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Adaptive Harvest Management 2007 Hunting Season U.S. Fish & Wildlife Service 1 Adaptive Harvest Management 2007 Hunting Season PREFACE The process of setting waterfowl hunting regulations is conducted annually in the United States (Blohm 1989). This process involves a number of meetings where the status of waterfowl is reviewed by the agencies responsible for setting hunting regulations. In addition, the U.S. Fish and Wildlife Service (USFWS) publishes proposed regulations in the Federal Register to allow public comment. This document is part of a series of reports intended to support development of harvest regulations for the 2007 hunting season. Specifically, this report is intended to provide waterfowl managers and the public with information about the use of adaptive harvest management (AHM) for setting waterfowl hunting regulations in the United States. This report provides the most current data, analyses, and decisionmaking protocols. However, adaptive management is a dynamic process and some information presented in this report will differ from that in previous reports. ACKNOWLEDGMENTS A working group comprised of representatives from the USFWS, the U.S. Geological Survey (USGS), the Canadian Wildlife Service (CWS), and the four Flyway Councils (Appendix A) was established in 1992 to review the scientific basis for managing waterfowl harvests. The working group, supported by technical experts from the waterfowl management and research communities, subsequently proposed a framework for adaptive harvest management, which was first implemented in 1995. The USFWS expresses its gratitude to the AHM Working Group and to the many other individuals, organizations, and agencies that have contributed to the development and implementation of AHM. This report was prepared by the USFWS Division of Migratory Bird Management. F. A. Johnson and G. S. Boomer were the principal authors. Individuals that provided essential information or otherwise assisted with report preparation were P. Garrettson (USFWS), T. Liddick (USGS), M. Otto (USFWS), R. Raftovich (USFWS), A. Royle (USGS), M. Runge (USGS), T. Sanders (USFWS), and K. Wilkins (USFWS). Comments regarding this document should be sent to the Chief, Division of Migratory Bird Management  USFWS, 4401 North Fairfax Drive, MS MSP4107, Arlington, VA 22203. Citation: U.S. Fish and Wildlife Service. 2007. Adaptive Harvest Management: 2007 Hunting Season. U.S. Dept. Interior, Washington, D.C. 44pp. Available online at http://www.fws.gov/migratorybirds/mgmt/AHM/AHMintro.htm U.S. Fish & Wildlife Service 2 TABLE OF CONTENTS Executive Summary .............................................................................................................3 Background ..........................................................................................................................4 Mallard Stocks and Flyway Management............................................................................5 Mallard Population Dynamics..............................................................................................6 HarvestManagement Objectives .......................................................................................13 Regulatory Alternatives .....................................................................................................13 Optimal Regulatory Strategies ...........................................................................................16 Application of AHM Concepts to Other Stocks ................................................................17 Literature Cited ..................................................................................................................29 Appendix A: AHM Working Group .................................................................................31 Appendix B: Midcontinent Mallard Models ...................................................................34 Appendix C: Eastern Mallard Models ..............................................................................38 Appendix D: Modeling Mallard Harvest Rates ................................................................41 3 EXECUTIVE SUMMARY In 1995 the U.S. Fish and Wildlife Service (USFWS) implemented the Adaptive Harvest Management (AHM) program for setting duck hunting regulations in the United States. The AHM approach provides a framework for making objective decisions in the face of incomplete knowledge concerning waterfowl population dynamics and regulatory impacts. The current AHM protocol is based on the population dynamics and status of two mallard (Anas platyrhynchos) stocks. Midcontinent mallards are defined as those breeding in the socalled traditional survey area, plus the states of Michigan, Minnesota, and Wisconsin. The prescribed regulatory alternative for the Mississippi, Central, and Pacific Flyways depends exclusively on the status of these mallards. Eastern mallards are defined as those breeding in the states of Virginia northward into Vermont, and in survey strata located on the Canadian side of the St. Lawrence River. The regulatory choice for the Atlantic Flyway depends exclusively on the status of these mallards. Investigations of the population dynamics of western mallards and their potential effect on hunting regulations in the West are ongoing. Mallard population models account for an apparent positive bias in estimates of survival and reproductive rates, and also allow for alternative hypotheses concerning the effects of harvest and the environment in regulating population size. Modelspecific weights reflect the relative confidence in alternative hypotheses, and are updated annually using comparisons of predicted and observed population sizes. For midcontinent mallards, current model weights favor the weakly densitydependent reproductive hypothesis (90%). Evidence for the additivemortality hypothesis remains equivocal (60%). For eastern mallards, virtually all of the weight is on models that have corrections for bias in estimates of survival or reproductive rates. Model weights also favor the strongly densitydependent reproductive hypothesis (59%). By consensus, hunting mortality is assumed to be additive in eastern mallards. For the 2007 hunting season, the USFWS is considering the same regulatory alternatives as last year. The nature of the restrictive, moderate, and liberal alternatives has remained essentially unchanged since 1997, except that extended framework dates have been offered in the moderate and liberal alternatives since 2002. Harvest rates associated with each of the regulatory alternatives have been updated based on bandreporting rate studies conducted since 1998. Estimated harvest rates of adult males from the 20022006 liberal hunting seasons have averaged 0.113 (SE = 0.001) and 0.136 (SE = 0.010) for midcontinent and eastern mallards, respectively. The estimated marginal effect of frameworkdate extensions has been an increase in harvest rate of 0.009 (SD = 0.008) and 0.005 (SD = 0.010) for midcontinent and eastern mallards, respectively. Optimal regulatory strategies for the 2007 hunting season were calculated using: (1) harvestmanagement objectives specific to each mallard stock; (2) the 2007 regulatory alternatives; and (3) current population models and associated weights for midcontinent and eastern mallards. Based on this year’s survey results of 9.05 million midcontinent mallards, 5.04 million ponds in Prairie Canada, and 907 thousand eastern mallards, the optimal choice for all four Flyways is the liberal regulatory alternative. AHM concepts and tools are also being applied to help improve harvest management for several other waterfowl stocks. In the last year, significant progress has been made in understanding the harvest potential of American black ducks (Anas rubripes), the Atlantic Population of Canada geese (Branta canadensis), northern pintails (Anas acuta), and scaup (Aythya affinis, A. marila). While these biological assessments are ongoing, they are already proving valuable in helping focus debate on the social aspects of harvesting policy, including management objectives and the nature of regulatory alternatives. 4 BACKGROUND The annual process of setting duckhunting regulations in the United States is based on a system of resource monitoring, data analyses, and rulemaking (Blohm 1989). Each year, monitoring activities such as aerial surveys and hunter questionnaires provide information on population size, habitat conditions, and harvest levels. Data collected from this monitoring program are analyzed each year, and proposals for duckhunting regulations are developed by the Flyway Councils, States, and USFWS. After extensive public review, the USFWS announces regulatory guidelines within which States can set their hunting seasons. In 1995, the USFWS adopted the concept of adaptive resource management (Walters 1986) for regulating duck harvests in the United States. This approach explicitly recognizes that the consequences of hunting regulations cannot be predicted with certainty, and provides a framework for making objective decisions in the face of that uncertainty (Williams and Johnson 1995). Inherent in the adaptive approach is an awareness that management performance can be maximized only if regulatory effects can be predicted reliably. Thus, adaptive management relies on an iterative cycle of monitoring, assessment, and decisionmaking to clarify the relationships among hunting regulations, harvests, and waterfowl abundance. In regulating waterfowl harvests, managers face four fundamental sources of uncertainty (Nichols et al. 1995, Johnson et al. 1996, Williams et al. 1996): (1) environmental variation  the temporal and spatial variation in weather conditions and other key features of waterfowl habitat; an example is the annual change in the number of ponds in the Prairie Pothole Region, where water conditions influence duck reproductive success; (2) partial controllability  the ability of managers to control harvest only within limits; the harvest resulting from a particular set of hunting regulations cannot be predicted with certainty because of variation in weather conditions, timing of migration, hunter effort, and other factors; (3) partial observability  the ability to estimate key population attributes (e.g., population size, reproductive rate, harvest) only within the precision afforded by extant monitoring programs; and (4) structural uncertainty  an incomplete understanding of biological processes; a familiar example is the longstanding debate about whether harvest is additive to other sources of mortality or whether populations compensate for hunting losses through reduced natural mortality. Structural uncertainty increases contentiousness in the decisionmaking process and decreases the extent to which managers can meet longterm conservation goals. AHM was developed as a systematic process for dealing objectively with these uncertainties. The key components of AHM include (Johnson et al. 1993, Williams and Johnson 1995): (1) a limited number of regulatory alternatives, which describe Flywayspecific season lengths, bag limits, and framework dates; (2) a set of population models describing various hypotheses about the effects of harvest and environmental factors on waterfowl abundance; (3) a measure of reliability (probability or "weight") for each population model; and (4) a mathematical description of the objective(s) of harvest management (i.e., an "objective function"), by which alternative regulatory strategies can be compared. These components are used in a stochastic optimization procedure to derive a regulatory strategy. A regulatory strategy specifies the optimal regulatory choice, with respect to the stated management objectives, for each possible combination of breeding population size, environmental conditions, and model weights (Johnson et al. 1997). The setting of annual hunting regulations then involves an iterative process: (1) each year, an optimal regulatory choice is identified based on resource and environmental conditions, and on current model weights; 5 (2) after the regulatory decision is made, modelspecific predictions for subsequent breeding population size are determined; (3) when monitoring data become available, model weights are increased to the extent that observations of population size agree with predictions, and decreased to the extent that they disagree; and (4) the new model weights are used to start another iteration of the process. By iteratively updating model weights and optimizing regulatory choices, the process should eventually identify which model is the best overall predictor of changes in population abundance. The process is optimal in the sense that it provides the regulatory choice each year necessary to maximize management performance. It is adaptive in the sense that the harvest strategy “evolves” to account for new knowledge generated by a comparison of predicted and observed population sizes. MALLARD STOCKS AND FLYWAY MANAGEMENT Since its inception AHM has focused on the population dynamics and harvest potential of mallards, especially those breeding in midcontinent North America. Mallards constitute a large portion of the total U.S. duck harvest, and traditionally have been a reliable indicator of the status of many other species. As management capabilities have grown, there has been increasing interest in the ecology and management of breeding mallards that occur outside the midcontinent region. Geographic differences in the reproduction, mortality, and migrations of mallard stocks suggest that there may be corresponding differences in optimal levels of sport harvest. The ability to regulate harvests of mallards originating from various breeding areas is complicated, however, by the fact that a large degree of mixing occurs during the hunting season. The challenge for managers, then, is to vary hunting regulations among Flyways in a manner that recognizes each Flyway’s unique breedingground derivation of mallards. Of course, no Flyway receives mallards exclusively from one breeding area, and so Flywayspecific harvest strategies ideally must account for multiple breeding stocks that are exposed to a common harvest. The optimization procedures used in AHM can account for breeding populations of mallards beyond the midcontinent region, and for the manner in which these ducks distribute themselves among the Flyways during the hunting season. An optimal approach would allow for Flywayspecific regulatory strategies, which in a sense represent for each Flyway an average of the optimal harvest strategies for each contributing breeding stock, weighted by the relative size of each stock in the fall flight. This joint optimization of multiple mallard stocks requires: (1) models of population dynamics for all recognized stocks of mallards; (2) an objective function that accounts for harvestmanagement goals for all mallard stocks in the aggregate; and (3) decision rules allowing Flywayspecific regulatory choices. Currently, two stocks of mallards are officially recognized for the purposes of AHM (Fig. 1). We continue to use a constrained approach to the optimization of these stocks’ harvest, in which the Atlantic Flyway regulatory strategy is based exclusively on the status of eastern mallards, and the regulatory strategy for the remaining Flyways is based exclusively on the status of midcontinent mallards. This approach has been determined to perform nearly as well as a jointoptimization because mixing of the two stocks during the hunting season is limited. 6 Fig 1. Survey areas currently assigned to the midcontinent and eastern stocks of mallards for the purposes of AHM. Delineation of the westernmallard stock is pending further development and review of population models and monitoring programs. MALLARD POPULATION DYNAMICS MidContinent Stock Midcontinent mallards are defined as those breeding in federal survey strata 118, 2050, and 7577 (i.e., the “traditional” survey area), and in Minnesota, Wisconsin, and Michigan (Fig. 1). Estimates of the size of this population are available since 1992, and have varied from 6.6 to 11.8 million (Table 1, Fig. 2). Estimated breedingpopulation size in 2007 was 9.053 (SE = 0.291 million), including 8.307 million (SE = 0.285 million) from the traditional survey area and 746 thousand (SE = 56 thousand) from the Great Lakes region. Details concerning the set of population models for midcontinent mallards are provided in Appendix B. The set consists of four alternatives, formed by the combination of two survival hypotheses (additive vs. compensatory hunting mortality) and two reproductive hypotheses (strongly vs. weakly density dependent). Relative weights for the alternative models of midcontinent mallards changed little until all models underpredicted the change in population size from 1998 to 1999, perhaps indicating there is a significant factor affecting population dynamics that is absent from all four models (Fig. 3). Updated model weights suggest some preference for the additivemortality models (60%) over those describing hunting mortality as compensatory (40%). For most of the time frame, model weights have strongly favored the weakly densitydependent reproductive models over the strongly densitydependent ones, with current model weights of 90% and 10%, respectively. The reader is cautioned, however, that models can sometimes make reliable predictions of population size for reasons having little to do with the biological hypotheses expressed therein (Johnson et al. 2002b). 7 Table 1. Estimates (N) and standard errors (SE) of mallards (in millions) in spring in the traditional survey area (strata 118, 2050, and 7577) and the states of Minnesota, Wisconsin, and Michigan. Traditional survey area Great Lakes region Total Year N SE N SE N SE 1992 5.9761 0.2410 0.9946 0.1597 6.9706 0.2891 1993 5.7083 0.2089 0.9347 0.1457 6.6430 0.2547 1994 6.9801 0.2828 1.1505 0.1163 8.1306 0.3058 1995 8.2694 0.2875 1.1214 0.1965 9.3908 0.3482 1996 7.9413 0.2629 1.0251 0.1443 8.9664 0.2999 1997 9.9397 0.3085 1.0777 0.1445 11.0174 0.3407 1998 9.6404 0.3016 1.1224 0.1792 10.7628 0.3508 1999 10.8057 0.3445 1.0591 0.2122 11.8648 0.4046 2000 9.4702 0.2902 1.2350 0.1761 10.7052 0.3395 2001 7.9040 0.2269 0.8622 0.1086 8.7662 0.2516 2002 7.5037 0.2465 1.0820 0.1152 8.5857 0.2721 2003 7.9497 0.2673 0.8360 0.0734 8.7857 0.2772 2004 7.4253 0.2820 0.9333 0.0748 8.3586 0.2917 2005 6.7553 0.2808 0.7862 0.0650 7.5415 0.2883 2006 7.2765 0.2237 0.5881 0.0465 7.8646 0.2284 2007 8.3073 0.2858 0.7459 0.0565 9.0532 0.2913 Fig. 2. Population estimates of midcontinent mallards in the traditional survey area (TSA) and the Great Lakes region. Error bars represent one standard error. Year 1995 2000 2005 Population size (millions) 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 TSA Great Lakes Total 8 Fig 3. Weights for models of midcontinent mallards (ScRs = compensatory mortality and strongly densitydependent reproduction, ScRw = compensatory mortality and weakly densitydependent reproduction, SaRs = additive mortality and strongly densitydependent reproduction, and SaRw = additive mortality and weakly densitydependent reproduction). Model weights were assumed to be equal in 1995. Eastern Stock Eastern mallards are defined as those breeding in southern Ontario and Quebec (federal survey strata 5154 and 56) and in the northeastern U.S. (state plot surveys; Heusman and Sauer 2000) (Fig. 1). Estimates of population size have varied from 856 thousand to 1.1 million since 1990, with the majority of the population accounted for in the northeastern U.S. (Table 3, Fig. 4). For 2007, the estimated breedingpopulation size of eastern mallards was 907 thousand (SE = 58 thousand), including 688 thousand (SE = 47 thousand) from the northeastern U.S. and 219 thousand (SE = 34 thousand) from the Canadian survey strata. The reader is cautioned that these estimates differ from those reported in the USFWS annual waterfowl trend and status reports, which include composite estimates based on more fixedwing strata in eastern Canada and helicopter surveys conducted by CWS. Details concerning the set of population models for eastern mallards are provided in Appendix C. The set consists of six alternatives, formed by the combination of two reproductive hypotheses (strongly vs. weakly density dependent) and three hypotheses concerning bias in estimates of survival and reproductive rates (no bias vs. biased survival rates vs. biased reproductive rates). With respect to model weights, there is no single model that is clearly favored over the others at the current time. Collectively, the models with strong densitydependent reproduction are slightly better predictors of changes in population size than those with weak density dependence, with current model weights of 59% and 41%, respectively (Fig. 5). In addition, there is overwhelming evidence of bias in extant estimates of survival or reproductive rates (100%), assuming that survey estimates are unbiased. Year 1996 1998 2000 2002 2004 2006 Model weight 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ScRs ScRw SaRs SaRw 9 Table 3. Estimates (N) and associated standard errors (SE) of mallards (in thousands) in spring in the northeastern U.S. (state plot surveys) and eastern Canada (federal survey strata 5154 and 56). Northeastern U.S. Canadian survey strata Total Year N SE N SE N SE 1990 665.1 78.3 190.7 47.2 855.8 91.4 1991 779.2 88.3 152.8 33.7 932.0 94.5 1992 562.2 47.9 320.3 53.0 882.5 71.5 1993 683.1 49.7 292.1 48.2 975.2 69.3 1994 853.1 62.7 219.5 28.2 1072.5 68.7 1995 862.8 70.2 184.4 40.0 1047.2 80.9 1996 848.4 61.1 283.1 55.7 1131.5 82.6 1997 795.1 49.6 212.1 39.6 1007.2 63.4 1998 775.1 49.7 263.8 67.2 1038.9 83.6 1999 879.7 60.2 212.5 36.9 1092.2 70.6 2000 757.8 48.5 132.3 26.4 890.0 55.2 2001 807.5 51.4 200.2 35.6 1007.7 62.5 2002 834.1 56.2 171.3 30.0 1005.4 63.8 2003 731.8 47.0 308.3 55.4 1040.1 72.6 2004 809.1 51.8 301.5 53.3 1110.7 74.3 2005 753.6 53.6 293.4 53.1 1047.0 75.5 2006 725.2 47.9 174.0 28.4 899.2 55.7 2007 687.6 46.7 219.3 33.6 906.9 57.6 10 Fig. 4. Population estimates of eastern mallards in the northeastern U.S. (NE plot survey) and in federal surveys in southern Ontario and Quebec (FWS survey). Error bars represent one standard error. Fig. 5. Weights for models of eastern mallards (Rw0 = weak densitydependent reproduction and no model bias, Rs0 = strong dependent reproduction and no model bias, RwS = weak densitydependent reproduction and biased survival rates, RsS = strong densitydependent reproduction and biased survival rates, RwR = weak densitydependent reproduction and biased reproductive ates, and RsR = strong densitydependent reproduction and biased reproductive rates). Model weights were assumed to be equal in 1996. Year 1990 1995 2000 2005 Populations size (millions) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 NE plot survey 1.4 FWS survey Total Year 1996 1998 2000 2002 2004 2006 Model weight 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 Rw0 Rs0 RwS RsS RwR RsR 11 Western Stock Recent efforts to develop Flywayspecific harvest strategies have focused on mallards breeding in the states of the Pacific Flyway (including Alaska), British Columbia, and the Yukon Territory. Efforts to understand and model the population dynamics of western mallards have been underway for several years and the Pacific Flyway States, the USFWS, and the Canadian Wildlife Service have been collaborating to improve survey and banding programs. We summarize the most recent results concerning the dynamics of these mallards, as well as some implications for harvest management. A more detailed report is available online at http://www.fws.gov/migratorybirds/mgmt/ahm/specialtopics.htm. Western mallards are distributed over a large area and we have had continuing concerns about our ability to determine changes in population size based on the collection of surveys conducted independently by Pacific Flyway States and the Province of British Columbia. These surveys tend to vary in design and intensity, and in some cases lack measures of precision. Therefore, we reviewed extant surveys to determine their adequacy for supporting a westernmallard AHM protocol and ultimately selected Alaska, California, and Oregon for modeling purposes. These three states likely harbor about 75% of the westernmallard breeding population. Nonetheless, this geographic delineation is considered temporary until surveys in other areas can be brought up to similar standards and an adequate record of population estimates is available for analysis. To predict changes in abundance we relied on a discrete logistic model, which combines reproduction and natural mortality into a single parameter r, the intrinsic rate of growth. This model assumes densitydependent growth, which is regulated by the ratio of population size, N, to the carrying capacity of the environment, K (i.e., population size in the absence of harvest). In the traditional formulation of the logistic model, harvest mortality is completely additive and any compensation for hunting losses occurs as a result of densitydependent responses beginning in the subsequent breeding season. To increase the model’s generality we included a scaling parameter for harvest that allows for the possibility of compensation prior to the breeding season. It is important to note, however, that this parameterization does not incorporate any hypothesized mechanism for harvest compensation and, therefore, must be interpreted cautiously. We modeled Alaska mallards independently of those in California and Oregon because of differing population trajectories (Fig. 6) and substantial differences in the distribution of band recoveries. We used Bayesian estimation methods in combination with a statespace model that accounts explicitly for both process and observation error in breeding population size (Meyer and Millar 1999). Breeding population estimates of mallards in Alaska are available since 1955, but we had to limit the timeseries to 19902005 because of changes in survey methodology and insufficient bandrecovery data. The logistic model and associated posterior parameter estimates provided a reasonable fit to the observed timeseries of Alaska population estimates. The estimated carrying capacity was 1.2 million, the intrinsic rate of growth was 0.31, and harvest mortality acted in an additive fashion. Breeding population and harvestrate data were available for CaliforniaOregon mallards for the period 19922006. The logistic model also provided a reasonable fit to these data, suggesting a carrying capacity of 0.7 million, an intrinsic rate of growth 0.34, and harvest mortality that acted in only a partially additive manner. For the purpose of understanding general patterns in optimal harvest rates, we assumed perfect control over harvest and evaluated statedependent harvest rates from 0.0 to 0.25 in increments of 0.05. We examined two different management objectives conditioned on this set of harvest rates: (1) maximize longterm cumulative yield; and (2) attain approximately 90% of the maximum longterm cumulative yield. For an objective to maximize longterm cumulative harvest, there were many combinations of stock sizes that had harvestrate prescriptions of either 0 or 25 percent. Very few stock sizes had intermediate harvestrate prescriptions. In contrast, an objective to attain 90% of the maximum yield produced an optimal strategy with a more even distribution of optimal harvest rates, and very few prescriptions for closed seasons. Empirical estimates of harvest rates showed no obvious response to changes in regulations, based on extensive 12 Fig. 6. Estimated abundances of mallards breeding in Alaska and CaliforniaOregon as derived from federal and state surveys, respectively. Error bars represent one standard error. analyses using a variety of regulatory metrics, including season length, mallard bag limits, and framework opening and closing dates (singly and in combination). We were forced to conclude that changes in regulations in the Pacific Flyway since 1980 have not resulted in significant changes in the harvest rates of western mallards. It appears that more extreme regulatory changes than those used in the past may be needed to effect substantive changes in harvest rates. To help understand the implications of this apparent lack of control over harvest rates, we assumed the most extreme case of two regulatory options: a closed season and an open season. We assumed that an open season would produce a harvest rate of 0.1259 (the mean of all our estimates) and that a closed season would produce a harvest rate of 0.0. We then conducted an optimization to determine the population thresholds for season closures assuming minimal control over harvest rates. Generally, as long as both stocks are above about 350k, then the optimal choice is an open season. Below that, the lower one stock is, the higher the other has to be to maintain an open season. We believe that the models developed thus far provide a sufficient basis for developing an initial AHM protocol. Moreover, extant monitoring of mallard abundance and harvest rates in Alaska and CaliforniaOregon will provide the necessary basis for updating estimates of model parameters and their variances. Similarly, we believe that sufficient information is available to inform the choice of an objective function for western mallards. For example, an objective to attain 90% of the maximum longterm cumulative harvest provides for levels of hunting opportunity that are similar to those now in effect for a wide range of stock sizes. On a more pessimistic note, we were unable to establish a viable set of regulatory alternatives with which to effect changes in harvest rate. Therefore, an essential task is consideration of hunting regulations beyond the realm of experience that might be expected to have a meaningful effect on harvest rates. Ideally, the development of AHM protocols for mallards would consider how different breeding stocks distribute themselves among the four flyways so that Flywayspecific harvest strategies could account for the mixing of birds during the hunting season. At present, however, a joint optimization of western, midcontinent, and eastern stocks is not feasible due to computational hurdles. Therefore, the initial AHM protocol for western mallards may need to be structured similarly to that used for eastern mallards, in which an optimal harvest strategy is based on Year 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Population size 2e+5 3e+5 4e+5 5e+5 6e+5 7e+5 8e+5 9e+5 1e+6 2e+5 3e+5 4e+5 5e+5 6e+5 7e+5 8e+5 9e+5 1e+6 AK CAOR 13 the status of a single breeding stock and harvest regulations in a single flyway. Although the contribution of midcontinent mallards to the Pacific Flyway harvest is significant, we believe an independent harvest strategy for western mallards poses little risk to the midcontinent stock. Further analyses will be needed to confirm this conclusion, as well as to better understand the potential effect of midcontinent mallard status on sustainable hunting opportunities in the Pacific Flyway. HARVESTMANAGEMENT OBJECTIVES The basic harvestmanagement objective for midcontinent mallards is to maximize cumulative harvest over the long term, which inherently requires perpetuation of a viable population. Moreover, this objective is constrained to avoid regulations that could be expected to result in a subsequent population size below the goal of the North American Waterfowl Management Plan (NAWMP). According to this constraint, the value of harvest decreases proportionally as the difference between the goal and expected population size increases. This balance of harvest and population objectives results in a regulatory strategy that is more conservative than that for maximizing longterm harvest, but more liberal than a strategy to attain the NAWMP goal (regardless of effects on hunting opportunity). The current objective uses a population goal of 8.8 million mallards, which is based on 8.2 million mallards in the traditional survey area (from the 1998 update of the NAWMP) and a goal of 0.6 million for the combined states of Minnesota, Wisconsin, and Michigan. For eastern mallards, there is no NAWMP goal or other established target for desired population size. Accordingly, the management objective for eastern mallards is simply to maximize longterm cumulative (i.e., sustainable) harvest. REGULATORY ALTERNATIVES Evolution of Alternatives When AHM was first implemented in 1995, three regulatory alternatives characterized as liberal, moderate, and restrictive were defined based on regulations used during 197984, 198587, and 198893, respectively. These regulatory alternatives also were considered for the 1996 hunting season. In 1997, the regulatory alternatives were modified to include: (1) the addition of a veryrestrictive alternative; (2) additional days and a higher duck bag limit in the moderate and liberal alternatives; and (3) an increase in the bag limit of hen mallards in the moderate and liberal alternatives. In 2002 the USFWS further modified the moderate and liberal alternatives to include extensions of approximately one week in both the opening and closing framework dates. In 2003 the veryrestrictive alternative was eliminated at the request of the Flyway Councils. Expected harvest rates under the veryrestrictive alternative did not differ significantly from those under the restrictive alternative, and the veryrestrictive alternative was expected to be prescribed for <5% of all hunting seasons. Also, at the request of the Flyway Councils the USFWS agreed to exclude closed duckhunting seasons from the AHM protocol when the population size of midcontinent mallards is ≥5.5 million (traditional survey area plus the Great Lakes region). Based on our assessment, closed hunting seasons do not appear to be necessary from the perspective of sustainable harvesting when the midcontinent mallard population exceeds this level. The impact of maintaining open seasons above this level also appears to be negligible for other midcontinent duck species, as based on population models developed by Johnson (2003). However, complete or partial seasonclosures for particular species or populations could still be deemed necessary in some situations regardless of the status of midcontinent mallards. Details of the regulatory alternatives for each Flyway are provided in Table 6. 14 Table 6. Regulatory alternatives for the 2007 duckhunting season. Flyway Regulation Atlantica Mississippi Centralb Pacificc Shooting hours onehalf hour before sunrise to sunset Framework dates Restrictive Oct 1  Jan 20 Saturday nearest Oct 1to the Sunday nearest Jan 20 Moderate and Liberal Saturday nearest September 24 to the last Sunday in January Season length (days) Restrictive 30 30 39 60 Moderate 45 45 60 86 Liberal 60 60 74 107 Bag limit (total / mallard / female mallard) Restrictive 3 / 3 / 1 3 / 2 / 1 3 / 3 / 1 4 / 3 / 1 Moderate 6 / 4 / 2 6 / 4 / 1 6 / 5 / 1 7 / 5 / 2 Liberal 6 / 4 / 2 6 / 4 / 2 6 / 5 / 2 7 / 7 / 2 a The states of Maine, Massachusetts, Connecticut, Pennsylvania, New Jersey, Maryland, Delaware, West Virginia, Virginia, and North Carolina are permitted to exclude Sundays, which are closed to hunting, from their total allotment of season days. b The High Plains Mallard Management Unit is allowed 8, 12, and 23 extra days in the restrictive, moderate, and liberal alternatives, respectively. c The Columbia Basin Mallard Management Unit is allowed seven extra days in the restrictive, and moderate alternatives. RegulationSpecific Harvest Rates Harvest rates of mallards associated with each of the openseason regulatory alternatives were initially predicted using harvestrate estimates from 197984, which were adjusted to reflect current hunter numbers and contemporary specifications of season lengths and bag limits. In the case of closed seasons in the U.S., we assumed rates of harvest would be similar to those observed in Canada during 198893, which was a period of restrictive regulations both in Canada and the U.S. All harvestrate predictions were based only in part on bandrecovery data, and relied heavily on models of hunting effort and success derived from hunter surveys (USFWS 2002: Appendix C). As such, these predictions had large sampling variances and their accuracy was uncertain. In 2002 we began relying on Bayesian statistical methods for improving regulationspecific predictions of harvest rates, including predictions of the effects of frameworkdate extensions. Essentially, the idea is to use existing (prior) information to develop initial harvestrate predictions (as above), to make regulatory decisions based on those predictions, and then to observe realized harvest rates. Those observed harvest rates, in turn, are treated as new sources of information for calculating updated (posterior) predictions. Bayesian methods are attractive because they provide a quantitative and formal, yet intuitive, approach to adaptive management. For midcontinent mallards, we have empirical estimates of harvest rate from the recent period of liberal hunting regulations (19982006). The Bayesian methods thus allow us to combine these estimates with our prior predictions to provide updated estimates of harvest rates expected under the liberal regulatory alternative. Moreover, in the absence of experience (so far) with the restrictive and moderate regulatory alternatives, we 15 reasoned that our initial predictions of harvest rates associated with those alternatives should be rescaled based on a comparison of predicted and observed harvest rates under the liberal regulatory alternative. In other words, if observed harvest rates under the liberal alternative were 10% less than predicted, then we might also expect that the mean harvest rate under the moderate alternative would be 10% less than predicted. The appropriate scaling factors currently are based exclusively on prior beliefs about differences in mean harvest rate among regulatory alternatives, but they will be updated once we have experience with something other than the liberal alternative. A detailed description of the analytical framework for modeling mallard harvest rates is provided in Appendix D. Our models of regulationspecific harvest rates also allow for the marginal effect of frameworkdate extensions in the moderate and liberal alternatives. A previous analysis by the USFWS (2001) suggested that implementation of frameworkdate extensions might be expected to increase the harvest rate of midcontinent mallards by about 15%, or in absolute terms by about 0.02 (SD = 0.01). Based on the observed harvest rates during the 20022006 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.009 (SD = 0.008). The estimated effect of the frameworkdate extension has been to increase harvest rate of midcontinent mallards by about 8% over what would otherwise be expected in the liberal alternative. However, the reader is strongly cautioned that reliable inference about the marginal effect of frameworkdate extensions ultimately depends on a rigorous experimental design (including controls and random application of treatments). Current predictions of harvest rates of adultmale midcontinent mallards associated with each of the regulatory alternatives are provided in Table 7. Predictions of harvest rates for the other agesex cohorts are based on the historical ratios of cohortspecific harvest rates to adultmale rates (Runge et al. 2002). These ratios are considered fixed at their longterm averages and are 1.5407, 0.7191, and 1.1175 for young males, adult females, and young females, respectively. We continued to make the simplifying assumption that the harvest rates of midcontinent mallards depend solely on the regulatory choice in the western three Flyways. This appears to be a reasonable assumption given the small proportion of midcontinent mallards wintering in the Atlantic Flyway (Munro and Kimball 1982), and harvestrate predictions that suggest a minimal effect of Atlantic Flyway regulations (USFWS 2000). Under this assumption, the optimal regulatory strategy for the western three Flyways can be derived by ignoring the harvest regulations imposed in the Atlantic Flyway. Table 7. Predictions of harvest rates of adultmale midcontinent mallards expected with application of the 2007 regulatory alternatives in the three western Flyways. Regulatory alternative Mean SD Closed (U.S.) 0.0088 0.0019 Restrictive 0.0578 0.0129 Moderate 0.1059 0.0216 Liberal 0.1225 0.0205 The predicted harvest rates of easternmallard are updated in the same fashion as that for midcontinent mallards based on reward banding conducted in eastern Canada and the northeastern U.S. (Appendix D). Like midcontinent mallards, harvest rates of age and sex cohorts other than adult male mallards are based on constant rates of differential vulnerability as derived from bandrecovery data. For eastern mallards, these constants are 1.153, 1.331, and 1.509 for adult females, young males, and young females, respectively (Johnson et al. 2002a). Regulationspecific predictions of harvest rates of adultmale eastern mallards are provided in Table 8. In contrast to midcontinent mallards, frameworkdate extensions were expected to increase the harvest rate of eastern mallards by only about 5% (USFWS 2001), or in absolute terms by about 0.01 (SD = 0.01). Based on the observed harvest rates during the 20022006 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.005 (SD = 0.010). The estimated effect of the frameworkdate extension has been to increase harvest rate of eastern mallards by about 3% over what would 16 otherwise be expected in the liberal alternative. Table 8. Predictions of harvest rates of adultmale eastern mallards expected with application of the 2007 regulatory alternatives in the Atlantic Flyway. Regulatory alternative Mean SD Closed (U.S.) 0.0798 0.0233 Restrictive 0.1202 0.0395 Moderate 0.1471 0.0474 Liberal 0.1578 0.0459 OPTIMAL REGULATORY STRATEGIES We calculated optimal regulatory strategies using stochastic dynamic programming (Lubow 1995, Johnson and Williams 1999). For the three western Flyways, we based this optimization on: (1) the 2007 regulatory alternatives, including the closedseason constraint; (2) current population models and associated weights for midcontinent mallards; and (3) the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.8 million midcontinent mallards. The resulting regulatory strategy (Table 9) is similar to that used last year. Note that prescriptions for closed seasons in this strategy represent resource conditions that are insufficient to support one of the current regulatory alternatives, given current harvestmanagement objectives and constraints. However, closed seasons under all of these conditions are not necessarily required for longterm resource protection, and simply reflect the NAWMP population goal and the nature of the current regulatory alternatives. Assuming that regulatory choices adhered to this strategy (and that current model weights accurately reflect population dynamics), breedingpopulation size would be expected to average 7.45 million (SD = 1.81 million). Based on an estimated population size of 9.05 million midcontinent mallards and 5.04 million ponds in Prairie Canada, the optimal choice for the Pacific, Central, and Mississippi Flyways in 2007 is the liberal regulatory alternative. We calculated an optimal regulatory strategy for the Atlantic Flyway based on: (1) the 2007 regulatory alternatives; (2) current population models and associated weights for eastern mallards; and (3) an objective to maximize longterm cumulative harvest. The resulting strategy suggests liberal regulations for all population sizes of record, and is characterized by a lack of intermediate regulations (Table 10). We simulated the use of this regulatory strategy to determine expected performance characteristics. Assuming that harvest management adhered to this strategy (and that current model weights accurately reflect population dynamics), breedingpopulation size would be expected to average 887 thousand (SD = 16 thousand). Based on an estimated breeding population size of 907 thousand mallards, the optimal choice for the Atlantic Flyway in 2007 is the liberal regulatory alternative. 17 Table 9. Optimal regulatory strategya for the three western Flyways for the 2007 hunting season. This strategy is based on current regulatory alternatives (including the closedseason constraint), on current midcontinent mallard models and weights, and on the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.8 million mallards. The shaded cell indicates the regulatory prescription for 2007. Pondsc Bpopb 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 ≤5.25 C C C C C C C C C C 5.506.25 R R R R R R R R R R 6.50 R R R R R R R R M M 6.75 R R R R R R M M M L 7.00 R R R R R M M L L L 7.25 R R R M L L L L L L 7.50 R M M L L L L L L L 7.75 M M M L L L L L L L 8.00 M L L L L L L L L L ≥8.25 L L L L L L L L L L a C = closed season, R = restrictive, M = moderate, L = liberal. b Mallard breeding population size (in millions) in the traditional survey area (survey strata 118, 2050, 7577) and Michigan, Minnesota, and Wisconsin. c Ponds (in millions) in Prairie Canada in May. Table 10. Optimal regulatory strategya for the Atlantic Flyway for the 2007 hunting season. This strategy is based on current regulatory alternatives, on current eastern mallard models and weights, and on an objective to maximize longterm cumulative harvest. The shaded cell indicates the regulatory prescription for 2007. Mallardsb Regulation <240 C 240 R >240 L a C = closed season, R = restrictive, M = moderate, and L = liberal. b Estimated number of mallards in eastern Canada (survey strata 5154, 56) and the northeastern U.S. (state plot surveys), in thousands. Application of AHM Concepts to Other Stocks The USFWS is striving to apply the principles and tools of AHM to improve decisionmaking for several other stocks of waterfowl. We report on four such efforts in which significant progress has been made since last year. American Black Ducks Beginning in 2003 the USFWS Division of Migratory Bird Management (DMBM) began investigating optimal 18 harvest strategies for black ducks based on models of population dynamics provided by Conroy et al (2002). As a result of that investigation DMBM concluded that recent harvest rates of black ducks have sometimes been at or above levels consistent with an objective to maximize sustainable harvest. That conclusion ultimately led to a DMBM recommendation in January 2006 to reduce the harvest rate of adult black ducks by 25%. However, the recommendation was subsequently withdrawn because of: (1) published information suggesting that the midwinter inventory (MWI) may be “capturing” a smaller proportion of the black duck population than in the past (Link et al. 2006); (2) concern about the U.S. acting unilaterally without the benefit of consultation with the CWS; and (3) the short amount of time available to communicate to the public the rationale and nature of restrictions on hunting opportunity. In November 2006 the international Black Duck Adaptive Harvest Management Working Group (BDAHMWG) met to discuss the most recent analysis by Drs. Mike Conroy and Jon Runge of the Georgia Cooperative Fish and Wildlife Research Unit. Their update of the original analysis by Conroy et al. (2002) suggests that black duck productivity has continued to decline for reasons that cannot be explained by changes in abundance of black ducks (through density dependence) or sympatric mallards (through interspecies competition). However, there were other differences in inferences based on the original and updated analyses that could not be reconciled. The focus of research has now turned to population models based on integrated fixedwing and helicopter surveys conducted during the breeding season. For the present, however, the question of whether current harvest rates of black ducks are consistent with black duck harvest potential and management objectives remains unanswered. Due to potential changes in the wintering distribution of black ducks, the BDAHMWG did not endorse a statedependent harvest strategy (i.e., one in which optimal harvest rates depend on annual black duck abundance) based on the MWI. However, it was suggested that a constant harvestrate strategy may perform nearly as well and might provide a basis for a joint CanadaU.S. harvest strategy until an assessment based on breedingseason surveys can be completed. The BDAHMWG agreed to investigate the performance of constant harvestrate strategies based on the original work of Conroy et al. (2002), recognizing that the original analysis was conducted prior to what may be significant changes in the wintering distribution of black ducks. Thus, it was agreed that the assessment by Conroy et al. (2002) might still provide a reasonable basis for investigating harvest impacts and for evaluating the expected performance of constant harvestrate strategies. We relied on the population models and corresponding weights provided by Conroy et al. (2002) to conduct an evaluation of constant harvestrate strategies. These models incorporate alternative hypotheses for reproduction (competition with mallards vs. no competition), survival (additive vs. compensatory hunting mortality), and estimation bias (positive bias in reproductive rates vs. survival rates). Both reproduction models incorporate a negative effect of year, presumably due to a longterm loss and/or degradation of habitat. For the purposes of this assessment we projected the loglinear decline in production rate through 2007. We believe this was justified because of evidence that the decline in productivity has continued to the present at about the same rate as that estimated from the 19611994 period. However, for the purposes of this assessment we had to assume that the decline in productivity halts in 2007. To project the decline indefinitely into the future would imply that no level of harvest is sustainable. Managers are currently considering plausible explanations for the productivity decline, and will need to be vigilant in assessing future trends in productivity. Because of the possible effect of mallard abundance on black duck productivity, it was necessary to include a dynamic model of mallard abundance (M). The model used by Conroy et al. (2002) was: t t t M = M λ +1 where λt is the finite rate of population growth. Estimates of λt from 19711994 were used to obtain an empirical distribution to specify random outcomes for λt. During 19711994 λt was highly variable, but with an average close to 1 (suggestive of a stable population). We were not completely satisfied with this model because it can produce biologically unrealistic changes in 19 population size (because population size and λt are uncorrelated) and because population size had to be constrained to an arbitrary maximum. Therefore, we described changes in mallard abundance as a 1storder autoregressive process using data from 19712000. The model is: M ( ) (M ( )) e t t = + + + 0.00001 1.801 0.494 0.00001 1 where e ~ Normal(0, 0.300). This model provided a satisfactory fit to the time series of observed population sizes and describes a stationary time series with M t = 355,826 . We evaluated constant harvestrate strategies using SDP (Lubow 1995) and by constraining the size of the harvest to be equal in Canada and the U.S. We specified fixed harvest rates of 0.00 to 0.16 in increments of 0.01. We simulated black duck and mallard population dynamics for 20,000 iterations under each of the fixed harvest rates, using starting values of 300k black ducks and 356k mallards. We then calculated the mean and standard deviation of black duck population size and harvest. For comparative purposes, we also derived an optimal statedependent harvest strategy using SDP and an objective to maximize longterm cumulative harvest. Based on simulated population dynamics, the black duck population averaged 546k (SD = 154k) in the absence of harvest (Fig. 7). An optimal statedependent strategy to maximize sustainable harvest (i.e., a strategy in which the harvest rate varies with black duck abundance) resulted in an average population size of 255k (SD = 50k) and an average harvest of 51K (SD = 41k). For a constant harvest rate, the maximum sustainable harvest was achieved at a harvest rate of 0.09 on adult males, resulting in an average of 240k (SD = 94k) black ducks in the MWI and a harvest of 47k (SD = 15k). Thus, the expected harvest under a constant harvestrate strategy was only 7% less than that which could be achieved under an optimal statedependent harvest strategy. However, population size was nearly twice as variable under the constant harvest rate of 0.09 as under the optimal statedependent strategy. For a target harvest rate of 0.09, the corresponding harvest rates in Canada and the U.S. to achieve parity in harvest are 0.045 and 0.048, respectively. By comparison, adult black duck harvest rates estimated from reward banding during the 20022006 hunting seasons averaged 0.0358 (SE = 0.00073) for Canada, 0.0584 (SE = 0.0017) for the U.S., and 0.0916 (SE = 0.0016) overall. The average population size in the MWI during 2003 2007 of 220k corresponds well with that predicted from the weighted models under an average harvest rate of 0.09 (240k). A harvest rate of 0.09 should be considered a maximum because it assumes that black duck productivity will not decline further. Moreover, a smaller harvest rate appears to be necessary to induce population growth. For example, attainment of the original North American Waterfowl Management Plan population objective of 385k black ducks in the MWI would require a constant harvest rate of approximately 0.05 under current environmental conditions. 20 Fig. 7. Black duck population sizes in winter (MWI) and harvests (both in thousands, with SD’s) expected under constant adultmale harvest rates of 0.00 (on the extreme right) to 0.16 (on the extreme left) in increments of 0.01. The datum depicted by the open circle is that expected under an optimal statedependent strategy with an objective to maximize longterm cumulative harvest. The vertical dashed line indicates the original North American Waterfowl Management Plan population goal of 385 thousand. Atlantic Population of Canada Geese For the purposes of this AHM application, Atlantic Population Canada Geese (APCG) are defined as those geese breeding on the Ungava Peninsula. By this delineation, we assume that geese in the Atlantic population outside this area are either few in number, similar in population dynamics to the Ungava birds, or both. To account for heterogeneity among individuals, we developed a base model consisting of a truncated timeinvariant agebased projection model to describe the dynamics of APCG: n(t+1)=An(t), where n(t) is a vector of the abundances of the ages in the population at time t, and A is the population projection matrix, whose ijth entry aij gives the contribution of an individual in stage j to stage i over 1 time step. The projection interval (from t to t+1) is one year, with the census being taken in midJune (i.e., this model has a prebreeding census). The life cycle diagram reflecting the transition sequence is: MWI 0 100 200 300 400 500 600 700 Harvest 0 20 40 60 80 100 21 where node 1 refers to oneyearold birds (N(1)), node 2 refers to twoyearold birds (N(2)), node B refers to adult breeders (N(B)), and node NB refers to adult nonbreeders N(NB). One immediate extension of the base model is to remove the assumption of timeinvariance, and express the parameters as timedependent quantities: Pt = proportion of adult birds in population in year t which breed; Rt = basic breeding productivity in year t (per capita); St (0) = annual survival rate of young from fledging in year t to the census point the next year; St (1) = annual survival rate of oneyearold birds in year t; etc. For APCG, only N(B), R and z are observable annually, where N(B) is the number of breeding adults, R is the per capita reproductive rate (ratio of fledged young to breeding adults), and z is an extrinsic, environmental variable (a function of timing of snow melt on the breeding grounds) that is used to predict R.. Note that at the time of the management decision in the United States (July), estimates for only the breeding population size and the environmental variable(s) are available; the ageratio isn’t estimated until later in the summer. Thus, in year t, the observable state variables are Nt (B), zt, and Rt–1. There are several other state variables of interest, however, namely, N(1), N(2), and N(NB). Because annual harvest decisions need to be made based on the total population size (Ntot), which is the sum of contributions from various nonbreeding age classes as well as the number of breeding individuals, abundance of nonbreeding individuals (N(NB), N(1), and N(2)) needs to be derived using populationreconstruction techniques. In most cases, population reconstruction involves estimating the most likely population projection matrix, given a time series of population vectors (where number of individuals in each age class at each time is known). However, in our case, only estimates of NB, R and z are available (not the complete population vector); in effect, we must estimate some of the population abundance values given the other parameters in the model. Extensions of Bayesian and nonlinear estimation methods to population reconstruction provide a reasonable solution. The time series of breeding population size, ageratio, and harvest rate were used to reconstruct the population structure from 1997 to the present, using a densityindependent model (Fig. 8). The estimated population 22 structure in 2007 is: 382,100 breeding adults, 99,300 nonbreeding adults, 235,900 secondyear birds, and 342,200 firstyear birds. Relative to the number of breeding adults, secondyear birds are 29.0% above the number expected from a stable stagedistribution, and firstyear birds are 35.9% above the expected number, reflecting the impact of the very successful 2005 and 2006 breeding seasons. The densityindependent model projects significant increases in the number of breeding pairs (~25% over the next two years) as these two sizeable cohorts come of age. Fig. 8. APCG breeding population size (in thousands), 19932007, with fitted values from reconstruction (Model 1: densityindependent). The diamonds show the observed estimates of breeding population size (not breeding pairs); errors bars are ±2 SE. The solid line shows the breeding population size estimated from the population reconstruction (also ±2 SE). The observed 1998 population size was smoothed because the survey conditions were poor. Based on the data available last year, we had postulated several alternative models to explain the apparent stabilization in the population trajectory. The three alternative models included mechanisms for densitydependent survival, densitydependent propensity to breed, and reporting rate bias. With the 2007 breeding survey data included, the differences among the trajectories for these models diminished, but the reconstructed agestructure of the current population differs substantially among them, as do the optimal harvest strategies. As an example, harvest strategies for two of the population models are shown in Fig. 9. With the densityindependent model, the strategy seeks an equilibrium breeding population size of 748,500. The strategy suggests a closed season this year, in order to increase more quickly toward the desired population size. On the other hand, the optimal strategy for the densitydependent model seeks an equilibrium breeding population size of 308,000 and because the current population size is above that, the recommended hunting regulations are liberal (20% harvest rate). Note that in both strategies, the recommended harvest rate is not strongly affected by the measure of current environmental conditions on the breeding grounds. Thus, harvest recommendations are strongly affected by uncertainty about the underlying population dynamics. We have not yet developed methods to weight the alternative models and produce a composite optimal policy; such development is a high priority. However, this population is in a very informative phase of its dynamics, such that each year of data greatly increases our ability to distinguish among alternative models. 0 50 100 150 200 250 300 350 400 450 500 1994 1996 1998 2000 2002 2004 2006 2008 23 Fig. 9. Examples of optimal harvest strategies for 2007 for models 1 (densityindependent) and 3 (densitydependent breeding propensity). These matrices show the breeding population size against the measure of breeding habitat conditions (principal component of the weather variables), with the other values of the population vector (NNB, N2, N1) fixed at their 2007 reconstructed values. The shaded areas represent the recommended harvest rate of adult males. Northern Pintails The Flyway Councils have long identified the northern pintail as a highpriority species for inclusion in the AHM process. In 1997, the USFWS adopted a pintail harvest strategy to help align harvest opportunity with population status, while providing a foundation upon which to develop a formal AHM framework. Since 1997, the harvest strategy has undergone a number of technical improvements and policy revisions. However, the strategy continues to be a set of regulatory prescriptions born out of consensus, rather than an optimal strategy derived from agreedupon population models, management objectives, regulatory alternatives, and measures of uncertainty. This year, the USFWS and Flyway Councils are taking a major step towards a truly adaptive approach by incorporating alternative models of population dynamics. Two models are being considered: one in which harvest is additive to natural mortality, and another in which harvest losses are compensated for by reductions in natural mortality. In the additive model, winter survival rate is a constant, whereas winter survival is densitydependent in the compensatory model. We here provide a summary of these recent modeling efforts. A detailed progress report is available online at http://www.fws.gov/migratorybirds/mgmt/ahm/specialtopics.htm. The predicted cBPOPt in year t + 1 ( t+1 cBPOP ) for the additive harvest mortality model is calculated as { } t t s R t t w cBPOP cBPOP s (1 Rˆ ) Hˆ /(1 c) s 1 = + − − + γ where t cBPOP is the latitudeadjusted breeding population size in year t, s s and w s are the summer and winter survival rates, respectively, R γ is a biascorrection constant for the ageratio, c is the crippling loss rate, t Rˆ is the predicted ageratio, and t Hˆ is the predicted continental harvest. Discussion of t Rˆ and t Hˆ submodels are found in the following sections. The model uses the following constants: s s = 0.07, w s = 0.93, R γ = 0.8, and c = 0.20. The compensatory harvest mortality model serves as a hypothesis that stands in contrast to the additive harvest 3 2 1 0 1 2 3 2 1 0 1 2 0.00 0.05 0.10 0.15 0.20 0 1 2 3 4 5 6 7 8 x 105 N(B) z (environmental variable) Warm May No June Snow Cold May Lots of June Snow Model 1: DI Model 3: DD Propensity 0 1 2 3 4 5 6 7 8 x 105 24 mortality model, positing a strong but realistic degree of compensation. The compensatory model assumes that the mechanism for compensation is densitydependent postharvest (winter) survival. The form is a logistic relationship between winter survival and postharvest population size, with the relationship anchored around the historic mean values for each variable. For the compensatory model then, predicted winter survival rate in year t ( t s ) is calculated as [ ( ( )) ] 1 0 1 0 ( )1 = + − + − a+b P −P − t s s s s e t , where 1 s (upper asymptote) is 1.0, 0 s (lower asymptote) is 0.7, b (slope term) is 1.0, t P is the postharvest population size in year t (expressed in millions), P is the mean postharvest population size (4.295 million from 1974 through 2005), and a = logit 0 1 0 s s s s ⎛ − ⎞ ⎜ − ⎟ ⎝ ⎠ or ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − − ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − = 1 0 0 1 0 log 0 log 1 s s s s s s a s s , where s is 0.93 (mean winter survival rate). At moderate population size and latitude, the compensatory model allows for greater harvest (Fig. 10) than does the additive model (note especially that the size of the restrictive region [seasonwithinaseason] is smaller and is invoked when the latitude is higher). Also, 2 and 3bird bag limits are called for under more circumstances. But, at high population sizes, the higher bag limits are called for less often, because the compensatory model predicts that growth of the population will be slower (densitydependence). The fit to historic data was used to compare the additive and compensatory harvest models. From the t cBPOP , t mLAT , and observed harvest ( t H ) for the period 1974–through year t, the subsequent year’s breeding population size (on the latitudeadjusted scale) was predicted with both the additive and compensatory model, and compared to the observed breeding population size (on the latitudeadjusted scale). The meansquared error of the predictions from the additive model ( add MSE ) was calculated as: Σ = − − + = t t add add t t cBPOP cBPOP t MSE 1975 ( )2 ( 1975) 1 1 and the meansquared error of the predictions from the compensatory model were calculated in a similar manner. The model weights for the additive and compensatory model were calculated from their relative meansquared errors. The model weight for the additive model ( add W ) was calculated as: add comp add add MSE MSE W MSE 1 1 1 + = . The model weight for the compensatory model was found in a corresponding manner, or by subtracting the 25 additive model weight from 1.0. As of 2006, the compensatory model did not fit the historic data as well as the additive model; the model weights were 0.597 for the additive model and 0.403 for the compensatory model. The 2006 average model calls for a strategy that is intermediate between the additive and compensatory models (Fig. 10). Fig. 10. Statedependent harvest strategy for northern pintails with (A) additive, (B) compensatory, and (C) 2006 weighted models. In each case the strategy assumes that the general duck hunting season is that prescribed under the liberal regulatory alternative. 1 2 3 4 5 6 7 Average Latitude of the BPOP Closed Restrictive Liberal (1 bird) Liberal (3 birds) L2 51 52 53 54 55 56 57 58 59 1 2 3 4 5 6 7 Average Latitude of the BPOP Closed Restrictive Liberal (1 bird) Liberal (3 birds) L2 51 52 53 54 55 56 57 58 59 1 2 3 4 5 6 7 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive Liberal (1 bird) Liberal (3 birds) L2 51 52 53 54 55 56 57 58 59 (A) Additive Model (B) Compensatory Model (C) 2006 Weighted Model 26 Scaup The continental scaup (greater and lesser combined) population has experienced a longterm decline (Austin et al. 2000, Afton and Anderson 2001, Austin et al. 2006). As a result, waterfowl managers are challenged with the issue of how to manage the harvest of this declining population in the absence of an objective harvest strategy. In response to this dilemma, the USFWS Migratory Bird Regulations Committee requested that a scaup harvest strategy be developed for the 2007 regulations cycle. Here, we report on the development of a proposed decisionmaking framework to guide scaup harvest management. A detailed report is available online at http://www.fws.gov/migratorybirds/mgmt/ahm/specialtopics.htm. The lack of scaup demographic information over a sufficient timeframe and at a continental scale precludes the use of a traditional balance equation to represent scaup population and harvest dynamics. As a result, we used a discretetime, stochastic, logisticgrowth population model to represent changes in scaup abundance: ( (1 / ) ) . t 1 1 1 1 N N rN N K qH e t t t t t ε − − − − = + − − With this formulation, annual changes in population size (N) are predicted by the intrinsic rate of increase (r), the carrying capacity (K), a scaled harvest (H), and a process error (ε). We use a Bayesian approach (Meyer and Millar 1999) to estimate the population parameters, and to characterize the uncertainty associated with the monitoring programs (observation error) and the ability of our model to predict actual changes in the system (process error). Our initial assessment relied on the critical assumption that data used to estimate population parameters were measured on the same absolute scale. Research conducted to model waterfowl populations from different sources of information has provided evidence of bias in waterfowl survey programs (Martin et al. 1979, Runge et al. 2002). While the source(s) of this bias are not yet known, it is possible to estimate correction factors to reconcile predictions based on disparate sources of information. To address this issue, we chose to include an additional parameter (q) in our assessment to function as a scaling factor that enables us to combine breeding population and harvest estimates in an expression of population change. It is important to note that this parameter represents the combined limitations and uncertainty of all the monitoring data and functional relationships used in our assessment framework. Although, our initial attempts to estimate a scaling parameter from population and harvest data yielded reasonable estimates, the variance estimates were large. We found that the inclusion of a limited amount of scaup banding and recovery data provided enough information to structure the harvest process and reduce the uncertainty in the scaling parameter estimate. As in past analyses, the state space formulation and Bayesian analysis framework provided reasonable fits to the observed breeding population and total harvest estimates with realistic measures of variation. The posterior mean harvest rate estimates ranged from 0.03 to 0.08. In general, harvest rates fluctuated over the first decade and then tracked the declining population trend until the early 1990’s, when harvest rate estimates increased significantly before dropping in 1999 (Fig. 11). The posterior mean estimate of the intrinsic rate of increase (r) is 0.110 while the posterior mean estimate of the carrying capacity (K) is 8.236 million birds (Table 1). The posterior mean estimate of the scaling parameter (q) is 0.541, ranging between 0.461 and 0.630 with 95% probability. Based on the estimated population parameters, the estimated average maximum sustainable yield (MSY) on the adjusted scale is 0.211 million scaup (0.389 million scaup on the observed scale). 27 Fig. 11. The posterior mean scaup population and harvest rate estimates derived from a Bayesian analysis of the modified logistic model. We used SDP software (Lubow 1995) to derive a statedependent harvest strategy under an objective to maximize longterm cumulative harvest (MSY) and an objective to attain a shoulder point (calculated as percentage of MSY) on the yield curve. We evaluated harvest levels from 0 to 5 million (in increments of 50,000) for population sizes of 1 to 10 million (in increments of 50,000) and harvest objectives ranging from 90 to 100% MSY (in 2 % increments). For each optimization we assumed perfect control over the harvest decision variable. We then simulated each strategy for 5000 iterations to characterize the management performance expected if the harvest strategy was followed and system dynamics did not change. Under an objective to maximize longterm cumulative harvest (MSY) the resulting strategy is extremely knifeedged (Fig. 12). This strategy prescribes zero harvests for population sizes less than 3.2 million and seeks to hold the population size at maximum productivity (one half the carrying capacity). In contrast to the MSY strategy, the harvest strategies necessary to achieve a shoulder point are considerably less knifeedged and would allow for harvest at lower population sizes. However, current scaup harvest levels (317,000) exceed the prescribed harvests resulting from optimizations with each of the objective functions we evaluated. The simulated management performance of each harvest strategy demonstrates the tradeoffs that arise when a shoulder point objective is used to derive an optimal harvest strategy. As the desired shoulder point moves away from MSY, average harvest levels decrease while the average population increases. The USFWS intends to work with the Flyways over the next year to determine an acceptable harvestmanagement objective and a set of regulatory alternatives that would be used in conjunction with our modeling framework to derive an optimal harvest strategy for scaup. 1975 1985 1995 2005 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Year Population X 10^6 Population 0.03 0.04 0.05 0.06 0.07 0.08 Harvest Rate Harvest Rate 28 Fig. 12. Optimal harvests of scaup as a function of the observed breeding population size derived under objective functions ranging from 90 to 100 percent of the maximum longterm yield (MSY). 0 1 2 3 4 5 6 7 8 9 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 BPOP Harvest MSY 98% MSY 96% MSY 95% MSY 94% MSY 92% MSY 90% MSY 29 LITERATURE CITED Afton, A. D., and M. G. Anderson. 2001. Declining scaup populations: a retrospective analysis of longterm population and harvest survey data. Journal of Wildlife Management 60:8393. Anderson, D. R., and K. P. Burnham. 1976. Population ecology of the mallard. VI. The effect of exploitation on survival. U.S. Fish and Wildlife Service Resource Publication No. 128. 66pp. Austin, J. E., A. D. Afton, M. G. Anderson, R. G. Clark, C. M. Custer, J. S. Lawrence, J. B. Pollard and J. K. Ringelman. 2000. Declining scaup populations: issues, hypotheses, and research needs. Wildlife Society Bulletin 28:254263. Austin, J. E., M. J. Anteau, J. S. Barclay, G. S. Boomer, F. C. Rohwer, and S. M. Slattery. 2006. Declining scaup populations: reassessment of the issues, hypotheses, and research directions. Consensus Report from the Second Scaup Workshop. 7pp. Blohm, R. J. 1989. Introduction to harvest  understanding surveys and season setting. Proceedings of the International Waterfowl Symposium 6:118133. Blohm, R. J., R. E. Reynolds, J. P. Bladen, J. D. Nichols, J. E. Hines, K. P. Pollock, and R. T. Eberhardt. 1987. Mallard mortality rates on key breeding and wintering areas. Transactions of the North American Wildlife and Natural Resources Conference 52:246263. Burnham, K. P., G. C. White, and D. R. Anderson. 1984. Estimating the effect of hunting on annual survival rates of adult mallards. Journal of Wildlife Management 48:350361. Conroy, M. J., M. W. Miller, and J. E. Hines. 2002. Identification and synthetic modeling of factors affecting American black duck populations. Wildlife Monographs 150. 64pp. Heusman, H W, and J. R. Sauer. 2000. The northeastern states’ waterfowl breeding population survey. Wildlife Society Bulletin 28:355364. Johnson, F. A. 2003. Population dynamics of ducks other than mallards in midcontinent North America. Draft. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 15pp. Johnson, F. A., J. A. Dubovsky, M. C. Runge, and D. R. Eggeman. 2002a. A revised protocol for the adaptive harvest management of eastern mallards. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 13pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/emalahm2002.pdf. Johnson, F. A., W. L. Kendall, and J. A. Dubovsky. 2002b. Conditions and limitations on learning in the adaptive management of mallard harvests. Wildlife Society Bulletin 30:176185. Johnson, F. A., C. T. Moore, W. L. Kendall, J. A. Dubovsky, D. F. Caithamer, J. R. Kelley, Jr., and B. K. Williams. 1997. Uncertainty and the management of mallard harvests. Journal of Wildlife Management 61:202216. Johnson, F. A., and B. K. Williams. 1999. Protocol and practice in the adaptive management of waterfowl harvests. Conservation Ecology 3(1): 8. [online] URL: http://www.consecol.org/vol3/iss1/art8. Johnson, F. A., B. K. Williams, J. D. Nichols, J. E. Hines, W. L. Kendall, G. W. Smith, and D. F. Caithamer. 1993. Developing an adaptive management strategy for harvesting waterfowl in North America. Transactions of the North American Wildlife and Natural Resources Conference 58:565583. 30 Johnson, F. A., B. K. Williams, and P. R. Schmidt. 1996. Adaptive decisionmaking in waterfowl harvest and habitat management. Proceedings of the International Waterfowl Symposium 7:2633. Link, W. A., J. R. Sauer, and D. K. Niven. 2006. A hierarchical model for regional analysis of population change using Christmas bird count data, with application to the American black duck. The Condor 108:1324. Lubow, B. C. 1995. SDP: Generalized software for solving stochastic dynamic optimization problems. Wildlife Society Bulletin 23:738742. Martin, F. W., R. S. Pospahala, and J. D. Nichols. 1979. Assessment and population management of North American migratory birds. Pages 187239 in J. Cairns, G. P. Patil, and W. E. Waters, eds., Environmental biomonitoring, assessment, prediction and management — certain case studies and related quantitative issues. Statistical Ecology, Vol. S11. International Cooperative Publishing House, Fairland, MD. Meyer, R., and R. B. Millar. 1999. BUGS in Bayesian stock assessments. Canadian Journal of Fisheries and Aquatic Sciences 56:10781086. Munro, R. E., and C. F. Kimball. 1982. Population ecology of the mallard. VII. Distribution and derivation of the harvest. U.S. Fish and Wildlife Service Resource Publication 147. 127pp. Nichols, J. D., F. A. Johnson, and B. K. Williams. 1995. Managing North American waterfowl in the face of uncertainty. Annual Review of Ecology and Systematics 26:177199. Runge, M. C., F. A. Johnson, J. A. Dubovsky, W. L. Kendall, J. Lawrence, and J. Gammonley. 2002. A revised protocol for the adaptive harvest management of midcontinent mallards. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 28pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/MCMrevise2002.pdf. U.S. Fish and Wildlife Service. 2000. Adaptive harvest management: 2000 duck hunting season. U.S. Dept. Interior, Washington. D.C. 43pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm00/ahm2000.pdf. U.S. Fish and Wildlife Service. 2001. Frameworkdate extensions for duck hunting in the United States: projected impacts & coping with uncertainty, U.S. Dept. Interior, Washington, D.C. 8pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm01/fwassess.pdf. U.S. Fish and Wildlife Service. 2002. Adaptive harvest management: 2002 duck hunting season. U.S. Dept. Interior, Washington. D.C. 34pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/2002AHMreport.pdf. Walters, C. J. 1986. Adaptive management of renewable resources. MacMillan Publ. Co., New York, N.Y. 374pp. Williams, B. K., and F. A. Johnson. 1995. Adaptive management and the regulation of waterfowl harvests. Wildlife Society Bulletin 23:430436. Williams, B. K., F. A. Johnson, and K. Wilkins. 1996. Uncertainty and the adaptive management of waterfowl harvests. Journal of Wildlife Management 60:223232. 31 APPENDIX A: AHM Working Group (Note: This list includes only permanent members of the AHM Working Group. Not listed here are numerous persons from federal and state agencies that assist the Working Group on an adhoc basis.) Coordinator: Fred Johnson U.S. Fish & Wildlife Service Bldg. 810, University of Florida P.O. Box 110485 Gainesville, FL 32611 phone: 3523925075 fax: 3528460841 email: fred_a_johnson@fws.gov USFWS Representatives: Bob Blohm (Region 9) U.S. Fish and Wildlife Service 4401 N Fairfax Drive MS MSP4107 Arlington, VA 22203 phone: 7033581966 fax: 7033582272 email: robert_blohm@fws.gov Brad Bortner (Region 1) U.S. Fish and Wildlife Service 911 NE 11th Ave. Portland, OR 972324181 phone: 5032316164 fax: 5032312364 email: brad_bortner@fws.gov Dave Case (contractor) D.J. Case & Associates 607 Lincolnway West Mishawaka, IN 46544 phone: 5742580100 fax: 5742580189 email: dave@djcase.com Jim Dubovsky (Region 6) U.S. Fish and Wildlife Service P.O. Box 25486DFC Denver, CO 802250486 phone: 3032364403 fax: 3032368680 email:james_dubovsky@fws.gov Jeff Haskins (Region 2) U.S. Fish and Wildlife Service P.O. Box 1306 Albuquerque, NM 87103 phone: 5052486827 (ext 30) fax: 5052487885 email: jeff_haskins@fws.gov Jim Kelley (Region 9) U.S. Fish and Wildlife Service 1 Federal Drive Fort Snelling, MN 551110458 phone: 6127135409 fax: 6127135393 email: james_r_kelley@fws.gov Sean Kelly (Region 3) U.S. Fish and Wildlife Service 1 Federal Drive Ft. Snelling, MN 551114056 phone: 6127135470 fax: 6127135393 email: sean_kelly@fws.gov Paul Padding (Region 9) U.S. Fish and Wildlife Service 11510 American Holly Drive Laurel, MD 20708 phone: 3014975851 fax: 3014975885 email: paul_padding@fws.gov 32 Diane Pence (Region 5) U.S. Fish and Wildlife Service 300 Westgate Center Drive Hadley, MA 010359589 phone: 4132538577 fax: 4132538424 email: diane_pence@fws.gov Russ Oates (Region 7) U.S. Fish and Wildlife Service 1011 East Tudor Road Anchorage, AK 995036119 phone: 9077863446 fax: 9077863641 email: russ_oates@fws.gov Dave Sharp (Region 9) U.S. Fish and Wildlife Service P.O. Box 25486, DFC Denver, CO 802250486 phone: 3032752386 fax: 3032752384 email: dave_sharp@fws.gov Bob Trost (Region 9) U.S. Fish and Wildlife Service 911 NE 11th Ave. Portland, OR 972324181 phone: 5032316162 fax: 5032316228 email: robert_trost@fws.gov David Viker (Region 4) U.S. Fish and Wildlife Service 1875 Century Blvd., Suite 345 Atlanta, GA 30345 phone: 4046797188 fax: 4046797285 email: david_viker@fws.gov Canadian Wildlife Service Representatives: Dale Caswell Canadian Wildlife Service 123 Main St. Suite 150 Winnepeg, Manitoba, Canada R3C 4W2 phone: 2049835260 fax: 2049835248 email: dale.caswell@ec.gc.ca Eric Reed Canadian Wildlife Service 351 St. Joseph Boulevard Hull, QC K1A OH3, Canada phone: 8199530294 fax: 8199536283 email: eric.reed@ec.gc.ca Flyway Council Representatives: Scott Baker (Mississippi Flyway) Mississippi Dept. of Wildlife, Fisheries, and Parks P.O. Box 378 Redwood, MS 39156 phone: 6016610294 fax: 6013642209 email: mahannah1@aol.com Diane Eggeman (Atlantic Flyway) Florida Fish and Wildlife Conservation Commission 8932 Apalachee Pkwy. Tallahassee, FL 32311 phone: 8504885878 fax: 8504885884 email: diane.eggeman@fwc.state.fl.us Mike Johnson (Central Flyway) North Dakota Game and Fish Department 100 North Bismarck Expressway Bismarck, ND 585015095 phone: 7013286319 fax: 7013286352 email: mjohnson@state.nd.us Don Kraege (Pacific Flyway) Washington Dept. of Fish and Wildlife 600 Capital Way North Olympia. WA 985011091 phone: 3609022509 fax: 3609022162 email: kraegdkk@dfw.wa.gov 33 Bryan Swift (Atlantic Flyway) Dept. Environmental Conservation 625 Broadway Albany, NY 122334754 phone: 5184028866 fax: 5184029027 or 4028925 email: blswift@gw.dec.state.ny.us Mark Vrtiska (Central Flyway) Nebraska Game and Parks Commission P.O. Box 30370 2200 North 33rd Street Lincoln, NE 685031417 phone: 4024715437 fax: 4024715528 email: mvrtiska@ngpc.state.ne.us Dan Yparraguirre (Pacific Flyway) California Dept. of Fish and Game 1812 Ninth Street Sacramento, CA 95814 phone: 9164453685 email: dyparraguirre@dfg.ca.gov Guy Zenner (Mississippi Flyway) Iowa Dept. of Natural Resources 1203 North Shore Drive Clear Lake, IA 50428 phone: 5153573517, ext. 23 fax: 5153575523 email: gzenner@netins.net 34 APPENDIX B: Midcontinent Mallard Models Model Structure In 2002 we extensively revised the set of alternative models describing the population dynamics of midcontinent mallards (Runge et al. 2002, USFWS 2002). Collectively, the models express uncertainty (or disagreement) about whether harvest is an additive or compensatory form of mortality (Burnham et al. 1984), and whether the reproductive process is weakly or strongly densitydependent (i.e., the degree to which reproductive rates decline with increasing population size). All population models for midcontinent mallards share a common “balance equation” to predict changes in breedingpopulation size as a function of annual survival and reproductive rates: Nt Nt (mSt AM ( m)(St AF Rt (St JF St JM F ))) sum M sum + = + − + + 1 , 1 , , , φ φ where: N = breeding population size, m = proportion of males in the breeding population, SAM, SAF, SJF, and SJM = survival rates of adult males, adult females, young females, and young males, respectively, R = reproductive rate, defined as the fall age ratio of females, φ F φ sum M sum = the ratio of female (F) to male (M) summer survival, and t = year. We assumed that m and φ F φ sum M sum are fixed and known. We also assumed, based in part on information provided by Blohm et al. (1987), the ratio of female to male summer survival was equivalent to the ratio of annual survival rates in the absence of harvest. Based on this assumption, we estimated φ F φ sum M sum = 0.897. To estimate m we expressed the balance equation in matrix form: N N S RS S RS N N t AM t AF AM JM F sum M sum AF JF t AM t AF + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 1 0 , , , , φ φ and substituted the constant ratio of summer survival and means of estimated survival and reproductive rates. The right eigenvector of the transition matrix is the stable sex structure that the breeding population eventually would attain with these constant demographic rates. This eigenvector yielded an estimate of m = 0.5246. Using estimates of annual survival and reproductive rates, the balance equation for midcontinent mallards overpredicted observed population sizes by 10.8% on average. The source of the bias is unknown, so we modified the balance equation to eliminate the bias by adjusting both survival and reproductive rates: Nt S Nt (mSt AM ( m)(St AF RRt (St JF St JM F ))) sum M sum + 1 = γ , + 1− , + γ , + , φ φ where γ denotes the biascorrection factors for survival (S) and reproduction (R). We used a least squares approach to estimate γS = 0.9479 and γR = 0.8620. 35 Survival Process We considered two alternative hypotheses for the relationship between annual survival and harvest rates. For both models, we assumed that survival in the absence of harvest was the same for adults and young of the same sex. In the model where harvest mortality is additive to natural mortality: St sex age s sex ( K ) A , , , t ,sex,age = − 0 1 and in the model where changes in natural mortality compensate for harvest losses (up to some threshold): S s if K s t sex age K if K s sex C t sex age sex C t sex age t sex age sex , , C , ,, , , , , , , = ≤ − − > − ⎧⎨ ⎪ ⎩⎪ 0 0 0 1 1 1 where s0 = survival in the absence of harvest under the additive (A) or compensatory (C) model, and K = harvest rate adjusted for crippling loss (20%, Anderson and Burnham 1976). We averaged estimates of s0 across banding reference areas by weighting by breedingpopulation size. For the additive model, s0 = 0.7896 and 0.6886 for males and females, respectively. For the compensatory model, s0 = 0.6467 and 0.5965 for males and females, respectively. These estimates may seem counterintuitive because survival in the absence of harvest should be the same for both models. However, estimating a common (but still sexspecific) s0 for both models leads to alternative models that do not fit available bandrecovery data equally well. More importantly, it suggests that the greatest uncertainty about survival rates is when harvest rate is within the realm of experience. By allowing s0 to differ between additive and compensatory models, we acknowledge that the greatest uncertainty about survival rate is its value in the absence of harvest (i.e., where we have no experience). Reproductive Process Annual reproductive rates were estimated from age ratios in the harvest of females, corrected using a constant estimate of differential vulnerability. Predictor variables were the number of ponds in May in Prairie Canada (P, in millions) and the size of the breeding population (N, in millions). We estimated the bestfitting linear model, and then calculated the 80% confidence ellipsoid for all model parameters. We chose the two points on this ellipsoid with the largest and smallest values for the effect of breedingpopulation size, and generated a weakly densitydependent model: Rt = 0.7166 + 0.1083Pt − 0.0373Nt and a strongly densitydependent model: Rt = 1.1390 + 0.1376Pt − 0.1131Nt Pond Dynamics We modeled annual variation in Canadian pond numbers as a firstorder autoregressive process. The estimated model was: P P t + t t = + + 1 2.2127 0.3420 ε where ponds are in millions and εt is normally distributed with mean = 0 and variance = 1.2567. 36 Variance of Prediction Errors Using the balance equation and submodels described above, predictions of breedingpopulation size in year t+1 depend only on specification of population size, pond numbers, and harvest rate in year t. For the period in which comparisons were possible, we compared these predictions with observed population sizes. We estimated the predictionerror variance by setting: ( ) ( ) ( ) [ ( ) ( )] ( ) e N N e N N N n t t obs t pre t t obs t pre t = − = Σ − − ln ln ~ , $ ln ln then assuming and estimating 0 1 σ 2 σ 2 2 where obs and pre are observed and predicted population sizes (in millions), respectively, and n = the number of years being compared. We were concerned about a variance estimate that was too small, either by chance or because the number of years in which comparisons were possible was small. Therefore, we calculated the upper 80% confidence limit for σ2 based on a Chisquared distribution for each combination of the alternative survival and reproductive submodels, and then averaged them. The final estimate of σ2 was 0.0243, equivalent to a coefficient of variation of about 17%. Model Implications The set of alternative population models suggests that carrying capacity (average population size in the absence of harvest) for an average number of Canadian ponds is somewhere between about 6 and 16 million mallards. The population model with additive hunting mortality and weakly densitydependent recruitment (SaRw) leads to the most conservative harvest strategy, whereas the model with compensatory hunting mortality and strongly densitydependent recruitment (ScRs) leads to the most liberal strategy. The other two models (SaRs and ScRw) lead to strategies that are intermediate between these extremes. Under the models with compensatory hunting mortality (ScRs and ScRw), the optimal strategy is to have a liberal regulation regardless of population size or number of ponds because at harvest rates achieved under the liberal alternative, harvest has no effect on population size. Under the strongly densitydependent model (ScRs), the densitydependence regulates the population and keeps it within narrow bounds. Under the weakly densitydependent model (ScRw), the densitydependence does not exert as strong a regulatory effect, and the population size fluctuates more. Model Weights Model weights are calculated as Bayesian probabilities, reflecting the relative ability of the individual alternative models to predict observed changes in population size. The Bayesian probability for each model is a function of the model’s previous (or prior) weight and the likelihood of the observed population size under that model. We used Bayes’ theorem to calculate model weights from a comparison of predicted and observed population sizes for the years 19962004, starting with equal model weights in 1995. For the purposes of updating, we predicted breedingpopulation size in the traditional survey area in year t + 1, from breedingpopulation size, Canadian ponds, and harvest rates in year t. 37 Inclusion of Mallards in the Great Lakes Region Model development originally did not include mallards breeding in the states of Wisconsin, Minnesota, and Michigan, primarily because full data sets were not available from these areas to permit the necessary analysis. However, mallards in the Great Lakes region have been included in the midcontinent mallard AHM protocol since 1997 by assuming that population dynamics for these mallards are similar to those in the traditional survey area. Based on that assumption, predictions of breeding population size are scaled to reflect inclusion of mallards in the Great Lakes region. From 1992 through 2007, when population estimates were available for all three states, the average proportion of the total midcontinent mallard population that was in the Great Lakes region was 0.1099 (SD = 0.0207). We assumed a normal distribution with these parameter values to make the conversion between the traditional survey area and total breedingpopulation size. 38 APPENDIX C: Eastern Mallard Models Model Structure We also revised the population models for eastern mallards in 2002 (Johnson et al. 2002a, USFWS 2002). The current set of six models: (1) relies solely on federal and state waterfowl surveys (rather than the Breeding Bird Survey) to estimate abundance; (2) allows for the possibility of a positive bias in estimates of survival or reproductive rates; (3) incorporates competing hypotheses of strongly and weakly densitydependent reproduction; and (4) assumes that hunting mortality is additive to other sources of mortality. As with midcontinent mallards, all population models for eastern mallards share a common balance equation to predict changes in breedingpopulation size as a function of annual survival and reproductive rates: Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S )) am t af t m t ym t m t yf + = ⋅ ⋅ + − ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 1 1 ψ where: N = breedingpopulation size, p = proportion of males in the breeding population, Sam, Saf, Sym, and Syf = survival rates of adult males, adult females, young males, and young females, respectively, Am = ratio of young males to adult males in the harvest, d = ratio of young male to adult male direct recovery rates, ψ = the ratio of male to female summer survival, and t = year. In this balance equation, we assume that p, d, and ψ are fixed and known. The parameter ψ is necessary to account for the difference in anniversary date between the breedingpopulation survey (May) and the survival and reproductive rate estimates (August). This model also assumes that the sex ratio of fledged young is 1:1; hence Am/d appears twice in the balance equation. We estimated d = 1.043 as the median ratio of young:adult male bandrecovery rates in those states from which wing receipts were obtained. We estimated ψ = 1.216 by regressing through the origin estimates of male survival against female survival in the absence of harvest, assuming that differences in natural mortality between males and females occur principally in summer. To estimate p, we used a population projection matrix of the form: ( ) ( ) ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ ⋅ + ⋅ = ⎥⎦ ⎤ ⎢⎣ ⎡ + + t t m yf af am m ym t t F M A d S S S A d S F M ψ 0 1 1 where M and F are the relative number of males and females in the breeding populations, respectively. To parameterize the projection matrix we used average annual survival rate and age ratio estimates, and the estimates of d and ψ provided above. The right eigenvector of the projection matrix is the stable proportion of males and females the breeding population eventually would attain in the face of constant demographic rates. This eigenvector yielded an estimate of p = 0.544. We also attempted to determine whether estimates of survival and reproductive rates were unbiased. We relied on the balance equation provided above, except that we included additional parameters to correct for any bias that might exist. Because we were unsure of the source(s) of potential bias, we alternatively assumed that any bias resided solely in survival rates: Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S )) am t af t m t ym t m t yf + = ⋅ ⋅ ⋅ + − ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 1 Ω 1 ψ 39 (where Ω is the biascorrection factor for survival rates), or solely in reproductive rates: Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S )) am t af t m t ym t m t yf + = ⋅ ⋅ + − ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ 1 1 α α ψ (where α is the biascorrection factor for reproductive rates). We estimated Ω and α by determining the values of these parameters that minimized the sum of squared differences between observed and predicted population sizes. Based on this analysis, Ω = 0.836 and α = 0.701, suggesting a positive bias in survival or reproductive rates. However, because of the limited number of years available for comparing observed and predicted population sizes, we also retained the balance equation that assumes estimates of survival and reproductive rates are unbiased. Survival Process For purposes of AHM, annual survival rates must be predicted based on the specification of regulationspecific harvest rates (and perhaps on other uncontrolled factors). Annual survival for each age (i) and sex (j) class under a given regulatory alternative is: ( ) ( ) S h v c t i j j t am i j , , = ⋅ − ⋅ − ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ θ 1 1 where: S = annual survival, θ j = mean survival from natural causes, ham = harvest rate of adult males, and v = harvest vulnerability relative to adult males, c = rate of crippling (unretrieved harvest). This model assumes that annual variation in survival is due solely to variation in harvest rates, that relative harvest vulnerability of the different agesex classes is fixed and known, and that survival from natural causes is fixed at its sample mean. We estimated θ j = 0.7307 and 0.5950 for males and females, respectively. Reproductive process As with survival, annual reproductive rates must be predicted in advance of setting regulations. We relied on the apparent relationship between breedingpopulation size and reproductive rates: Rt = a ⋅ exp(b ⋅ Nt ) where Rt is the reproductive rate (i.e., Am d t ), Nt is breedingpopulation size in millions, and a and b are model parameters. The leastsquares parameter estimates were a = 2.508 and b = 0.875. Because of both the importance and uncertainty of the relationship between population size and reproduction, we specified two alternative models in which the slope (b) was fixed at the leastsquares estimate ± one standard error, and in which the intercepts (a) were subsequently reestimated. This provided alternative hypotheses of strongly densitydependent (a = 4.154, b = 1.377) and weakly densitydependent reproduction (a = 1.518, b = 0.373). 40 Variance of Prediction Errors Using the balance equations and submodels provided above, predictions of breedingpopulation size in year t+1 depend only on the specification of a regulatory alternative and on an estimate of population size in year t. For the period in which comparisons were possible (199196), we were interested in how well these predictions corresponded with observed population sizes. In making these comparisons, we were primarily concerned with how well the biascorrected balance equations and reproductive and survival submodels performed. Therefore, we relied on estimates of harvest rates rather than regulations as model inputs. We estimated the predictionerror variance by setting: ( ) ( ) ( ) [ ( ) ( )] e N N e N N N n t t obs t pre t t obs t pre t = − = Σ ��� ln ln ~ , $ ln ln then assuming and estimating 0 σ 2 σ 2 2 where obs and pre are observed and predicted population sizes (in millions), respectively, and n = 6. Variance estimates were similar regardless of whether we assumed that the bias was in reproductive rates or in survival, or whether we assumed that reproduction was strongly or weakly densitydependent. Thus, we averaged variance estimates to provide a final estimate of σ2 = 0.006, which is equivalent to a coefficient of variation (CV) of 8.0%. We were concerned, however, about the small number of years available for estimating this variance. Therefore, we estimated an 80% confidence interval for σ2 based on a Chisquared distribution and used the upper limit for σ2 = 0.018 (i.e., CV = 14.5%) to express the additional uncertainty about the magnitude of prediction errors attributable to potentially important environmental effects not expressed by the models. Model Implications Modelspecific regulatory strategies based on the hypothesis of weakly densitydependent reproduction are considerably more conservative than those based on the hypothesis of strongly densitydependent reproduction. The three models with weakly densitydependent reproduction suggest a carrying capacity (i.e., average population size in the absence of harvest) >2.0 million mallards, and prescribe extremely restrictive regulations for population size <1.0 million. The three models with strongly densitydependent reproduction suggest a carrying capacity of about 1.5 million mallards, and prescribe liberal regulations for population sizes >300 thousand. Optimal regulatory strategies are relatively insensitive to whether models include a bias correction or not. All modelspecific regulatory strategies are “knifeedged,” meaning that large differences in the optimal regulatory choice can be precipitated by only small changes in breedingpopulation size. This result is at least partially due to the small differences in predicted harvest rates among the current regulatory alternatives (see the section on Regulatory Alternatives later in this report). Model Weights We used Bayes’ theorem to calculate model weights from a comparison of predicted and observed population sizes for the years 19962006. We calculated weights for the alternative models based on an assumption of equal model weights in 1996 (the last year data was used to develop most model components) and on estimates of yearspecific harvest rates (Appendix D). 41 APPENDIX D: Modeling Mallard Harvest Rates We modeled harvest rates of midcontinent mallards within a Bayesian hierarchical framework. We developed a set of models to predict harvest rates under each regulatory alternative as a function of the harvest rates observed under the liberal alternative, using historical information relating harvest rates to various regulatory alternatives. We modeled the probability of regulationspecific harvest rates (h) based on normal distributions with the following parameterizations: Closed: Restrictive: Moderate: Liberal: p h N p h N p h N p h N C C C R R L R M M L f M L L f L ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) μ ν γ μ ν γ μ δ ν μ δ ν 2 2 2 2 + + For the restrictive and moderate alternatives we introduced the parameter γ to represent the relative difference between the harvest rate observed under the liberal alternative and the moderate or restrictive alternatives. Based on this parameterization, we are making use of the information that has been gained (under the liberal alternative) and are modeling harvest rates for the restrictive and moderate alternatives as a function of the mean harvest rate observed under the liberal alternative. For the harvestrate distributions assumed under the restrictive and moderate regulatory packages, we specified that γR and γM are equal to the prior estimates of the predicted mean harvest rates under the restrictive and moderate alternatives divided by the prior estimates of the predicted mean harvest rates observed under the liberal alternative. Thus, these parameters act to scale the mean of the restrictive and moderate distributions in relation to the mean harvest rate observed under the liberal regulatory alternative. We also considered the marginal effect of frameworkdate extensions under the moderate and liberal alternatives by including the parameter δf. In order to update the probability distributions of harvest rates realized under each regulatory alternative, we first needed to specify a prior probability distribution for each of the model parameters. These distributions represent prior beliefs regarding the relationship between each regulatory alternative and the expected harvest rates. We used a normal distribution to represent the mean and a scaled inversechisquare distribution to represent the variance of the normal distribution of the likelihood. For the mean (μ) of each harvestrate distribution associated with each regulatory alternative, we use the predicted mean harvest rates provided in USFWS (2000a:1314), assuming uniformity of regulatory prescriptions across flyways. We set prior values of each standard deviation (ν) equal to 20% of the mean (CV = 0.2) based on an analysis by Johnson et al. (1997). We then specified the following prior distributions and parameter values under each regulatory package: Closed (in U.S. only): p N p ScaledInv C C ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 0 0088 0 0018 6 6 0 0018 2 2 2 2 − These closedseason parameter values are based on observed harvest rates in Canada during the 198893 seasons, which was a period of restrictive regulations in both Canada and the United States. For the restrictive and moderate alternatives, we specified that the standard error of the normal distribution of the scaling parameter is based on a coefficient of variation for the mean equal to 0.3. The scale parameter of the inversechisquare distribution was set equal to the standard deviation of the harvest rate mean under the restrictive and moderate regulation alternatives (i.e., CV = 0.2). 42 Restrictive: p N p Scaled Inv R R ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 051 015 6 6 0 0133 2 2 2 2 − Moderate: p N p ScaledInv M M ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 085 026 6 6 0 0223 2 2 2 2 − Liberal: p N p ScaledInv L L ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 01305 0 0261 6 6 0 0261 2 2 2 2 − The prior distribution for the marginal effect of the frameworkdate extension was specified as: p( ) N( ) f δ ~ 0.02,0.012 The prior distributions were multiplied by the likelihood functions based on the last seven years of data under liberal regulations, and the resulting posterior distributions were evaluated with Markov Chain Monte Carlo simulation. Posterior estimates of model parameters and of annual harvest rates are provided in the following table: Parameter Estimate SD Parameter Estimate SD μC 0.0088 0.0022 h1998 0.1098 0.0113 νC 0.0019 0.0005 h1999 0.1002 0.0076 γR 0.5090 0.0617 h2000 0.1252 0.0099 νR 0.0129 0.0033 h2001 0.1068 0.0112 γM 0.8530 0.1062 h2002 0.1145 0.0057 νM 0.0216 0.0055 h2003 0.1100 0.0064 μL 0.1139 0.0070 h2004 0.11188 0.0098 νL 0.0205 0.0055 h2005 0.1158 0.0081 δf 0.0087 0.0079 h2006 0.1061 0.0073 43 We modeled harvest rates of eastern mallards using the same parameterizations as those for midcontinent mallards: Closed: Restrictive: Moderate: Liberal: p h N p h N p h N p h N C C C R R L R M M L f M L L f L ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) μ ν γ μ ν γ μ δ ν μ δ ν 2 2 2 2 + + We set prior values of each standard deviation (ν) equal to 30% of the mean (CV = 0.3) to account for additional variation due to changes in regulations in the other Flyways and their unpredictable effects on the harvest rates of eastern mallards. We then specified the following prior distribution and parameter values for the liberal regulatory alternative: Liberal: p N p ScaledInv L L ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 01771 0 0531 6 6 0 0531 2 2 2 2 − Moderate: p N p ScaledInv M M ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 092 028 6 6 0 0488 2 2 2 2 − Restrictive: p N p ScaledInv R R ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 076 028 6 6 0 0406 2 2 2 2 − Closed (in U.S. only): p N p ScaledInv C C ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 0 0800 0 0240 6 6 0 0240 2 2 2 2 − A previous analysis suggested that the effect of the frameworkdate extension on eastern mallards would be of lower magnitude and more variable than on midcontinent mallards (USFWS 2000). Therefore, we specified the following prior distribution for the marginal effect of the frameworkdate extension for eastern mallards as: p( ) N( ) f δ ~ 0.01,0.012 44 The prior distributions were multiplied by the likelihood functions based on the last four years of data under liberal regulations, and the resulting posterior distributions were evaluated with Markov Chain Monte Carlo simulation. Posterior estimates of model parameters and of annual harvest rates are provided in the following table: Parameter Estimate SD Parameter Estimate SD μC 0.0798 0.0262 h2002 0.1627 0.0129 νC 0.0233 0.0059 h2003 0.1462 0.0104 γR 0.7642 0.1144 h2004 0.1364 0.0114 νR 0.0395 0.0104 h2005 0.1311 0.0120 γM 0.9187 0.1148 h2006 0.1048 0.0134 νM 0.0474 0.0121 μL 0.1532 0.0166 νL 0.0459 0.0099 δf 0.0046 0.0096
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Title  Adaptive harvest management 2007 duck hunting season 
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Description  adaptharvestmallards2007.pdf 
FWS Resource Links  http://library.fws.gov 
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Document Birds 
Publisher  U.S. Fish and Wildlife Service 
Date of Original  2007 
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Transcript  Adaptive Harvest Management 2007 Hunting Season U.S. Fish & Wildlife Service 1 Adaptive Harvest Management 2007 Hunting Season PREFACE The process of setting waterfowl hunting regulations is conducted annually in the United States (Blohm 1989). This process involves a number of meetings where the status of waterfowl is reviewed by the agencies responsible for setting hunting regulations. In addition, the U.S. Fish and Wildlife Service (USFWS) publishes proposed regulations in the Federal Register to allow public comment. This document is part of a series of reports intended to support development of harvest regulations for the 2007 hunting season. Specifically, this report is intended to provide waterfowl managers and the public with information about the use of adaptive harvest management (AHM) for setting waterfowl hunting regulations in the United States. This report provides the most current data, analyses, and decisionmaking protocols. However, adaptive management is a dynamic process and some information presented in this report will differ from that in previous reports. ACKNOWLEDGMENTS A working group comprised of representatives from the USFWS, the U.S. Geological Survey (USGS), the Canadian Wildlife Service (CWS), and the four Flyway Councils (Appendix A) was established in 1992 to review the scientific basis for managing waterfowl harvests. The working group, supported by technical experts from the waterfowl management and research communities, subsequently proposed a framework for adaptive harvest management, which was first implemented in 1995. The USFWS expresses its gratitude to the AHM Working Group and to the many other individuals, organizations, and agencies that have contributed to the development and implementation of AHM. This report was prepared by the USFWS Division of Migratory Bird Management. F. A. Johnson and G. S. Boomer were the principal authors. Individuals that provided essential information or otherwise assisted with report preparation were P. Garrettson (USFWS), T. Liddick (USGS), M. Otto (USFWS), R. Raftovich (USFWS), A. Royle (USGS), M. Runge (USGS), T. Sanders (USFWS), and K. Wilkins (USFWS). Comments regarding this document should be sent to the Chief, Division of Migratory Bird Management  USFWS, 4401 North Fairfax Drive, MS MSP4107, Arlington, VA 22203. Citation: U.S. Fish and Wildlife Service. 2007. Adaptive Harvest Management: 2007 Hunting Season. U.S. Dept. Interior, Washington, D.C. 44pp. Available online at http://www.fws.gov/migratorybirds/mgmt/AHM/AHMintro.htm U.S. Fish & Wildlife Service 2 TABLE OF CONTENTS Executive Summary .............................................................................................................3 Background ..........................................................................................................................4 Mallard Stocks and Flyway Management............................................................................5 Mallard Population Dynamics..............................................................................................6 HarvestManagement Objectives .......................................................................................13 Regulatory Alternatives .....................................................................................................13 Optimal Regulatory Strategies ...........................................................................................16 Application of AHM Concepts to Other Stocks ................................................................17 Literature Cited ..................................................................................................................29 Appendix A: AHM Working Group .................................................................................31 Appendix B: Midcontinent Mallard Models ...................................................................34 Appendix C: Eastern Mallard Models ..............................................................................38 Appendix D: Modeling Mallard Harvest Rates ................................................................41 3 EXECUTIVE SUMMARY In 1995 the U.S. Fish and Wildlife Service (USFWS) implemented the Adaptive Harvest Management (AHM) program for setting duck hunting regulations in the United States. The AHM approach provides a framework for making objective decisions in the face of incomplete knowledge concerning waterfowl population dynamics and regulatory impacts. The current AHM protocol is based on the population dynamics and status of two mallard (Anas platyrhynchos) stocks. Midcontinent mallards are defined as those breeding in the socalled traditional survey area, plus the states of Michigan, Minnesota, and Wisconsin. The prescribed regulatory alternative for the Mississippi, Central, and Pacific Flyways depends exclusively on the status of these mallards. Eastern mallards are defined as those breeding in the states of Virginia northward into Vermont, and in survey strata located on the Canadian side of the St. Lawrence River. The regulatory choice for the Atlantic Flyway depends exclusively on the status of these mallards. Investigations of the population dynamics of western mallards and their potential effect on hunting regulations in the West are ongoing. Mallard population models account for an apparent positive bias in estimates of survival and reproductive rates, and also allow for alternative hypotheses concerning the effects of harvest and the environment in regulating population size. Modelspecific weights reflect the relative confidence in alternative hypotheses, and are updated annually using comparisons of predicted and observed population sizes. For midcontinent mallards, current model weights favor the weakly densitydependent reproductive hypothesis (90%). Evidence for the additivemortality hypothesis remains equivocal (60%). For eastern mallards, virtually all of the weight is on models that have corrections for bias in estimates of survival or reproductive rates. Model weights also favor the strongly densitydependent reproductive hypothesis (59%). By consensus, hunting mortality is assumed to be additive in eastern mallards. For the 2007 hunting season, the USFWS is considering the same regulatory alternatives as last year. The nature of the restrictive, moderate, and liberal alternatives has remained essentially unchanged since 1997, except that extended framework dates have been offered in the moderate and liberal alternatives since 2002. Harvest rates associated with each of the regulatory alternatives have been updated based on bandreporting rate studies conducted since 1998. Estimated harvest rates of adult males from the 20022006 liberal hunting seasons have averaged 0.113 (SE = 0.001) and 0.136 (SE = 0.010) for midcontinent and eastern mallards, respectively. The estimated marginal effect of frameworkdate extensions has been an increase in harvest rate of 0.009 (SD = 0.008) and 0.005 (SD = 0.010) for midcontinent and eastern mallards, respectively. Optimal regulatory strategies for the 2007 hunting season were calculated using: (1) harvestmanagement objectives specific to each mallard stock; (2) the 2007 regulatory alternatives; and (3) current population models and associated weights for midcontinent and eastern mallards. Based on this year’s survey results of 9.05 million midcontinent mallards, 5.04 million ponds in Prairie Canada, and 907 thousand eastern mallards, the optimal choice for all four Flyways is the liberal regulatory alternative. AHM concepts and tools are also being applied to help improve harvest management for several other waterfowl stocks. In the last year, significant progress has been made in understanding the harvest potential of American black ducks (Anas rubripes), the Atlantic Population of Canada geese (Branta canadensis), northern pintails (Anas acuta), and scaup (Aythya affinis, A. marila). While these biological assessments are ongoing, they are already proving valuable in helping focus debate on the social aspects of harvesting policy, including management objectives and the nature of regulatory alternatives. 4 BACKGROUND The annual process of setting duckhunting regulations in the United States is based on a system of resource monitoring, data analyses, and rulemaking (Blohm 1989). Each year, monitoring activities such as aerial surveys and hunter questionnaires provide information on population size, habitat conditions, and harvest levels. Data collected from this monitoring program are analyzed each year, and proposals for duckhunting regulations are developed by the Flyway Councils, States, and USFWS. After extensive public review, the USFWS announces regulatory guidelines within which States can set their hunting seasons. In 1995, the USFWS adopted the concept of adaptive resource management (Walters 1986) for regulating duck harvests in the United States. This approach explicitly recognizes that the consequences of hunting regulations cannot be predicted with certainty, and provides a framework for making objective decisions in the face of that uncertainty (Williams and Johnson 1995). Inherent in the adaptive approach is an awareness that management performance can be maximized only if regulatory effects can be predicted reliably. Thus, adaptive management relies on an iterative cycle of monitoring, assessment, and decisionmaking to clarify the relationships among hunting regulations, harvests, and waterfowl abundance. In regulating waterfowl harvests, managers face four fundamental sources of uncertainty (Nichols et al. 1995, Johnson et al. 1996, Williams et al. 1996): (1) environmental variation  the temporal and spatial variation in weather conditions and other key features of waterfowl habitat; an example is the annual change in the number of ponds in the Prairie Pothole Region, where water conditions influence duck reproductive success; (2) partial controllability  the ability of managers to control harvest only within limits; the harvest resulting from a particular set of hunting regulations cannot be predicted with certainty because of variation in weather conditions, timing of migration, hunter effort, and other factors; (3) partial observability  the ability to estimate key population attributes (e.g., population size, reproductive rate, harvest) only within the precision afforded by extant monitoring programs; and (4) structural uncertainty  an incomplete understanding of biological processes; a familiar example is the longstanding debate about whether harvest is additive to other sources of mortality or whether populations compensate for hunting losses through reduced natural mortality. Structural uncertainty increases contentiousness in the decisionmaking process and decreases the extent to which managers can meet longterm conservation goals. AHM was developed as a systematic process for dealing objectively with these uncertainties. The key components of AHM include (Johnson et al. 1993, Williams and Johnson 1995): (1) a limited number of regulatory alternatives, which describe Flywayspecific season lengths, bag limits, and framework dates; (2) a set of population models describing various hypotheses about the effects of harvest and environmental factors on waterfowl abundance; (3) a measure of reliability (probability or "weight") for each population model; and (4) a mathematical description of the objective(s) of harvest management (i.e., an "objective function"), by which alternative regulatory strategies can be compared. These components are used in a stochastic optimization procedure to derive a regulatory strategy. A regulatory strategy specifies the optimal regulatory choice, with respect to the stated management objectives, for each possible combination of breeding population size, environmental conditions, and model weights (Johnson et al. 1997). The setting of annual hunting regulations then involves an iterative process: (1) each year, an optimal regulatory choice is identified based on resource and environmental conditions, and on current model weights; 5 (2) after the regulatory decision is made, modelspecific predictions for subsequent breeding population size are determined; (3) when monitoring data become available, model weights are increased to the extent that observations of population size agree with predictions, and decreased to the extent that they disagree; and (4) the new model weights are used to start another iteration of the process. By iteratively updating model weights and optimizing regulatory choices, the process should eventually identify which model is the best overall predictor of changes in population abundance. The process is optimal in the sense that it provides the regulatory choice each year necessary to maximize management performance. It is adaptive in the sense that the harvest strategy “evolves” to account for new knowledge generated by a comparison of predicted and observed population sizes. MALLARD STOCKS AND FLYWAY MANAGEMENT Since its inception AHM has focused on the population dynamics and harvest potential of mallards, especially those breeding in midcontinent North America. Mallards constitute a large portion of the total U.S. duck harvest, and traditionally have been a reliable indicator of the status of many other species. As management capabilities have grown, there has been increasing interest in the ecology and management of breeding mallards that occur outside the midcontinent region. Geographic differences in the reproduction, mortality, and migrations of mallard stocks suggest that there may be corresponding differences in optimal levels of sport harvest. The ability to regulate harvests of mallards originating from various breeding areas is complicated, however, by the fact that a large degree of mixing occurs during the hunting season. The challenge for managers, then, is to vary hunting regulations among Flyways in a manner that recognizes each Flyway’s unique breedingground derivation of mallards. Of course, no Flyway receives mallards exclusively from one breeding area, and so Flywayspecific harvest strategies ideally must account for multiple breeding stocks that are exposed to a common harvest. The optimization procedures used in AHM can account for breeding populations of mallards beyond the midcontinent region, and for the manner in which these ducks distribute themselves among the Flyways during the hunting season. An optimal approach would allow for Flywayspecific regulatory strategies, which in a sense represent for each Flyway an average of the optimal harvest strategies for each contributing breeding stock, weighted by the relative size of each stock in the fall flight. This joint optimization of multiple mallard stocks requires: (1) models of population dynamics for all recognized stocks of mallards; (2) an objective function that accounts for harvestmanagement goals for all mallard stocks in the aggregate; and (3) decision rules allowing Flywayspecific regulatory choices. Currently, two stocks of mallards are officially recognized for the purposes of AHM (Fig. 1). We continue to use a constrained approach to the optimization of these stocks’ harvest, in which the Atlantic Flyway regulatory strategy is based exclusively on the status of eastern mallards, and the regulatory strategy for the remaining Flyways is based exclusively on the status of midcontinent mallards. This approach has been determined to perform nearly as well as a jointoptimization because mixing of the two stocks during the hunting season is limited. 6 Fig 1. Survey areas currently assigned to the midcontinent and eastern stocks of mallards for the purposes of AHM. Delineation of the westernmallard stock is pending further development and review of population models and monitoring programs. MALLARD POPULATION DYNAMICS MidContinent Stock Midcontinent mallards are defined as those breeding in federal survey strata 118, 2050, and 7577 (i.e., the “traditional” survey area), and in Minnesota, Wisconsin, and Michigan (Fig. 1). Estimates of the size of this population are available since 1992, and have varied from 6.6 to 11.8 million (Table 1, Fig. 2). Estimated breedingpopulation size in 2007 was 9.053 (SE = 0.291 million), including 8.307 million (SE = 0.285 million) from the traditional survey area and 746 thousand (SE = 56 thousand) from the Great Lakes region. Details concerning the set of population models for midcontinent mallards are provided in Appendix B. The set consists of four alternatives, formed by the combination of two survival hypotheses (additive vs. compensatory hunting mortality) and two reproductive hypotheses (strongly vs. weakly density dependent). Relative weights for the alternative models of midcontinent mallards changed little until all models underpredicted the change in population size from 1998 to 1999, perhaps indicating there is a significant factor affecting population dynamics that is absent from all four models (Fig. 3). Updated model weights suggest some preference for the additivemortality models (60%) over those describing hunting mortality as compensatory (40%). For most of the time frame, model weights have strongly favored the weakly densitydependent reproductive models over the strongly densitydependent ones, with current model weights of 90% and 10%, respectively. The reader is cautioned, however, that models can sometimes make reliable predictions of population size for reasons having little to do with the biological hypotheses expressed therein (Johnson et al. 2002b). 7 Table 1. Estimates (N) and standard errors (SE) of mallards (in millions) in spring in the traditional survey area (strata 118, 2050, and 7577) and the states of Minnesota, Wisconsin, and Michigan. Traditional survey area Great Lakes region Total Year N SE N SE N SE 1992 5.9761 0.2410 0.9946 0.1597 6.9706 0.2891 1993 5.7083 0.2089 0.9347 0.1457 6.6430 0.2547 1994 6.9801 0.2828 1.1505 0.1163 8.1306 0.3058 1995 8.2694 0.2875 1.1214 0.1965 9.3908 0.3482 1996 7.9413 0.2629 1.0251 0.1443 8.9664 0.2999 1997 9.9397 0.3085 1.0777 0.1445 11.0174 0.3407 1998 9.6404 0.3016 1.1224 0.1792 10.7628 0.3508 1999 10.8057 0.3445 1.0591 0.2122 11.8648 0.4046 2000 9.4702 0.2902 1.2350 0.1761 10.7052 0.3395 2001 7.9040 0.2269 0.8622 0.1086 8.7662 0.2516 2002 7.5037 0.2465 1.0820 0.1152 8.5857 0.2721 2003 7.9497 0.2673 0.8360 0.0734 8.7857 0.2772 2004 7.4253 0.2820 0.9333 0.0748 8.3586 0.2917 2005 6.7553 0.2808 0.7862 0.0650 7.5415 0.2883 2006 7.2765 0.2237 0.5881 0.0465 7.8646 0.2284 2007 8.3073 0.2858 0.7459 0.0565 9.0532 0.2913 Fig. 2. Population estimates of midcontinent mallards in the traditional survey area (TSA) and the Great Lakes region. Error bars represent one standard error. Year 1995 2000 2005 Population size (millions) 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 TSA Great Lakes Total 8 Fig 3. Weights for models of midcontinent mallards (ScRs = compensatory mortality and strongly densitydependent reproduction, ScRw = compensatory mortality and weakly densitydependent reproduction, SaRs = additive mortality and strongly densitydependent reproduction, and SaRw = additive mortality and weakly densitydependent reproduction). Model weights were assumed to be equal in 1995. Eastern Stock Eastern mallards are defined as those breeding in southern Ontario and Quebec (federal survey strata 5154 and 56) and in the northeastern U.S. (state plot surveys; Heusman and Sauer 2000) (Fig. 1). Estimates of population size have varied from 856 thousand to 1.1 million since 1990, with the majority of the population accounted for in the northeastern U.S. (Table 3, Fig. 4). For 2007, the estimated breedingpopulation size of eastern mallards was 907 thousand (SE = 58 thousand), including 688 thousand (SE = 47 thousand) from the northeastern U.S. and 219 thousand (SE = 34 thousand) from the Canadian survey strata. The reader is cautioned that these estimates differ from those reported in the USFWS annual waterfowl trend and status reports, which include composite estimates based on more fixedwing strata in eastern Canada and helicopter surveys conducted by CWS. Details concerning the set of population models for eastern mallards are provided in Appendix C. The set consists of six alternatives, formed by the combination of two reproductive hypotheses (strongly vs. weakly density dependent) and three hypotheses concerning bias in estimates of survival and reproductive rates (no bias vs. biased survival rates vs. biased reproductive rates). With respect to model weights, there is no single model that is clearly favored over the others at the current time. Collectively, the models with strong densitydependent reproduction are slightly better predictors of changes in population size than those with weak density dependence, with current model weights of 59% and 41%, respectively (Fig. 5). In addition, there is overwhelming evidence of bias in extant estimates of survival or reproductive rates (100%), assuming that survey estimates are unbiased. Year 1996 1998 2000 2002 2004 2006 Model weight 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ScRs ScRw SaRs SaRw 9 Table 3. Estimates (N) and associated standard errors (SE) of mallards (in thousands) in spring in the northeastern U.S. (state plot surveys) and eastern Canada (federal survey strata 5154 and 56). Northeastern U.S. Canadian survey strata Total Year N SE N SE N SE 1990 665.1 78.3 190.7 47.2 855.8 91.4 1991 779.2 88.3 152.8 33.7 932.0 94.5 1992 562.2 47.9 320.3 53.0 882.5 71.5 1993 683.1 49.7 292.1 48.2 975.2 69.3 1994 853.1 62.7 219.5 28.2 1072.5 68.7 1995 862.8 70.2 184.4 40.0 1047.2 80.9 1996 848.4 61.1 283.1 55.7 1131.5 82.6 1997 795.1 49.6 212.1 39.6 1007.2 63.4 1998 775.1 49.7 263.8 67.2 1038.9 83.6 1999 879.7 60.2 212.5 36.9 1092.2 70.6 2000 757.8 48.5 132.3 26.4 890.0 55.2 2001 807.5 51.4 200.2 35.6 1007.7 62.5 2002 834.1 56.2 171.3 30.0 1005.4 63.8 2003 731.8 47.0 308.3 55.4 1040.1 72.6 2004 809.1 51.8 301.5 53.3 1110.7 74.3 2005 753.6 53.6 293.4 53.1 1047.0 75.5 2006 725.2 47.9 174.0 28.4 899.2 55.7 2007 687.6 46.7 219.3 33.6 906.9 57.6 10 Fig. 4. Population estimates of eastern mallards in the northeastern U.S. (NE plot survey) and in federal surveys in southern Ontario and Quebec (FWS survey). Error bars represent one standard error. Fig. 5. Weights for models of eastern mallards (Rw0 = weak densitydependent reproduction and no model bias, Rs0 = strong dependent reproduction and no model bias, RwS = weak densitydependent reproduction and biased survival rates, RsS = strong densitydependent reproduction and biased survival rates, RwR = weak densitydependent reproduction and biased reproductive ates, and RsR = strong densitydependent reproduction and biased reproductive rates). Model weights were assumed to be equal in 1996. Year 1990 1995 2000 2005 Populations size (millions) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 NE plot survey 1.4 FWS survey Total Year 1996 1998 2000 2002 2004 2006 Model weight 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 Rw0 Rs0 RwS RsS RwR RsR 11 Western Stock Recent efforts to develop Flywayspecific harvest strategies have focused on mallards breeding in the states of the Pacific Flyway (including Alaska), British Columbia, and the Yukon Territory. Efforts to understand and model the population dynamics of western mallards have been underway for several years and the Pacific Flyway States, the USFWS, and the Canadian Wildlife Service have been collaborating to improve survey and banding programs. We summarize the most recent results concerning the dynamics of these mallards, as well as some implications for harvest management. A more detailed report is available online at http://www.fws.gov/migratorybirds/mgmt/ahm/specialtopics.htm. Western mallards are distributed over a large area and we have had continuing concerns about our ability to determine changes in population size based on the collection of surveys conducted independently by Pacific Flyway States and the Province of British Columbia. These surveys tend to vary in design and intensity, and in some cases lack measures of precision. Therefore, we reviewed extant surveys to determine their adequacy for supporting a westernmallard AHM protocol and ultimately selected Alaska, California, and Oregon for modeling purposes. These three states likely harbor about 75% of the westernmallard breeding population. Nonetheless, this geographic delineation is considered temporary until surveys in other areas can be brought up to similar standards and an adequate record of population estimates is available for analysis. To predict changes in abundance we relied on a discrete logistic model, which combines reproduction and natural mortality into a single parameter r, the intrinsic rate of growth. This model assumes densitydependent growth, which is regulated by the ratio of population size, N, to the carrying capacity of the environment, K (i.e., population size in the absence of harvest). In the traditional formulation of the logistic model, harvest mortality is completely additive and any compensation for hunting losses occurs as a result of densitydependent responses beginning in the subsequent breeding season. To increase the model’s generality we included a scaling parameter for harvest that allows for the possibility of compensation prior to the breeding season. It is important to note, however, that this parameterization does not incorporate any hypothesized mechanism for harvest compensation and, therefore, must be interpreted cautiously. We modeled Alaska mallards independently of those in California and Oregon because of differing population trajectories (Fig. 6) and substantial differences in the distribution of band recoveries. We used Bayesian estimation methods in combination with a statespace model that accounts explicitly for both process and observation error in breeding population size (Meyer and Millar 1999). Breeding population estimates of mallards in Alaska are available since 1955, but we had to limit the timeseries to 19902005 because of changes in survey methodology and insufficient bandrecovery data. The logistic model and associated posterior parameter estimates provided a reasonable fit to the observed timeseries of Alaska population estimates. The estimated carrying capacity was 1.2 million, the intrinsic rate of growth was 0.31, and harvest mortality acted in an additive fashion. Breeding population and harvestrate data were available for CaliforniaOregon mallards for the period 19922006. The logistic model also provided a reasonable fit to these data, suggesting a carrying capacity of 0.7 million, an intrinsic rate of growth 0.34, and harvest mortality that acted in only a partially additive manner. For the purpose of understanding general patterns in optimal harvest rates, we assumed perfect control over harvest and evaluated statedependent harvest rates from 0.0 to 0.25 in increments of 0.05. We examined two different management objectives conditioned on this set of harvest rates: (1) maximize longterm cumulative yield; and (2) attain approximately 90% of the maximum longterm cumulative yield. For an objective to maximize longterm cumulative harvest, there were many combinations of stock sizes that had harvestrate prescriptions of either 0 or 25 percent. Very few stock sizes had intermediate harvestrate prescriptions. In contrast, an objective to attain 90% of the maximum yield produced an optimal strategy with a more even distribution of optimal harvest rates, and very few prescriptions for closed seasons. Empirical estimates of harvest rates showed no obvious response to changes in regulations, based on extensive 12 Fig. 6. Estimated abundances of mallards breeding in Alaska and CaliforniaOregon as derived from federal and state surveys, respectively. Error bars represent one standard error. analyses using a variety of regulatory metrics, including season length, mallard bag limits, and framework opening and closing dates (singly and in combination). We were forced to conclude that changes in regulations in the Pacific Flyway since 1980 have not resulted in significant changes in the harvest rates of western mallards. It appears that more extreme regulatory changes than those used in the past may be needed to effect substantive changes in harvest rates. To help understand the implications of this apparent lack of control over harvest rates, we assumed the most extreme case of two regulatory options: a closed season and an open season. We assumed that an open season would produce a harvest rate of 0.1259 (the mean of all our estimates) and that a closed season would produce a harvest rate of 0.0. We then conducted an optimization to determine the population thresholds for season closures assuming minimal control over harvest rates. Generally, as long as both stocks are above about 350k, then the optimal choice is an open season. Below that, the lower one stock is, the higher the other has to be to maintain an open season. We believe that the models developed thus far provide a sufficient basis for developing an initial AHM protocol. Moreover, extant monitoring of mallard abundance and harvest rates in Alaska and CaliforniaOregon will provide the necessary basis for updating estimates of model parameters and their variances. Similarly, we believe that sufficient information is available to inform the choice of an objective function for western mallards. For example, an objective to attain 90% of the maximum longterm cumulative harvest provides for levels of hunting opportunity that are similar to those now in effect for a wide range of stock sizes. On a more pessimistic note, we were unable to establish a viable set of regulatory alternatives with which to effect changes in harvest rate. Therefore, an essential task is consideration of hunting regulations beyond the realm of experience that might be expected to have a meaningful effect on harvest rates. Ideally, the development of AHM protocols for mallards would consider how different breeding stocks distribute themselves among the four flyways so that Flywayspecific harvest strategies could account for the mixing of birds during the hunting season. At present, however, a joint optimization of western, midcontinent, and eastern stocks is not feasible due to computational hurdles. Therefore, the initial AHM protocol for western mallards may need to be structured similarly to that used for eastern mallards, in which an optimal harvest strategy is based on Year 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Population size 2e+5 3e+5 4e+5 5e+5 6e+5 7e+5 8e+5 9e+5 1e+6 2e+5 3e+5 4e+5 5e+5 6e+5 7e+5 8e+5 9e+5 1e+6 AK CAOR 13 the status of a single breeding stock and harvest regulations in a single flyway. Although the contribution of midcontinent mallards to the Pacific Flyway harvest is significant, we believe an independent harvest strategy for western mallards poses little risk to the midcontinent stock. Further analyses will be needed to confirm this conclusion, as well as to better understand the potential effect of midcontinent mallard status on sustainable hunting opportunities in the Pacific Flyway. HARVESTMANAGEMENT OBJECTIVES The basic harvestmanagement objective for midcontinent mallards is to maximize cumulative harvest over the long term, which inherently requires perpetuation of a viable population. Moreover, this objective is constrained to avoid regulations that could be expected to result in a subsequent population size below the goal of the North American Waterfowl Management Plan (NAWMP). According to this constraint, the value of harvest decreases proportionally as the difference between the goal and expected population size increases. This balance of harvest and population objectives results in a regulatory strategy that is more conservative than that for maximizing longterm harvest, but more liberal than a strategy to attain the NAWMP goal (regardless of effects on hunting opportunity). The current objective uses a population goal of 8.8 million mallards, which is based on 8.2 million mallards in the traditional survey area (from the 1998 update of the NAWMP) and a goal of 0.6 million for the combined states of Minnesota, Wisconsin, and Michigan. For eastern mallards, there is no NAWMP goal or other established target for desired population size. Accordingly, the management objective for eastern mallards is simply to maximize longterm cumulative (i.e., sustainable) harvest. REGULATORY ALTERNATIVES Evolution of Alternatives When AHM was first implemented in 1995, three regulatory alternatives characterized as liberal, moderate, and restrictive were defined based on regulations used during 197984, 198587, and 198893, respectively. These regulatory alternatives also were considered for the 1996 hunting season. In 1997, the regulatory alternatives were modified to include: (1) the addition of a veryrestrictive alternative; (2) additional days and a higher duck bag limit in the moderate and liberal alternatives; and (3) an increase in the bag limit of hen mallards in the moderate and liberal alternatives. In 2002 the USFWS further modified the moderate and liberal alternatives to include extensions of approximately one week in both the opening and closing framework dates. In 2003 the veryrestrictive alternative was eliminated at the request of the Flyway Councils. Expected harvest rates under the veryrestrictive alternative did not differ significantly from those under the restrictive alternative, and the veryrestrictive alternative was expected to be prescribed for <5% of all hunting seasons. Also, at the request of the Flyway Councils the USFWS agreed to exclude closed duckhunting seasons from the AHM protocol when the population size of midcontinent mallards is ≥5.5 million (traditional survey area plus the Great Lakes region). Based on our assessment, closed hunting seasons do not appear to be necessary from the perspective of sustainable harvesting when the midcontinent mallard population exceeds this level. The impact of maintaining open seasons above this level also appears to be negligible for other midcontinent duck species, as based on population models developed by Johnson (2003). However, complete or partial seasonclosures for particular species or populations could still be deemed necessary in some situations regardless of the status of midcontinent mallards. Details of the regulatory alternatives for each Flyway are provided in Table 6. 14 Table 6. Regulatory alternatives for the 2007 duckhunting season. Flyway Regulation Atlantica Mississippi Centralb Pacificc Shooting hours onehalf hour before sunrise to sunset Framework dates Restrictive Oct 1  Jan 20 Saturday nearest Oct 1to the Sunday nearest Jan 20 Moderate and Liberal Saturday nearest September 24 to the last Sunday in January Season length (days) Restrictive 30 30 39 60 Moderate 45 45 60 86 Liberal 60 60 74 107 Bag limit (total / mallard / female mallard) Restrictive 3 / 3 / 1 3 / 2 / 1 3 / 3 / 1 4 / 3 / 1 Moderate 6 / 4 / 2 6 / 4 / 1 6 / 5 / 1 7 / 5 / 2 Liberal 6 / 4 / 2 6 / 4 / 2 6 / 5 / 2 7 / 7 / 2 a The states of Maine, Massachusetts, Connecticut, Pennsylvania, New Jersey, Maryland, Delaware, West Virginia, Virginia, and North Carolina are permitted to exclude Sundays, which are closed to hunting, from their total allotment of season days. b The High Plains Mallard Management Unit is allowed 8, 12, and 23 extra days in the restrictive, moderate, and liberal alternatives, respectively. c The Columbia Basin Mallard Management Unit is allowed seven extra days in the restrictive, and moderate alternatives. RegulationSpecific Harvest Rates Harvest rates of mallards associated with each of the openseason regulatory alternatives were initially predicted using harvestrate estimates from 197984, which were adjusted to reflect current hunter numbers and contemporary specifications of season lengths and bag limits. In the case of closed seasons in the U.S., we assumed rates of harvest would be similar to those observed in Canada during 198893, which was a period of restrictive regulations both in Canada and the U.S. All harvestrate predictions were based only in part on bandrecovery data, and relied heavily on models of hunting effort and success derived from hunter surveys (USFWS 2002: Appendix C). As such, these predictions had large sampling variances and their accuracy was uncertain. In 2002 we began relying on Bayesian statistical methods for improving regulationspecific predictions of harvest rates, including predictions of the effects of frameworkdate extensions. Essentially, the idea is to use existing (prior) information to develop initial harvestrate predictions (as above), to make regulatory decisions based on those predictions, and then to observe realized harvest rates. Those observed harvest rates, in turn, are treated as new sources of information for calculating updated (posterior) predictions. Bayesian methods are attractive because they provide a quantitative and formal, yet intuitive, approach to adaptive management. For midcontinent mallards, we have empirical estimates of harvest rate from the recent period of liberal hunting regulations (19982006). The Bayesian methods thus allow us to combine these estimates with our prior predictions to provide updated estimates of harvest rates expected under the liberal regulatory alternative. Moreover, in the absence of experience (so far) with the restrictive and moderate regulatory alternatives, we 15 reasoned that our initial predictions of harvest rates associated with those alternatives should be rescaled based on a comparison of predicted and observed harvest rates under the liberal regulatory alternative. In other words, if observed harvest rates under the liberal alternative were 10% less than predicted, then we might also expect that the mean harvest rate under the moderate alternative would be 10% less than predicted. The appropriate scaling factors currently are based exclusively on prior beliefs about differences in mean harvest rate among regulatory alternatives, but they will be updated once we have experience with something other than the liberal alternative. A detailed description of the analytical framework for modeling mallard harvest rates is provided in Appendix D. Our models of regulationspecific harvest rates also allow for the marginal effect of frameworkdate extensions in the moderate and liberal alternatives. A previous analysis by the USFWS (2001) suggested that implementation of frameworkdate extensions might be expected to increase the harvest rate of midcontinent mallards by about 15%, or in absolute terms by about 0.02 (SD = 0.01). Based on the observed harvest rates during the 20022006 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.009 (SD = 0.008). The estimated effect of the frameworkdate extension has been to increase harvest rate of midcontinent mallards by about 8% over what would otherwise be expected in the liberal alternative. However, the reader is strongly cautioned that reliable inference about the marginal effect of frameworkdate extensions ultimately depends on a rigorous experimental design (including controls and random application of treatments). Current predictions of harvest rates of adultmale midcontinent mallards associated with each of the regulatory alternatives are provided in Table 7. Predictions of harvest rates for the other agesex cohorts are based on the historical ratios of cohortspecific harvest rates to adultmale rates (Runge et al. 2002). These ratios are considered fixed at their longterm averages and are 1.5407, 0.7191, and 1.1175 for young males, adult females, and young females, respectively. We continued to make the simplifying assumption that the harvest rates of midcontinent mallards depend solely on the regulatory choice in the western three Flyways. This appears to be a reasonable assumption given the small proportion of midcontinent mallards wintering in the Atlantic Flyway (Munro and Kimball 1982), and harvestrate predictions that suggest a minimal effect of Atlantic Flyway regulations (USFWS 2000). Under this assumption, the optimal regulatory strategy for the western three Flyways can be derived by ignoring the harvest regulations imposed in the Atlantic Flyway. Table 7. Predictions of harvest rates of adultmale midcontinent mallards expected with application of the 2007 regulatory alternatives in the three western Flyways. Regulatory alternative Mean SD Closed (U.S.) 0.0088 0.0019 Restrictive 0.0578 0.0129 Moderate 0.1059 0.0216 Liberal 0.1225 0.0205 The predicted harvest rates of easternmallard are updated in the same fashion as that for midcontinent mallards based on reward banding conducted in eastern Canada and the northeastern U.S. (Appendix D). Like midcontinent mallards, harvest rates of age and sex cohorts other than adult male mallards are based on constant rates of differential vulnerability as derived from bandrecovery data. For eastern mallards, these constants are 1.153, 1.331, and 1.509 for adult females, young males, and young females, respectively (Johnson et al. 2002a). Regulationspecific predictions of harvest rates of adultmale eastern mallards are provided in Table 8. In contrast to midcontinent mallards, frameworkdate extensions were expected to increase the harvest rate of eastern mallards by only about 5% (USFWS 2001), or in absolute terms by about 0.01 (SD = 0.01). Based on the observed harvest rates during the 20022006 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.005 (SD = 0.010). The estimated effect of the frameworkdate extension has been to increase harvest rate of eastern mallards by about 3% over what would 16 otherwise be expected in the liberal alternative. Table 8. Predictions of harvest rates of adultmale eastern mallards expected with application of the 2007 regulatory alternatives in the Atlantic Flyway. Regulatory alternative Mean SD Closed (U.S.) 0.0798 0.0233 Restrictive 0.1202 0.0395 Moderate 0.1471 0.0474 Liberal 0.1578 0.0459 OPTIMAL REGULATORY STRATEGIES We calculated optimal regulatory strategies using stochastic dynamic programming (Lubow 1995, Johnson and Williams 1999). For the three western Flyways, we based this optimization on: (1) the 2007 regulatory alternatives, including the closedseason constraint; (2) current population models and associated weights for midcontinent mallards; and (3) the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.8 million midcontinent mallards. The resulting regulatory strategy (Table 9) is similar to that used last year. Note that prescriptions for closed seasons in this strategy represent resource conditions that are insufficient to support one of the current regulatory alternatives, given current harvestmanagement objectives and constraints. However, closed seasons under all of these conditions are not necessarily required for longterm resource protection, and simply reflect the NAWMP population goal and the nature of the current regulatory alternatives. Assuming that regulatory choices adhered to this strategy (and that current model weights accurately reflect population dynamics), breedingpopulation size would be expected to average 7.45 million (SD = 1.81 million). Based on an estimated population size of 9.05 million midcontinent mallards and 5.04 million ponds in Prairie Canada, the optimal choice for the Pacific, Central, and Mississippi Flyways in 2007 is the liberal regulatory alternative. We calculated an optimal regulatory strategy for the Atlantic Flyway based on: (1) the 2007 regulatory alternatives; (2) current population models and associated weights for eastern mallards; and (3) an objective to maximize longterm cumulative harvest. The resulting strategy suggests liberal regulations for all population sizes of record, and is characterized by a lack of intermediate regulations (Table 10). We simulated the use of this regulatory strategy to determine expected performance characteristics. Assuming that harvest management adhered to this strategy (and that current model weights accurately reflect population dynamics), breedingpopulation size would be expected to average 887 thousand (SD = 16 thousand). Based on an estimated breeding population size of 907 thousand mallards, the optimal choice for the Atlantic Flyway in 2007 is the liberal regulatory alternative. 17 Table 9. Optimal regulatory strategya for the three western Flyways for the 2007 hunting season. This strategy is based on current regulatory alternatives (including the closedseason constraint), on current midcontinent mallard models and weights, and on the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.8 million mallards. The shaded cell indicates the regulatory prescription for 2007. Pondsc Bpopb 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 ≤5.25 C C C C C C C C C C 5.506.25 R R R R R R R R R R 6.50 R R R R R R R R M M 6.75 R R R R R R M M M L 7.00 R R R R R M M L L L 7.25 R R R M L L L L L L 7.50 R M M L L L L L L L 7.75 M M M L L L L L L L 8.00 M L L L L L L L L L ≥8.25 L L L L L L L L L L a C = closed season, R = restrictive, M = moderate, L = liberal. b Mallard breeding population size (in millions) in the traditional survey area (survey strata 118, 2050, 7577) and Michigan, Minnesota, and Wisconsin. c Ponds (in millions) in Prairie Canada in May. Table 10. Optimal regulatory strategya for the Atlantic Flyway for the 2007 hunting season. This strategy is based on current regulatory alternatives, on current eastern mallard models and weights, and on an objective to maximize longterm cumulative harvest. The shaded cell indicates the regulatory prescription for 2007. Mallardsb Regulation <240 C 240 R >240 L a C = closed season, R = restrictive, M = moderate, and L = liberal. b Estimated number of mallards in eastern Canada (survey strata 5154, 56) and the northeastern U.S. (state plot surveys), in thousands. Application of AHM Concepts to Other Stocks The USFWS is striving to apply the principles and tools of AHM to improve decisionmaking for several other stocks of waterfowl. We report on four such efforts in which significant progress has been made since last year. American Black Ducks Beginning in 2003 the USFWS Division of Migratory Bird Management (DMBM) began investigating optimal 18 harvest strategies for black ducks based on models of population dynamics provided by Conroy et al (2002). As a result of that investigation DMBM concluded that recent harvest rates of black ducks have sometimes been at or above levels consistent with an objective to maximize sustainable harvest. That conclusion ultimately led to a DMBM recommendation in January 2006 to reduce the harvest rate of adult black ducks by 25%. However, the recommendation was subsequently withdrawn because of: (1) published information suggesting that the midwinter inventory (MWI) may be “capturing” a smaller proportion of the black duck population than in the past (Link et al. 2006); (2) concern about the U.S. acting unilaterally without the benefit of consultation with the CWS; and (3) the short amount of time available to communicate to the public the rationale and nature of restrictions on hunting opportunity. In November 2006 the international Black Duck Adaptive Harvest Management Working Group (BDAHMWG) met to discuss the most recent analysis by Drs. Mike Conroy and Jon Runge of the Georgia Cooperative Fish and Wildlife Research Unit. Their update of the original analysis by Conroy et al. (2002) suggests that black duck productivity has continued to decline for reasons that cannot be explained by changes in abundance of black ducks (through density dependence) or sympatric mallards (through interspecies competition). However, there were other differences in inferences based on the original and updated analyses that could not be reconciled. The focus of research has now turned to population models based on integrated fixedwing and helicopter surveys conducted during the breeding season. For the present, however, the question of whether current harvest rates of black ducks are consistent with black duck harvest potential and management objectives remains unanswered. Due to potential changes in the wintering distribution of black ducks, the BDAHMWG did not endorse a statedependent harvest strategy (i.e., one in which optimal harvest rates depend on annual black duck abundance) based on the MWI. However, it was suggested that a constant harvestrate strategy may perform nearly as well and might provide a basis for a joint CanadaU.S. harvest strategy until an assessment based on breedingseason surveys can be completed. The BDAHMWG agreed to investigate the performance of constant harvestrate strategies based on the original work of Conroy et al. (2002), recognizing that the original analysis was conducted prior to what may be significant changes in the wintering distribution of black ducks. Thus, it was agreed that the assessment by Conroy et al. (2002) might still provide a reasonable basis for investigating harvest impacts and for evaluating the expected performance of constant harvestrate strategies. We relied on the population models and corresponding weights provided by Conroy et al. (2002) to conduct an evaluation of constant harvestrate strategies. These models incorporate alternative hypotheses for reproduction (competition with mallards vs. no competition), survival (additive vs. compensatory hunting mortality), and estimation bias (positive bias in reproductive rates vs. survival rates). Both reproduction models incorporate a negative effect of year, presumably due to a longterm loss and/or degradation of habitat. For the purposes of this assessment we projected the loglinear decline in production rate through 2007. We believe this was justified because of evidence that the decline in productivity has continued to the present at about the same rate as that estimated from the 19611994 period. However, for the purposes of this assessment we had to assume that the decline in productivity halts in 2007. To project the decline indefinitely into the future would imply that no level of harvest is sustainable. Managers are currently considering plausible explanations for the productivity decline, and will need to be vigilant in assessing future trends in productivity. Because of the possible effect of mallard abundance on black duck productivity, it was necessary to include a dynamic model of mallard abundance (M). The model used by Conroy et al. (2002) was: t t t M = M λ +1 where λt is the finite rate of population growth. Estimates of λt from 19711994 were used to obtain an empirical distribution to specify random outcomes for λt. During 19711994 λt was highly variable, but with an average close to 1 (suggestive of a stable population). We were not completely satisfied with this model because it can produce biologically unrealistic changes in 19 population size (because population size and λt are uncorrelated) and because population size had to be constrained to an arbitrary maximum. Therefore, we described changes in mallard abundance as a 1storder autoregressive process using data from 19712000. The model is: M ( ) (M ( )) e t t = + + + 0.00001 1.801 0.494 0.00001 1 where e ~ Normal(0, 0.300). This model provided a satisfactory fit to the time series of observed population sizes and describes a stationary time series with M t = 355,826 . We evaluated constant harvestrate strategies using SDP (Lubow 1995) and by constraining the size of the harvest to be equal in Canada and the U.S. We specified fixed harvest rates of 0.00 to 0.16 in increments of 0.01. We simulated black duck and mallard population dynamics for 20,000 iterations under each of the fixed harvest rates, using starting values of 300k black ducks and 356k mallards. We then calculated the mean and standard deviation of black duck population size and harvest. For comparative purposes, we also derived an optimal statedependent harvest strategy using SDP and an objective to maximize longterm cumulative harvest. Based on simulated population dynamics, the black duck population averaged 546k (SD = 154k) in the absence of harvest (Fig. 7). An optimal statedependent strategy to maximize sustainable harvest (i.e., a strategy in which the harvest rate varies with black duck abundance) resulted in an average population size of 255k (SD = 50k) and an average harvest of 51K (SD = 41k). For a constant harvest rate, the maximum sustainable harvest was achieved at a harvest rate of 0.09 on adult males, resulting in an average of 240k (SD = 94k) black ducks in the MWI and a harvest of 47k (SD = 15k). Thus, the expected harvest under a constant harvestrate strategy was only 7% less than that which could be achieved under an optimal statedependent harvest strategy. However, population size was nearly twice as variable under the constant harvest rate of 0.09 as under the optimal statedependent strategy. For a target harvest rate of 0.09, the corresponding harvest rates in Canada and the U.S. to achieve parity in harvest are 0.045 and 0.048, respectively. By comparison, adult black duck harvest rates estimated from reward banding during the 20022006 hunting seasons averaged 0.0358 (SE = 0.00073) for Canada, 0.0584 (SE = 0.0017) for the U.S., and 0.0916 (SE = 0.0016) overall. The average population size in the MWI during 2003 2007 of 220k corresponds well with that predicted from the weighted models under an average harvest rate of 0.09 (240k). A harvest rate of 0.09 should be considered a maximum because it assumes that black duck productivity will not decline further. Moreover, a smaller harvest rate appears to be necessary to induce population growth. For example, attainment of the original North American Waterfowl Management Plan population objective of 385k black ducks in the MWI would require a constant harvest rate of approximately 0.05 under current environmental conditions. 20 Fig. 7. Black duck population sizes in winter (MWI) and harvests (both in thousands, with SD’s) expected under constant adultmale harvest rates of 0.00 (on the extreme right) to 0.16 (on the extreme left) in increments of 0.01. The datum depicted by the open circle is that expected under an optimal statedependent strategy with an objective to maximize longterm cumulative harvest. The vertical dashed line indicates the original North American Waterfowl Management Plan population goal of 385 thousand. Atlantic Population of Canada Geese For the purposes of this AHM application, Atlantic Population Canada Geese (APCG) are defined as those geese breeding on the Ungava Peninsula. By this delineation, we assume that geese in the Atlantic population outside this area are either few in number, similar in population dynamics to the Ungava birds, or both. To account for heterogeneity among individuals, we developed a base model consisting of a truncated timeinvariant agebased projection model to describe the dynamics of APCG: n(t+1)=An(t), where n(t) is a vector of the abundances of the ages in the population at time t, and A is the population projection matrix, whose ijth entry aij gives the contribution of an individual in stage j to stage i over 1 time step. The projection interval (from t to t+1) is one year, with the census being taken in midJune (i.e., this model has a prebreeding census). The life cycle diagram reflecting the transition sequence is: MWI 0 100 200 300 400 500 600 700 Harvest 0 20 40 60 80 100 21 where node 1 refers to oneyearold birds (N(1)), node 2 refers to twoyearold birds (N(2)), node B refers to adult breeders (N(B)), and node NB refers to adult nonbreeders N(NB). One immediate extension of the base model is to remove the assumption of timeinvariance, and express the parameters as timedependent quantities: Pt = proportion of adult birds in population in year t which breed; Rt = basic breeding productivity in year t (per capita); St (0) = annual survival rate of young from fledging in year t to the census point the next year; St (1) = annual survival rate of oneyearold birds in year t; etc. For APCG, only N(B), R and z are observable annually, where N(B) is the number of breeding adults, R is the per capita reproductive rate (ratio of fledged young to breeding adults), and z is an extrinsic, environmental variable (a function of timing of snow melt on the breeding grounds) that is used to predict R.. Note that at the time of the management decision in the United States (July), estimates for only the breeding population size and the environmental variable(s) are available; the ageratio isn’t estimated until later in the summer. Thus, in year t, the observable state variables are Nt (B), zt, and Rt–1. There are several other state variables of interest, however, namely, N(1), N(2), and N(NB). Because annual harvest decisions need to be made based on the total population size (Ntot), which is the sum of contributions from various nonbreeding age classes as well as the number of breeding individuals, abundance of nonbreeding individuals (N(NB), N(1), and N(2)) needs to be derived using populationreconstruction techniques. In most cases, population reconstruction involves estimating the most likely population projection matrix, given a time series of population vectors (where number of individuals in each age class at each time is known). However, in our case, only estimates of NB, R and z are available (not the complete population vector); in effect, we must estimate some of the population abundance values given the other parameters in the model. Extensions of Bayesian and nonlinear estimation methods to population reconstruction provide a reasonable solution. The time series of breeding population size, ageratio, and harvest rate were used to reconstruct the population structure from 1997 to the present, using a densityindependent model (Fig. 8). The estimated population 22 structure in 2007 is: 382,100 breeding adults, 99,300 nonbreeding adults, 235,900 secondyear birds, and 342,200 firstyear birds. Relative to the number of breeding adults, secondyear birds are 29.0% above the number expected from a stable stagedistribution, and firstyear birds are 35.9% above the expected number, reflecting the impact of the very successful 2005 and 2006 breeding seasons. The densityindependent model projects significant increases in the number of breeding pairs (~25% over the next two years) as these two sizeable cohorts come of age. Fig. 8. APCG breeding population size (in thousands), 19932007, with fitted values from reconstruction (Model 1: densityindependent). The diamonds show the observed estimates of breeding population size (not breeding pairs); errors bars are ±2 SE. The solid line shows the breeding population size estimated from the population reconstruction (also ±2 SE). The observed 1998 population size was smoothed because the survey conditions were poor. Based on the data available last year, we had postulated several alternative models to explain the apparent stabilization in the population trajectory. The three alternative models included mechanisms for densitydependent survival, densitydependent propensity to breed, and reporting rate bias. With the 2007 breeding survey data included, the differences among the trajectories for these models diminished, but the reconstructed agestructure of the current population differs substantially among them, as do the optimal harvest strategies. As an example, harvest strategies for two of the population models are shown in Fig. 9. With the densityindependent model, the strategy seeks an equilibrium breeding population size of 748,500. The strategy suggests a closed season this year, in order to increase more quickly toward the desired population size. On the other hand, the optimal strategy for the densitydependent model seeks an equilibrium breeding population size of 308,000 and because the current population size is above that, the recommended hunting regulations are liberal (20% harvest rate). Note that in both strategies, the recommended harvest rate is not strongly affected by the measure of current environmental conditions on the breeding grounds. Thus, harvest recommendations are strongly affected by uncertainty about the underlying population dynamics. We have not yet developed methods to weight the alternative models and produce a composite optimal policy; such development is a high priority. However, this population is in a very informative phase of its dynamics, such that each year of data greatly increases our ability to distinguish among alternative models. 0 50 100 150 200 250 300 350 400 450 500 1994 1996 1998 2000 2002 2004 2006 2008 23 Fig. 9. Examples of optimal harvest strategies for 2007 for models 1 (densityindependent) and 3 (densitydependent breeding propensity). These matrices show the breeding population size against the measure of breeding habitat conditions (principal component of the weather variables), with the other values of the population vector (NNB, N2, N1) fixed at their 2007 reconstructed values. The shaded areas represent the recommended harvest rate of adult males. Northern Pintails The Flyway Councils have long identified the northern pintail as a highpriority species for inclusion in the AHM process. In 1997, the USFWS adopted a pintail harvest strategy to help align harvest opportunity with population status, while providing a foundation upon which to develop a formal AHM framework. Since 1997, the harvest strategy has undergone a number of technical improvements and policy revisions. However, the strategy continues to be a set of regulatory prescriptions born out of consensus, rather than an optimal strategy derived from agreedupon population models, management objectives, regulatory alternatives, and measures of uncertainty. This year, the USFWS and Flyway Councils are taking a major step towards a truly adaptive approach by incorporating alternative models of population dynamics. Two models are being considered: one in which harvest is additive to natural mortality, and another in which harvest losses are compensated for by reductions in natural mortality. In the additive model, winter survival rate is a constant, whereas winter survival is densitydependent in the compensatory model. We here provide a summary of these recent modeling efforts. A detailed progress report is available online at http://www.fws.gov/migratorybirds/mgmt/ahm/specialtopics.htm. The predicted cBPOPt in year t + 1 ( t+1 cBPOP ) for the additive harvest mortality model is calculated as { } t t s R t t w cBPOP cBPOP s (1 Rˆ ) Hˆ /(1 c) s 1 = + − − + γ where t cBPOP is the latitudeadjusted breeding population size in year t, s s and w s are the summer and winter survival rates, respectively, R γ is a biascorrection constant for the ageratio, c is the crippling loss rate, t Rˆ is the predicted ageratio, and t Hˆ is the predicted continental harvest. Discussion of t Rˆ and t Hˆ submodels are found in the following sections. The model uses the following constants: s s = 0.07, w s = 0.93, R γ = 0.8, and c = 0.20. The compensatory harvest mortality model serves as a hypothesis that stands in contrast to the additive harvest 3 2 1 0 1 2 3 2 1 0 1 2 0.00 0.05 0.10 0.15 0.20 0 1 2 3 4 5 6 7 8 x 105 N(B) z (environmental variable) Warm May No June Snow Cold May Lots of June Snow Model 1: DI Model 3: DD Propensity 0 1 2 3 4 5 6 7 8 x 105 24 mortality model, positing a strong but realistic degree of compensation. The compensatory model assumes that the mechanism for compensation is densitydependent postharvest (winter) survival. The form is a logistic relationship between winter survival and postharvest population size, with the relationship anchored around the historic mean values for each variable. For the compensatory model then, predicted winter survival rate in year t ( t s ) is calculated as [ ( ( )) ] 1 0 1 0 ( )1 = + − + − a+b P −P − t s s s s e t , where 1 s (upper asymptote) is 1.0, 0 s (lower asymptote) is 0.7, b (slope term) is 1.0, t P is the postharvest population size in year t (expressed in millions), P is the mean postharvest population size (4.295 million from 1974 through 2005), and a = logit 0 1 0 s s s s ⎛ − ⎞ ⎜ − ⎟ ⎝ ⎠ or ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − − ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − = 1 0 0 1 0 log 0 log 1 s s s s s s a s s , where s is 0.93 (mean winter survival rate). At moderate population size and latitude, the compensatory model allows for greater harvest (Fig. 10) than does the additive model (note especially that the size of the restrictive region [seasonwithinaseason] is smaller and is invoked when the latitude is higher). Also, 2 and 3bird bag limits are called for under more circumstances. But, at high population sizes, the higher bag limits are called for less often, because the compensatory model predicts that growth of the population will be slower (densitydependence). The fit to historic data was used to compare the additive and compensatory harvest models. From the t cBPOP , t mLAT , and observed harvest ( t H ) for the period 1974–through year t, the subsequent year’s breeding population size (on the latitudeadjusted scale) was predicted with both the additive and compensatory model, and compared to the observed breeding population size (on the latitudeadjusted scale). The meansquared error of the predictions from the additive model ( add MSE ) was calculated as: Σ = − − + = t t add add t t cBPOP cBPOP t MSE 1975 ( )2 ( 1975) 1 1 and the meansquared error of the predictions from the compensatory model were calculated in a similar manner. The model weights for the additive and compensatory model were calculated from their relative meansquared errors. The model weight for the additive model ( add W ) was calculated as: add comp add add MSE MSE W MSE 1 1 1 + = . The model weight for the compensatory model was found in a corresponding manner, or by subtracting the 25 additive model weight from 1.0. As of 2006, the compensatory model did not fit the historic data as well as the additive model; the model weights were 0.597 for the additive model and 0.403 for the compensatory model. The 2006 average model calls for a strategy that is intermediate between the additive and compensatory models (Fig. 10). Fig. 10. Statedependent harvest strategy for northern pintails with (A) additive, (B) compensatory, and (C) 2006 weighted models. In each case the strategy assumes that the general duck hunting season is that prescribed under the liberal regulatory alternative. 1 2 3 4 5 6 7 Average Latitude of the BPOP Closed Restrictive Liberal (1 bird) Liberal (3 birds) L2 51 52 53 54 55 56 57 58 59 1 2 3 4 5 6 7 Average Latitude of the BPOP Closed Restrictive Liberal (1 bird) Liberal (3 birds) L2 51 52 53 54 55 56 57 58 59 1 2 3 4 5 6 7 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive Liberal (1 bird) Liberal (3 birds) L2 51 52 53 54 55 56 57 58 59 (A) Additive Model (B) Compensatory Model (C) 2006 Weighted Model 26 Scaup The continental scaup (greater and lesser combined) population has experienced a longterm decline (Austin et al. 2000, Afton and Anderson 2001, Austin et al. 2006). As a result, waterfowl managers are challenged with the issue of how to manage the harvest of this declining population in the absence of an objective harvest strategy. In response to this dilemma, the USFWS Migratory Bird Regulations Committee requested that a scaup harvest strategy be developed for the 2007 regulations cycle. Here, we report on the development of a proposed decisionmaking framework to guide scaup harvest management. A detailed report is available online at http://www.fws.gov/migratorybirds/mgmt/ahm/specialtopics.htm. The lack of scaup demographic information over a sufficient timeframe and at a continental scale precludes the use of a traditional balance equation to represent scaup population and harvest dynamics. As a result, we used a discretetime, stochastic, logisticgrowth population model to represent changes in scaup abundance: ( (1 / ) ) . t 1 1 1 1 N N rN N K qH e t t t t t ε − − − − = + − − With this formulation, annual changes in population size (N) are predicted by the intrinsic rate of increase (r), the carrying capacity (K), a scaled harvest (H), and a process error (ε). We use a Bayesian approach (Meyer and Millar 1999) to estimate the population parameters, and to characterize the uncertainty associated with the monitoring programs (observation error) and the ability of our model to predict actual changes in the system (process error). Our initial assessment relied on the critical assumption that data used to estimate population parameters were measured on the same absolute scale. Research conducted to model waterfowl populations from different sources of information has provided evidence of bias in waterfowl survey programs (Martin et al. 1979, Runge et al. 2002). While the source(s) of this bias are not yet known, it is possible to estimate correction factors to reconcile predictions based on disparate sources of information. To address this issue, we chose to include an additional parameter (q) in our assessment to function as a scaling factor that enables us to combine breeding population and harvest estimates in an expression of population change. It is important to note that this parameter represents the combined limitations and uncertainty of all the monitoring data and functional relationships used in our assessment framework. Although, our initial attempts to estimate a scaling parameter from population and harvest data yielded reasonable estimates, the variance estimates were large. We found that the inclusion of a limited amount of scaup banding and recovery data provided enough information to structure the harvest process and reduce the uncertainty in the scaling parameter estimate. As in past analyses, the state space formulation and Bayesian analysis framework provided reasonable fits to the observed breeding population and total harvest estimates with realistic measures of variation. The posterior mean harvest rate estimates ranged from 0.03 to 0.08. In general, harvest rates fluctuated over the first decade and then tracked the declining population trend until the early 1990’s, when harvest rate estimates increased significantly before dropping in 1999 (Fig. 11). The posterior mean estimate of the intrinsic rate of increase (r) is 0.110 while the posterior mean estimate of the carrying capacity (K) is 8.236 million birds (Table 1). The posterior mean estimate of the scaling parameter (q) is 0.541, ranging between 0.461 and 0.630 with 95% probability. Based on the estimated population parameters, the estimated average maximum sustainable yield (MSY) on the adjusted scale is 0.211 million scaup (0.389 million scaup on the observed scale). 27 Fig. 11. The posterior mean scaup population and harvest rate estimates derived from a Bayesian analysis of the modified logistic model. We used SDP software (Lubow 1995) to derive a statedependent harvest strategy under an objective to maximize longterm cumulative harvest (MSY) and an objective to attain a shoulder point (calculated as percentage of MSY) on the yield curve. We evaluated harvest levels from 0 to 5 million (in increments of 50,000) for population sizes of 1 to 10 million (in increments of 50,000) and harvest objectives ranging from 90 to 100% MSY (in 2 % increments). For each optimization we assumed perfect control over the harvest decision variable. We then simulated each strategy for 5000 iterations to characterize the management performance expected if the harvest strategy was followed and system dynamics did not change. Under an objective to maximize longterm cumulative harvest (MSY) the resulting strategy is extremely knifeedged (Fig. 12). This strategy prescribes zero harvests for population sizes less than 3.2 million and seeks to hold the population size at maximum productivity (one half the carrying capacity). In contrast to the MSY strategy, the harvest strategies necessary to achieve a shoulder point are considerably less knifeedged and would allow for harvest at lower population sizes. However, current scaup harvest levels (317,000) exceed the prescribed harvests resulting from optimizations with each of the objective functions we evaluated. The simulated management performance of each harvest strategy demonstrates the tradeoffs that arise when a shoulder point objective is used to derive an optimal harvest strategy. As the desired shoulder point moves away from MSY, average harvest levels decrease while the average population increases. The USFWS intends to work with the Flyways over the next year to determine an acceptable harvestmanagement objective and a set of regulatory alternatives that would be used in conjunction with our modeling framework to derive an optimal harvest strategy for scaup. 1975 1985 1995 2005 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Year Population X 10^6 Population 0.03 0.04 0.05 0.06 0.07 0.08 Harvest Rate Harvest Rate 28 Fig. 12. Optimal harvests of scaup as a function of the observed breeding population size derived under objective functions ranging from 90 to 100 percent of the maximum longterm yield (MSY). 0 1 2 3 4 5 6 7 8 9 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 BPOP Harvest MSY 98% MSY 96% MSY 95% MSY 94% MSY 92% MSY 90% MSY 29 LITERATURE CITED Afton, A. D., and M. G. Anderson. 2001. Declining scaup populations: a retrospective analysis of longterm population and harvest survey data. Journal of Wildlife Management 60:8393. Anderson, D. R., and K. P. Burnham. 1976. Population ecology of the mallard. VI. The effect of exploitation on survival. U.S. Fish and Wildlife Service Resource Publication No. 128. 66pp. Austin, J. E., A. D. Afton, M. G. Anderson, R. G. Clark, C. M. Custer, J. S. Lawrence, J. B. Pollard and J. K. Ringelman. 2000. Declining scaup populations: issues, hypotheses, and research needs. Wildlife Society Bulletin 28:254263. Austin, J. E., M. J. Anteau, J. S. Barclay, G. S. Boomer, F. C. Rohwer, and S. M. Slattery. 2006. Declining scaup populations: reassessment of the issues, hypotheses, and research directions. Consensus Report from the Second Scaup Workshop. 7pp. Blohm, R. J. 1989. Introduction to harvest  understanding surveys and season setting. Proceedings of the International Waterfowl Symposium 6:118133. Blohm, R. J., R. E. Reynolds, J. P. Bladen, J. D. Nichols, J. E. Hines, K. P. Pollock, and R. T. Eberhardt. 1987. Mallard mortality rates on key breeding and wintering areas. Transactions of the North American Wildlife and Natural Resources Conference 52:246263. Burnham, K. P., G. C. White, and D. R. Anderson. 1984. Estimating the effect of hunting on annual survival rates of adult mallards. Journal of Wildlife Management 48:350361. Conroy, M. J., M. W. Miller, and J. E. Hines. 2002. Identification and synthetic modeling of factors affecting American black duck populations. Wildlife Monographs 150. 64pp. Heusman, H W, and J. R. Sauer. 2000. The northeastern states’ waterfowl breeding population survey. Wildlife Society Bulletin 28:355364. Johnson, F. A. 2003. Population dynamics of ducks other than mallards in midcontinent North America. Draft. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 15pp. Johnson, F. A., J. A. Dubovsky, M. C. Runge, and D. R. Eggeman. 2002a. A revised protocol for the adaptive harvest management of eastern mallards. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 13pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/emalahm2002.pdf. Johnson, F. A., W. L. Kendall, and J. A. Dubovsky. 2002b. Conditions and limitations on learning in the adaptive management of mallard harvests. Wildlife Society Bulletin 30:176185. Johnson, F. A., C. T. Moore, W. L. Kendall, J. A. Dubovsky, D. F. Caithamer, J. R. Kelley, Jr., and B. K. Williams. 1997. Uncertainty and the management of mallard harvests. Journal of Wildlife Management 61:202216. Johnson, F. A., and B. K. Williams. 1999. Protocol and practice in the adaptive management of waterfowl harvests. Conservation Ecology 3(1): 8. [online] URL: http://www.consecol.org/vol3/iss1/art8. Johnson, F. A., B. K. Williams, J. D. Nichols, J. E. Hines, W. L. Kendall, G. W. Smith, and D. F. Caithamer. 1993. Developing an adaptive management strategy for harvesting waterfowl in North America. Transactions of the North American Wildlife and Natural Resources Conference 58:565583. 30 Johnson, F. A., B. K. Williams, and P. R. Schmidt. 1996. Adaptive decisionmaking in waterfowl harvest and habitat management. Proceedings of the International Waterfowl Symposium 7:2633. Link, W. A., J. R. Sauer, and D. K. Niven. 2006. A hierarchical model for regional analysis of population change using Christmas bird count data, with application to the American black duck. The Condor 108:1324. Lubow, B. C. 1995. SDP: Generalized software for solving stochastic dynamic optimization problems. Wildlife Society Bulletin 23:738742. Martin, F. W., R. S. Pospahala, and J. D. Nichols. 1979. Assessment and population management of North American migratory birds. Pages 187239 in J. Cairns, G. P. Patil, and W. E. Waters, eds., Environmental biomonitoring, assessment, prediction and management — certain case studies and related quantitative issues. Statistical Ecology, Vol. S11. International Cooperative Publishing House, Fairland, MD. Meyer, R., and R. B. Millar. 1999. BUGS in Bayesian stock assessments. Canadian Journal of Fisheries and Aquatic Sciences 56:10781086. Munro, R. E., and C. F. Kimball. 1982. Population ecology of the mallard. VII. Distribution and derivation of the harvest. U.S. Fish and Wildlife Service Resource Publication 147. 127pp. Nichols, J. D., F. A. Johnson, and B. K. Williams. 1995. Managing North American waterfowl in the face of uncertainty. Annual Review of Ecology and Systematics 26:177199. Runge, M. C., F. A. Johnson, J. A. Dubovsky, W. L. Kendall, J. Lawrence, and J. Gammonley. 2002. A revised protocol for the adaptive harvest management of midcontinent mallards. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 28pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/MCMrevise2002.pdf. U.S. Fish and Wildlife Service. 2000. Adaptive harvest management: 2000 duck hunting season. U.S. Dept. Interior, Washington. D.C. 43pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm00/ahm2000.pdf. U.S. Fish and Wildlife Service. 2001. Frameworkdate extensions for duck hunting in the United States: projected impacts & coping with uncertainty, U.S. Dept. Interior, Washington, D.C. 8pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm01/fwassess.pdf. U.S. Fish and Wildlife Service. 2002. Adaptive harvest management: 2002 duck hunting season. U.S. Dept. Interior, Washington. D.C. 34pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/2002AHMreport.pdf. Walters, C. J. 1986. Adaptive management of renewable resources. MacMillan Publ. Co., New York, N.Y. 374pp. Williams, B. K., and F. A. Johnson. 1995. Adaptive management and the regulation of waterfowl harvests. Wildlife Society Bulletin 23:430436. Williams, B. K., F. A. Johnson, and K. Wilkins. 1996. Uncertainty and the adaptive management of waterfowl harvests. Journal of Wildlife Management 60:223232. 31 APPENDIX A: AHM Working Group (Note: This list includes only permanent members of the AHM Working Group. Not listed here are numerous persons from federal and state agencies that assist the Working Group on an adhoc basis.) Coordinator: Fred Johnson U.S. Fish & Wildlife Service Bldg. 810, University of Florida P.O. Box 110485 Gainesville, FL 32611 phone: 3523925075 fax: 3528460841 email: fred_a_johnson@fws.gov USFWS Representatives: Bob Blohm (Region 9) U.S. Fish and Wildlife Service 4401 N Fairfax Drive MS MSP4107 Arlington, VA 22203 phone: 7033581966 fax: 7033582272 email: robert_blohm@fws.gov Brad Bortner (Region 1) U.S. Fish and Wildlife Service 911 NE 11th Ave. Portland, OR 972324181 phone: 5032316164 fax: 5032312364 email: brad_bortner@fws.gov Dave Case (contractor) D.J. Case & Associates 607 Lincolnway West Mishawaka, IN 46544 phone: 5742580100 fax: 5742580189 email: dave@djcase.com Jim Dubovsky (Region 6) U.S. Fish and Wildlife Service P.O. Box 25486DFC Denver, CO 802250486 phone: 3032364403 fax: 3032368680 email:james_dubovsky@fws.gov Jeff Haskins (Region 2) U.S. Fish and Wildlife Service P.O. Box 1306 Albuquerque, NM 87103 phone: 5052486827 (ext 30) fax: 5052487885 email: jeff_haskins@fws.gov Jim Kelley (Region 9) U.S. Fish and Wildlife Service 1 Federal Drive Fort Snelling, MN 551110458 phone: 6127135409 fax: 6127135393 email: james_r_kelley@fws.gov Sean Kelly (Region 3) U.S. Fish and Wildlife Service 1 Federal Drive Ft. Snelling, MN 551114056 phone: 6127135470 fax: 6127135393 email: sean_kelly@fws.gov Paul Padding (Region 9) U.S. Fish and Wildlife Service 11510 American Holly Drive Laurel, MD 20708 phone: 3014975851 fax: 3014975885 email: paul_padding@fws.gov 32 Diane Pence (Region 5) U.S. Fish and Wildlife Service 300 Westgate Center Drive Hadley, MA 010359589 phone: 4132538577 fax: 4132538424 email: diane_pence@fws.gov Russ Oates (Region 7) U.S. Fish and Wildlife Service 1011 East Tudor Road Anchorage, AK 995036119 phone: 9077863446 fax: 9077863641 email: russ_oates@fws.gov Dave Sharp (Region 9) U.S. Fish and Wildlife Service P.O. Box 25486, DFC Denver, CO 802250486 phone: 3032752386 fax: 3032752384 email: dave_sharp@fws.gov Bob Trost (Region 9) U.S. Fish and Wildlife Service 911 NE 11th Ave. Portland, OR 972324181 phone: 5032316162 fax: 5032316228 email: robert_trost@fws.gov David Viker (Region 4) U.S. Fish and Wildlife Service 1875 Century Blvd., Suite 345 Atlanta, GA 30345 phone: 4046797188 fax: 4046797285 email: david_viker@fws.gov Canadian Wildlife Service Representatives: Dale Caswell Canadian Wildlife Service 123 Main St. Suite 150 Winnepeg, Manitoba, Canada R3C 4W2 phone: 2049835260 fax: 2049835248 email: dale.caswell@ec.gc.ca Eric Reed Canadian Wildlife Service 351 St. Joseph Boulevard Hull, QC K1A OH3, Canada phone: 8199530294 fax: 8199536283 email: eric.reed@ec.gc.ca Flyway Council Representatives: Scott Baker (Mississippi Flyway) Mississippi Dept. of Wildlife, Fisheries, and Parks P.O. Box 378 Redwood, MS 39156 phone: 6016610294 fax: 6013642209 email: mahannah1@aol.com Diane Eggeman (Atlantic Flyway) Florida Fish and Wildlife Conservation Commission 8932 Apalachee Pkwy. Tallahassee, FL 32311 phone: 8504885878 fax: 8504885884 email: diane.eggeman@fwc.state.fl.us Mike Johnson (Central Flyway) North Dakota Game and Fish Department 100 North Bismarck Expressway Bismarck, ND 585015095 phone: 7013286319 fax: 7013286352 email: mjohnson@state.nd.us Don Kraege (Pacific Flyway) Washington Dept. of Fish and Wildlife 600 Capital Way North Olympia. WA 985011091 phone: 3609022509 fax: 3609022162 email: kraegdkk@dfw.wa.gov 33 Bryan Swift (Atlantic Flyway) Dept. Environmental Conservation 625 Broadway Albany, NY 122334754 phone: 5184028866 fax: 5184029027 or 4028925 email: blswift@gw.dec.state.ny.us Mark Vrtiska (Central Flyway) Nebraska Game and Parks Commission P.O. Box 30370 2200 North 33rd Street Lincoln, NE 685031417 phone: 4024715437 fax: 4024715528 email: mvrtiska@ngpc.state.ne.us Dan Yparraguirre (Pacific Flyway) California Dept. of Fish and Game 1812 Ninth Street Sacramento, CA 95814 phone: 9164453685 email: dyparraguirre@dfg.ca.gov Guy Zenner (Mississippi Flyway) Iowa Dept. of Natural Resources 1203 North Shore Drive Clear Lake, IA 50428 phone: 5153573517, ext. 23 fax: 5153575523 email: gzenner@netins.net 34 APPENDIX B: Midcontinent Mallard Models Model Structure In 2002 we extensively revised the set of alternative models describing the population dynamics of midcontinent mallards (Runge et al. 2002, USFWS 2002). Collectively, the models express uncertainty (or disagreement) about whether harvest is an additive or compensatory form of mortality (Burnham et al. 1984), and whether the reproductive process is weakly or strongly densitydependent (i.e., the degree to which reproductive rates decline with increasing population size). All population models for midcontinent mallards share a common “balance equation” to predict changes in breedingpopulation size as a function of annual survival and reproductive rates: Nt Nt (mSt AM ( m)(St AF Rt (St JF St JM F ))) sum M sum + = + − + + 1 , 1 , , , φ φ where: N = breeding population size, m = proportion of males in the breeding population, SAM, SAF, SJF, and SJM = survival rates of adult males, adult females, young females, and young males, respectively, R = reproductive rate, defined as the fall age ratio of females, φ F φ sum M sum = the ratio of female (F) to male (M) summer survival, and t = year. We assumed that m and φ F φ sum M sum are fixed and known. We also assumed, based in part on information provided by Blohm et al. (1987), the ratio of female to male summer survival was equivalent to the ratio of annual survival rates in the absence of harvest. Based on this assumption, we estimated φ F φ sum M sum = 0.897. To estimate m we expressed the balance equation in matrix form: N N S RS S RS N N t AM t AF AM JM F sum M sum AF JF t AM t AF + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 1 0 , , , , φ φ and substituted the constant ratio of summer survival and means of estimated survival and reproductive rates. The right eigenvector of the transition matrix is the stable sex structure that the breeding population eventually would attain with these constant demographic rates. This eigenvector yielded an estimate of m = 0.5246. Using estimates of annual survival and reproductive rates, the balance equation for midcontinent mallards overpredicted observed population sizes by 10.8% on average. The source of the bias is unknown, so we modified the balance equation to eliminate the bias by adjusting both survival and reproductive rates: Nt S Nt (mSt AM ( m)(St AF RRt (St JF St JM F ))) sum M sum + 1 = γ , + 1− , + γ , + , φ φ where γ denotes the biascorrection factors for survival (S) and reproduction (R). We used a least squares approach to estimate γS = 0.9479 and γR = 0.8620. 35 Survival Process We considered two alternative hypotheses for the relationship between annual survival and harvest rates. For both models, we assumed that survival in the absence of harvest was the same for adults and young of the same sex. In the model where harvest mortality is additive to natural mortality: St sex age s sex ( K ) A , , , t ,sex,age = − 0 1 and in the model where changes in natural mortality compensate for harvest losses (up to some threshold): S s if K s t sex age K if K s sex C t sex age sex C t sex age t sex age sex , , C , ,, , , , , , , = ≤ − − > − ⎧⎨ ⎪ ⎩⎪ 0 0 0 1 1 1 where s0 = survival in the absence of harvest under the additive (A) or compensatory (C) model, and K = harvest rate adjusted for crippling loss (20%, Anderson and Burnham 1976). We averaged estimates of s0 across banding reference areas by weighting by breedingpopulation size. For the additive model, s0 = 0.7896 and 0.6886 for males and females, respectively. For the compensatory model, s0 = 0.6467 and 0.5965 for males and females, respectively. These estimates may seem counterintuitive because survival in the absence of harvest should be the same for both models. However, estimating a common (but still sexspecific) s0 for both models leads to alternative models that do not fit available bandrecovery data equally well. More importantly, it suggests that the greatest uncertainty about survival rates is when harvest rate is within the realm of experience. By allowing s0 to differ between additive and compensatory models, we acknowledge that the greatest uncertainty about survival rate is its value in the absence of harvest (i.e., where we have no experience). Reproductive Process Annual reproductive rates were estimated from age ratios in the harvest of females, corrected using a constant estimate of differential vulnerability. Predictor variables were the number of ponds in May in Prairie Canada (P, in millions) and the size of the breeding population (N, in millions). We estimated the bestfitting linear model, and then calculated the 80% confidence ellipsoid for all model parameters. We chose the two points on this ellipsoid with the largest and smallest values for the effect of breedingpopulation size, and generated a weakly densitydependent model: Rt = 0.7166 + 0.1083Pt − 0.0373Nt and a strongly densitydependent model: Rt = 1.1390 + 0.1376Pt − 0.1131Nt Pond Dynamics We modeled annual variation in Canadian pond numbers as a firstorder autoregressive process. The estimated model was: P P t + t t = + + 1 2.2127 0.3420 ε where ponds are in millions and εt is normally distributed with mean = 0 and variance = 1.2567. 36 Variance of Prediction Errors Using the balance equation and submodels described above, predictions of breedingpopulation size in year t+1 depend only on specification of population size, pond numbers, and harvest rate in year t. For the period in which comparisons were possible, we compared these predictions with observed population sizes. We estimated the predictionerror variance by setting: ( ) ( ) ( ) [ ( ) ( )] ( ) e N N e N N N n t t obs t pre t t obs t pre t = − = Σ − − ln ln ~ , $ ln ln then assuming and estimating 0 1 σ 2 σ 2 2 where obs and pre are observed and predicted population sizes (in millions), respectively, and n = the number of years being compared. We were concerned about a variance estimate that was too small, either by chance or because the number of years in which comparisons were possible was small. Therefore, we calculated the upper 80% confidence limit for σ2 based on a Chisquared distribution for each combination of the alternative survival and reproductive submodels, and then averaged them. The final estimate of σ2 was 0.0243, equivalent to a coefficient of variation of about 17%. Model Implications The set of alternative population models suggests that carrying capacity (average population size in the absence of harvest) for an average number of Canadian ponds is somewhere between about 6 and 16 million mallards. The population model with additive hunting mortality and weakly densitydependent recruitment (SaRw) leads to the most conservative harvest strategy, whereas the model with compensatory hunting mortality and strongly densitydependent recruitment (ScRs) leads to the most liberal strategy. The other two models (SaRs and ScRw) lead to strategies that are intermediate between these extremes. Under the models with compensatory hunting mortality (ScRs and ScRw), the optimal strategy is to have a liberal regulation regardless of population size or number of ponds because at harvest rates achieved under the liberal alternative, harvest has no effect on population size. Under the strongly densitydependent model (ScRs), the densitydependence regulates the population and keeps it within narrow bounds. Under the weakly densitydependent model (ScRw), the densitydependence does not exert as strong a regulatory effect, and the population size fluctuates more. Model Weights Model weights are calculated as Bayesian probabilities, reflecting the relative ability of the individual alternative models to predict observed changes in population size. The Bayesian probability for each model is a function of the model’s previous (or prior) weight and the likelihood of the observed population size under that model. We used Bayes’ theorem to calculate model weights from a comparison of predicted and observed population sizes for the years 19962004, starting with equal model weights in 1995. For the purposes of updating, we predicted breedingpopulation size in the traditional survey area in year t + 1, from breedingpopulation size, Canadian ponds, and harvest rates in year t. 37 Inclusion of Mallards in the Great Lakes Region Model development originally did not include mallards breeding in the states of Wisconsin, Minnesota, and Michigan, primarily because full data sets were not available from these areas to permit the necessary analysis. However, mallards in the Great Lakes region have been included in the midcontinent mallard AHM protocol since 1997 by assuming that population dynamics for these mallards are similar to those in the traditional survey area. Based on that assumption, predictions of breeding population size are scaled to reflect inclusion of mallards in the Great Lakes region. From 1992 through 2007, when population estimates were available for all three states, the average proportion of the total midcontinent mallard population that was in the Great Lakes region was 0.1099 (SD = 0.0207). We assumed a normal distribution with these parameter values to make the conversion between the traditional survey area and total breedingpopulation size. 38 APPENDIX C: Eastern Mallard Models Model Structure We also revised the population models for eastern mallards in 2002 (Johnson et al. 2002a, USFWS 2002). The current set of six models: (1) relies solely on federal and state waterfowl surveys (rather than the Breeding Bird Survey) to estimate abundance; (2) allows for the possibility of a positive bias in estimates of survival or reproductive rates; (3) incorporates competing hypotheses of strongly and weakly densitydependent reproduction; and (4) assumes that hunting mortality is additive to other sources of mortality. As with midcontinent mallards, all population models for eastern mallards share a common balance equation to predict changes in breedingpopulation size as a function of annual survival and reproductive rates: Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S )) am t af t m t ym t m t yf + = ⋅ ⋅ + − ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 1 1 ψ where: N = breedingpopulation size, p = proportion of males in the breeding population, Sam, Saf, Sym, and Syf = survival rates of adult males, adult females, young males, and young females, respectively, Am = ratio of young males to adult males in the harvest, d = ratio of young male to adult male direct recovery rates, ψ = the ratio of male to female summer survival, and t = year. In this balance equation, we assume that p, d, and ψ are fixed and known. The parameter ψ is necessary to account for the difference in anniversary date between the breedingpopulation survey (May) and the survival and reproductive rate estimates (August). This model also assumes that the sex ratio of fledged young is 1:1; hence Am/d appears twice in the balance equation. We estimated d = 1.043 as the median ratio of young:adult male bandrecovery rates in those states from which wing receipts were obtained. We estimated ψ = 1.216 by regressing through the origin estimates of male survival against female survival in the absence of harvest, assuming that differences in natural mortality between males and females occur principally in summer. To estimate p, we used a population projection matrix of the form: ( ) ( ) ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ ⋅ + ⋅ = ⎥⎦ ⎤ ⎢⎣ ⎡ + + t t m yf af am m ym t t F M A d S S S A d S F M ψ 0 1 1 where M and F are the relative number of males and females in the breeding populations, respectively. To parameterize the projection matrix we used average annual survival rate and age ratio estimates, and the estimates of d and ψ provided above. The right eigenvector of the projection matrix is the stable proportion of males and females the breeding population eventually would attain in the face of constant demographic rates. This eigenvector yielded an estimate of p = 0.544. We also attempted to determine whether estimates of survival and reproductive rates were unbiased. We relied on the balance equation provided above, except that we included additional parameters to correct for any bias that might exist. Because we were unsure of the source(s) of potential bias, we alternatively assumed that any bias resided solely in survival rates: Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S )) am t af t m t ym t m t yf + = ⋅ ⋅ ⋅ + − ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 1 Ω 1 ψ 39 (where Ω is the biascorrection factor for survival rates), or solely in reproductive rates: Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S )) am t af t m t ym t m t yf + = ⋅ ⋅ + − ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ 1 1 α α ψ (where α is the biascorrection factor for reproductive rates). We estimated Ω and α by determining the values of these parameters that minimized the sum of squared differences between observed and predicted population sizes. Based on this analysis, Ω = 0.836 and α = 0.701, suggesting a positive bias in survival or reproductive rates. However, because of the limited number of years available for comparing observed and predicted population sizes, we also retained the balance equation that assumes estimates of survival and reproductive rates are unbiased. Survival Process For purposes of AHM, annual survival rates must be predicted based on the specification of regulationspecific harvest rates (and perhaps on other uncontrolled factors). Annual survival for each age (i) and sex (j) class under a given regulatory alternative is: ( ) ( ) S h v c t i j j t am i j , , = ⋅ − ⋅ − ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ θ 1 1 where: S = annual survival, θ j = mean survival from natural causes, ham = harvest rate of adult males, and v = harvest vulnerability relative to adult males, c = rate of crippling (unretrieved harvest). This model assumes that annual variation in survival is due solely to variation in harvest rates, that relative harvest vulnerability of the different agesex classes is fixed and known, and that survival from natural causes is fixed at its sample mean. We estimated θ j = 0.7307 and 0.5950 for males and females, respectively. Reproductive process As with survival, annual reproductive rates must be predicted in advance of setting regulations. We relied on the apparent relationship between breedingpopulation size and reproductive rates: Rt = a ⋅ exp(b ⋅ Nt ) where Rt is the reproductive rate (i.e., Am d t ), Nt is breedingpopulation size in millions, and a and b are model parameters. The leastsquares parameter estimates were a = 2.508 and b = 0.875. Because of both the importance and uncertainty of the relationship between population size and reproduction, we specified two alternative models in which the slope (b) was fixed at the leastsquares estimate ± one standard error, and in which the intercepts (a) were subsequently reestimated. This provided alternative hypotheses of strongly densitydependent (a = 4.154, b = 1.377) and weakly densitydependent reproduction (a = 1.518, b = 0.373). 40 Variance of Prediction Errors Using the balance equations and submodels provided above, predictions of breedingpopulation size in year t+1 depend only on the specification of a regulatory alternative and on an estimate of population size in year t. For the period in which comparisons were possible (199196), we were interested in how well these predictions corresponded with observed population sizes. In making these comparisons, we were primarily concerned with how well the biascorrected balance equations and reproductive and survival submodels performed. Therefore, we relied on estimates of harvest rates rather than regulations as model inputs. We estimated the predictionerror variance by setting: ( ) ( ) ( ) [ ( ) ( )] e N N e N N N n t t obs t pre t t obs t pre t = − = Σ ��� ln ln ~ , $ ln ln then assuming and estimating 0 σ 2 σ 2 2 where obs and pre are observed and predicted population sizes (in millions), respectively, and n = 6. Variance estimates were similar regardless of whether we assumed that the bias was in reproductive rates or in survival, or whether we assumed that reproduction was strongly or weakly densitydependent. Thus, we averaged variance estimates to provide a final estimate of σ2 = 0.006, which is equivalent to a coefficient of variation (CV) of 8.0%. We were concerned, however, about the small number of years available for estimating this variance. Therefore, we estimated an 80% confidence interval for σ2 based on a Chisquared distribution and used the upper limit for σ2 = 0.018 (i.e., CV = 14.5%) to express the additional uncertainty about the magnitude of prediction errors attributable to potentially important environmental effects not expressed by the models. Model Implications Modelspecific regulatory strategies based on the hypothesis of weakly densitydependent reproduction are considerably more conservative than those based on the hypothesis of strongly densitydependent reproduction. The three models with weakly densitydependent reproduction suggest a carrying capacity (i.e., average population size in the absence of harvest) >2.0 million mallards, and prescribe extremely restrictive regulations for population size <1.0 million. The three models with strongly densitydependent reproduction suggest a carrying capacity of about 1.5 million mallards, and prescribe liberal regulations for population sizes >300 thousand. Optimal regulatory strategies are relatively insensitive to whether models include a bias correction or not. All modelspecific regulatory strategies are “knifeedged,” meaning that large differences in the optimal regulatory choice can be precipitated by only small changes in breedingpopulation size. This result is at least partially due to the small differences in predicted harvest rates among the current regulatory alternatives (see the section on Regulatory Alternatives later in this report). Model Weights We used Bayes’ theorem to calculate model weights from a comparison of predicted and observed population sizes for the years 19962006. We calculated weights for the alternative models based on an assumption of equal model weights in 1996 (the last year data was used to develop most model components) and on estimates of yearspecific harvest rates (Appendix D). 41 APPENDIX D: Modeling Mallard Harvest Rates We modeled harvest rates of midcontinent mallards within a Bayesian hierarchical framework. We developed a set of models to predict harvest rates under each regulatory alternative as a function of the harvest rates observed under the liberal alternative, using historical information relating harvest rates to various regulatory alternatives. We modeled the probability of regulationspecific harvest rates (h) based on normal distributions with the following parameterizations: Closed: Restrictive: Moderate: Liberal: p h N p h N p h N p h N C C C R R L R M M L f M L L f L ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) μ ν γ μ ν γ μ δ ν μ δ ν 2 2 2 2 + + For the restrictive and moderate alternatives we introduced the parameter γ to represent the relative difference between the harvest rate observed under the liberal alternative and the moderate or restrictive alternatives. Based on this parameterization, we are making use of the information that has been gained (under the liberal alternative) and are modeling harvest rates for the restrictive and moderate alternatives as a function of the mean harvest rate observed under the liberal alternative. For the harvestrate distributions assumed under the restrictive and moderate regulatory packages, we specified that γR and γM are equal to the prior estimates of the predicted mean harvest rates under the restrictive and moderate alternatives divided by the prior estimates of the predicted mean harvest rates observed under the liberal alternative. Thus, these parameters act to scale the mean of the restrictive and moderate distributions in relation to the mean harvest rate observed under the liberal regulatory alternative. We also considered the marginal effect of frameworkdate extensions under the moderate and liberal alternatives by including the parameter δf. In order to update the probability distributions of harvest rates realized under each regulatory alternative, we first needed to specify a prior probability distribution for each of the model parameters. These distributions represent prior beliefs regarding the relationship between each regulatory alternative and the expected harvest rates. We used a normal distribution to represent the mean and a scaled inversechisquare distribution to represent the variance of the normal distribution of the likelihood. For the mean (μ) of each harvestrate distribution associated with each regulatory alternative, we use the predicted mean harvest rates provided in USFWS (2000a:1314), assuming uniformity of regulatory prescriptions across flyways. We set prior values of each standard deviation (ν) equal to 20% of the mean (CV = 0.2) based on an analysis by Johnson et al. (1997). We then specified the following prior distributions and parameter values under each regulatory package: Closed (in U.S. only): p N p ScaledInv C C ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 0 0088 0 0018 6 6 0 0018 2 2 2 2 − These closedseason parameter values are based on observed harvest rates in Canada during the 198893 seasons, which was a period of restrictive regulations in both Canada and the United States. For the restrictive and moderate alternatives, we specified that the standard error of the normal distribution of the scaling parameter is based on a coefficient of variation for the mean equal to 0.3. The scale parameter of the inversechisquare distribution was set equal to the standard deviation of the harvest rate mean under the restrictive and moderate regulation alternatives (i.e., CV = 0.2). 42 Restrictive: p N p Scaled Inv R R ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 051 015 6 6 0 0133 2 2 2 2 − Moderate: p N p ScaledInv M M ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 085 026 6 6 0 0223 2 2 2 2 − Liberal: p N p ScaledInv L L ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 01305 0 0261 6 6 0 0261 2 2 2 2 − The prior distribution for the marginal effect of the frameworkdate extension was specified as: p( ) N( ) f δ ~ 0.02,0.012 The prior distributions were multiplied by the likelihood functions based on the last seven years of data under liberal regulations, and the resulting posterior distributions were evaluated with Markov Chain Monte Carlo simulation. Posterior estimates of model parameters and of annual harvest rates are provided in the following table: Parameter Estimate SD Parameter Estimate SD μC 0.0088 0.0022 h1998 0.1098 0.0113 νC 0.0019 0.0005 h1999 0.1002 0.0076 γR 0.5090 0.0617 h2000 0.1252 0.0099 νR 0.0129 0.0033 h2001 0.1068 0.0112 γM 0.8530 0.1062 h2002 0.1145 0.0057 νM 0.0216 0.0055 h2003 0.1100 0.0064 μL 0.1139 0.0070 h2004 0.11188 0.0098 νL 0.0205 0.0055 h2005 0.1158 0.0081 δf 0.0087 0.0079 h2006 0.1061 0.0073 43 We modeled harvest rates of eastern mallards using the same parameterizations as those for midcontinent mallards: Closed: Restrictive: Moderate: Liberal: p h N p h N p h N p h N C C C R R L R M M L f M L L f L ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) ( )~ ( , ) μ ν γ μ ν γ μ δ ν μ δ ν 2 2 2 2 + + We set prior values of each standard deviation (ν) equal to 30% of the mean (CV = 0.3) to account for additional variation due to changes in regulations in the other Flyways and their unpredictable effects on the harvest rates of eastern mallards. We then specified the following prior distribution and parameter values for the liberal regulatory alternative: Liberal: p N p ScaledInv L L ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 01771 0 0531 6 6 0 0531 2 2 2 2 − Moderate: p N p ScaledInv M M ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 092 028 6 6 0 0488 2 2 2 2 − Restrictive: p N p ScaledInv R R ( )~ ( . , . ) ( )~ ( , . ) γ ν χ 076 028 6 6 0 0406 2 2 2 2 − Closed (in U.S. only): p N p ScaledInv C C ( )~ ( . , . ) ( )~ ( , . ) μ ν χ 0 0800 0 0240 6 6 0 0240 2 2 2 2 − A previous analysis suggested that the effect of the frameworkdate extension on eastern mallards would be of lower magnitude and more variable than on midcontinent mallards (USFWS 2000). Therefore, we specified the following prior distribution for the marginal effect of the frameworkdate extension for eastern mallards as: p( ) N( ) f δ ~ 0.01,0.012 44 The prior distributions were multiplied by the likelihood functions based on the last four years of data under liberal regulations, and the resulting posterior distributions were evaluated with Markov Chain Monte Carlo simulation. Posterior estimates of model parameters and of annual harvest rates are provided in the following table: Parameter Estimate SD Parameter Estimate SD μC 0.0798 0.0262 h2002 0.1627 0.0129 νC 0.0233 0.0059 h2003 0.1462 0.0104 γR 0.7642 0.1144 h2004 0.1364 0.0114 νR 0.0395 0.0104 h2005 0.1311 0.0120 γM 0.9187 0.1148 h2006 0.1048 0.0134 νM 0.0474 0.0121 μL 0.1532 0.0166 νL 0.0459 0.0099 δf 0.0046 0.0096 
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Date created  20130123 
Date modified  20130306 



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