greater scaup *Aythya marila*, and emperor goose *Anser canagicus* (Schmutz et al. 1997; Flint et al. 1998, 2006). Even within a given study site, life-history parameter estimates vary widely based on analytical approach. Both Sedinger and Nicolai (2011) and Leach et al. (2017) estimated juvenile annual survival for the same study area and subset of years (i.e., 1990–2006). Sedinger and Nicolai (2011) used only band recoveries, and Leach et al. (2017) used both recoveries and resightings. Although both analyses show similar patterns, annual estimates of first-year survival differed substantially. Thus, it appears that demographic information collected on localized study areas is useful for understanding patterns of variation and local population processes; however, extrapolation of local site-derived parameter estimates to the larger population levels should be done with caution.

## Management Implications

Given that the Izembek adult index provided reasonable estimates of autocorrelation and better fit with annual demographic rates than other indices, it seems to be most useful for tracking population processes and thereby informing management decisions. This requires continuation of the fall aerial survey and age-ratio sampling (Amundson et al. 2020). The aerial survey could be improved by developing techniques to reduce intrayear variation in replicate counts. Additional information on subpopulation representation and seasonal timing of juvenile mortality would further enhance interpretation of these annual survey data. The Lincoln–Petersen approach seems to generate some estimates that are biologically unrealistic, which may lead to incorrect management decisions. Compared with midwinter or Izembek indices, use of the Lincoln–Petersen estimates would have resulted in more liberal harvest regulations before 2005 and restrictive regulations or closure between 2009 and 2012. Both the Izembek survey estimate of adults and the Lincoln–Petersen estimate require similar assumptions regarding population sampling and representation, but the Lincoln–Petersen method requires additional analytical assumptions for model parameters. The fall Izembek survey and associated age-ratio estimation may be more feasible than maintaining geographically extensive long-term banding and mark–resighting or mark–recapture programs required to generate Lincoln–Petersen estimates.

## Supplemental Material

Please note: The *Journal of Fish and Wildlife Management* is not responsible for the content of functionality of any supplemental material. Queries should be directed to the corresponding author.

**Text S1.** Analysis of the effect of fall ocular aerial survey timing on Izembek Lagoon total index of Pacific black brant *Branta bernicla nigricans*. This analysis considers whether the observed increase in the Izembek index of the total numbers of brant across years (from 1976 to 2017) is spurious, resulting from changes in aerial survey timing. Results indicate that the minor trend of earlier aerial survey timing should lead to a decrease in counts of black brant using the lagoon. Therefore, there is no relationship between the observed increase and survey timing.

Available: https://doi.org/10.3996/JFWM-21-088.S1 (14 KB DOCX)

**Data S1.** Annual data points (and associated error, if available) for Lincoln–Petersen estimates of the adult population of Pacific black brant *Branta bernicla nigricans* from 1992 to 2017; midwinter counts of brant from Izembek Lagoon, Alaska, through Baja California, Mexico, from 1976 to 2017; and replicate counts of fall Izembek Lagoon, Alaska, staging brant from 1976 to 2017. Also included are derived estimates of brant Izembek adult and juvenile indices.

Available: https://doi.org/10.3996/JFWM-21-088.S2 (33 KB XLSX)

**Reference S1.** Olson SM. 2017. Pacific Flyway Data Book, 2017. Vancouver, Washington: U.S. Department of Interior, U.S. Fish and Wildlife Service, Division of Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S3 (2.598 MB PDF)

**Reference S2.**Pacific Flyway Council. 2002. Pacific Flyway management plan for Pacific brant. Pacific Flyway Study Committee, care of U.S. Fish and Wildlife Service, Division of Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S4 (823 KB PDF)

**Reference S3.**Pacific Flyway Council. 2018. Management plan for the Pacific population of brant. Vancouver, Washington: Pacific Flyway Council, care of U.S. Fish and Wildlife Service, Division of Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S5 (2.518 MB PDF)

**Reference S4.** Shults BS, Zeller TK. 2017. Abundance and distribution of molting geese in the vicinity of Teshekpuk Lake, Alaska, July 2017. Unpublished report. Anchorage, Alaska: U.S. Fish and Wildlife Service, Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S6 (478 KB PDF)

**Reference S5.** Stehn RA, Platte RM, Wilson HW, Fischer JB. 2011. Monitoring the nesting population of Pacific black brant. Report to the Pacific Flyway Study Committee. Anchorage, Alaska: U.S. Fish and Wildlife Service, Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S7 (509 KB PDF)

**Reference S6.** Wilson HM. 2017. Aerial survey of emperor geese and other waterbirds in southwestern Alaska, fall 2015. 2017. Unpublished report. Anchorage, Alaska: U.S. Fish and Wildlife Service, Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S8 (597 KB PDF)

**Reference S7.** Wilson HM. 2018. Aerial photographic survey of brant colonies on the Yukon-Kuskokwim Delta, Alaska, 2017. Unpublished report. Anchorage, Alaska: U.S. Fish and Wildlife Service, Migratory Bird Management.

Available: https://doi.org/10.3996/JFWM-21-088.S9 (485 KB PDF)

## Acknowledgments

The initial idea for these analyses stems from discussion with C. Lensink. Credit for the data used in these analyses goes to the USFWS for maintaining these surveys over long periods of time. C. Amundson, D. Derksen, J. Fischer, J. Pearce, and T. Riecke commented on an earlier draft of this manuscript. The manuscript was improved by comments from anonymous reviewers, and I thank the Associate Editor D. Haukos for his detailed guidance and comments.

Any use of trade, product, website, or firm names in this publication is for descriptive purposes only and does not imply endorsement by the U.S. Government.

## References

*Branta bernicla nigricans*) fall age ratio surveys at Izembek Lagoon, Alaska

*Branta bernicla*

*Branta bernicla nigricans*

*Branta bernicla*), version 1.0

*Branta bernicla nigricans*, metapopulation

The findings and conclusions in this article are those of the author(s) and do not necessarily represent the views of the U.S. Fish and Wildlife Service.

## Author notes

Citation: Flint PL. 2022. Comparison of indices to infer population dynamics of black brant. *Journal of Fish and Wildlife Management* 13(2):344–358; e1944-687X. https://doi.org/10.3996/JFWM-21-088