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Adaptive Harvest Management 22000066 Hunttiingg Seeaassoon U.S. Fish & Wildlife Service 1 Adaptive Harvest Management 2006 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 2006 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 duckhunting 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 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 community, 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 D. Case (D.J. Case & Assoc.), M. Conroy (U.S. Geological Survey [USGS]), E. Cooch (Cornell University), P. Garrettson (USFWS), W. Harvey (Maryland Dept. of Natural Resources), R. Raftovich (USFWS), E. Reed (Canadian Wildlife Service), K. Richkus (USFWS), J. Royle (USGS), M. Runge (USGS), J. Serie, (USFWS), S. Sheaffer (Cornell University), 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. 2006. Adaptive Harvest Management: 2006 Hunting Season. U.S. Dept. Interior, Washington, D.C. 45pp. 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 .......................................................................................21 Regulatory Alternatives .....................................................................................................22 Optimal Regulatory Strategies ...........................................................................................25 Application of AHM Concepts to Species of Concern......................................................27 Literature Cited ..................................................................................................................37 Appendix A: AHM Working Group .................................................................................39 Appendix B: Modeling Mallard Harvest Rates.................................................................42 This report and others regarding Adaptive Harvest Management are available online at http://migratorybirds.fws.gov 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 original AHM protocol was based solely on the dynamics of midcontinent mallards, but efforts are being made to account for mallards breeding eastward and westward of the midcontinent region. The challenge for managers is to vary hunting regulations among Flyways in a manner that recognizes each Flyway’s unique breedingground derivation of mallards. For the 2006 hunting season, the USFWS will continue to consider a regulatory choice for the Atlantic Flyway that depends exclusively on the status of eastern mallards. The prescribed regulatory choice for the Mississippi, Central, and Pacific Flyways continues to depend exclusively on the status of midcontinent mallards. Investigations of the dynamics of western mallards (and their potential effect on regulations in the West) are continuing and the USFWS is not yet prepared to recommend an AHM protocol for this mallard stock. The mallard population models that are the basis for prescribing hunting regulations were revised extensively in 2002. These revised 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 (91%). Evidence for the additivemortality hypothesis remains equivocal (58%). For eastern mallards, current model weights favor the strongly densitydependent reproductive hypothesis (65%). By consensus, hunting mortality is assumed to be additive in eastern mallards. For the 2006 hunting season, the USFWS is continuing to consider 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. Also, at the request of the Flyway Councils in 2003 the USFWS agreed to exclude closed duckhunting seasons from the AHM protocol when the breedingpopulation size of midcontinent mallards is >5.5 million (traditional survey area plus the Great Lakes region). Harvest rates associated with each of the regulatory alternatives are predicted using Bayesian statistical methods. Essentially, the idea is to use historical information to develop initial harvestrate predictions, to make regulatory decisions based on those predictions, and then to observe realized harvest rates. Those observed harvest rates, in turn, are used to update the predictions. Using this approach, predictions of harvest rates of mallards under the regulatory alternatives have been updated based on bandreporting rate studies conducted since 1998. Estimated harvest rates from the 20022005 liberal hunting seasons have averaged 0.12 (SD = 0.01) and 0.14 (SD = 0.01) for adult male midcontinent and eastern mallards, respectively. The estimated marginal effect of frameworkdate extensions has been an increase in harvest rate of 0.012 (SD = 0.008) and 0.006 (SD = 0.010) for midcontinent and eastern mallards, respectively. Optimal regulatory strategies for the 2006 hunting season were calculated using: (1) harvestmanagement objectives specific to each mallard stock; (2) the 2006 regulatory alternatives; and (3) current population models and associated weights for midcontinent and eastern mallards. Based on this year’s survey results of 7.86 million midcontinent mallards (traditional survey area plus MN, WI, and MI), 4.45 million ponds in Prairie Canada, and 899 thousand eastern mallards, the optimal regulatory choice for all four Flyways is the liberal alternative. 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. Joint optimization of multiple stocks presents many challenges in terms of population modeling, parameter estimation, and computation of regulatory strategies. These challenges cannot always be overcome due to limitations in monitoring and assessment programs and in access to sufficient computing resources. In some cases, it may be possible to impose constraints or assumptions that simplify the problem. Although suboptimal by design, these constrained regulatory strategies may perform nearly as well as those that are optimal, particularly in cases where breeding stocks differ little in their ability to support harvest, where Flyways do not receive significant numbers of birds from more than one breeding stock, or where management outcomes are highly uncertain. 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 stock’s harvest, whereby 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 approach 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 review and development of population models and monitoring programs. MALLARD POPULATION DYNAMICS Midcontinent Mallards Population size.For the purposes of AHM, midcontinent mallards currently 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. Estimates of the abundance of this midcontinent population are available only since 1992 (Table 1, Fig. 2). Population models.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). midcontinent eastern 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 surveys State surveys 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 0.4645 7.8646 0.2284 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 1992 1994 1996 1998 2000 2002 2004 2006 Population size (millions) 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 TSA Great Lakes Total 8 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. 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 9 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. 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 10 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. Model weights 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). Throughout the period of updating model weights, there has been no clear preference for either the additive (58%) or compensatory (42%) mortality models. For most of the time frame, model weights have strongly favored the weakly densitydependent (91%) reproductive model over the strongly densitydependent (9%) one. 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). 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 2006, 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 0.1117 (SD = 0.0200). We assumed a normal distribution with these parameter values to make the conversion between the traditional survey area and total breedingpopulation size. 11 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 Mallards Population size.For purposes of AHM, 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) (see 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). 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. Population models.–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. 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 12 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). State surveys Federal surveys 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 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. 13 Fig. 4. Population estimates of eastern mallards in the northeastern U.S. (NE state survey) and in federal surveys in southern Ontario and Quebec. Error bars represent one standard error. 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 Year 1990 1992 1994 1996 1998 2000 2002 2004 2006 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 statesurvey 1.4 Federal survey Total 14 (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). 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 15 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 B). There is no single model that is clearly favored over the others at the end of the time frame, although collectively the models with strongly densitydependent reproduction (65%) are better predictors of changes in population size than those with weak density dependence (35%) (Fig. 5). In addition, there is substantial evidence of bias in extant estimates of survival and/or reproductive rates (99%). 16 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. Western Mallards Substantial numbers of mallards occur in the states of the Pacific Flyway (including Alaska), British Columbia, and the Yukon Territory during the breeding season. The distribution of these mallards during fall and winter is centered in the Pacific Flyway (Munro and Kimball 1982). Unfortunately, datacollection programs for understanding and monitoring the dynamics of this mallard stock are highly fragmented in both time and space. This makes it difficult to aggregate monitoring instruments in a way that can be used to reliably model this stock’s dynamics and, thus, to establish criteria for regulatory decisionmaking under AHM. Another complicating factor is that federal survey strata 112 in Alaska and the Yukon are within the current geographic bounds of midcontinent mallards. The AHM Working Group is continuing its investigations of western mallards and while it is not prepared to recommend an AHM protocol at this time, progress is being made on a number of issues: Breeding populations surveys – The development of AHM for western mallards continues to present technical challenges that make implementation much more difficult than with either midcontinent or eastern mallards. In particular, we remain concerned about our ability to reliably determine changes in the population size of western mallards based on a 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 (i.e., sampling error). For example, methods for estimating mallard abundance in British Columbia are still in the development and evaluation phase, and there are as yet unanswered questions about how mallard 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 17 abundance will be determined there on an operational basis. Helicopters are currently being evaluated for use in surveys that eventually could cover the majority of key waterfowl habitats in British Columbia. During the last year we reviewed extant surveys to determine their adequacy for supporting a westernmallard AHM protocol. We were principally interested in whether the surveys: (a) estimate total birds (rather than breeding pairs); (b) have a sound sampling design (and SEs available); (c) consider imperfect detection of birds; and (d) require data augmentation (i.e., filling missing years). Based on these criteria, Alaska, California, and Oregon were selected for modeling purposes. These three states likely harbor about 75% of the westernmallard breeding population (Fig. 6). Nonetheless, this geographic focus is temporary until such time that surveys in other areas can be brought up to similar standards and an adequate record of population estimates is available for analysis. Fig. 6. Status of surveys in the range of western mallards. States with solid shading represent those that currently are being used to model westernmallard population dynamics. 18 Population modeling – For modeling purposes we were hesitant to pool Alaska mallards with those in California and Oregon because of differing population trajectories (Fig. 7), and because we believed it likely that different environmental driving variables were at play during the breeding season in northern and southern latitudes. Mallards banded in Alaska and in Californian/Oregon also had different recovery distributions, suggesting that these two groups of mallards may be subject to differences in mortality rates during the nonbreeding season. The separation of western mallards into northern and southern groups is for exploratory purposes. It remains unclear whether the dynamics of these two groups are different enough to be meaningful in a harvestmanagement context, especially given that many of these birds are subject to a common hunting season. Fig. 7. Population estimates of two groups of western mallards. Surveys were not conducted in Oregon in 1992, 1993, and 2001 so we imputed estimates based on the correlation between estimates from Oregon and California. Error bars represent one standard error. We used a discrete logistic model to characterize population dynamics because it requires a minimum of data to parameterize. The traditional approach of constructing a population balance equation (i.e., that used for midcontinent and eastern mallards) was deemed impractical because of the paucity of banding data to estimate survival rates and because of the difficulty of estimating reproductive rates from a collection of wings aggregated from several mallard stocks. The logistic model took the form: and where: N = breedingpopulation size, r = the intrinsic rate of population growth, K = carrying capacity, hAM = ( )d c where h K N N r N K N N N r N AM t t t t t t t t t t − = + ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ − + ⎛ − ⎟⎠ ⎞ ⎜⎝ = + ⎛ − + 1 1 1 2 1 α α σ 19 adultmale harvest rate, c = crippling loss, d = a scaling factor, and σ2 = process error. As model inputs, we used breeding population estimates (derived from breeding surveys) and harvest rates of adult males (derived from bandrecovery data corrected for reporting rates). We assumed a 20% crippling loss and scaled adultmale harvest rates (d) to represent a harvestrate for the population as a whole. The magnitude of d depends on the differential vulnerability of the agesex cohorts, as well as their relative abundance in the population. Based on values from better understood mallard stocks we specified d = 1.4, but also explicitly allowed for considerable variation in this parameter throughout the analyses. The model we used assumes that both the Alaska and CaliforniaOregon breeding stocks are closed. While this is a tenuous assumption, breedingground fidelity is difficult to investigate using dead recoveries of banded birds, even where large samples are available from all stocks of interest (in this case including midcontinent mallards). Low fidelity could lead to poor model fit; however, this lack of fit should be absorbed in the processerror term. The logistic model also assumes densitydependent growth, but does not specify whether it occurs in the mortality process, the reproductive process, or both. However, some assumptions about the seasonality of events had to be assumed in order to reconcile the different timing of population surveys and preseason banding. We used a “statespace model” that allows the partitioning of observation error, which is specified by the sampling error of population estimates, and process error, which specifies the discrepancy between predicted and observed population sizes (Meyer and Millar 1999). Estimates of model parameters were calculated via Markov Chain Monte Carlo simulations using the WinBugs publicdomain software. We specified uninformative or vague prior distributions for all model parameters. Once parameter estimates were available, we derived optimal harvest strategies with stochastic dynamic programming and simulated their use via the publicdomain software ASDP (Lubow 1995). We assumed perfect control over harvest rates, but explicitly considered estimation error in key model parameters. We examined a number of models in which K was either constant or allowed to vary over time. We present here only the results for the model with constant K. This model provides a reasonable fit to the data from both areas (Fig. 8) and suggests a similar K (Table 4). The intrinsic rate of growth r appears to be higher in California/Oregon and thus these mallards should be able to support slightly more harvest pressure over the long term than those originating from Alaska. However, we note there is a high degree of uncertainty associated with all parameter estimates. Table 4. Estimates of carrying capacity K and the intrinsic rate of growth r from the logistic model for two groups of western mallards. (CI = Bayesian credibility intervals). Stock K 95% CI (K) r 95% CI (r) AK 0.95 0.68 – 1.09 0.36 0.14 – 0.64 CA/OR 0.88 0.60 – 1.09 0.46 0.20 – 0.94 20 Fig. 8. Mallard breeding population size in Alaska (top) and California/Oregon (bottom) as estimated from surveys (obs pop) and those predicted from a logistic model that assumes a constant carrying capacity K (M0). Year 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 Bpop (millions) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 obs pop M0 Year 1992 1994 1996 1998 2000 2002 2004 2006 Bpop 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 obs pop M0 21 Harvest rates – We estimated harvest rates of adult male mallards in Alaska and California/Oregon directly from recoveries of reward bands placed on mallards prior to the hunting seasons in 20022005 (Table 5). Generally, these rates are similar to those for midcontinent mallards. Table 5. Harvest rates (h, and standard errors, se) of adultmale mallards banded in Alaska and California/Oregon as based on reward banding. (There was an insufficient number of reward bandings in Alaska in 2005). AK CA/OR Year h se h se 2002 0.1121 0.0306 0.1049 0.0109 2003 0.1000 0.0391 0.0970 0.0124 2004 0.0968 0.0379 0.1239 0.0175 2005 0.1086 0.0098 mean 0.1030 0.0057 0.1099 0.0082 Ultimately, the ability to predict stockspecific harvest rates as a function of flywayspecific regulations involves: (a) accounting for movements of breeding stocks to various harvest areas (flyways); (b) estimating harvest rates on mallards wintering in the various flyways; and (c) correlating these harvest rates with flyway framework regulations. This work is currently underway and may be completed prior to the 2007 regulations cycle. 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) (Fig. 9). 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. Population size expected next year (millions) 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 Harvest value (%) 0 20 40 60 80 100 0 20 40 60 80 100 NAWMP goal (8.8 million) Fig. 9. The relative value of midcontinent mallard harvest, expressed as a function of breedingpopulation size expected in the subsequent year. 22 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 breedingpopulation 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 (scaup, gadwall, wigeon, greenwinged teal, bluewinged teal, shoveler, pintail, redhead, and canvasbacks), 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. RegulationSpecific Harvest Rates Initially, harvest rates of mallards associated with each of the openseason regulatory alternatives were 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. 23 Table 6. Regulatory alternatives for the 2006 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. For midcontinent mallards, we have empirical estimates of harvest rate from the recent period of liberal hunting regulations (19982005). 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 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 B. 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 rate during the 20022005 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.012 (SD = 0.008). Therefore, the estimated effect of the frameworkdate extension has been to increase harvest rate of midcontinent mallards by about 10% 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 24 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 and Fig. 9. 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 2006 regulatory alternatives in the three western Flyways. Regulatory alternative Mean SD Closed (U.S.) 0.0088 0.0019 Restrictive 0.0583 0.0128 Moderate 0.1107 0.0216 Liberal 0.1269 0.0213 Fig. 9. Probability distributions of harvest rates of adult male midcontinent mallards expected with application of the 2006 regulatory alternatives in the three western Flyways. Harvest rate 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 pdf(harvest rate) 0 10 20 30 200 210 25 Until last year, predictions of harvest rates for eastern mallards depended exclusively on historical (prior) information because more contemporary estimates of harvest rate were unavailable. However, we can now update the predictions of easternmallard harvest rates in the same fashion as that for midcontinent mallards based on reward banding conducted in eastern Canada and the northeastern U.S. (Appendix B). 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 and Fig. 10. 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 rate during the 20022005 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.006 (SD = 0.010). Therefore, the estimated effect of the frameworkdate extension has been to increase harvest rate of eastern mallards by about 4% over what would otherwise be expected in the liberal alternative. Table 8. Predictions of harvest rates of adultmale eastern mallards expected with application of the 2006 regulatory alternatives in the Atlantic Flyway. Regulatory alternative Mean SD Closed (U.S.) 0.0797 0.0230 Restrictive 0.1209 0.0392 Moderate 0.1417 0.0473 Liberal 0.1636 0.0460 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 2006 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. Assuming that regulatory choices adhered to this strategy (and that current model weights accurately reflect population dynamics), breedingpopulation size would be expected to average 7.37 million (SD = 1.77). 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. Based on an observed population size of 7.86 million midcontinent mallards (traditional surveys plus MN, MI, and WI) and 4.45 million ponds in Prairie Canada, the optimal regulatory choice for the Pacific, Central, and Mississippi Flyways in 2006 is the liberal alternative. 26 Fig. 10. Probability distributions of harvest rates of adult male eastern mallards expected with application of the 2005 regulatory alternatives in the Atlantic Flyway. Table 9. Optimal regulatory strategya for the three western Flyways for the 2006 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 2006. 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 R M M L 7.00 R R R R M M M L L L 7.25 R R R M M L L L L L 7.50 R R M M L L L L L L 7.75 M L L 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. Harvest rate 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 pdf(harvest rate) 0 10 20 27 We calculated an optimal regulatory strategy for the Atlantic Flyway based on: (1) the 2006 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). The strategy exhibits this behavior in part because of the small differences in harvest rate among regulatory alternatives (Fig. 10). Table 10. Optimal regulatory strategya for the Atlantic Flyway for the 2006 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 2006. Mallardsb Regulation <225 C 225 R >225 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. We simulated the use of the regulatory strategy in Table 10 to determine expected performance characteristics. Assuming that harvest management adhered to this strategy (and that current model weights accurately reflect population dynamics), the annual breedingpopulation size would be expected to average 872 thousand (SD = 16 thousand). Based on a breeding population size of 899 thousand mallards, the optimal regulatory choice for the Atlantic Flyway in 2006 is the liberal alternative. Application of AHM Concepts to Species of Concern The USFWS is striving to apply the principles and tools of AHM to improve decisionmaking for several species of special concern. We here report on four such efforts in which progress has been made since last year. This work is being conducted as a joint effort with USGS and we particularly appreciate the technical assistance provided M. C. Runge (Patuxent Wildlife Research Center) and M. J. Conroy (Georgia Cooperative Fish and Wildlife Research Unit). Pintails Pursuant to requests from the Service Regulations Committee and the Pacific Flyway Council, we reviewed available information about northern pintail (Anas acuta) demography, population dynamics, and harvest (http://www.fws.gov/migratorybirds/reports/ahm05/NOPI%202005%20Report%202.pdf). Based on this review, several technical improvements in our ability to model pintail harvest dynamics have been adopted. In addition, we undertook an effort to evaluate pintail harvest potential based on these model improvements and to explore the impacts of these improvements on past and future pintail harvest management policy. Breeding population survey corrections.There is general agreement among waterfowl biologists that the May breeding population survey undercounts pintails in dry years when pintails tend to settle farther north on the breeding grounds. We developed a method to correct the observed breeding population estimates for this bias. The effect of this correction is to remove some of the apparent sharp drops in pintail numbers during dry years. Further, this correction suggests that in recent years, there were 3060% more pintails in the breeding population 28 than the May surveys indicated. Updated recruitment, harvest, and population Models.We developed improved methods to predict recruitment and harvest, and included these components in an updated population model for use in the pintail harvest strategy. The recruitment model uses latitude of the pintail population and the corrected breeding population size estimates as predictors. The harvest models identify a “seasonwithinaseason” effect in the Central and Mississippi flyways. The new population model predicts population change better than the previous model. Pintail harvest potential.Using the new pintail population model, we were able to analyze the harvest potential of the pintail population. There is evidence that the pintail population is settling, on average, about 2.4° of latitude farther north now than it did prior to 1975, possibly as a result of largescale changes in habitat. This more northern distribution has resulted in lower reproduction, a 3045% decrease in carrying capacity, and a 4065% decrease in sustainable harvest potential. Incorporating the technical improvements described above into the population model, we can calculate and depict the current harvest strategy (Fig. 11). The season would be closed when the observed BPOP is less than ~1 million (which is roughly equivalent to a corrected fall flight of 2 million), restrictive when the observed BPOP is less than 2.5 million with a high latitude of the BPOP, and liberal otherwise (this graph assumes the AHM package is liberal; the restrictive section of the graph implies a seasonwithinaseason). More than one bird in the bag is allowed when the population growth is expected to be greater than 6%. The corresponding statedependent harvest policies when the AHM package is moderate or restrictive are shown in Fig. 12. When the AHM package is moderate, a restrictive seasonwithinaseason is possible. When the AHM package is restrictive, the pintail season is either also restrictive, or else closed. 1 2 3 4 5 6 7 51 52 53 54 55 56 57 58 59 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive Liberal (1bird) Liberal (3birds) L2 Fig. 11. Pintail harvest strategy for a liberal duck season, based on the overflight bias correction, new recruitment model, and updated harvest models. For a given value of the observed (not corrected) pintail BPOP and average latitude of the BPOP, the resulting regulatory decision is shown. Note that this graph assumes the AHM package is “liberal”, thus the “restrictive” regulation implies a seasonwithinaseason. 29 Black Ducks We continued to examine the harvest potential of black ducks using models of population dynamics described by Conroy et al. (2002). These models incorporate the most controversial hypotheses about reproductive and survival processes in black ducks, and also allow for the possibility that extant estimates of reproductive and survival rates are positively biased. Using empirically based model weights (from 196293) in conjunction with deterministic dynamic programming, we can derive combinations of equilibrium population size and harvest for a range of adult harvest rates. Because of evidence that the reproductive rate of black ducks declines with increasing numbers of mallards, the carrying capacity and harvestable surplus of black ducks are smaller with higher numbers of sympatric mallards. There also appears be a temporal decline in the reproductive rate of black ducks that cannot be explained by changes in black duck and mallard abundance. The cause is unknown but may be related to declines in the quantity and/or quality of breeding habitat, wintering habitat, or both. Whatever the cause, the management implications are profound, suggesting that carrying capacity and maximum sustainable harvest of black ducks have decreased by 35% and 60%, respectively, in the past two decades (Fig. 13). Since 1983, the U.S. Fish and Wildlife Service has been operating under guidance provided in an Environmental Assessment that specified states harvesting significant numbers of black ducks achieve at least a 25% reduction in harvest from 1977 81 levels. Although this level of harvest reduction has been achieved, black duck harvest rates appear to have increased with the return of 5060 day duck hunting seasons associated with implementation of AHM (Fig. 14). Whether these harvest rates are appropriate given current population status is unclear; based on black duck counts in the midwinter survey, current rates might be at or above maximum sustainable levels. However, a recent publication by Link et al. (2006), which compared the U.S. midwinter survey with the Christmas Bird Count, suggests that a larger portion of the black duck population may now be wintering in Canada than in the past. If this is the case, then regulatory prescriptions based on the U.S. midwinter survey could be overly conservative. Therefore, the USFWS, USGS, the Atlantic and Mississippi Flyways, and CWS are aggressively pursuing efforts to develop an adaptive framework based on breedingpopulation surveys and internationally agreedupon management objectives. 1 2 3 4 5 6 7 51 52 53 54 55 56 57 58 59 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive Moderate (1bird) Moderate (3birds) M2 1 2 3 4 5 6 7 51 52 53 54 55 56 57 58 59 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive (1bird) Restrictive (3birds) R2 Fig. 12. Statedependent pintail harvest strategy, given a moderate AHM alternative (left) and a restrictive AHM alternative (right). For details see Fig. 11. 30 Fig 13. Collapsing yield curves of black ducks as result of declining productivity. Yield curves were based on population models provided by Conroy et al. (2002). For each period, we used the median year to represent black duck productivity and fixed the number of mallards at their average midwinter count. The diagonal lines intersect the indicated adult harvest rate on each yield curve. Fig 14. Estimates of harvest rates of adultmale black ducks based on recoveries of standard bands, adjusted for preliminary estimates of bandreporting rates (P. Garrettson, unpubl. data). Black duck MWI (k) 0 100 200 300 400 500 600 700 800 Harvest (k) 0 20 40 60 80 100 0.05 0.15 0.10 0.20 198084 199094 200004 Year 1970 1975 1980 1985 1990 1995 2000 2005 Harvest rate 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 31 Scaup We continued to evaluate the harvest potential of the continental scaup (greater Aythya marila and lesser Aythya affinis) population using a discrete, logistic population model and available monitoring information on scaup population and harvest dynamics (http://www.fws.gov/migratorybirds/reports/ahm05/scaupharvestpotential.pdf). We used a fully Bayesian approach to estimate scaup population parameters and to characterize the uncertainty related to scaup harvest potential (Table 11.). We plotted mean scaup equilibrium population sizes and corresponding sustainable harvests (Fig. 15) along with 95% credibility intervals (gray shading). This yield curve, in combination with observed harvests and breeding population sizes from 1994 – 2005, suggests that current harvests may be at maximum sustainable levels. Table 11. Estimates of model and management parameters (posterior means and 95% credibility intervals) derived from fitting a logistic population model to continental scaup populations using a Bayesian hierarchical approach. (r = intrinsic rate of growth, K = carrying capacity, MSY = maximum sustainable yield, hMSY = harvest rate at MSY, and BPOPMSY = breeding population size at MSY) Parameter mean 2.50% 97.50% r 0.1763 0.0978 0.2974 K 8.6259 6.4830 11.450 MSY 0.3694 0.2271 0.5377 hMSY 0.0881 0.0489 0.1487 BPOPMSY 4.3129 3.2420 5.7230 Fig. 15. Equilibrium population sizes and harvests (shaded area represents 95% credibility interval) estimated for continental scaup populations from a logistic model using a Bayesian hierarchical approach. The years represent recent combinations of observed population size and harvest. 32 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: where node 1 refers to oneyearold birds, node 2 refers to twoyearold birds, node B refers to adult breeders, and node NB refers to adult nonbreeders. 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. 33 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 variable (a function of timing of snow melt on the breeding grounds). 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. Recent extensions of Bayesian statistical methods to population reconstruction may provide an adequate solution (Fig. 16). Fig.16. Observed breeding population size of Atlantic Population Canada Geese (estimates and 95% confidence bounds) compared with the population trajectory based on our population model and population reconstruction techniques Management of the APCG has, in recent years, been focused on achieving the minimum population needed to sustain some level of sport harvest. However, there is growing concern over the potential problems caused by overabundant goose species, and management objectives for goose species are increasingly considering population control as an important objective. Specification of an explicit, mathematical objective function for APCG will require careful deliberation among the appropriate stakeholders. Since formal AHM is an exercise in optimization, the objective often not only drives the outcome, but also strongly influences the development of the other components of the decision framework (e.g., the decision variables, the projection model, etc.). As a starting point for our work in developing an AHM application for APCG, and as a starting point for discussions about the management objectives for this resource, we developed a candidate objective function. We propose that the management objective needs to 34 reflect the simultaneous problem of maximizing opportunity for harvest, while minimizing the risk that the population will become either too large (i.e., beyond human tolerance in terms of impacts on habitat or other species), or too small (i.e., requiring season closure for political reasons). We believe that the critical components governing the dynamics of APCG, unlike those governing ducks, are generally densityindependent over the range of population sizes that likely characterize management objectives; as such, harvest represents an imposed regulatory mechanism on the dynamics of the population. This requires specification of a desired range for the population size. Let NMTP represent the maximum tolerable population size that stakeholders would accept, given the potential for negative impacts of overabundant APCG on stakeholder interests. Let NMin be the minimum tolerable population size, below which season closure is the only politically viable management option. The management objective is to maintain the population in the range between the maximum and minimum values, while simultaneously maximizing opportunity for sport harvest. There is another implicit dynamic that may interact with this objective: there may be a limit to the amount of harvest that could be induced with traditional harvest regulations. Let NMCP represent the maximum controllable population level that could be regulated by harvest (a function of a finite number of goose hunters or hunting effort; this is currently an unknown quantity for APCG). We think it’s most likely that NMTP < NMCP, although this assumption won’t affect the development of any other aspect of the AHM protocol. NMCP might strongly affect the optimal policy, however, as the policy should avoid letting the population reach an uncontrollable level, especially if that level is also intolerable. Thus, the objective should implicitly minimize the risk of losing the ability to control the population. Note that NMCP should be calculated from biological considerations in conjunction with information about the limits to harvest. NMTP, however, is a purely sociological constraint. We think this objective will hold the population as close to the maximum tolerable population size as possible (thus, allowing the greatest harvest), while guarding against the risk of the population getting out of control. Mathematically, these objectives can be expressed as max ( ) , 0 u N Ht t t Σ ∞ = that is, maximizing the longterm cumulative harvest utility, where the value (utility) of harvest is decremented relative to the bounds of the constraint (i.e., the maximum and minimum bounds). One possible form of the utility function u is a ‘squarewave’, where utility of the harvest is 0 when the population size is above and below NMTP and NMin, respectively. The Atlantic Flyway Canada Goose Committee (AFCGC) has proposed values for NMTP and NMin (Fig. 17). Fig. 17. Proposed utility of APCG harvest as a function of breedingpopulation size. 35 The AFCGC would also like to consider a management objective to avoid overly restrictive regulations when populations are close to goal, as well as abrupt changes in regulations with relatively small changes in population size or spring weather conditions (i.e., a “knifeedged” strategy). One way this may be accomplished is by adding a cost that increases with increasing harvest rates to the objective function. The addition of this cost is sufficient to induce more intermediate harvest rates in the optimal harvest strategy (Fig. 18). Development of a preliminary AHM framework for APCG is nearing completion and, pending sufficient review and evaluation, may be ready for implementation in 2007. 36 Fig. 18. Optimal harvest rates for APCG at a stable stage distribution where there is no cost (above) and a relatively high cost (below) for a knifeedged harvest strategy. 37 LITERATURE CITED 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. 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. 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. 38 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. 39 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 McCarty C 420, University of Florida P.O. Box 110339 Gainesville, FL 32611 phone: 3523923052 fax: 3523928555 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 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 Dave Case (contractor) D.J. Case & Associates 607 Lincolnway West Mishawaka, IN 46544 phone: 5742580100 fax: 5742580189 email: dave@djcase.com John Cornely (Region 6) U.S. Fish and Wildlife Service P.O. Box 25486, DFC Denver, CO 80225 phone: 3032368155 (ext 259) fax: 3032368680 email: john_cornely@fws.gov Ken Gamble (Region 9) U.S. Fish and Wildlife Service 101 Park DeVille Drive, Suite B Columbia, MO 65203 phone: 5732341473 fax: 5732341475 email: ken_gamble@fws.gov 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 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 40 Bob Leedy (Region 7) U.S. Fish and Wildlife Service 1011 East Tudor Road Anchorage, AK 995036119 phone: 9077863446 fax: 9077863641 email: robert_leedy@fws.gov Jerry Serie (Region 9) U.S. Fish and Wildlife Service 11510 American Holly Drive Laurel, MD 20708 phone: 3014975851 fax: 3014975885 email: jerry_serie@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 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 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 41 Jim Gammonley (Central Flyway) Colorado Division of Wildlife 317 West Prospect Fort Collins, CO 80526 phone: 9704724379 fax: 9704724457 email: jim.gammonley@state.co.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 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 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: 515/3573517, ext. 23 fax: 5153575523 email: gzenner@netins.net 42 APPENDIX B: 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). 43 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 0.0020 h1998 0.1093 0.0114 νC 0.0019 0.0005 h1999 0.1000 0.0077 γR 0.5105 0.0593 h2000 0.1261 0.0101 νR 0.0128 0.0032 h2001 0.1071 0.0111 γM 0.8501 0.1113 h2002 0.1132 0.0059 νM 0.0216 0.0054 h2003 0.1131 0.0084 μL 0.1154 0.0071 h2004 0.1245 0.0108 νL 0.0213 0.0042 h2005 0.1174 0.0082 δf 0.0115 0.0082 44 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 ( )~ ( . , . ) ( )~ ( , . ) �� 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 45 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 0.0797 0.0251 h2002 0.1630 0.0129 νC 0.0230 0.0055 h2003 0.1466 0.0105 γR 0.7695 0.1175 h2004 0.1373 0.0115 νR 0.0392 0.0098 h2005 0.1282 0.0118 γM 0.9074 0.1139 νM 0.0473 0.0119 μL 0.1576 0.0169 νL 0.0460 0.0106 δf 0.0060 0.0098
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Title  Adaptive harvest management 2006 duck hunting season 
Description  adaptharvestmallards2006.pdf 
FWS Resource Links  http://library.fws.gov 
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Document Birds 
Publisher  U.S. Fish and Wildlife Service 
Date of Original  2006 
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Transcript  Adaptive Harvest Management 22000066 Hunttiingg Seeaassoon U.S. Fish & Wildlife Service 1 Adaptive Harvest Management 2006 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 2006 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 duckhunting 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 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 community, 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 D. Case (D.J. Case & Assoc.), M. Conroy (U.S. Geological Survey [USGS]), E. Cooch (Cornell University), P. Garrettson (USFWS), W. Harvey (Maryland Dept. of Natural Resources), R. Raftovich (USFWS), E. Reed (Canadian Wildlife Service), K. Richkus (USFWS), J. Royle (USGS), M. Runge (USGS), J. Serie, (USFWS), S. Sheaffer (Cornell University), 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. 2006. Adaptive Harvest Management: 2006 Hunting Season. U.S. Dept. Interior, Washington, D.C. 45pp. 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 .......................................................................................21 Regulatory Alternatives .....................................................................................................22 Optimal Regulatory Strategies ...........................................................................................25 Application of AHM Concepts to Species of Concern......................................................27 Literature Cited ..................................................................................................................37 Appendix A: AHM Working Group .................................................................................39 Appendix B: Modeling Mallard Harvest Rates.................................................................42 This report and others regarding Adaptive Harvest Management are available online at http://migratorybirds.fws.gov 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 original AHM protocol was based solely on the dynamics of midcontinent mallards, but efforts are being made to account for mallards breeding eastward and westward of the midcontinent region. The challenge for managers is to vary hunting regulations among Flyways in a manner that recognizes each Flyway’s unique breedingground derivation of mallards. For the 2006 hunting season, the USFWS will continue to consider a regulatory choice for the Atlantic Flyway that depends exclusively on the status of eastern mallards. The prescribed regulatory choice for the Mississippi, Central, and Pacific Flyways continues to depend exclusively on the status of midcontinent mallards. Investigations of the dynamics of western mallards (and their potential effect on regulations in the West) are continuing and the USFWS is not yet prepared to recommend an AHM protocol for this mallard stock. The mallard population models that are the basis for prescribing hunting regulations were revised extensively in 2002. These revised 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 (91%). Evidence for the additivemortality hypothesis remains equivocal (58%). For eastern mallards, current model weights favor the strongly densitydependent reproductive hypothesis (65%). By consensus, hunting mortality is assumed to be additive in eastern mallards. For the 2006 hunting season, the USFWS is continuing to consider 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. Also, at the request of the Flyway Councils in 2003 the USFWS agreed to exclude closed duckhunting seasons from the AHM protocol when the breedingpopulation size of midcontinent mallards is >5.5 million (traditional survey area plus the Great Lakes region). Harvest rates associated with each of the regulatory alternatives are predicted using Bayesian statistical methods. Essentially, the idea is to use historical information to develop initial harvestrate predictions, to make regulatory decisions based on those predictions, and then to observe realized harvest rates. Those observed harvest rates, in turn, are used to update the predictions. Using this approach, predictions of harvest rates of mallards under the regulatory alternatives have been updated based on bandreporting rate studies conducted since 1998. Estimated harvest rates from the 20022005 liberal hunting seasons have averaged 0.12 (SD = 0.01) and 0.14 (SD = 0.01) for adult male midcontinent and eastern mallards, respectively. The estimated marginal effect of frameworkdate extensions has been an increase in harvest rate of 0.012 (SD = 0.008) and 0.006 (SD = 0.010) for midcontinent and eastern mallards, respectively. Optimal regulatory strategies for the 2006 hunting season were calculated using: (1) harvestmanagement objectives specific to each mallard stock; (2) the 2006 regulatory alternatives; and (3) current population models and associated weights for midcontinent and eastern mallards. Based on this year’s survey results of 7.86 million midcontinent mallards (traditional survey area plus MN, WI, and MI), 4.45 million ponds in Prairie Canada, and 899 thousand eastern mallards, the optimal regulatory choice for all four Flyways is the liberal alternative. 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. Joint optimization of multiple stocks presents many challenges in terms of population modeling, parameter estimation, and computation of regulatory strategies. These challenges cannot always be overcome due to limitations in monitoring and assessment programs and in access to sufficient computing resources. In some cases, it may be possible to impose constraints or assumptions that simplify the problem. Although suboptimal by design, these constrained regulatory strategies may perform nearly as well as those that are optimal, particularly in cases where breeding stocks differ little in their ability to support harvest, where Flyways do not receive significant numbers of birds from more than one breeding stock, or where management outcomes are highly uncertain. 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 stock’s harvest, whereby 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 approach 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 review and development of population models and monitoring programs. MALLARD POPULATION DYNAMICS Midcontinent Mallards Population size.For the purposes of AHM, midcontinent mallards currently 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. Estimates of the abundance of this midcontinent population are available only since 1992 (Table 1, Fig. 2). Population models.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). midcontinent eastern 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 surveys State surveys 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 0.4645 7.8646 0.2284 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 1992 1994 1996 1998 2000 2002 2004 2006 Population size (millions) 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 TSA Great Lakes Total 8 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. 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 9 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. 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 10 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. Model weights 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). Throughout the period of updating model weights, there has been no clear preference for either the additive (58%) or compensatory (42%) mortality models. For most of the time frame, model weights have strongly favored the weakly densitydependent (91%) reproductive model over the strongly densitydependent (9%) one. 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). 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 2006, 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 0.1117 (SD = 0.0200). We assumed a normal distribution with these parameter values to make the conversion between the traditional survey area and total breedingpopulation size. 11 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 Mallards Population size.For purposes of AHM, 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) (see 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). 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. Population models.–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. 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 12 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). State surveys Federal surveys 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 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. 13 Fig. 4. Population estimates of eastern mallards in the northeastern U.S. (NE state survey) and in federal surveys in southern Ontario and Quebec. Error bars represent one standard error. 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 Year 1990 1992 1994 1996 1998 2000 2002 2004 2006 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 statesurvey 1.4 Federal survey Total 14 (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). 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 15 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 B). There is no single model that is clearly favored over the others at the end of the time frame, although collectively the models with strongly densitydependent reproduction (65%) are better predictors of changes in population size than those with weak density dependence (35%) (Fig. 5). In addition, there is substantial evidence of bias in extant estimates of survival and/or reproductive rates (99%). 16 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. Western Mallards Substantial numbers of mallards occur in the states of the Pacific Flyway (including Alaska), British Columbia, and the Yukon Territory during the breeding season. The distribution of these mallards during fall and winter is centered in the Pacific Flyway (Munro and Kimball 1982). Unfortunately, datacollection programs for understanding and monitoring the dynamics of this mallard stock are highly fragmented in both time and space. This makes it difficult to aggregate monitoring instruments in a way that can be used to reliably model this stock’s dynamics and, thus, to establish criteria for regulatory decisionmaking under AHM. Another complicating factor is that federal survey strata 112 in Alaska and the Yukon are within the current geographic bounds of midcontinent mallards. The AHM Working Group is continuing its investigations of western mallards and while it is not prepared to recommend an AHM protocol at this time, progress is being made on a number of issues: Breeding populations surveys – The development of AHM for western mallards continues to present technical challenges that make implementation much more difficult than with either midcontinent or eastern mallards. In particular, we remain concerned about our ability to reliably determine changes in the population size of western mallards based on a 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 (i.e., sampling error). For example, methods for estimating mallard abundance in British Columbia are still in the development and evaluation phase, and there are as yet unanswered questions about how mallard 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 17 abundance will be determined there on an operational basis. Helicopters are currently being evaluated for use in surveys that eventually could cover the majority of key waterfowl habitats in British Columbia. During the last year we reviewed extant surveys to determine their adequacy for supporting a westernmallard AHM protocol. We were principally interested in whether the surveys: (a) estimate total birds (rather than breeding pairs); (b) have a sound sampling design (and SEs available); (c) consider imperfect detection of birds; and (d) require data augmentation (i.e., filling missing years). Based on these criteria, Alaska, California, and Oregon were selected for modeling purposes. These three states likely harbor about 75% of the westernmallard breeding population (Fig. 6). Nonetheless, this geographic focus is temporary until such time that surveys in other areas can be brought up to similar standards and an adequate record of population estimates is available for analysis. Fig. 6. Status of surveys in the range of western mallards. States with solid shading represent those that currently are being used to model westernmallard population dynamics. 18 Population modeling – For modeling purposes we were hesitant to pool Alaska mallards with those in California and Oregon because of differing population trajectories (Fig. 7), and because we believed it likely that different environmental driving variables were at play during the breeding season in northern and southern latitudes. Mallards banded in Alaska and in Californian/Oregon also had different recovery distributions, suggesting that these two groups of mallards may be subject to differences in mortality rates during the nonbreeding season. The separation of western mallards into northern and southern groups is for exploratory purposes. It remains unclear whether the dynamics of these two groups are different enough to be meaningful in a harvestmanagement context, especially given that many of these birds are subject to a common hunting season. Fig. 7. Population estimates of two groups of western mallards. Surveys were not conducted in Oregon in 1992, 1993, and 2001 so we imputed estimates based on the correlation between estimates from Oregon and California. Error bars represent one standard error. We used a discrete logistic model to characterize population dynamics because it requires a minimum of data to parameterize. The traditional approach of constructing a population balance equation (i.e., that used for midcontinent and eastern mallards) was deemed impractical because of the paucity of banding data to estimate survival rates and because of the difficulty of estimating reproductive rates from a collection of wings aggregated from several mallard stocks. The logistic model took the form: and where: N = breedingpopulation size, r = the intrinsic rate of population growth, K = carrying capacity, hAM = ( )d c where h K N N r N K N N N r N AM t t t t t t t t t t − = + ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ − + ⎛ − ⎟⎠ ⎞ ⎜⎝ = + ⎛ − + 1 1 1 2 1 α α σ 19 adultmale harvest rate, c = crippling loss, d = a scaling factor, and σ2 = process error. As model inputs, we used breeding population estimates (derived from breeding surveys) and harvest rates of adult males (derived from bandrecovery data corrected for reporting rates). We assumed a 20% crippling loss and scaled adultmale harvest rates (d) to represent a harvestrate for the population as a whole. The magnitude of d depends on the differential vulnerability of the agesex cohorts, as well as their relative abundance in the population. Based on values from better understood mallard stocks we specified d = 1.4, but also explicitly allowed for considerable variation in this parameter throughout the analyses. The model we used assumes that both the Alaska and CaliforniaOregon breeding stocks are closed. While this is a tenuous assumption, breedingground fidelity is difficult to investigate using dead recoveries of banded birds, even where large samples are available from all stocks of interest (in this case including midcontinent mallards). Low fidelity could lead to poor model fit; however, this lack of fit should be absorbed in the processerror term. The logistic model also assumes densitydependent growth, but does not specify whether it occurs in the mortality process, the reproductive process, or both. However, some assumptions about the seasonality of events had to be assumed in order to reconcile the different timing of population surveys and preseason banding. We used a “statespace model” that allows the partitioning of observation error, which is specified by the sampling error of population estimates, and process error, which specifies the discrepancy between predicted and observed population sizes (Meyer and Millar 1999). Estimates of model parameters were calculated via Markov Chain Monte Carlo simulations using the WinBugs publicdomain software. We specified uninformative or vague prior distributions for all model parameters. Once parameter estimates were available, we derived optimal harvest strategies with stochastic dynamic programming and simulated their use via the publicdomain software ASDP (Lubow 1995). We assumed perfect control over harvest rates, but explicitly considered estimation error in key model parameters. We examined a number of models in which K was either constant or allowed to vary over time. We present here only the results for the model with constant K. This model provides a reasonable fit to the data from both areas (Fig. 8) and suggests a similar K (Table 4). The intrinsic rate of growth r appears to be higher in California/Oregon and thus these mallards should be able to support slightly more harvest pressure over the long term than those originating from Alaska. However, we note there is a high degree of uncertainty associated with all parameter estimates. Table 4. Estimates of carrying capacity K and the intrinsic rate of growth r from the logistic model for two groups of western mallards. (CI = Bayesian credibility intervals). Stock K 95% CI (K) r 95% CI (r) AK 0.95 0.68 – 1.09 0.36 0.14 – 0.64 CA/OR 0.88 0.60 – 1.09 0.46 0.20 – 0.94 20 Fig. 8. Mallard breeding population size in Alaska (top) and California/Oregon (bottom) as estimated from surveys (obs pop) and those predicted from a logistic model that assumes a constant carrying capacity K (M0). Year 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 Bpop (millions) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 obs pop M0 Year 1992 1994 1996 1998 2000 2002 2004 2006 Bpop 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 obs pop M0 21 Harvest rates – We estimated harvest rates of adult male mallards in Alaska and California/Oregon directly from recoveries of reward bands placed on mallards prior to the hunting seasons in 20022005 (Table 5). Generally, these rates are similar to those for midcontinent mallards. Table 5. Harvest rates (h, and standard errors, se) of adultmale mallards banded in Alaska and California/Oregon as based on reward banding. (There was an insufficient number of reward bandings in Alaska in 2005). AK CA/OR Year h se h se 2002 0.1121 0.0306 0.1049 0.0109 2003 0.1000 0.0391 0.0970 0.0124 2004 0.0968 0.0379 0.1239 0.0175 2005 0.1086 0.0098 mean 0.1030 0.0057 0.1099 0.0082 Ultimately, the ability to predict stockspecific harvest rates as a function of flywayspecific regulations involves: (a) accounting for movements of breeding stocks to various harvest areas (flyways); (b) estimating harvest rates on mallards wintering in the various flyways; and (c) correlating these harvest rates with flyway framework regulations. This work is currently underway and may be completed prior to the 2007 regulations cycle. 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) (Fig. 9). 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. Population size expected next year (millions) 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 Harvest value (%) 0 20 40 60 80 100 0 20 40 60 80 100 NAWMP goal (8.8 million) Fig. 9. The relative value of midcontinent mallard harvest, expressed as a function of breedingpopulation size expected in the subsequent year. 22 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 breedingpopulation 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 (scaup, gadwall, wigeon, greenwinged teal, bluewinged teal, shoveler, pintail, redhead, and canvasbacks), 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. RegulationSpecific Harvest Rates Initially, harvest rates of mallards associated with each of the openseason regulatory alternatives were 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. 23 Table 6. Regulatory alternatives for the 2006 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. For midcontinent mallards, we have empirical estimates of harvest rate from the recent period of liberal hunting regulations (19982005). 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 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 B. 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 rate during the 20022005 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.012 (SD = 0.008). Therefore, the estimated effect of the frameworkdate extension has been to increase harvest rate of midcontinent mallards by about 10% 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 24 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 and Fig. 9. 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 2006 regulatory alternatives in the three western Flyways. Regulatory alternative Mean SD Closed (U.S.) 0.0088 0.0019 Restrictive 0.0583 0.0128 Moderate 0.1107 0.0216 Liberal 0.1269 0.0213 Fig. 9. Probability distributions of harvest rates of adult male midcontinent mallards expected with application of the 2006 regulatory alternatives in the three western Flyways. Harvest rate 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 pdf(harvest rate) 0 10 20 30 200 210 25 Until last year, predictions of harvest rates for eastern mallards depended exclusively on historical (prior) information because more contemporary estimates of harvest rate were unavailable. However, we can now update the predictions of easternmallard harvest rates in the same fashion as that for midcontinent mallards based on reward banding conducted in eastern Canada and the northeastern U.S. (Appendix B). 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 and Fig. 10. 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 rate during the 20022005 hunting seasons, the updated (posterior) estimate of the marginal change in harvest rate attributable to the frameworkdate extension is 0.006 (SD = 0.010). Therefore, the estimated effect of the frameworkdate extension has been to increase harvest rate of eastern mallards by about 4% over what would otherwise be expected in the liberal alternative. Table 8. Predictions of harvest rates of adultmale eastern mallards expected with application of the 2006 regulatory alternatives in the Atlantic Flyway. Regulatory alternative Mean SD Closed (U.S.) 0.0797 0.0230 Restrictive 0.1209 0.0392 Moderate 0.1417 0.0473 Liberal 0.1636 0.0460 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 2006 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. Assuming that regulatory choices adhered to this strategy (and that current model weights accurately reflect population dynamics), breedingpopulation size would be expected to average 7.37 million (SD = 1.77). 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. Based on an observed population size of 7.86 million midcontinent mallards (traditional surveys plus MN, MI, and WI) and 4.45 million ponds in Prairie Canada, the optimal regulatory choice for the Pacific, Central, and Mississippi Flyways in 2006 is the liberal alternative. 26 Fig. 10. Probability distributions of harvest rates of adult male eastern mallards expected with application of the 2005 regulatory alternatives in the Atlantic Flyway. Table 9. Optimal regulatory strategya for the three western Flyways for the 2006 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 2006. 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 R M M L 7.00 R R R R M M M L L L 7.25 R R R M M L L L L L 7.50 R R M M L L L L L L 7.75 M L L 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. Harvest rate 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 pdf(harvest rate) 0 10 20 27 We calculated an optimal regulatory strategy for the Atlantic Flyway based on: (1) the 2006 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). The strategy exhibits this behavior in part because of the small differences in harvest rate among regulatory alternatives (Fig. 10). Table 10. Optimal regulatory strategya for the Atlantic Flyway for the 2006 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 2006. Mallardsb Regulation <225 C 225 R >225 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. We simulated the use of the regulatory strategy in Table 10 to determine expected performance characteristics. Assuming that harvest management adhered to this strategy (and that current model weights accurately reflect population dynamics), the annual breedingpopulation size would be expected to average 872 thousand (SD = 16 thousand). Based on a breeding population size of 899 thousand mallards, the optimal regulatory choice for the Atlantic Flyway in 2006 is the liberal alternative. Application of AHM Concepts to Species of Concern The USFWS is striving to apply the principles and tools of AHM to improve decisionmaking for several species of special concern. We here report on four such efforts in which progress has been made since last year. This work is being conducted as a joint effort with USGS and we particularly appreciate the technical assistance provided M. C. Runge (Patuxent Wildlife Research Center) and M. J. Conroy (Georgia Cooperative Fish and Wildlife Research Unit). Pintails Pursuant to requests from the Service Regulations Committee and the Pacific Flyway Council, we reviewed available information about northern pintail (Anas acuta) demography, population dynamics, and harvest (http://www.fws.gov/migratorybirds/reports/ahm05/NOPI%202005%20Report%202.pdf). Based on this review, several technical improvements in our ability to model pintail harvest dynamics have been adopted. In addition, we undertook an effort to evaluate pintail harvest potential based on these model improvements and to explore the impacts of these improvements on past and future pintail harvest management policy. Breeding population survey corrections.There is general agreement among waterfowl biologists that the May breeding population survey undercounts pintails in dry years when pintails tend to settle farther north on the breeding grounds. We developed a method to correct the observed breeding population estimates for this bias. The effect of this correction is to remove some of the apparent sharp drops in pintail numbers during dry years. Further, this correction suggests that in recent years, there were 3060% more pintails in the breeding population 28 than the May surveys indicated. Updated recruitment, harvest, and population Models.We developed improved methods to predict recruitment and harvest, and included these components in an updated population model for use in the pintail harvest strategy. The recruitment model uses latitude of the pintail population and the corrected breeding population size estimates as predictors. The harvest models identify a “seasonwithinaseason” effect in the Central and Mississippi flyways. The new population model predicts population change better than the previous model. Pintail harvest potential.Using the new pintail population model, we were able to analyze the harvest potential of the pintail population. There is evidence that the pintail population is settling, on average, about 2.4° of latitude farther north now than it did prior to 1975, possibly as a result of largescale changes in habitat. This more northern distribution has resulted in lower reproduction, a 3045% decrease in carrying capacity, and a 4065% decrease in sustainable harvest potential. Incorporating the technical improvements described above into the population model, we can calculate and depict the current harvest strategy (Fig. 11). The season would be closed when the observed BPOP is less than ~1 million (which is roughly equivalent to a corrected fall flight of 2 million), restrictive when the observed BPOP is less than 2.5 million with a high latitude of the BPOP, and liberal otherwise (this graph assumes the AHM package is liberal; the restrictive section of the graph implies a seasonwithinaseason). More than one bird in the bag is allowed when the population growth is expected to be greater than 6%. The corresponding statedependent harvest policies when the AHM package is moderate or restrictive are shown in Fig. 12. When the AHM package is moderate, a restrictive seasonwithinaseason is possible. When the AHM package is restrictive, the pintail season is either also restrictive, or else closed. 1 2 3 4 5 6 7 51 52 53 54 55 56 57 58 59 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive Liberal (1bird) Liberal (3birds) L2 Fig. 11. Pintail harvest strategy for a liberal duck season, based on the overflight bias correction, new recruitment model, and updated harvest models. For a given value of the observed (not corrected) pintail BPOP and average latitude of the BPOP, the resulting regulatory decision is shown. Note that this graph assumes the AHM package is “liberal”, thus the “restrictive” regulation implies a seasonwithinaseason. 29 Black Ducks We continued to examine the harvest potential of black ducks using models of population dynamics described by Conroy et al. (2002). These models incorporate the most controversial hypotheses about reproductive and survival processes in black ducks, and also allow for the possibility that extant estimates of reproductive and survival rates are positively biased. Using empirically based model weights (from 196293) in conjunction with deterministic dynamic programming, we can derive combinations of equilibrium population size and harvest for a range of adult harvest rates. Because of evidence that the reproductive rate of black ducks declines with increasing numbers of mallards, the carrying capacity and harvestable surplus of black ducks are smaller with higher numbers of sympatric mallards. There also appears be a temporal decline in the reproductive rate of black ducks that cannot be explained by changes in black duck and mallard abundance. The cause is unknown but may be related to declines in the quantity and/or quality of breeding habitat, wintering habitat, or both. Whatever the cause, the management implications are profound, suggesting that carrying capacity and maximum sustainable harvest of black ducks have decreased by 35% and 60%, respectively, in the past two decades (Fig. 13). Since 1983, the U.S. Fish and Wildlife Service has been operating under guidance provided in an Environmental Assessment that specified states harvesting significant numbers of black ducks achieve at least a 25% reduction in harvest from 1977 81 levels. Although this level of harvest reduction has been achieved, black duck harvest rates appear to have increased with the return of 5060 day duck hunting seasons associated with implementation of AHM (Fig. 14). Whether these harvest rates are appropriate given current population status is unclear; based on black duck counts in the midwinter survey, current rates might be at or above maximum sustainable levels. However, a recent publication by Link et al. (2006), which compared the U.S. midwinter survey with the Christmas Bird Count, suggests that a larger portion of the black duck population may now be wintering in Canada than in the past. If this is the case, then regulatory prescriptions based on the U.S. midwinter survey could be overly conservative. Therefore, the USFWS, USGS, the Atlantic and Mississippi Flyways, and CWS are aggressively pursuing efforts to develop an adaptive framework based on breedingpopulation surveys and internationally agreedupon management objectives. 1 2 3 4 5 6 7 51 52 53 54 55 56 57 58 59 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive Moderate (1bird) Moderate (3birds) M2 1 2 3 4 5 6 7 51 52 53 54 55 56 57 58 59 Average Latitude of the BPOP Pintail BPOP (millions) Closed Restrictive (1bird) Restrictive (3birds) R2 Fig. 12. Statedependent pintail harvest strategy, given a moderate AHM alternative (left) and a restrictive AHM alternative (right). For details see Fig. 11. 30 Fig 13. Collapsing yield curves of black ducks as result of declining productivity. Yield curves were based on population models provided by Conroy et al. (2002). For each period, we used the median year to represent black duck productivity and fixed the number of mallards at their average midwinter count. The diagonal lines intersect the indicated adult harvest rate on each yield curve. Fig 14. Estimates of harvest rates of adultmale black ducks based on recoveries of standard bands, adjusted for preliminary estimates of bandreporting rates (P. Garrettson, unpubl. data). Black duck MWI (k) 0 100 200 300 400 500 600 700 800 Harvest (k) 0 20 40 60 80 100 0.05 0.15 0.10 0.20 198084 199094 200004 Year 1970 1975 1980 1985 1990 1995 2000 2005 Harvest rate 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 31 Scaup We continued to evaluate the harvest potential of the continental scaup (greater Aythya marila and lesser Aythya affinis) population using a discrete, logistic population model and available monitoring information on scaup population and harvest dynamics (http://www.fws.gov/migratorybirds/reports/ahm05/scaupharvestpotential.pdf). We used a fully Bayesian approach to estimate scaup population parameters and to characterize the uncertainty related to scaup harvest potential (Table 11.). We plotted mean scaup equilibrium population sizes and corresponding sustainable harvests (Fig. 15) along with 95% credibility intervals (gray shading). This yield curve, in combination with observed harvests and breeding population sizes from 1994 – 2005, suggests that current harvests may be at maximum sustainable levels. Table 11. Estimates of model and management parameters (posterior means and 95% credibility intervals) derived from fitting a logistic population model to continental scaup populations using a Bayesian hierarchical approach. (r = intrinsic rate of growth, K = carrying capacity, MSY = maximum sustainable yield, hMSY = harvest rate at MSY, and BPOPMSY = breeding population size at MSY) Parameter mean 2.50% 97.50% r 0.1763 0.0978 0.2974 K 8.6259 6.4830 11.450 MSY 0.3694 0.2271 0.5377 hMSY 0.0881 0.0489 0.1487 BPOPMSY 4.3129 3.2420 5.7230 Fig. 15. Equilibrium population sizes and harvests (shaded area represents 95% credibility interval) estimated for continental scaup populations from a logistic model using a Bayesian hierarchical approach. The years represent recent combinations of observed population size and harvest. 32 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: where node 1 refers to oneyearold birds, node 2 refers to twoyearold birds, node B refers to adult breeders, and node NB refers to adult nonbreeders. 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. 33 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 variable (a function of timing of snow melt on the breeding grounds). 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. Recent extensions of Bayesian statistical methods to population reconstruction may provide an adequate solution (Fig. 16). Fig.16. Observed breeding population size of Atlantic Population Canada Geese (estimates and 95% confidence bounds) compared with the population trajectory based on our population model and population reconstruction techniques Management of the APCG has, in recent years, been focused on achieving the minimum population needed to sustain some level of sport harvest. However, there is growing concern over the potential problems caused by overabundant goose species, and management objectives for goose species are increasingly considering population control as an important objective. Specification of an explicit, mathematical objective function for APCG will require careful deliberation among the appropriate stakeholders. Since formal AHM is an exercise in optimization, the objective often not only drives the outcome, but also strongly influences the development of the other components of the decision framework (e.g., the decision variables, the projection model, etc.). As a starting point for our work in developing an AHM application for APCG, and as a starting point for discussions about the management objectives for this resource, we developed a candidate objective function. We propose that the management objective needs to 34 reflect the simultaneous problem of maximizing opportunity for harvest, while minimizing the risk that the population will become either too large (i.e., beyond human tolerance in terms of impacts on habitat or other species), or too small (i.e., requiring season closure for political reasons). We believe that the critical components governing the dynamics of APCG, unlike those governing ducks, are generally densityindependent over the range of population sizes that likely characterize management objectives; as such, harvest represents an imposed regulatory mechanism on the dynamics of the population. This requires specification of a desired range for the population size. Let NMTP represent the maximum tolerable population size that stakeholders would accept, given the potential for negative impacts of overabundant APCG on stakeholder interests. Let NMin be the minimum tolerable population size, below which season closure is the only politically viable management option. The management objective is to maintain the population in the range between the maximum and minimum values, while simultaneously maximizing opportunity for sport harvest. There is another implicit dynamic that may interact with this objective: there may be a limit to the amount of harvest that could be induced with traditional harvest regulations. Let NMCP represent the maximum controllable population level that could be regulated by harvest (a function of a finite number of goose hunters or hunting effort; this is currently an unknown quantity for APCG). We think it’s most likely that NMTP < NMCP, although this assumption won’t affect the development of any other aspect of the AHM protocol. NMCP might strongly affect the optimal policy, however, as the policy should avoid letting the population reach an uncontrollable level, especially if that level is also intolerable. Thus, the objective should implicitly minimize the risk of losing the ability to control the population. Note that NMCP should be calculated from biological considerations in conjunction with information about the limits to harvest. NMTP, however, is a purely sociological constraint. We think this objective will hold the population as close to the maximum tolerable population size as possible (thus, allowing the greatest harvest), while guarding against the risk of the population getting out of control. Mathematically, these objectives can be expressed as max ( ) , 0 u N Ht t t Σ ∞ = that is, maximizing the longterm cumulative harvest utility, where the value (utility) of harvest is decremented relative to the bounds of the constraint (i.e., the maximum and minimum bounds). One possible form of the utility function u is a ‘squarewave’, where utility of the harvest is 0 when the population size is above and below NMTP and NMin, respectively. The Atlantic Flyway Canada Goose Committee (AFCGC) has proposed values for NMTP and NMin (Fig. 17). Fig. 17. Proposed utility of APCG harvest as a function of breedingpopulation size. 35 The AFCGC would also like to consider a management objective to avoid overly restrictive regulations when populations are close to goal, as well as abrupt changes in regulations with relatively small changes in population size or spring weather conditions (i.e., a “knifeedged” strategy). One way this may be accomplished is by adding a cost that increases with increasing harvest rates to the objective function. The addition of this cost is sufficient to induce more intermediate harvest rates in the optimal harvest strategy (Fig. 18). Development of a preliminary AHM framework for APCG is nearing completion and, pending sufficient review and evaluation, may be ready for implementation in 2007. 36 Fig. 18. Optimal harvest rates for APCG at a stable stage distribution where there is no cost (above) and a relatively high cost (below) for a knifeedged harvest strategy. 37 LITERATURE CITED 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. 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. 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. 38 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. 39 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 McCarty C 420, University of Florida P.O. Box 110339 Gainesville, FL 32611 phone: 3523923052 fax: 3523928555 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 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 Dave Case (contractor) D.J. Case & Associates 607 Lincolnway West Mishawaka, IN 46544 phone: 5742580100 fax: 5742580189 email: dave@djcase.com John Cornely (Region 6) U.S. Fish and Wildlife Service P.O. Box 25486, DFC Denver, CO 80225 phone: 3032368155 (ext 259) fax: 3032368680 email: john_cornely@fws.gov Ken Gamble (Region 9) U.S. Fish and Wildlife Service 101 Park DeVille Drive, Suite B Columbia, MO 65203 phone: 5732341473 fax: 5732341475 email: ken_gamble@fws.gov 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 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 40 Bob Leedy (Region 7) U.S. Fish and Wildlife Service 1011 East Tudor Road Anchorage, AK 995036119 phone: 9077863446 fax: 9077863641 email: robert_leedy@fws.gov Jerry Serie (Region 9) U.S. Fish and Wildlife Service 11510 American Holly Drive Laurel, MD 20708 phone: 3014975851 fax: 3014975885 email: jerry_serie@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 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 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 41 Jim Gammonley (Central Flyway) Colorado Division of Wildlife 317 West Prospect Fort Collins, CO 80526 phone: 9704724379 fax: 9704724457 email: jim.gammonley@state.co.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 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 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: 515/3573517, ext. 23 fax: 5153575523 email: gzenner@netins.net 42 APPENDIX B: 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). 43 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 0.0020 h1998 0.1093 0.0114 νC 0.0019 0.0005 h1999 0.1000 0.0077 γR 0.5105 0.0593 h2000 0.1261 0.0101 νR 0.0128 0.0032 h2001 0.1071 0.0111 γM 0.8501 0.1113 h2002 0.1132 0.0059 νM 0.0216 0.0054 h2003 0.1131 0.0084 μL 0.1154 0.0071 h2004 0.1245 0.0108 νL 0.0213 0.0042 h2005 0.1174 0.0082 δf 0.0115 0.0082 44 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 ( )~ ( . , . ) ( )~ ( , . ) �� 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 45 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 0.0797 0.0251 h2002 0.1630 0.0129 νC 0.0230 0.0055 h2003 0.1466 0.0105 γR 0.7695 0.1175 h2004 0.1373 0.0115 νR 0.0392 0.0098 h2005 0.1282 0.0118 γM 0.9074 0.1139 νM 0.0473 0.0119 μL 0.1576 0.0169 νL 0.0460 0.0106 δf 0.0060 0.0098 
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