Adaptive harvest management 2002 duck hunting season

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Cover art: Joseph Hautman’s rendering of a black scoter (Melanitta nigra), which was selected for the 2002 federal “duck
stamp.” (The image has been reversed to function better as cover art for this report.)
U. S. Fish & Wildlife Service
Adaptive
Harvest
Management
2002 Duck Hunting Season
PREFACE
The process of setting waterfowl hunting regulations is conducted annually in the United States. 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
2002 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 duck-hunting regulations in the United States. This report
provides the most current data, analyses, and decision-making protocols. However, adaptive management is a dynamic
process, and information presented in this report may differ from that in previous reports.
ACKNOWLEDGMENTS
A working group comprised of technical representatives from the USFWS, the four Flyway Councils, and the U.S. Geological
Survey (USGS) (Appendix A) was established in 1992 to review the scientific basis for managing waterfowl harvests. The
working group subsequently proposed a framework of adaptive harvest management, which was first implemented in 1995.
The USFWS expresses its gratitude to the AHM Working Group and other individuals, organizations, and agencies that have
contributed to the development and implementation of AHM. We especially thank D. J. Case and Associates for help with
information and education efforts.
This report was prepared by the USFWS Division of Migratory Bird Management. F. A. Johnson (USFWS) was the principal
author, but significant contributions to the report were made by M. C. Runge (USGS Patuxent Wildlife Research Center),
J. A. Dubovsky (USFWS), W. L. Kendall (USGS Patuxent Wildlife Research Center), and J. A. Royle (USFWS). D. J. Case
(D.J. Case & Assoc.), R. Raftovich (USFWS), G. W. Smith (USFWS), and K. A. Wilkins (USFWS) provided essential
information or otherwise assisted with report preparation. Comments or questions regarding this document should be sent
to the Chief, Division of Migratory Bird Management - USFWS, Arlington Square, Room 634, 4401 North Fairfax Drive,
Arlington, VA 22203.
Citation: U.S. Fish and Wildlife Service. 2002. Adaptive Harvest Management: 2002 Duck Hunting Season. U.S. Dept.
Interior, Washington, D.C. 34pp.
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This report and others regarding Adaptive Harvest Management are available online at:
http://migratorybirds.fws.gov
TABLE OF CONTENTS
Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Mallard Stocks and Flyway Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Mallard Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Harvest-Management Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Regulatory Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Optimal Regulatory Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Strategic Issues in AHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Appendix A: AHM Working Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Appendix B: Regulatory Alternatives for the 1995 and 1996 Hunting Seasons . . . . . . . . . . . 27
Appendix C: Predicting Harvest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Appendix D: Updating Predictions of Harvest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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EXECUTIVE SUMMARY
In 1995, the USFWS adopted the concept of adaptive resource management for regulating duck harvests in the United States.
The adaptive 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.
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 breeding-ground derivation of mallards. For
the 2002 hunting season, the USFWS will continue to consider a regulatory choice for the Atlantic Flyway that depends
exclusively on the status of eastern mallards. This arrangement continues to be considered provisional, however, until the
implications of this approach are better understood. The recommended regulatory choice for the Mississippi, Central, and
Pacific Flyways continues to depend exclusively on the status of midcontinent mallards.
The population models upon which harvest regulations for midcontinent and eastern mallards are based have been in place
since 1995 and 2000, respectively. However, the basic structure of the models, alternative hypotheses of population
dynamics, and evidence associated with each hypothesis (i.e., model “weights”) are subject to continuous review by parties
both internal and external to the AHM process. This year, some important revisions have been made to these protocols. Most
notably, empirical corrections have been made for the +11% and +16% bias in estimated growth rates of midcontinent and
eastern mallards, respectively. The procedure for updating model weights also has been revised to include all sources of
prediction error, including those not accounted for by the models of survival and reproductive rates. In addition, the eastern-mallard
protocol now relies solely on federal and state waterfowl surveys to index breeding-population size, and includes
competing hypotheses of strongly and weakly density-dependent reproduction.
For the 2002 season, the USFWS is maintaining the same regulatory alternatives as those used during 1997-2001, except that
opening and closing framework dates have been extended in the moderate and liberal alternatives. Based on a
recommendation from the AHM Working Group, the USFWS has adopted Bayesian statistical methods for generating
predictions of harvest rates associated with these alternatives. Essentially, the idea is to use existing (“prior”) information
to develop initial harvest-rate predictions, 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. Using this approach, predictions of harvest rates of midcontinent mallards under the liberal
regulatory alternative have been updated based on reward banding during 1998-2001. The initial prediction about extended
framework dates is that they will increase harvest rates by 15% and 5% for midcontinent and eastern mallards, respectively.
Optimal regulatory choices for the 2002 hunting season were calculated using: (1) stock-specific harvest-management
objectives; (2) the revised regulatory alternatives for 2002; and (3) the revised population models and associated weights for
midcontinent and eastern mallards. Based on this year’s survey results of 8.53 million midcontinent mallards (traditional
surveys plus state surveys in MN, WI, and MI), 1.44 million ponds in Prairie Canada, and 1.0 million eastern mallards, the
recommended regulatory choice for all Flyways is the liberal alternative.
The USFWS is continuing discussions with the AHM Working Group, Flyway Councils, States, and others about future
development and application of AHM. The USFWS has decided to convene a task force, comprised of recognized state and
federal leaders in waterfowl management, to help provide policy guidance regarding the nature of harvest management
objectives and regulatory alternatives. Moreover, it is apparent that the future success of AHM will depend on how managers
account for variation in the ability of different duck species to support harvest without adverse impact. An effective means
to account for these differences in the face of a common duck-hunting season is a high priority for the task force and AHM
Working Group.
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BACKGROUND
The annual process of setting duck-hunting regulations in the United States is based on a system of resource monitoring, data
analyses, and rule making (Blohm 1989). Each year, monitoring activities such as aerial surveys and hunter questionnaires
provide information on harvest levels, population size, and habitat conditions. Data collected from this monitoring program
are analyzed each year, and proposals for duck-hunting regulations are developed by the Flyway Councils, States, and
USFWS. After extensive public review, the USFWS announces a regulatory framework 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. The adaptive 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 decision making 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 existing monitoring programs; and
(4) structural uncertainty - an incomplete understanding of biological processes; a familiar example is the long-standing
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 decision-making
process and decreases the extent to which managers can meet long-term conservation goals.
AHM was developed as a systematic process for dealing objectively with these uncertainties. The key components of AHM
(Johnson et al. 1993, Williams and Johnson 1995) include:
(1) a limited number of regulatory alternatives, which describe Flyway-specific 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 evaluated.
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 objectives of management, 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 alternative is identified based on resource and environmental conditions, and on
current model weights;
(2) after the regulatory decision is made, model-specific predictions for subsequent breeding population size are
determined;
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(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 abundance of the managed population. 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
Significant numbers of breeding mallards occur from the northern U.S. through Canada and into Alaska. Geographic
differences in the reproduction, mortality, and migrations of these mallards suggest that there are 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 breeding-ground
derivation of mallards. Of course, no Flyway receives mallards exclusively from one breeding area, and so Flyway-specific
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 Flyway-specific 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 harvest-management goals for all mallard stocks in the aggregate; and
(3) modification of the decision rules to allow independent regulatory choices among the Flyways.
Joint optimization of multiple stocks presents many challenges in terms of 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 is possible to impose constraints or assumptions
that simplify the problem. Although sub-optimal 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 don��t 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 stocks’ 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 joint-optimization
approach because mixing of the two stocks during the hunting season is limited. However, the approach continues to be
considered provisional until its implications are better understood.
MALLARD POPULATION DYNAMICS
Midcontinent Mallards
For the purposes of AHM, midcontinent mallards are defined as those breeding in federal survey strata 1-18, 20-50, and 75-77
(i.e., the “traditional” survey area), and in Minnesota, Wisconsin, and Michigan. Estimates of the midcontinent population
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1
6
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12 13
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15
17
16
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26
27
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30
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76 77
75
28
29 33
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35
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39 38
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50 51
57
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68
56 62
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65
67
66
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midcontinent
eastern
Fig. 1. Survey areas currently assigned to the midcontinent and eastern stocks of
mallards for the purposes of AHM. Delineation of the western-mallard stock is pending a
review of population monitoring programs.
so defined are available only since 1992 (Table 1).
Table 1. Estimates (N) and associated standard errors (SE) of mallards (in thousands) in spring in the
traditional survey area (strata 1-18, 20-50, and 75-77) and the states of Minnesota, Wisconsin, and Michigan.
Traditional surveys State surveys Total
Year N SE N SE N SE
1992 5976.1 241.0 977.9 118.7 6954.0 268.6
1993 5708.3 208.9 863.5 100.5 6571.8 231.8
1994 6980.1 282.8 1103.0 138.8 8083.1 315.0
1995 8269.4 287.5 1052.2 130.6 9321.6 304.5
1996 7941.3 262.9 945.7 81.0 8887.0 275.1
1997 9939.7 308.5 1026.1 91.2 10965.8 321.7
1998 9640.4 301.6 979.6 88.4 10620.0 314.3
1999 10805.7 344.5 957.5 100.6 11763.1 358.9
2000 9470.2 290.2 1031.1 85.3 10501.3 302.5
2001 7904.0 226.9 779.7 59.0 8683.7 234.5
2002 7503.7 246.5 1029.6 78.0 8533.3 258.5
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4 5 6 7 8 9 10 11 12
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5
6
7
8
9
10
11
12
BPOP (observed, millions)
BPOP (predicted, millions)
Fig. 2. Predicted versus observed breeding-population size (BPOP) for midcontinent
mallards in the traditional survey area, 1962-1996. Predictions were generated using the
balance equation and uncorrected estimates of survival and reproductive rates.
The dynamics of midcontinent mallards are described by four alternative models, which result from combining two mortality
and two reproductive hypotheses (Johnson et al. 1997). 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 density dependent (i.e., the degree to which reproductive rates decline with increasing population size).
The model with additive hunting mortality and weakly density-dependent recruitment (SaRw) leads to the most conservative
harvest strategy, whereas the model with compensatory hunting mortality and strongly density-dependent recruitment (ScRs)
leads to the most liberal strategy. The other two models (SaRs and ScRw) lead to strategies that are intermediate between
these extremes.
The population models upon which harvest regulations for midcontinent mallards are based have been in place since
implementation of AHM in 1995. However, the basic structure of the models, the alternative hypotheses of population
dynamics, and the evidence associated with each hypothesis (i.e., model “weights”) are subject to continuous review by
parties both internal and external to the AHM process. Over the last two years the AHM Working Group has focused on two
especially important concerns about the models for midcontinent mallards:
(1) Apparent bias in estimates of reproductive or survival rates – The population models for midcontinent mallards share a
common structure referred to as the balance equation. The balance equation is essentially an accounting tool, which predicts
population size in a given year based on population size (N), reproduction (R), and survival (S) from the previous year. In
theory, N, R, and S from a given year should perfectly predict N the next year. In fact, they do not (Fig. 2). Predicted
population sizes are 11% higher on average than those observed in the population surveys. The bias is not related to the nature
of any predictive models, but to estimates of survival and reproductive rates derived from resource monitoring programs.
Those programs are now being carefully scrutinized for the source and cause of the bias.
(2) Updating model weights – The purpose of annually updating model weights is to eventually identify the model providing
the most accurate predictions over time, based on a comparison of the observed population size with those predicted under
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each alternative model. Model weights are highly influential in determining optimal harvest strategies because they determine
the degree to which a single set of biological hypotheses (i.e., a particular model) dominates the optimization. The AHM
Working Group has determined that the procedure for updating model weights used since 1995 is inadequate because of the
omission of certain random errors common to all predictive models. The inclusion of these prediction errors in the updating
procedure will minimize the chances of major shifts of model weights in any single year, and help ensure that model weights
more accurately reflect the evidence in support of the alternative hypotheses (Johnson et al. 2002b).
Based on recommendations from the AHM Working Group, the USFWS has made revisions to the AHM protocol for
midcontinent mallards to address the concerns described above. These revisions involve empirical corrections to all models
for the +11% bias in population growth rates, and modification of the updating procedure to include all sources of prediction
error, including those not accounted for by the sub-models of survival and reproductive rates. In addition, a number of other
noteworthy revisions were made:
(a) The alternative survival hypotheses were re-parameterized to better reflect the uncertainty about survival rates in the
absence of hunting.
(b) The alternative hypotheses of reproduction were chosen so that they were equally supported by the available data (this
was not the case for the models specified by Johnson et al. [1997]).
(c) The dynamic model for Canadian ponds was revised to predict ponds in year t+1 as a function of ponds in year t and
random error from all sources. Annual precipitation was dropped as a predictor and its contribution to variation in
ponds is now included in a residual error term.
We derived optimal harvest strategies for each of the revised models using stochastic dynamic programming (Lubow 1995),
conditioning on the 2001 set of regulatory alternatives (i.e., without framework-date extensions) and the current objective
function (see the sections on Harvest-Management Objectives and Regulatory Alternatives later in this report). We also
simulated these strategies to understand their expected dynamics.
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 (Fig. 3). Under the strongly density-dependent model (ScRs), the density-dependence regulates
the population and keeps it within narrow bounds. Under the weakly density-dependent model (ScRw), the density-dependence
does not exert as strong a regulatory effect, and the population size fluctuates more.
The optimal strategies associated with the models with additive hunting mortality (SaRs and SaRw) are more conservative
because hunting regulations can have a pronounced effect on population size and, thus, on the ability to maintain the mallard
population above the goal of the North American Waterfowl Management Plan (NAWMP). The strategy associated with the
strongly density-dependent reproductive model (SaRs) is considerably more liberal, however, than that associated with the
weakly density-dependent model (SaRw). Average population size is expected to be slightly higher under the model with
weakly density-dependent reproduction (SaRw).
The revisions to the AHM protocol for midcontinent mallards have produced different conclusions about the best predictive
models than those published previously (USFWS 2001, Johnson et al. 2002b). Using the revised protocol, model weights
suggest much less evidence for the hypotheses of additive hunting mortality and strongly density-dependent reproduction than
had been the case with the old protocol (Table 2). Under the old protocol, the model that accumulated the most weight (SaRs)
was the one that best compensated for the positive bias in projected growth rates (i.e., the model that predicted the lowest
growth rate provided the most accurate predictions because growth rates from all of the old models were biased high). This
self-regulating mechanism helped ensure that regulations in the past were commensurate with resource status, even in the face
of an uncorrected bias in growth rates. This example also illustrates why 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).
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01 2 3 4 5 6 7
5
10
15
Ponds
BPOP
01 2 3 4 5 6 7
5
10
15
Ponds
BPOP
01 2 3 4 5 6 7
5
10
15
Ponds
BPOP
}
01 2 3 4 5 6 7
5
10
15
Ponds
BPOP
ScRs
SaRs SaRw
ScRw
C
L
C
L
VR
RM
}VR
RM
C
L
C
L
Fig. 3. Model-specific, optimal regulatory strategies for midcontinent mallards conditioned
on population size (BPOP) and ponds in Prairie Canada. These strategies are based on
the current harvest-management objective and the 2001 set of regulatory alternatives
(ScRs = compensatory mortality and strongly density-dependent reproduction, ScRw =
compensatory mortality and weakly density-dependent reproduction, SaRs = additive
mortality and strongly density-dependent reproduction, and SaRw = additive mortality and
weakly density-dependent reproduction) (C = closed season, VR = very restrictive, R =
restrictive, M = moderate, and L = liberal). The dot in the center of each figure represents
predicted mean population size and pond numbers under the model-specific strategy, and
the ellipse represents conditions expected in 95% of all years.
Table 2. Weights for the revised models of midcontinent mallards (ScRs = compensatory mortality and strongly density-dependent
reproduction, ScRw = compensatory mortality and weakly density-dependent reproduction, SaRs = additive mortality
and strongly density-dependent reproduction, and SaRw = additive mortality and weakly density-dependent reproduction).
Model weights were assumed to be equal in 1995.
Model
Year ScRs ScRw SaRs SaRw
1996 0.2469 0.2525 0.2481 0.2524
1997 0.2299 0.2659 0.2353 0.2688
1998 0.2244 0.2733 0.2220 0.2804
1999 0.0600 0.3826 0.0872 0.4702
2000 0.0551 0.4030 0.0910 0.4509
2001 0.0517 0.4052 0.0863 0.4568
2002 0.0461 0.4056 0.0823 0.4660
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The revised models and weights for midcontinent mallards have good predictive ability (Table 3). Although the variance of
predicted population sizes is large, the point predictions differ from the observations by only 6% on average. The relatively
large discrepancies in some years (1997 and 1999) likely are indicative of important environmental factors influencing
population dynamics that are not included in the current set of models.
Table 3. Observed and predicted population sizes (N, in millions) of midcontinent mallards in the traditional survey area.
Predicted population sizes were calculated using the revised models of population dynamics to generate model-specific
predictions, and then these predictions were averaged using annual model weights.
N in subsequent year
Predictor variables Predicted Observed
Year N Ponds Harvest
rate N SE(N) N % difference
1995 8.269 3.892 0.120 7.854 1.324 7.941 -1.10
1996 7.941 5.003 0.118 8.126 1.370 9.940 -18.25
1997 9.940 5.061 0.131 9.466 1.596 9.640 -1.80
1998 9.640 2.522 0.112 8.093 1.364 10.806 -25.11
1999 10.806 3.862 0.101 9.971 1.681 9.470 +5.29
2000 9.470 2.422 0.127 8.143 1.373 7.904 +3.02
2001 7.904 2.747 0.109 7.299 1.231 7.504 -2.73
A complete description of the revisions to the AHM protocol for midcontinent mallards is beyond the scope of this document,
so we encourage the reader to examine the technical report prepared by Runge et al. (2002). A copy of this report can be
obtained online at http://migratorybirds.fws.gov/reports/ahm02/MCMrevise2002.pdf.
Eastern Mallards
For purposes of AHM, eastern mallards are defined as those breeding in southern Ontario and Quebec (federal survey strata
51-54 and 56) and in the northeastern U.S. (state plot surveys; Heusmann and Sauer 2000) (Fig. 1). Estimates of population
size have varied from 856 thousand to 1.1 million since 1990, with the majority of the population accounted for in the
northeastern U.S. (Table 4).
Last year, the USFWS (2001) described several reasons that revisions to the existing AHM protocol for eastern mallards might
be warranted. First, it was believed that the number of existing models might be reduced because differences in model-specific
regulatory strategies were relatively minor. Another motivation concerned the tendency for empirical estimates of
survival and reproductive rates of eastern mallards to imply annual growth rates that are higher than those observed in surveys
of population size (as in midcontinent mallards). Finally, the Breeding Bird Survey (BBS) index used to predict reproductive
success of eastern mallards appeared to be biased low in years of high spring precipitation.
Based on recommendations from the AHM Working Group, the USFWS has revised the AHM protocol for eastern mallards.
A revised set of six models: (1) rely solely on federal and state waterfowl surveys rather than the BBS to predict reproductive
rates; (2) allow for the possibility of a positive bias in estimates of survival or reproductive rates; (3) incorporate competing
hypotheses of strongly and weakly density-dependent reproduction; and (4) assume that hunting mortality is additive to other
sources of mortality. We retained models with no bias-correction in the final model set because the time-series available for
comparing observed and predicted population sizes was relatively short. For the bias-corrected models, we allowed for the
possibility that the bias resides either in estimates of survival or reproductive rates.
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Table 4. 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 51-54 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
We derived optimal harvest strategies for each of the six models using stochastic dynamic programming (Lubow 1995), and
by conditioning on the current set of regulatory alternatives and a management objective to maximize long-term cumulative
harvest. Equilibrium population sizes were determined by simulating the optimal strategy with the model that gave rise to
it. The model-specific strategies based on the hypothesis of weakly density-dependent reproduction appear to be considerably
more conservative than those based on the hypothesis of strongly density-dependent reproduction (Table 5). All harvest
strategies are “knife-edged,” meaning that large differences in the optimal regulatory alternative can be precipitated by only
small changes in breeding-population size. This result is largely due to the small differences in predicted harvest rates among
the current regulatory alternatives (see the section on Regulatory Alternatives later in this report).
For the first time, we have calculated weights for the alternative models of eastern mallard population dynamics, based on
an assumption of equal model weights in 1996 (the last year data was used to develop most model components) and on
predictions of year-specific harvest rates. The model best predicting observed population size has varied among years;
accordingly, there is no single model that is clearly favored over the others at the end of the time frame (Table 6). However,
we note that the two models with no bias correction performed poorly compared to the models with a bias correction.
As with midcontinent mallards, a detailed description of the revisions to the AHM protocol for eastern mallards is beyond
the scope of this document. Therefore, we encourage the reader to examine the technical report prepared by Johnson et al.
(2002a). A copy of this report can be obtained online at http://migratorybirds.fws.gov/reports/ahm02/emal-ahm-2002.pdf.
Western Mallards
Western 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, data-collection programs for understanding and monitoring the dynamics of this mallard
stock are highly fragmented in both time and space. This fact is making it difficult to aggregate monitoring instruments in
12
Table 5. Model-specific, optimal regulatory strategies for eastern mallards based on the current set of regulatory alternatives
and an objective to maximize long-term cumulative harvest. The average population size expected under each model and
accompanying strategy is represented by the shaded cells. (BPOP = breeding-population size in millions) (BnRw = no bias-correction
and weakly density-dependent reproduction, BnRs = no bias-correction and strongly density-dependent reproduction,
BsRw = bias-corrected survival rates and weakly density-dependent reproduction, BsRs = bias-corrected survival rates and
strongly density-dependent reproduction, BrRw = bias-corrected reproductive rates and weakly density-dependent reproduction,
and BrRs = bias-corrected reproductive rates and strongly density-dependent reproduction).
Model
BPOP BnRw BnRs BsRw BsRs BrRw BrRs
0.1 C C C C C C
0.2 C L C L C VR
0.3 C L C L C L
0.4 C L C L C L
0.5 C L C L C L
0.6 C L C L C L
0.7 C L C L C L
0.8 C L C L C L
0.9 C L C L C L
1.0 C L VR L C L
1.1 C L M L VR L
1.2 R L L L R L
1.3 M L L L L L
1.4 L L L L L L
1.5 L L L L L L
Table 6. Weights for the revised models of eastern mallards. (BnRw = no bias-correction and weakly density-dependent
reproduction, BnRs = no bias-correction and strongly density-dependent reproduction, BsRw = bias-corrected survival rates
and weakly density-dependent reproduction, BsRs = bias-corrected survival rates and strongly density-dependent reproduction,
BrRw = bias-corrected reproductive rates and weakly density-dependent reproduction, and BrRs = bias-corrected reproductive
rates and strongly density-dependent reproduction). Model weights were assumed to be equal in 1996.
Model
Year BnRw BnRs BsRw BsRs BrRw BrRs
1997 0.0565 0.1100 0.2053 0.2129 0.1996 0.2157
1998 0.0775 0.1515 0.1855 0.1897 0.1913 0.2045
1999 0.1257 0.2489 0.1552 0.1344 0.1732 0.1627
2000 0.0297 0.1066 0.2042 0.2068 0.2153 0.2374
2001 0.0553 0.1932 0.1270 0.2303 0.1408 0.2533
2002 0.0585 0.2062 0.1223 0.2190 0.1416 0.2524
13
Population size expected next year (millions)
0 1 2 3 4 5 6 7 8 9 10 11
Harvest value (%)
0
20
40
60
80
100
NAWMP goal
(8.799 million)
a way that can be used to reliably model this stock’s dynamics and, thus, to establish criteria for regulatory decision-making
under AHM (USFWS 2001). Another complicating factor is that federal survey strata 1-12 in Alaska and the Yukon are
within the current geographic bounds of midcontinent mallards. Therefore, the AHM Working Group is continuing its
investigations of western mallards and is not prepared to recommend an AHM protocol at this time.
HARVEST MANAGEMENT OBJECTIVES
The basic harvest-management 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 NAWMP (Fig. 4). 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 long-term 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 8.799 million mallards, which is based on 8.199 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 long-term cumulative harvest.
REGULATORY ALTERNATIVES
Evolution of Alternatives
When AHM was first implemented in 1995, three regulatory alternatives characterized as liberal, moderate, and restrictive
14
were defined based on regulations used during 1979-84, 1985-87, and 1988-93, respectively (Appendix B). 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 very restrictive 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. This year, the
USFWS further modified the moderate and liberal alternatives to include extensions of approximately one week in both the
opening and closing framework dates (Table 7).
Table 7. Regulatory alternatives considered for the 2002 duck-hunting season.
Flyway
Regulation Atlantica Mississippi Centralb Pacificc
Shooting hours one-half hour before sunrise to sunset
Framework dates
Very restrictive
and Restrictive
Oct 1 - Jan 20 Saturday nearest Oct 1 - Sunday nearest Jan 20
Moderate and
Liberal
Saturday nearest Sep 24 - last Sunday in Jan
Season length (days)
Very restrictive 20 20 25 38
Restrictive 30 30 39 60
Moderate 45 45 60 86
Liberal 60 60 74 107
Bag limit (total / mallard / female mallard)
Very restrictive 3 / 3 / 1 3 / 2 / 1 3 / 3 / 1 4 / 3 / 1
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, 23, and 23 extra days in the very restrictive, restrictive, moderate,
and liberal alternatives, respectively.
c The Columbia Basin Mallard Management Unit is allowed seven extra days in the very restrictive, restrictive, and moderate
alternatives.
Predictions of Mallard Harvest Rates
Since 1995, harvest rates of adult male mallards associated with the regulatory alternatives have been predicted using harvest-rate
estimates from 1979-84, which have been adjusted to reflect current specifications of season lengths and bag limits, and
for contemporary numbers of hunters. These predictions are based only in part on band-recovery data, and rely heavily on
models of hunting effort and success derived from hunter surveys (Appendix C). As such, these predictions have large
sampling variances, and their accuracy is uncertain. Moreover, these predictions rely implicitly on an assumption that the
historic relationship between hunting regulations (and harvest rates) in the U.S. and Canada will remain unchanged in the
future. Currently, we have no way to judge whether this is a reasonable assumption. We also assumed that if hunting seasons
15
were closed in the U.S., rates of harvest in Canada would be similar to those observed during 1988-93, which is the most
recent period for which reliable estimates from Canada are available. This is a conservative approach given that we cannot
be sure Canada would close its hunting season at the same time as the U.S. Fortunately, optimal regulatory strategies do not
appear to be very sensitive to what we believe to be a realistic range of harvest-rate values associated with closed seasons
in the U.S.
We have adopted standard Bayesian statistical methods for improving regulation-specific predictions of harvest rates,
including predictions of the effects of framework-date extensions (Appendix D). Essentially, the idea is to use existing
(“prior”) information to develop initial harvest-rate 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, model with which to express an adaptive approach to management. In this approach,
we rely on the following model structure:
ht ~ Normal      ,    2
where ht is the harvest rate realized under a particular regulatory alternative in any year t. These rates are assumed to be
normally distributed with mean   +   and variance  2. In this statistical model,   is the mean harvest rate expected in the
absence of framework-date extensions,   is the marginal change in mean harvest rate associated with extended framework
dates, and  2 is the amount of annual variability in harvest rate due to uncontrolled environmental factors and changes in
hunter effort. For prior estimates of  , we relied on the regulation-specific estimates of mean harvest rates provided by the
USFWS (2000a:13-14) under the assumption of uniform regulatory choices across Flyways. Based on analyses by Johnson
et al. (1997), we assumed that   averages 20% of the mean. Updated predictions of midcontinent mallard harvest rates under
the liberal alternative (without framework-date extensions) were calculated based on observed harvest rates during the 1998-
2001 hunting seasons that were estimated with data from a small-scale reward-band study (Appendix D).
A previous analysis by the USFWS (2000b) suggested that implementation of framework-date extensions might be expected
to increase harvest by 15% and 5% for midcontinent and eastern mallards, respectively. Accordingly, we used a prior mean
of   = 0.02 for midcontinent mallards and   = 0.01for eastern mallards. However, there is a great deal of uncertainty about
the magnitude of the increases because of a lack of prior experience with nationwide extensions. Therefore, we explicitly
recognized this uncertainty in deriving optimal harvest strategies. The measures of uncertainty (i.e., the variances of  ) we
used were large enough to admit the possibility that extensions will result in no increase in mean harvest rates. After
framework-date extensions have been implemented, estimates of harvest rate derived from band-recovery data will be used
to update the estimate of   (the marginal effect of extensions). However, strong inference about   depends on reliable
knowledge of   (which is lacking in some cases), and on an experimental study design (which is likely not feasible).
Predicted harvest rates of adult male mallards associated with each of the regulatory alternatives are provided in Tables 8 and
9 and Figs. 5 and 6. We made the simplifying assumption that the harvest rate of midcontinent mallards depends solely on
the regulatory choice in the western three Flyways. This appears to be a reasonable assumption given the the small proportion
of midcontinent mallards wintering in the Atlantic Flyway (Munro and Kimball 1982), and harvest-rate predictions that
suggest a minimal effect of Atlantic Flyway regulations (USFWS 2000a). 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.
However, the harvest rate of eastern mallards is affected significantly by regulatory choices beyond the Atlantic Flyway
USFWS 2000a). To avoid making the regulatory choice in the Atlantic Flyway conditional on regulations elsewhere, we
inflated the variance of predicted harvest rates of eastern mallards to account for uncontrolled changes in regulations in the
three western Flyways (Johnson et al. 2002a).
Harvest rates of age and sex cohorts other than adult male mallards are based on constant rates of differential vulnerability
as derived from band-recovery data. For midcontinent mallards, these constants are 0.719, 1.541, and 1.118 for adult females,
young males, and young females, respectively. For eastern mallards, these constants are 1.153, 1.331, and 1.509 for adult
females, young males, and young females, respectively.
16
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
40
220 C
VR
R
M L
Fig. 5. Probability density functions (pdf) of harvest rates of adult male midcontinent
mallards under current regulatory alternatives in the three western Flyways (C = closed
season, VR = very restrictive, R = restrictive, M = moderate, L = liberal).
Table 8. Predicted harvest rates of adult male midcontinent mallards based on regulations in the three western Flyways. The
parameter   is the mean harvest rate expected in the absence of framework-date extensions and   is the marginal change
in mean harvest rate expected with extended framework dates. Standard errors are in parentheses.
Regulatory
alternative       +  
Closed (U.S.) 0.0088 (0.0018) N/A
Very restrictive 0.0526 (0.0105) N/A
Restrictive 0.0665 (0.0133) N/A
Moderate 0.1114 (0.0223) 0.02 (0.01) 0.1314 (0.0244)
Liberal 0.1210 (0.0222) 0.02 (0.01) 0.1410 (0.0243)
17
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
40
C
VR R M L
Fig. 6. Probability density functions (pdf) of harvest rates of adult male eastern mallards
under current regulatory alternatives in the Atlantic Flyway (C = closed, VR = very
restrictive, R = restrictive, M = moderate, L = liberal).
Table 9. Predicted harvest rates of adult male eastern mallards based on regulations in the Atlantic Flyway. The parameter
  is the mean harvest rate expected in the absence of framework-date extensions and   is the marginal change in mean
harvest rate expected with extended framework dates. Standard errors are in parentheses.
Regulatory
alternative       +  
Closed (U.S.) 0.0800 (0.0240) N/A
Very restrictive 0.1212 (0.0364) N/A
Restrictive 0.1352 (0.0406) N/A
Moderate 0.1625 (0.0488) 0.01 (0.01) 0.1725 (0.0498)
Liberal 0.1771 (0.0531) 0.01 (0.01) 0.1871 (0.0540)
18
1 2 3 4 5 6
4
5
6
7
8
9
10
BPOP
Ponds
L
C
VR
R
M
Fig. 7. Optimal regulatory choices in the three western Flyways for the 2002 hunting
season conditioned on population size (BPOP) and pond numbers in Prairie Canada.
This strategy is based on current regulatory alternatives (with framework-date
extensions), on the revised midcontinent-mallard models and weights, and on the dual
objectives of maximizing long-term cumulative harvest and achieving a population goal of
8.799 million mallards. The dot in the center of the figure represents predicted mean
population size and pond numbers under this strategy and combination of model weights,
and the ellipse represents conditions expected in 95% of all years. (C = closed season,
VR = very restrictive, R = restrictive, M = moderate, L = liberal)
OPTIMAL REGULATORY STRATEGIES
The optimal regulatory strategy for the three western Flyways was derived using: (1) the 2002 regulatory alternatives; (2) the
revised population models and associated weights for midcontinent mallards; and (3) the dual objectives to maximize long-term
cumulative harvest and achieve a population goal of 8.799 million midcontinent mallards. The resulting regulatory
strategy (Fig. 7 and Table 10) is somewhat more conservative and knife-edged than that used last year (USFWS 2001).
Assuming that regulatory choices adhered to this strategy, the harvest value and breeding-population size would be expected
to average 1.02 (SD = 0.85) million and 7.10 (SD = 1.56) million, respectively. Note that prescriptions for closed seasons
in this strategy and others in this report represent resource conditions that are insufficient to support one of the current
regulatory alternatives, given current harvest-management objectives. However, closed seasons under all of these conditions
are not necessarily required for long-term resource protection, and simply reflect the constraints of the NAWMP population
goal and the current regulatory alternatives.
Based on a midcontinent population size of 8.5 million mallards (traditional surveys plus MN, MI, and WI) and 1.44 million
ponds in Prairie Canada, the optimal regulatory choice for the Pacific, Central, and Mississippi Flyways in 2002 is the liberal
alternative.
19
Table 10. Optimal regulatory choicesa in the three western Flyways for the 2002 hunting season. This strategy is based on
current regulatory alternatives (with framework-date extensions), on the revised midcontinent-mallard models and weights,
and on the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.799 million
mallards. The shaded cell represents the prescribed regulatory choice for 2002.
Pondsb
Mallardsc 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
 5.5 C C C C C C C C C C C
6.0 C C C C C C C VR VR VR VR
6.5 C C C VR R R R R R M M
7.0 R R R R R M M M M L L
7.5 R R M M M L L L L L L
8.0 M L L L L L L L L L L
 8.5 L L L L L L L L L L L
a C = closed season, VR = very restrictive, R = restrictive, M = moderate, and L = liberal.
b Estimated number of ponds in Prairie Canada in May, in millions.
c Estimated number of midcontinent mallards during May, in millions.
We optimized the regulatory choice for the Atlantic Flyway based on: (1) current regulatory alternatives; (2) the revised
population models and associated weights for eastern mallards; and (3) an objective to maximize long-term cumulative
harvest. The resulting strategy suggests liberal regulations for all population sizes of record, and is characterized by a lack
of intermediate regulations (Table 11). The strategy exhibits this behavior largely because of the small differences in harvest
rate among regulatory alternatives (Fig. 6).
Table 11. Optimal regulatory choicesa in the Atlantic Flyway for the 2002 hunting season. This strategy is based on current
regulatory alternatives (with framework-date extensions), on the revised eastern-mallard models and weights, and on an
objective to maximize long-term cumulative harvest. The shaded cell represents the prescribed regulatory choice for 2002.
Mallardsb Regulation
 200 C
225 VR
250 R
275 M
 300 L
a C = closed season, VR = very restrictive, R = restrictive, M = moderate, and L = liberal.
b Estimated number of eastern mallards in the combined federal and state surveys, in thousands.
We simulated the use of the regulatory strategy in Table 11 to determine expected performance characteristics. Assuming
that harvest management adhered to this strategy, the annual harvest and breeding population size would be expected to
average 478 (SD = 128) thousand and 880 (SD = 175) thousand, respectively.
Based on a breeding population size of 1.0 million mallards, the optimal regulatory choice for the Atlantic Flyway in 2002
is the liberal alternative.
20
Strategic Issues in AHM
Policy Questions
The AHM Working Group has begun a strategic discussion about future development and application of AHM. This
discussion was motivated in part by the special session on AHM that was held at the 2000 North American Wildlife and
Natural Resources Conference (Humburg et al. 2000, Johnson and Case 2000, Nichols 2000). That session offered a
retrospective on the development of AHM, and described a number of policy issues affecting future progress. Relevant
questions that need to be addressed include:
(a) Should AHM account explicitly for hunter “satisfaction” and, if so, how would it be measured and monitored? What
constitutes a “fair” distribution of harvest or hunting opportunity, and should regulations be used to achieve this
fairness?
(b) To what extent should the population goals of the North American Waterfowl Management Plan influence hunting
opportunity?
(c) To date, AHM has been based on mallards. To what extent should the USFWS try to account for differences in the
harvest potential of various duck stocks in setting hunting regulations? How does the USFWS distinguish what is
desirable from what is practical?
(d) How many regulatory alternatives should there be? Among the alternatives, what are desirable or acceptable ranges
of season lengths, bag limits, and framework dates? How often should the set of regulatory alternatives be reviewed?
The USFWS has decided to convene a task force, comprised of recognized state and federal leaders in waterfowl management,
to help address these and other questions related to future application of AHM. This task force will need to work closely with
the USFWS, the AHM Working Group, and the Wildlife Management Institute (WMI). The WMI has received federal aid
to help explore the relationship between waterfowl hunting regulations and hunter satisfaction and participation, and to
recommend how such information might be used in the AHM process.
Multi-Species Harvest Management
Variation is a defining feature of ecological systems. Virtually all ecological systems exhibit a broad range of variation on
temporal, spatial, and organizational (e.g., taxonomic) scales, ultimately as a function of how individual organisms respond
to their environment (Levin 1992). The scales at which individuals are aggregated for management purposes is an arbitrary
decision, but one that can strongly influence both the benefits and costs of management (Johnson and Williams 1999).
Management systems that account for important sources of ecological variation are expected to yield the highest benefits,
but also are likely to be characterized by relatively high monitoring and assessment costs (Babcock and Sparrowe 1989,
Sparrowe 1990).
One of the most difficult scale issues confronting AHM concerns that of multiple duck species. The problem is characterized
by the following features:
(1) duck species vary in their potential to support sport harvest;
(2) multiple species generally are exposed to a common hunting season (although species-specific harvests can be
regulated within limits by stratifying hunting regulations on spatial, temporal, and organizational scales);
(3) species-specific harvest returns and population trajectories are subject to considerable uncertainty, whose sources
include uncontrolled environmental variation, random effects of regulations (i.e., partial controllability), uncertainties
in population dynamics, and errors and biases in data-collection programs (i.e., partial observability); and
(4) management objectives are complex, in that they must account for species-specific values (i.e., not all species will
be equally valued by hunters) and for the legal mandate to prevent over-exploitation of any particular species.
It is increasingly apparent that the future development and success of AHM will depend heavily on how we address the multi-species
problem. There are several challenges, however, that won’t be overcome easily. First, although AHM provides a
coherent framework for exploring the effects of various approaches to multi-species harvesting, those explorations can be
21
conducted only to the extent that information on the dynamics of potential stocks is available. And while AHM provides an
effective means for coping with uncertainty, high levels of uncertainty in management outcomes will severely reduce the
benefits expected from an explicit recognition of variation in harvest potential. Second, more objective decisions about the
appropriate multi-species approach will require a full accounting of monitoring and assessment costs. There also may be
social costs to consider as regulations become increasingly complex to account for variation in harvest potential. Third, there
will be difficult decisions about harvest-management objectives, including the importance of population goals, the relative
value of species among hunters, and what constitutes fair allocations of hunting opportunity.
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harvests. Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 10pp.
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23
APPENDIX A: AHM Working Group
Scott Baker
Dept. of Wildlife, Fisheries, & Parks
P.O. Box 378
Redwood, MS 39156
phone: 601-661-0294
fax: 601-364-2209
e-mail: mahannah1@aol.com
Bob Blohm
U.S. Fish and Wildlife Service
Arlington Square, Room 634
4401 North Fairfax Drive,
Arlington, VA 22203
phone: 703-358-1966
fax: 703-358-2272
e-mail: robert_blohm@fws.gov
Brad Bortner
U.S. Fish and Wildlife Service
911 NE 11th Ave.
Portland, OR 97232-4181
phone: 503-231-6164
fax: 503-231-2364
e-mail: brad_bortner@fws.gov
Frank Bowers
U.S. Fish and Wildlife Service
1875 Century Blvd., Suite 345
Atlanta, GA 30345
phone: 404-679-7188
fax: 404-679-7285
e-mail: frank_bowers@fws.gov
Dave Case
D.J. Case & Associates
607 Lincolnway West
Mishawaka, IN 46544
phone: 574-258-0100
fax: 574-258-0189
e-mail: dave@djcase.com
Dale Caswell
Canadian Wildlife Service
123 Main St. Suite 150
Winnepeg, Manitoba, CANADA R3C 4W2
phone: 204-983-5260
fax: 204-983-5248
e-mail: dale.caswell@ec.gc.ca
John Cornely
U.S. Fish and Wildlife Service
P.O. Box 25486, DFC
Denver, CO 80225
phone: 303-236-8155 (ext 259)
fax: 303-236-8680
e-mail: john_cornely@fws.gov
Gary Costanzo
Dept. of Game and Inland Fisheries
5806 Mooretown Road
Williamsburg, VA 23188
phone: 757-253-4180
fax: 757-253-4182
e-mail: gcostanzo@dgif.state.va.us
Jim Dubovsky
U.S. Fish & Wildlife Service
P.O. Box 25486 DFC
Denver, CO 80225-0486
phone: 303-236-8155 (ext 238)
fax: 303-236-8680
e-mail: james_dubovsky@fws.gov
Diane Eggeman
Fish & Wildlife Conservation Commission
8932 Apalachee Pkwy.
Tallahassee, FL 32311
phone: 850-488-5878
fax: 850-488-5884
e-mail: eggemadr@fwc.state.fl.us
Ken Gamble
U.S. Fish and Wildlife Service
608 Cherry Street, Room 119
Columbia, MO 65201
phone: 573-876-1915
fax: 573-876-1917
e-mail: ken_gamble@fws.gov
Jim Gammonley
Division of Wildlife
317 West Prospect
Fort Collins, CO 80526
phone: 970-472-4379
fax: 970-472-4457
e-mail: jim.gammonley@state.co.us
24
Pam Garrettson
U.S. Fish & Wildlife service
11500 American Holly Drive
Laurel, MD 20708-4016
phone: 301-497-5865
fax: 301-497-5871
e-mail: pam_garrettson@fws.gov
George Haas
U.S. Fish and Wildlife Service
300 Westgate Center Drive
Hadley, MA 01035-9589
phone: 413-253-8576
fax: 413-253-8480
e-mail: george_haas@fws.gov
Jeff Haskins
U.S. Fish and Wildlife Service
P.O. Box 1306
Albuquerque, NM 87103
phone: 505-248-6827 (ext 30)
fax: 505-248-7885
e-mail: jeff_haskins@fws.gov
Mark Herzog
Dept. Environmental & Resource Sciences
University of Nevada
Reno, NV 89512
phone: 775-784-6558
e-mail: mherzog@unr.edu
Dale Humburg
Dept. of Conservation
Fish & Wildlife Research Center
1110 South College Ave.
Columbia, MO 65201
phone: 573-882-9880 (ext 3246)
fax: 573-882-4517
e-mail: humbud@mail.conservation.state.mo.us
Fred Johnson
U.S. Fish & Wildlife Service
7920 NW 71st Street
Gainesville, FL 32653
phone: 352-378-8181 (ext 372)
fax: 352-378-4956
e-mail: fred_a_johnson@fws.gov
Mike Johnson
Game and Fish Department
100 North Bismarck Expressway
Bismarck, ND 58501-5095
phone: 701-328-6319
fax: 701-328-6352
e-mail: mjohnson@state.nd.us
Jim Kelley
U.S. Fish and Wildlife Service
BH Whipple Federal Building, 1 Federal Drive
Fort Snelling, MN 55111-4056
phone: 612-713-5409
fax: 612-713-5286
e-mail: james_r_kelley@fws.gov
Bill Kendall
U.S.G.S. Patuxent Wildlife Research Center
11510 American Holly Drive
Laurel, MD 20708-4017
phone: 301-497-5868
fax: 301-497-5666
e-mail: william_kendall@usgs.gov
Don Kraege
Dept. of Fish & Wildlife
600 Capital Way North
Olympia. WA 98501-1091
phone: 360-902-2509
fax: 360-902-2162
e-mail: kraegdkk@dfw.wa.gov
Jeff Lawrence
Dept. of Natural Resources
102 23rd St. NE
Bemidji, MN 56601
phone: 218-755-3910
fax: 218-755-2604
e-mail: jeff.lawrence@dnr.state.mn.us
Bob Leedy
U.S. Fish and Wildlife Service
1011 East Tudor Road
Anchorage, AK 99503-6119
phone: 907-786-3446
fax: 907-786-3641
e-mail: robert_leedy@fws.gov
25
Mary Moore
U.S. Fish & Wildlife Service
206 Concord Drive
Watkinsville, GA 30677
phone: 706-769-2359
fax: 706-769-2359
e-mail: mary_moore@fws.gov
Jim Nichols
U.S.G.S. Patuxent Wildlife Research Center
11510 American Holly Drive
Laurel, MD 20708-4017
phone: 301-497-5660
fax: 301-497-5666
e-mail: jim_nichols@usgs.gov
Mark Otto
U.S. Fish & Wildlife Service
11500 American Holly Drive
Laurel, MD 20708-4016
phone: 301-497-5872
fax: 301-497-5871
e-mail: mark_otto@fws.gov
Paul Padding
U.S. Fish & Wildlife Service
10815 Loblolly Pine Drive
Laurel, MD 20708-4028
phone: 301-497-5980
fax: 301-497-5981
e-mail: paul_padding@fws.gov
Michael C. Runge
U.S.G.S. Patuxent Wildlife Research Center
11510 American Holly Drive
Laurel, MD 20708-4017
phone: 301-497-5748
fax: 301-497-5666
e-mail: michael_runge@usgs.gov
Jerry Serie
U.S. Fish & Wildlife Service
12100 Beech Forest Road
Laurel, MD 20708-4038
phone: 301-497-5851
fax: 301-497-5885
e-mail: jerry_serie@fws.gov
Dave Sharp
U.S. Fish and Wildlife Service
P.O. Box 25486, DFC
Denver, CO 80225-0486
phone: 303-275-2386
fax: 303-275-2384
e-mail: dave_sharp@fws.gov
Sue Sheaffer
Coop. Fish & Wildl. Research Unit
Fernow Hall, Cornell University
Ithaca, NY 14853
phone: 607-255-2837
fax: 607-255-1895
e-mail: ses11@cornell.edu
Graham Smith
U.S. Fish & Wildlife Service
11500 American Holly Drive
Laurel, MD 20708-4016
phone: 301-497-5860
fax: 301-497-5871
e-mail: graham_smith@fws.gov
Bob Trost
U.S. Fish and Wildlife Service
911 NE 11th Ave.
Portland, OR 97232-4181
phone: 503-231-6162
fax: 503-231-6228
e-mail: robert_trost@fws.gov
Khristi Wilkins
U.S. Fish & Wildlife Service
11500 American Holly Drive
Laurel, MD 20708-4016
phone: 301-497-5557
fax: 301-497-5971
e-mail: khristi_a_wilkins@fws.gov
Ken Williams
U.S.G.S. Cooperative Research Units
12201 Sunrise Valley Drive
Mail Stop 303
Reston, VA 20192
phone: 703-648-4260
fax: 703-648-4269
e-mail: byron_ken_williams@usgs.gov
26
Dan Yparraguirre
Dept. of Fish & Game
1812 Ninth Street
Sacramento, CA 95814
phone: 916-445-3685
e-mail: dyparraguirre@dfg.ca.gov
27
Appendix B: Regulatory Alternatives for the 1995 and
1996 Hunting Seasons
Flyway
Regulation Atlantic Mississippi Centrala Pacificb
Shooting hours one-half hour before sunrise to sunset
Framework dates Oct 1 - Jan 20 Saturday closest to October 1 and Sunday closest to January
20
Season length (days)
Restrictive 30 30 39 59
Moderate 40 40 51 79
Liberal 50 50 60 93
Bag limit (total / mallard / female mallard)
Restrictive 3 / 3 / 1 3 / 2 / 1 3 / 3 / 1 4 / 3 / 1
Moderate 4 / 4 / 1 4 / 3 / 1 4 / 4 / 1 5 / 4 / 1
Liberal 5 / 5 / 1 5 / 4 / 1 5 / 5 / 1 6-7c / 6-7c / 1
a The High Plains Mallard Management Unit was allowed 12, 16, and 23 extra days under the restrictive, moderate, and liberal
alternatives, respectively.
b The Columbia Basin Mallard Management Unit was allowed seven extra days under all three alternatives.
c The limits were 6 in 1995 and 7 in 1996.
28
APPENDIX C: Predicting Harvest Rates
This procedure involves: (1) linear models that predict total seasonal mallard harvest for varying regulations (daily bag limit
and season length), while accounting for trends in numbers of successful duck hunters; and (2) use of these models to adjust
historical estimates of mallard harvest rates to reflect differences in bag limit, season length and trends in hunter numbers.
Using historical data from both the U.S. Waterfowl Mail Questionnaire and Parts Collection Surveys, and with the use of
several key assumptions, the resulting models allowed us to predict total seasonal mallard harvest and associated predicted
harvest rates for varying combinations of season length and daily bag limits.
Total seasonal mallard harvest is predicted using two separate models: the “harvest” model which predicts average daily
mallard harvest per successful duck hunter for each day of the hunting season (Table C-1), and the “hunter” model which
predicts the number of successful duck hunters (Table C-2). The “harvest” model uses as the dependent variable the square
root of the average daily mallard harvest (per successful duck hunter). The independent variables include the consecutive
day of the hunting season (splits were ignored), daily mallard bag limit, season length, and the interaction of bag limit and
season length. Also included is an effect representing the opening day (of the first split), an effect representing a week (7
day) effect, and several other interaction terms. Seasonal mallard harvest per successful duck hunter is obtained by back-transforming
the predicted values that resulted from the model, and summing the average daily harvest over the season length.
The “hunter” model uses information on the numbers of successful duck hunters (based on duck stamp sales information)
from 1981-95. Using daily bag limit and season length as independent variables, the number of successful duck hunters is
predicted for each state.
Both the “harvest” and “hunter” models were developed for each of seven management areas: the Atlantic Flyway portion
with compensatory days (AF-COMP); the Atlantic Flyway portion without compensatory days (AF-NOCOMP); the
Mississippi Flyway (MF); the low plains portion of Central Flyway (CF-lp); the High Plains Mallard Management Unit in
the Central Flyway (CF-HP); the Columbia Basin Mallard Management Unit in the Pacific Flyway (PF-CB); and the
remainder of the Pacific Flyway excluding Alaska (PF). The numbers of successful hunters predicted at the state level are
summed to obtain a total number (H) for each management area. Likewise, the “harvest” model results in a seasonal mallard
harvest per successful duck hunter (A) for each management area. Total seasonal mallard harvest (T) is formed by the product
of H and A.
To compare total seasonal mallard harvest under different regulatory alternatives, ratios of T are formed for each management
area and then combined into a weighted mean. Under the key assumption that the ratio of harvest rates realized under two
different regulatory alternatives is equal to the expected ratio of total harvest obtained under the same two alternatives, the
harvest rate experienced under the historic “liberal” package (1979-84) was adjusted by T to produce predicted harvest rates
for the current regulatory alternatives.
The models developed here were not designed, nor are able, to predict mallard harvest rates directly. The procedure relies
heavily on statistical and conceptual models that must meet certain assumptions. We have no way to verify these
assumptions, nor can we gauge their effects should they not be met. The use of this procedure for predicting mallard harvest
rates for regulations alternatives for which we have little or no experience warrants considerable caution.
29
Table C-1. Parameter estimates by management area for models of seasonal harvest per successful hunter.a
Model effecta AF- COMP. AF-NOCOMP MF CF-lp CF-HP PF-CB PF
INTERCEPT 0.378359 0.555790 0.485971 0.554667 0.593799 0.736258 0.543791
(SE) (0.061477) (0.134516) (0.037175) (0.041430) (0.059649) (0.154315) (0.054712)
OPEN 0.194945 0.263793 0.175012 0.092507 0.113074 0.361696 0.322255
(SE) (0.010586) (0.018365) (0.011258) (0.015623) (0.018530) (0.040605) (0.012730)
WEEK 0.024232 0.040392 -0.016479 -0.108472 -0.074895 -0.063422 -0.060477
(SE) (0.006561) (0.011436) (0.006965) (0.008860) (0.009437) (0.018220) (0.006118)
WEEK2 -0.003586 -0.006823 0.000422 0.010472 0.006782 0.003573 0.004893
(SE) (0.000796) (0.001392) (0.000847) (0.001075) (0.001150) (0.002266) (0.000746)
WK*SDAY -0.001245 -0.001395 -0.000073 0.002578 0.001222 -0.000102 0.000116
(SE) (0.000231) (0.000407) (0.000248) (0.000260) (0.000215) (0.000289) (0.000120)
WK2*SDAY 0.000163 0.000219 0.000052 -0.000271 -0.000109 0.000045 0.000007
(SE) (0.000028) (0.000050) (0.000030) (0.000032) (0.000026) (0.000037) (0.000015)
SEASDAY 0.000419 -0.001034 -0.002559 -0.006322 -0.003174 -0.000615 -0.000909
(SE) (0.000407) (0.000712) (0.000434) (0.000464) (0.000382) (0.000476) (0.000209)
MALBAG -0.025557 -0.062755 0.026729 0.016049 -0.029753 -0.049532 -0.021774
(SE) (0.019282) (0.043020) (0.015007) (0.010766) (0.013918) (0.047903) (0.017457)
SEASLEN -0.004852 -0.008836 -0.004869 -0.001250 -0.003089 0.001562 -0.001931
(SE) (0.001260) (0.002750) (0.000768) (0.000833) (0.000995) (0.001682) (0.000591)
BAG*SEAS 0.000926 0.002018 0.000332 -0.000033 0.000732 0.000024 0.000328
(SE) (0.000393) (0.000877) (0.000310) (0.000202) (0.000216) (0.000464) (0.000184)
aModel effect Description
—————����� —————
INTERCEPT Intercept
OPEN Opening Day of First Split (Y,N)
WEEK Day of Week (1,2,3,4,5,6,7)
WEEK2 Week * Week (Quadratic Effect)
WK*SDAY Week * Day of Season Interaction
WK2*SDAY Week * Week * Day of Season Interaction
SEASDAY Day of Season (Consecutive)
MALBAG Daily Mallard Bag Limit
SEASLEN Season Length
BAG*SEAS Daily Mallard Bag Limit * Season Length Interaction
30
Table C-2. Parameter estimates by management area for models to predict hunter numbers.
Mgmt Area Effect State/Zone Estimate SE Mgmt Area Effect State/Zone Estimate SE
AF-Comp. MALBAG -229.854 320.613 CF - lp MALBAG 577.848 715.617
SEASLEN 119.595 28.473 SEASLEN 317.973 100.931
Intercepts: CT 925.275 823.888 Intercepts: KS -6,006.131 3,108.375
DE 376.732 829.784 NE -4,997.796 3,114.451
ME 3,581.062 825.956 ND -3,930.604 3,021.002
MD 10,712.000 809.333 OK -8,010.002 3,208.936
NJ 5,940.028 813.652 SD -4,053.537 3,021.002
NC 12,798.000 836.186 TX 33,480.000 3,021.002
PA 17,683.000 822.566 CF - HP MALBAG 734.041 181.624
VA 7,276.371 809.333 SEASLEN -1.332 16.318
WV -2,884.782 818.825 Intercepts: CO 12,354.000 687.696
MA_3 1,679.885 818.507 KS -973.654 688.526
MA_R -336.288 843.081 MT 482.197 699.176
AF-No comp. MALBAG 71.885 188.301 NE 3,222.880 688.526
SEASLEN 62.574 18.776 NM 447.280 688.526
Intercepts: FL 9,709.872 530.458 ND 4,559.079 541.659
GA 7,058.253 541.184 OK -2,299.609 687.696
RI -1,515.873 543.352 SD 748.221 695.658
SC 10,004.000 541.184 TX 2,817.864 695.658
VT 679.453 541.184 WY 1,639.613 688.526
NH_1 -1,536.280 541.184 PF - CB MALBAG 505.129 411.451
NH_2 201.430 536.395 SEASLEN 31.446 48.602
NY_1 336.305 537.703 Intercepts: OR -3,910.659 2,311.323
NY_2 -2,122.214 541.184 WA 5,433.261 2,334.479
NY_5 7,070.786 541.184
NY_R 8,650.966 538.322
OH_1 -2,426.542 535.906
31
Table C-2, continued.
Mgmt Area Effect State/Zone Estimate SE Mgmt Area Effect State/Zone Estimate SE
MF MALBAG -4,523.798 1,231.622 PF MALBAG 790.844 284.473
SEASLEN 897.413 120.583 SEASLEN 59.303 31.696
Intercepts: AL -15,044.000 2,361.763 Intercepts: AZ -3,958.814 1,402.487
AR 5,599.384 2,361.763 CO -4,832.461 1,400.722
IL 7,438.650 2,361.763 ID 6,285.454 1,384.878
IN -13,932.000 2,361.763 MT -887.114 1,458.939
IA -1,346.879 2,337.443 NV -2,483.897 1,369.116
KY -15,477.000 2,394.393 NM -7,588.133 1,395.432
LA 41,690.000 2,543.303 OR 11,687.000 1,397.194
MI 10,232.000 2,361.763 UT 6,803.640 1,415.495
MN 61,174.000 2,635.798 WY 9,398.653 1,402.487
MS -9,207.288 2,285.436 CA_1 -3,696.948 1,385.102
MO -2,225.616 2,361.763 CA_2 -5,421.502 1,427.980
TN -6,958.016 2,361.763 CA_3 3,580.319 1,385.102
WI 27,254.000 2,361.763 CA_4 -6,475.400 1,378.069
OH_R -9,163.989 2,635.798 CA_5 29,744.000 1,385.102
32
APPENDIX D: Updating Predictions of Harvest Rates
Statistical Procedures
We rely on standard Bayesian statistical techniques for updating predictions of regulation-specific harvest rates of mallards
(Johnson et al. 2002c). We first specified the following model structure:
data y Normal h
truth h Normal
t t t
t
: ~ ( , )
: ~ ( , )
 
   
2
2
In this model, yt represents an estimate (with sampling variance  2) of the annual harvest rate ht under a given regulatory
alternative. In turn, we assume that ht is drawn from a normal distribution with mean   and process error  2. The process
error represents the amount of variability in harvest rates under the same regulatory alternative due to annual variation in
weather, habitat conditions, timing of migration, etc.
The problem is to estimate   and  2, given our prior beliefs about these parameters and any estimates of harvest rates yt that
become available from subsequent experience. To do this, we must specify prior distributions for the parameters   and  2.
These distributions represent a quantitative statement of our prior beliefs about the mean and variance of the harvest rate
expected under a given regulatory alternative. For the prior distribution of  , we use:
   
~ Normal ,  
n
  2
   
 
   
where
  2      2 02
6
   
 
.
n
The parameter   must be specified for each regulatory alternative. We use the predicted mean harvest rates derived from the
procedures described in Appendix C and provided by USFWS (2000a:13-14). Those procedures, however, provide only a
combined estimate of  2 +  2, which is not useful for our purpose. Therefore, we relied on information provided by Johnson
et al. (1997), who estimated that   is typically about 20% of the mean (i.e., CV = 0.2). We set n = 6 to reflect the empirical
estimates of harvest rate during 1979-84 that were used to help derive initial predictions under the current regulatory
alternatives.
Based on theoretical considerations, we suggest a scaled inverse gamma with n degrees of freedom and scale parameter  2
for the prior distribution of  2:
  2 ~ Inverse Gamma n,      0.2  2  
Once prior distributions are fully specified for each regulatory alternative, they can be updated using estimates of harvest rates
(based on band-recovery data) that become available after a particular regulatory alternative is implemented. Using standard
Bayesian methodology, these prior distributions are converted to posterior distributions, from which sample posterior means
and variances are derived. These posterior means and variances provide updated measures of   and  2, which are used for
predictive purposes in the next cycle of regulation-setting.
33
Next, we modified the model structure to account for the marginal effect of framework-date extensions:
ht  ~ Normal      ,    2
where   is the absolute change in mean harvest rate. In this model we assume that the process error remains unaffected.
As before, a key question is what to use as a prior distribution for the parameter  . We use:
  ~ Normal       ,   2  
where   is the expected proportional change in mean harvest rate, and the variance  2 is a measure of uncertainty about ( × ).
A previous assessment suggests that   = 0.15 for midcontinent mallards and   0.05 for eastern mallards (USFWS 2000b).
In the absence of other relevant assessments, we use these values for making initial predictions about the effects of
framework-date extensions. Specifying  2 is more difficult because information provided by USFWS (2000b) is based on
two critical assumptions: (1) that changes in harvest rate are proportional to changes in harvest; and (2) that past experience
with an early opening date in Iowa and a late closing date in Mississippi accurately reflects the expected effect in other States
where extensions have never been offered. Because of the tenuousness of these assumptions, we suggest that variance
estimates presented by USFWS (2000b) do not adequately characterize the level of uncertainty about  . Therefore, we use
values of  2 that bound a lower confidence limit for ( × ) by zero. This is an explicit recognition that our prior beliefs
include the possibility that   = 0.
When framework-date extensions are implemented, estimates of harvest rate derived from band-recovery data would be used
to update the prior distribution for  . We note, however, that any inference about the causal relationship between   and
framework extensions will be very weak because changes due to extensions will be confounded with any other uncontrolled
changes in harvest rates (i.e., there will be no experimental controls).
Application to Midcontinent Mallards
We here demonstrate application of the Bayesian methods described above for updating predictions of harvest rates of
midcontinent mallards under the liberal regulatory alternative without framework-date extensions. Based on analyses and
assumptions described previously, we first specified prior distributions for   and  2 :
   
 
 
~ . , .
~ ,.
Normal
Inverse Gamma
01305 0 0261
6
6 0 0261
2
2 2
 
   
 
   
We have estimates of harvest rate realized under the liberal alternative for the 1998 through 2001 hunting seasons. Reward
banding in banding reference areas 2, 4, and 5 provided the basis for these estimates (USFWS 2001:40). Harvest rates in un-sampled
reference areas were treated as missing, and conventional data augmentation techniques were used (Schafer 1997).
Estimates of harvest rate were first computed for each reference area, and then these estimates were averaged using breeding-population
estimates in each reference area as weights.
34
This procedure resulted in the following estimates:
y1998 = 0.108 ( 2 = 0.0132)
y1999 = 0.098 ( 2 = 0.0082)
y2000 = 0.129 ( 2 = 0.0112)
y2001 = 0.104 ( 2 = 0.0132)
Combining our prior distributions and these harvest-rate estimates, we calculated posterior (updated) means and standard
deviations for the following parameters of interest:
h1998 = 0.112 (SD = 0.0111)
h1999 = 0.101 (SD = 0.0076)
h2000 = 0.127 (SD = 0.0100)
h2001 = 0.109 (SD = 0.0112)
  = 0.121 (SD = 0.0082)
 2 = 0.02222 (SD = 0.00031)
Thus, our best estimate of the mean harvest rate ( ) under the liberal regulatory alternative (without framework-date
extensions) decreased from 0.130 to 0.121 to become more consistent with the four years of observation. Also, the estimate
of process error ( 2) decreased slightly to reflect the relatively low variability in harvest rates observed among the last four
years.
NOTES