Adaptive Harvest Management 2000 duck hunting season

              Citation: U.S. Fish and Wildlife Service. 2000. Adaptive Harvest Management: 2000 Hunting Season. U.S. Dept.
Interior, Washington, D.C. 40pp.
Cover art: Adam Grimm’s painting of a mottled duck, which was selected for the 2000 federal “duck stamp.”
U. S. Fish & Wildlife Service
Adaptive
Harvest
Management
2000 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 2000 hunting season. Specifically, this report is intended to provide waterfowl managers and the public with
information about the use of adaptive harvest management 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 herein may differ from that published previously.
ACKNOWLEDGEMENTS
A working group comprised of technical representatives from the USFWS, the four Flyway Councils, and the USGS
Biological Resources Division (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 (AHM),
which was first implemented in 1995. The USFWS expresses its gratitude to the 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 Adaptive Management & Assessment Team, which is administered by the
Divisions of Migratory Bird Management and North American Waterfowl and Wetlands. F. A. Johnson (USFWS) was
the principal author, but significant contributions to the report were made by J. A. Dubovsky (USFWS), W. L. Kendall
(USGS Patuxent Wildlife Research Center), and M. T. Moore (USFWS). J. P. Bladen (USFWS), D. J. Case (D.J. Case &
Assoc.), J. R. Kelley (USFWS), E. M. Martin (USFWS), M. C. Otto (USFWS), P. I. Padding (USFWS), G. W. Smith
(USFWS), and K. A. Wilkins (USFWS) provided information or otherwise assisted with report preparation. Comments
regarding this document should be sent to Jon Andrew, Chief, Division of Migratory Bird Management - USFWS,
Arlington Square, Room 634, 4401 North Fairfax Drive, Arlington, VA 22203
Annual reports and other information regarding adaptive harvest management are available on the Internet at:
www.migratorybirds.fws.gov/reports/reports.html
TABLE OF CONTENTS
Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Mallard Stocks and Flyway Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Mallard Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Harvest Management Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Regulatory Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Optimal Harvest Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Current AHM Priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Appendix A: AHM Working Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Appendix B: Mallard Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Appendix C: Updating Model Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Appendix D: Predicting Harvest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Appendix E: Estimating the Mallard Harvest Rate for the 1999-00 Hunting Season . . . . . . . 36
Appendix F: Past Regulations and Harvest Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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EXECUTIVE SUMMARY
In 1995, the U.S. Fish and Wildlife Service (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.
To date, adaptive harvest management (AHM) has been based midcontinent mallards, but efforts are being made to modify
the decision-making protocol to account for mallards breeding eastward and westward of the midcontinent region The ability
to regulate harvest on 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 is to vary hunting regulations among Flyways in
a manner that recognizes each Flyway’s unique breeding-ground derivation of mallards. This year, the USFWS intends to
propose modifications to the current AHM protocol to account for eastern mallards. The USFWS has identified two basic
alternatives in this report. The first involves a single, joint optimization for midcontinent and eastern mallards. The
characteristic feature of this approach is that all regulatory choices, regardless of Flyway, would depend on the status of both
midcontinent and eastern mallards (with the degree of dependence based on each harvest area’s unique combination of the
two mallard populations). The second alternative would entail two separate optimizations, in which the Atlantic Flyway
regulation would be based exclusively on the status of eastern mallards, and the regulatory choice for the remainder of the
country would be based exclusively on the status of midcontinent mallards.
A critical need for successful implementation of AHM is a set of regulatory alternatives that remain fixed for an extended
period. For the 2000 season, the USFWS is maintaining the same regulatory alternatives as those used during1997-99.
However, this year, the prediction of harvest rates associated with these regulatory alternatives must account for the possibility
that the AHM protocol will be modified to allow a regulatory alternative in the Atlantic Flyway that is different from other
Flyways. Therefore, it was necessary to predict harvest rates for each mallard population for the 25 combinations of regulatory
alternatives (including the option of closed seasons) in the Atlantic Flyway and the remainder of the country. Based on this
analysis, harvest rates of eastern mallards depend not only on the regulation in the Atlantic Flyway, but on the regulation in
the remainder of the country. Harvest rates of midcontinent mallards depend almost completely on regulations in the three
western Flyways.
Using current regulatory alternatives and associated harvest rates, both the joint-optimization and separate-optimization
alternatives would be expected to greatly increase the frequency of liberal regulations in the Atlantic Flyway. Based on the
joint optimization, however, there seems to be no discernible influence of midcontinent mallard status on optimal regulatory
prescriptions for the Atlantic Flyway, nor does there seem to be any significant impact of eastern mallard status on optimal
regulations in the remainder of the country. The notable exception is the case in which midcontinent population size is below
goal and eastern population size is high; under these conditions the regulation in the three western Flyways would be slightly
more liberal than it would be in the absence of a consideration of eastern mallard status. These results seem to follow from
the high degree of spatial discrimination between the two mallard populations during the hunting season.
Optimal regulatory choices for the 2000 hunting season were calculated using: (1) objectives to maximize long-term
cumulative harvest utility (i.e., harvest conditioned on a population goal) and harvest of midcontinent and eastern mallards,
respectively; (2) all possible combinations of regulatory alternatives in the Atlantic Flyway and the remainder of the country;
and (3) four alternative population models and their updated weights for midcontinent mallards, and eight alternative models
of eastern mallards, equally weighted. Based on this year’s breeding survey results of 10.5 million midcontinent mallards,
2.4 million ponds in Prairie Canada, and 890 thousand eastern mallards, the optimal regulatory choice for all Flyways is the
liberal alternative (irrespective of whether the joint-optimization or separate-optimization alternative is applied).
A characteristic feature of AHM is the annual updating of model probabilities (“weights”) based on a comparison of observed
and predicted population sizes. This year, the weights associated with the midcontinent-mallard models reflect increased
support for the hypothesis of strongly density-dependent reproduction. Model weights continue to suggest that hunting
mortality is completely additive in midcontinent mallards. Weights associated with the models of eastern mallards will be
updated for the first time next year.
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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 U.S.
Fish and Wildlife Service (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. 1995a, Johnson et
al. 1996, Williams et al. 1996):
(1) environmental variation - 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 variables (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.
Adaptive harvest management (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 contain 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 harvest
strategies can be evaluated.
These components are used in an optimization procedure to derive a harvest strategy, which specifies the appropriate
regulatory choice 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
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current model weights;
(2) after the regulatory decision is made, model-specific 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 most appropriate to describe the dynamics 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 also differences in optimal
levels of sport harvest. The ability to regulate harvest on 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 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.
To date, AHM strategies have been based solely on the status of midcontinent mallards (Fig. 1). An optimal regulatory choice
for midcontinent mallards has been based on breeding population size and prairie water conditions, and on the weights
assigned to the alternative models of population dynamics. The same regulatory alternative has been applied in all four
Flyways, although season lengths and bag limits always have been Flyway-specific. Efforts are underway, however, to extend
the AHM process to account for mallards breeding westward and eastward of the midcontinent survey area. These mallard
stocks make significant contributions to the total mallard harvest, particularly in the Atlantic and Pacific Flyways (Munro and
Kimball 1982).
The optimization procedures currently employed in AHM can be extended to account for the population dynamics of eastern
and western mallards, and for the manner in which these ducks distribute themselves among the Flyways during the hunting
season. A globally optimal approach would allow for Flyway-specific regulatory strategies, which for each Flyway would
represent 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 involves:
(1) augmentation of the current decision criteria to include population and environmental variables relevant to eastern
and western mallards (as based on models of population dynamics);
(2) revision of the objective function to account for harvest-management goals for mallards breeding outside the
midcontinent region; 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 harvest strategies. These challenges cannot always be overcome due to limitations in monitoring and assessment programs,
and in access to sufficiently powerful computing resources. In these situations, however, it may be appropriate to impose
constraints or simplifying assumptions that reduce the dimensionality of the problem. Although sub-optimal by definition,
these constrained harvest strategies may perform nearly as well as those that are globally 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 due to poor ability to observe stock status,
6
Mallard stock:
midcontinent
eastern
western
Fig. 1. Survey areas currently assigned to the western, midcontinent, and eastern
stocks of mallards for the purpose of harvest management. Delineation of the western
stock is preliminary pending additional information from British Columbia and other
western areas with significant numbers of breeding mallards.
environmental variation, partial control of harvests, or limited understanding of stock dynamics.
MALLARD POPULATION DYNAMICS
Midcontinent Mallards
Midcontinent mallards are defined as those breeding in federal survey strata 1-18, 20-50, and 75-77, and in Minnesota,
Wisconsin, and Michigan. Estimates of the entire midcontinent population are available only since 1992. Since then, the
number of midcontinent mallards has grown by an average of 7.1 percent (SE = 1.2) per annum (Table 1).
The dynamics of midcontinent mallards are described by four alternative models, which result from combining two mortality
and two reproductive hypotheses. 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 habitat availability limits reproductive success). 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 leads to the most liberal strategy
(SCRS). The other two models (SARS and SCRW) lead to strategies that are intermediate between these extremes.
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Table 1. Estimatesa of midcontinent mallards breeding in the federal survey area (strata 1-18, 20-
50, and 75-77) and the states of Minnesota, Wisconsin, and Michigan.
Federal 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
a In thousands.
Two other sources of uncertainty in mallard harvest management are acknowledged. Uncertainty about future environmental
conditions is characterized by random variation in annual precipitation, which affects the number of ponds available during
May in Canada. There is also an accounting for partial controllability, in which the link between regulations and harvest rates
is imperfect due to uncontrollable factors (e.g., weather, timing of migration) that affect mallard harvest. A detailed
description of the population dynamics of midcontinent mallards and associated sources of uncertainty are provided by
Johnson et al. (1997) and in Appendix B.
A key component of the AHM process for midcontinent mallards is the annual updating of model weights (Appendix C).
These weights describe the relative ability of the alternative models to predict changes in population size, and they ultimately
influence the nature of the optimal harvest strategy. Model weights are based on a comparison of predicted and observed
population sizes, with the updating leading to higher weights for models that prove to be good predictors (i.e., models with
relatively small differences between predicted and observed population sizes) (Fig. 2). These comparisons account for
sampling error (i.e., partial observability) in population size and pond counts, as well as for partial observability and
controllability of harvest rates.
When the AHM process was initiated in 1995, the four alternative models of population dynamics were considered equally
likely, reflecting a high degree of uncertainty (or disagreement) about harvest and environmental impacts on mallard
abundance. This year, the updated weights reflect increased support for the hypothesis of strongly density-dependent
reproduction (Table 2). Model weights continue to suggest that hunting mortality is completely additive in midcontinent
mallards.
8
Year
1996 1997 1998 1999 2000
Population size (thousands)
8000
9000
10000
11000
12000
13000
14000
observed
ScRs
ScRw
SaRs
SaRw
Fig. 2. Estimates of observed mallard population size (line with open circles) compared with
predictions from four alternative models of population dynamics (ScRs = compensatory mortality and
strongly density-dependent reproduction; ScRw = compensatory mortality and weakly density-dependent
reproduction; SaRs = additive mortality and strongly density-dependent reproduction;
SaRw = additive mortality and weakly density-dependent reproduction). Vertical bars represent one
standard deviation on either side of the estimated population size.
Table 2. Temporal changes in probabilities ("weights") for alternative hypotheses of midcontinent
mallard population dynamics.
Model weights
Mortality
hypothesis
Reproductive
hypothesis 1995 1996 1997 1998 1999 2000
Additive Strong density
dependence
0.25000 0.65479 0.53015 0.61311 0.60883 0.92176
Additive Weak density
dependence
0.25000 0.34514 0.46872 0.38687 0.38416 0.07822
Compensatory Strong density
dependence
0.25000 0.00006 0.00112 0.00001 0.00001 0.00001
Compensatory Weak density
dependence
0.25000 0.00001 0.00001 0.00001 0.00700 0.00001
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Eastern Mallards
Eastern mallards are defined as those breeding in survey strata 51-54 and 56, and in New Hampshire, Vermont, Massachusetts,
Connecticut, Rhode Island, New York, Pennsylvania, New Jersey, Delaware, Maryland, and Virginia (Fig. 1). Midwinter
counts and the Breeding Bird Survey provide evidence of rapid growth in the eastern mallard population during the 1970s and
1980s. Since 1990, however, the mallard population in the fixed-wing (strata 51-54 and 56) and northeastern plot (New
Hampshire south through Virginia) surveys (Table 3) grew at an average rate of only 1.3 percent (SE = 0.8) per annum (Table
3).
Table 3. Estimatesa of mallards breeding in the northeastern U.S. (plot survey from New Hampshire
to Virginia) and eastern Canada (fixed-wing survey strata 51-54 and 56).
Plot survey Fixed-wing survey 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
a In thousands.
The population dynamics of eastern mallards were studied extensively by Sheaffer and Malecki (1996), but the USFWS has
not yet adopted a set of alternative models. A proposed model set for eastern mallards includes eight alternatives based on
key uncertainties in reproductive and survival processes. This model set captures uncertainty about the relationship between
fall age ratios (i.e., young/adult) and the Breeding Bird Survey (BBS) index, between the BBS index and actual population
size as measured by aerial and ground surveys, and between the BBS index and natural-mortality rates of females. Each of
the models is considered equally plausible given available data. In constructing this model set we chose to focus on the nature
of density-dependent population regulation because of its pivotal role in determining sustainable harvest strategies. There
continues to be a need for a more comprehensive examination of environmental variables (e.g., precipitation) that might
influence survival and reproductive rates irrespective of population size. Mathematical details of the alternative models for
eastern mallards are provided in Appendix B and in “Adaptive Harvest Management for Eastern Mallards: Progress Report -
January 13, 2000" (available on the Internet at www.migratorybirds.fws.gov/reports/reports.html).
The proposed model set suggests that in the absence of harvest the eastern mallard population would stop growing (i.e., reach
carrying capacity) somewhere between 1.23 and 3.49 million birds. All eight models suggest fairly liberal harvest strategies,
at least by historical standards. For a population size >1 million, seven of the eight models suggest an allowable harvest rate
in excess of that achieved under the most liberal regulatory alternative. All eight models suggest that hunting should be
curtailed when the breeding-population size is <400 thousand.
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Western Mallards
For purposes of this report, western mallards are defined as those breeding in the states of California, Oregon, and Washington.
This definition may be modified if monitoring and assessment information becomes available for other important breeding
areas of western mallards, such as British Columbia. A major effort to model the population dynamics of western mallards
was completed in 1999. Estimated natural mortality rates of western mallards were similar to those of midcontinent mallards.
As with midcontinent mallards, the relationship between harvest rates and annual survival rates was equivocal. Reproductive
rates appear to be related to the size of the breeding population and to the amount of winter precipitation in California, Oregon,
and Washington. A final set of population models, which describe how western mallards respond to harvest and uncontrolled
environmental factors, as well as key uncertainties associated with those relationships, may be proposed next year.
HARVEST MANAGEMENT OBJECTIVES
Midcontinent Mallards
The basic harvest management objective for midcontinent mallards is to maximize cumulative harvest over the long term,
which inherently requires conservation of a viable mallard population. Moreover, this objective is constrained to avoid
regulatory decisions that could be expected to result in a subsequent population size below the goal of the North American
Waterfowl Management Plan (NAWMP) (Fig. 3). According to this constraint, the value of harvest opportunity decreases
proportionally as the difference between the goal and expected population size increases. This balance of harvest and
population objectives results in a harvest 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 8.7 million mallards, which is based on the NAWMP goal of 8.1 million for the federal survey area
and a goal 0.6 million for the combined states of Minnesota, Wisconsin, and Michigan.
0 1 2 3 4 5 6 7 8 9 10
Harvest value (%)
100
80
60
40
20
0
Expected population size next year (in millions)
population
goal = 8.7
Fig. 3. The relative value of midcontinent mallard harvest, expressed as
a function of breeding-population size expected in the subsequent year.
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Eastern Mallards
The preliminary management objective for eastern mallards is to maximize long-term cumulative harvest. This objective is
subject to change once the implications for average population size, variability in annual regulations, and other performance
characteristics are better understood.
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 1979-84, 1985-87, and 1988-93, respectively (Appendix F, Table F-1). 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. The basic
structure of the regulatory alternatives has remain unchanged since 1997, although in 1998 the U.S. Congress intervened to
allow the option of extended framework dates and shorter seasons in some southern Mississippi Flyway States (Table 4).
Predictions of Mallard Harvest Rates
Since 1995, harvest rates of adult male mallards associated with the AHM regulatory alternatives have been predicted using
harvest-rate estimates from 1979-84, which have been adjusted to reflect current specification of season lengths and bag limits,
and for contemporary numbers of hunters. The prediction of mallard harvest rates is complicated this year by the possibility
that modification of the AHM protocol to account for eastern mallards could allow for a regulatory alternative in the Atlantic
Flyway that is different from the other Flyways. Therefore, it was necessary to predict harvest rates for midcontinent and
eastern mallards for the 25 combinations of regulatory alternatives (including the option of closed seasons) in the Atlantic
Flyway and the remainder of the country (Tables 5 and 6). As usual, 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 D). 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. As a conservative
measure, we assumed that when hunting seasons are closed in the U.S., then rates of harvest in Canada would be similar to
those observed during 1988-93, which is the most recent period for which reliable estimates are available. Fortunately,
optimal harvest 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.
Adult female mallards tend to be less vulnerable to harvest than adult males, while young are more vulnerable (Table 7).
Estimates of the relative vulnerability of adult females and young in the eastern mallard population tend to be higher and more
variable than in the midcontinent population.
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Table 4. Regulatory alternatives considered for the 2000 duck-hunting season.
Flyway
Regulation Atlantica Mississippib Centralc Pacificd
Shooting hours one-half hour before sunrise to sunset for all Flyways
Framework dates Oct 1 - Jan 20 Saturday closest to October 1 and Sunday closest to January
20
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 In the states of Alabama, Mississippi, and Tennessee, in the moderate and liberal alternatives, there is
an option for a framework closing date of January 31 and a season length of 38 days and 51 days,
respectively.
c The High Plains Mallard Management Unit is allowed 8, 12, 23, and 23 extra days under the very
restrictive, restrictive, moderate, and liberal alternatives, respectively.
d The Columbia Basin Mallard Management Unit is allowed seven extra days under the very restrictive,
restrictive, and moderate alternatives.
13
Table 5. Predicted harvest rates of adult male midcontinent mallards under the current regulatory
alternatives, and allowing for a regulatory choice in the Atlantic Flyway that could differ from the
remaining Flyways.
Regulatory
alternative in the
three western
Flyways
Regulatory
alternative in the
Atlantic Flyway Harvest rate SE
Closed Closed 0.0088 0.0030
Closed Very restrictive 0.0193 0.0052
Closed Restrictive 0.0197 0.0052
Closed Moderate 0.0203 0.0052
Closed Liberal 0.0207 0.0053
Very restrictive Closed 0.0521 0.0103
Very restrictive Very restrictive 0.0526 0.0106
Very restrictive Restrictive 0.0530 0.0107
Very restrictive Moderate 0.0536 0.0108
Very restrictive Liberal 0.0540 0.0109
Restrictive Closed 0.0658 0.0136
Restrictive Very restrictive 0.0662 0.0142
Restrictive Restrictive 0.0665 0.0142
Restrictive Moderate 0.0672 0.0143
Restrictive Liberal 0.0676 0.0144
Moderate Closed 0.1094 0.0250
Moderate Very restrictive 0.1104 0.0264
Moderate Restrictive 0.1108 0.0264
Moderate Moderate 0.1114 0.0266
Moderate Liberal 0.1118 0.0266
Liberal Closed 0.1282 0.0304
Liberal Very restrictive 0.1291 0.0320
Liberal Restrictive 0.1295 0.0321
Liberal Moderate 0.1301 0.0322
Liberal Liberal 0.1305 0.0323
14
Table 6. Predicted harvest rates of adult male eastern mallards under the current regulatory
alternatives, and allowing for a regulatory choice in the Atlantic Flyway that could differ from the
remaining Flyways.
Regulatory
alternative in the
Atlantic Flyway
Regulatory
alternative in the
three western
Flyways Harvest rate SE
Closed Closed 0.0248 0.0050
Closed Very restrictive 0.0927 0.0142
Closed Restrictive 0.0959 0.0138
Closed Moderate 0.1062 0.0131
Closed Liberal 0.1110 0.0130
Very restrictive Closed 0.1130 0.0211
Very restrictive Very restrictive 0.1212 0.0205
Very restrictive Restrictive 0.1245 0.0203
Very restrictive Moderate 0.1348 0.0201
Very restrictive Liberal 0.1395 0.0202
Restrictive Closed 0.1237 0.0025
Restrictive Very restrictive 0.1320 0.0220
Restrictive Restrictive 0.1352 0.0219
Restrictive Moderate 0.1455 0.0218
Restrictive Liberal 0.1502 0.0219
Moderate Closed 0.1407 0.0257
Moderate Very restrictive 0.1473 0.0252
Moderate Restrictive 0.1522 0.0253
Moderate Moderate 0.1625 0.0254
Moderate Liberal 0.1672 0.0255
Liberal Closed 0.1506 0.0282
Liberal Very restrictive 0.1588 0.0279
Liberal Restrictive 0.1621 0.0279
Liberal Moderate 0.1724 0.0280
Liberal Liberal 0.1771 0.0282
15
Table 7. Mean harvest vulnerability (SE) of adult female and young mallards, relative to adult
males, based on band-recovery data, 1979-95.
Age and sex
Mallard
population
Adult females Young females Young males
Midcontinent 0.748 (0.108) 1.188 (0.138) 1.361 (0.144)
Eastern 0.985 (0.145) 1.320 (0.264) 1.449 (0.211)
OPTIMAL HARVEST STRATEGIES
Joint Optimization of Midcontinent and Eastern Mallards
We derived an optimal regulatory strategy for the Flyways based on a joint optimization of the midcontinent and eastern
mallards. We specified the following conditions to derive this strategy:
(1) an objective function that maximizes the long-term cumulative sum of eastern-mallard harvest and midcontinent-mallard
harvest utility (where harvest utility is a function of both harvest and population size; see Fig.3);
(2) all possible combinations of current regulatory alternatives in the Atlantic Flyway and the remainder of the country
(Tables 5 and 6), and a simplifying assumption of perfect controllability (i.e., deterministic harvest rates); and
(3) current population models and associated weights for midcontinent mallards, and the eight alternative models of
eastern mallards, equally weighted.
The optimal regulatory choice for the Atlantic Flyway rarely diverges from the liberal alternative, even when the status of
midcontinent mallards is poor (Table 8). The status of eastern mallards has more effect on the optimal regulatory choice in
the remainder of the country, but the effect is minimal and observed only when midcontinent mallards fall below the
population goal. These results are consistent with the high degree of spatial discrimination between the two populations during
the hunting season.
Table 8. A Flyway-specific regulatory strategy, based on a joint optimization of midcontinent and
eastern mallards. The objective function, models of population dynamics, and harvest rates
associated with the regulatory alternatives are described elsewhere in this report.
Midcontinent
mallard
population (millions)
Ponds in
Prairie
Canada
(millions)
Eastern mallard
population (millions)
Regulation in
the three
western
Flyways
Regulation
in the
Atlantic
Flyway
3 1-5 0.5-1.5 L
3 6 0.5 M
3 6 0.6-1.5 L
3 7 0.5 M
3 7 0.6-1.5 L
4 1-6 0.5-1.5 L
16
Midcontinent
mallard
population (millions)
Ponds in
Prairie
Canada
(millions)
Eastern mallard
population (millions)
Regulation in
the three
western
Flyways
Regulation
in the
Atlantic
Flyway
4 7 0.5 M
4 7 0.6-1.5 L
5 1-5 0.5-1.5 L
5 6 0.5 L
5 6 0.6-1.5 VR L
5 7 0.5-0.6 VR L
5 7 0.7-1.5 R L
6 1-2 0.5-1.5 L
6 3 0.5-1.5 VR L
6 4 0.5-0.6 VR L
6 4 0.7-1.5 R L
6 5 0.5-0.9 R L
6 5 1.0-1.5 R L
6 6 0.5 M M
6 6 0.6-1.5 M L
6 7 0.5 M M
6 7 0.6-1.5 M L
7 1 0.5-0.7 VR L
7 1 0.8-1.5 R L
7 2 0.5-1.5 R L
7 3 0.5-0.7 R L
7 3 0.8-1.5 M L
7 4 0.5 M M
7 4 0.6-1.5 M L
7 5 0.5 L M
7 5 0.6-1.5 L L
7 6-7 0.5-1.5 L L
8 1 0.5-1.5 M L
8 2 0.5-1.0 M L
8 2 1.1-1.3 L L
8 2 1.4-1.5 M L
17
Midcontinent
mallard
population (millions)
Ponds in
Prairie
Canada
(millions)
Eastern mallard
population (millions)
Regulation in
the three
western
Flyways
Regulation
in the
Atlantic
Flyway
8 3 0.5 M L
8 3 0.6-1.5 L L
8 4-7 0.5-1.5 L L
9 1 0.5 L M
9 1 0.6-1.5 L L
9 2 0.5 L M
9 2 0.6-1.5 L L
9 3-7 0.5-1.5 L L
10 1 0.5 L M
10 1 0.6-1.5 L L
10 2-7 0.5-1.5 L L
11-12 1-7 0.5-1.5 L L
Blank cells in Table 8 (and in other harvest strategies in this report) represent combinations of population size and
environmental conditions that are insufficient to support an open season, given current regulatory alternatives. In the case
of midcontinent mallards, the prescriptions for closed seasons largely are a result of the harvest-management objective, which
emphasizes population growth at the expense of hunting opportunity when mallard numbers are below the NAWMP goal.
However, limited harvests at low population levels would not be expected to impact long-term population viability. Therefore,
the decision to actually close the hunting season would depend on both biological and sociological considerations.
Based on the harvest strategy in Table 8, the benefits of a joint optimization of midcontinent and eastern mallards appear to
be negligible (at least in terms of currently specified objectives) because regulations within each harvest area are affected
principally by a single stock of mallards. However, the computational costs associated with the joint optimization of
midcontinent and eastern mallards is considerable. We experienced severe limitations in our ability to fully explore the
implications of all sources of uncertainty (e.g., partial control of harvests), for all possible system states, even when using
state-of-the-art Pentium workstations.
Stock-specific Optimization
An alternative, constrained approach is to conduct two separate optimizations, in which the Atlantic Flyway regulation is based
exclusively on the status of eastern mallards, and the regulatory choice for the remainder of the country is based exclusively
on the status of midcontinent mallards. From the perspectives of hunting opportunity and resource conservation, there is little
apparent difference between the joint optimization and this constrained approach. Moreover, the advantage of the stock-specific
optimization is that it would make more computing resources available for use in joint optimizations of mallards and
other species important to recreational hunting (e.g., wood ducks, black ducks).
We used the the separate-optimization approach to optimize regulatory choices for the Atlantic Flyway and the remainder of
the country based on the status of eastern and midcontinent mallards, respectively. Subject to this constraint, the optimal
regulatory strategy for the western three Flyways was derived using: (1) current regulatory alternatives; (2) the four alternative
models and associated weights for midcontinent mallards; and (3) the dual objectives to maximize long-term cumulative
harvest and achieve a population goal of 8.7 million midcontinent mallards. We assumed that the regulatory choice in the
Atlantic Flyway would have no discernable effect on the overall harvest rate of midcontinent mallards or, if an effect existed,
that it was accounted for by the range of variation associated with harvest rates when regulatory choices are not Flyway-
18
specific. The resulting harvest strategy (Table 9) is substantially more liberal than that for midcontinent mallards in 1999,
due to the increase in probability associated with the hypothesis of strongly density-dependent reproduction. The optimal
harvest strategies based on midcontinent mallards for the 1995-99 seasons are provided in Appendix F (Tables F-2 to F-6)
so that the reader can assess how the harvest strategy has “evolved” over time.
Table 9. Optimal regulatory choicesa in the three western Flyways during the 2000 hunting season.
This strategy is based on current regulatory alternatives, on current midcontinent-mallard models
and weights, and on the dual objectives of maximizing long-term cumulative harvest and achieving
a population goal of 8.7 million midcontinent mallards.
Pondsb
Mallardsc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
<4.5
4.5 VR
5.0 VR VR VR R R
5.5 VR VR VR VR VR R R R M M
6.0 VR R R R R M M M L L
6.5 R R M M M M L L L L
7.0 M M M M L L L L L L
7.5 M M L L L L L L L L
8.0 L L L L L L L L L L
>8.0 L L L L L L L L L L
a 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 simulated the use of the harvest strategy in Table 9 with the four population models and current weights to determine
expected performance characteristics. Assuming that regulatory choices adhered to this strategy, the annual harvest and
breeding population size would average 1.29 (SE = 0.42) million and 8.05 (SE = 1.00) million, respectively.
Based on a midcontinent population size of 10.5 million mallards and 2.4 million ponds in Prairie Canada, the optimal
regulatory choice for the Pacific, Central, and Mississippi Flyways in 2000 is the liberal alternative.
We optimized the regulatory choice for the Atlantic Flyway based on: (1) current regulatory alternatives; (2) the eight
alternative models of population dynamics, equally weighted; and (3) an objective to maximize long-term cumulative harvest.
Unlike the situation with midcontinent mallards, however, the regulatory choice in the three western Flyways has a discernable
effect on the harvest rate of eastern mallards (see Table 6). Therefore, the optimal regulatory choice for the Atlantic Flyway
depends on the regulatory choice in the other Flyways. To avoid making the regulatory choice in the Atlantic Flyway
conditional on regulations elsewhere, we estimated the expected harvest rates of eastern mallards when managers lack a priori
knowledge of the chosen regulation in the western three Flyways. We did this by taking a weighted average of the estimated
harvest rates associated with each of the possible regulatory alternatives in the western Flyways, for each possible regulatory
alternative in the Atlantic Flyway (see Table 6). The weights were derived using simulations of the midcontinent-mallard
strategy described above to determine the expected frequency of regulatory choices in the western Flyways. We estimated
the variances associated with each regulatory alternative in the Atlantic Flyway using Monte Carlo simulations, based on the
variances in Table 6 and their associated weights. The resulting regulatory strategy (Table 10) is very liberal (at least by
historical standards), and is characterized by a lack of intermediate regulations. The strategy exhibits this behavior largely
because of the small differences in harvest rate among regulatory alternatives.
19
Table 10. Optimal regulatory choicesa for the Atlantic Flyway during the 2000 hunting season. This
strategy is based on current regulatory alternatives, on eight alternative models of eastern mallards
(equally weighted), and on an objective to maximize long-term cumulative harvest.
Mallardsb Regulation
<500
500 R
550 L
>550 L
a VR = very restrictive, R = restrictive, M = moderate, and L = liberal.
b Estimated number of eastern mallards in the combined fixed-wing and northeastern plot surveys, in
thousands.
We simulated the use of the harvest strategy in Table 10 to determine expected performance characteristics. Assuming that
harvest management adhered to this strategy, the annual harvest and breeding population size would average 387 (SE = 99)
thousand and 1.06 (SE = 0.2) million, respectively.
Based on a breeding population size of 890 thousand mallards, the optimal regulatory choice for the Atlantic Flyway in 2000
is the liberal alternative.
CURRENT AHM PRIORITIES
Midcontinent Mallards
The current AHM specifications for midcontinent mallards have been in place since 1995. Therefore, the AHM technical
working group is reviewing all aspects of these specifications to determine if revisions are warranted. This review is focusing
principally on the set of models describing population dynamics, and on the method by which the weights associated with
alternative models are updated. Unfortunately, efforts to develop mechanistic models of density-dependent survival have been
stymied by programming problems and a lack of demographic and environmental data at the appropriate scales. Patuxent
Wildlife Research Center has recently acquired additional staff to help address the problems. On a more promising note, it
appears that some of the variability in reproductive success can be explained by the distribution of ponds in the Prairie Pothole
Region. The AHM working group is exploring the implications of this spatial effect, as well as alternative forms of the
relationships among reproductive success and important environmental factors. With respect to the updating of model
weights, there is an agreed-upon need to account for all sources of variation in the updating procedure, whether or not the
variation is explained by the models. This will be more straightforward once the model set for mallards has been revised.
The inclusion of additional variance components in the updating procedure likely will slow the movement of model weights,
and perhaps be more reflective of actual rates of learning.
The ability to accurately determine model performance also is influenced by our ability to estimate actual harvest rates. Since
1994 there has been a systematic effort to increase the rate at which hunters report band recoveries, and this effort has made
it temporarily difficult to estimate the harvest rate of mallards. A large-scale study to evaluate recent changes in band-reporting
rate is in the planning stage, with implementation to occur in 2001 or 2002. As part of that planning effort, a pilot
study was conducted in southern Saskatchewan in 1998 and again in 1999 to obtain a preliminary estimate of band-reporting
rates for adult male mallards. Results of preliminary analyses suggested a constant reporting rate of 0.84 (SE = 0.05) for the
1998 and 1999 hunting seasons. The pilot study will continue for the 2000 hunting season.
Eastern Mallards
There are many possible approaches to modifying the AHM protocol to account for eastern mallards, but we have identified
20
two basic alternatives in this report. The first involves a single, joint optimization for midcontinent and eastern mallards. This
approach would result in optimal regulatory choices for the Atlantic Flyway and for the remainder of the country, for each
possible combination of midcontinent population size, pond numbers in Canada, and eastern mallard population size. The
characteristic feature of this approach is that all regulatory choices, regardless of Flyway, would depend on the status of both
midcontinent and eastern mallards (with the degree of dependence based on each harvest area’s unique combination of the
two mallard populations). This joint-optimization approach is globally optimal, in the sense that it would be expected to
outperform all other alternatives in terms of harvest-management objectives for midcontinent and eastern mallards. The
second alternative would entail two separate optimizations, in which the Atlantic Flyway regulation would be based
exclusively on the status of eastern mallards, and the regulatory choice for the remainder of the country would be based
exclusively on the status of midcontinent mallards. This approach is sub-optimal by definition because neither the Atlantic
Flyway nor the three western Flyways (as a unit) derive their harvest exclusively from one mallard population. However, the
joint-optimization alternative appears to provide little advantage over the separate-optimization alternative in terms of meeting
harvest-management objectives for mallards. This result follows from the high degree of spatial discrimination between the
two mallard populations during the hunting season.
The USFWS intends to make appropriate modifications to the existing AHM protocol to account for eastern mallards.
However, the USFWS seeks a discussion and review of the relevant issues among the Flyway Councils and States prior to
any changes. The USFWS will propose modifications to the current AHM protocol when late-season regulatory frameworks
are proposed in August 2000.
Western Mallards
The Study Committee of the Pacific Flyway Council has assumed responsibility for recommending a set of alternative models
for western mallards. These models will be based on the assessment prepared by the New York Cooperative Fish and Wildlife
Research Unit, but also must satisfactorily address several modeling issues raised by the AHM technical working group (see
the latest report from the working group on the Internet at www.migratorybirds.fws.gov/reports/reports.html). The USFWS
will assume responsibility for proposing modifications to AHM protocols once a final model set is agreed upon.
Pintails
The Study Committee of the Pacific Flyway Council and Patuxent Wildlife Research Center are exploring alternative
population models for pintails, based on a recent assessment by the New York Cooperative Fish and Wildlife Research Unit.
As with western mallards, however, a number of critical modeling issues must be addressed before AHM can be implemented
for pintails. Also, additional analyses will be required to model the relationship between hunting regulations and pintail
harvests. While these analyses are being conducted, the management community should begin discussion about whether to:
(a) optimize the selection of a regulatory alternative based on the joint status of pintails and mallards; (b) set hunting
regulations for pintails independent of mallards; or (c) set pintail bag limits (or other regulatory tools) conditioned on the
choice of regulatory alternatives prescribed for mallards.
Information and Education Needs
There is growing concern that the technical complexity of AHM is preventing some waterfowl managers from fully
participating in the development of AHM protocols. The AHM technical working group will take the following actions to
help address this concern: (a) a “refresher” workshop on AHM will be held in December 2000; the workshop will be
conducted principally for members of the AHM working group, although other technical personnel may be invited; and (b)
the AHM working group will develop 1-day and 2-hour team-taught courses, which could be offered on short notice
throughout the country; course work and applications also may be presented on the Internet.
AHM and Considerations of Hunter Preferences
In spite of significant progress in defining harvest-management objectives, there continue to be unresolved disagreements
among stakeholders about how to value harvest benefits and how those benefits should be shared. Resolution of these
disagreements might be facilitated by a better understanding of how regulations affect hunter satisfaction. Most Flyway
Council members have expressed an interested in coordinated, nationwide hunter surveys to monitor the opinions of waterfowl
21
hunters. Despite agreement on the need for better data, however, the specific use and application of these data in the AHM
process remain unclear. Therefore, the AHM working group has suggested that managers engage in a dialogue to better define
“hunter satisfaction,” and how it might be affected by variation in hunting regulations. To facilitate this dialogue, the AHM
working will: (a) investigate regional differences in harvest and other metrics of hunter activity and success; and (b) consider
facilitated focus groups among technicians and administrators to explore expectations related to the effect of regulations on
hunter satisfaction.
Revision of the Set of Regulatory Alternatives
There currently are no guidelines describing when changes to the set of regulatory alternatives might be considered. Therefore,
the AHM working group is considering the development of a schedule for making periodic revisions, as well as criteria
governing the nature of any changes. The AHM working group currently is soliciting input from the Flyway Councils, and
will address the issue during their meeting in April 2001.
LITERATURE CITED
Anderson, D. R., and C. J. Henny. 1972. Population ecology of the mallard. I. A review of previous studies and the
distribution and migration from breeding areas. U. S. Fish and Wildl. Serv. Resour. Pub. 105. 166pp.
Blohm, R. J. 1989. Introduction to harvest - understanding surveys and season setting. Proc. Inter. Waterfowl Symp. 6:118-
133.
Burnham, K. P., G. C. White, and D. R. Anderson. 1984. Estimating the effect of hunting on annual survival rates of adult
mallards. J. Wildl. Manage. 48:350-361 .
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. J. Wildl. Manage. 61:202-216.
_____, 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. Trans. North Am. Wildl. Nat. Resour.
Conf. 58:565-583.
_____, _____, and P. R. Schmidt. 1996. Adaptive decision-making in waterfowl harvest and habitat management. Proc.
Inter. Waterfowl Symp. 7:26-33.
Munro, R. E., and C. F. Kimball. 1982. Population ecology of the mallard. VII. Distribution and derivation of the harvest.
U.S. Fish and Wildl. Serv. Resour. Pub. 147. 127pp.
Nichols, J. D., F. A. Johnson, and B. K. Williams. 1995a. Managing North American waterfowl in the face of uncertainty.
Ann. Rev. Ecol. Syst. 26:177-199.
Sheaffer, S. E., and R. A. Malecki. 1996. Quantitative models for adaptive harvest management of mallards in eastern North
America. New York Coop. Fish and Wildl. Res. Unit, Cornell Univ., Ithaca, unpubl. rep. 116pp.
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. Wildl. Soc. Bull.
23:430-436.
_____, _____, and K. Wilkins. 1996. Uncertainty and the adaptive management of waterfowl harvests. J. Wildl. Manage.
60:223-232.
22
APPENDIX A: AHM Working Group
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
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
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: 219-258-0100
fax: 219-258-0189
e-mail: djcase@csi.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: 303-236-8676
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: 301-497-5870
fax: 301-497-5706
e-mail: james_dubovsky@fws.gov
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
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
23
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
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
11510 American Holly Drive
Laurel, MD 20708-4017
phone: 301-497-5861
fax: 301-497-5706
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
Division of Migratory Bird Management
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
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
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
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
24
Michael C. Runge
Field of Natural Resources
Cornell University
Ithaca, NY 14850
phone: 607-273-8127
e-mail: mcr5@cornell.edu
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: 303-275-2385
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
Bryan Swift
Dept.of Environmental Conservation
108 Game Farm Road, Building 9
Delmar, NY 12054-9767
phone: 518-78-3022
fax: 518-478-3004
e-mail: bl.swift@gw.dec.state.ny.us
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
Dan Yparraguirre
Dept. of Fish & Game
1416 Ninth Street
Sacramento, CA 94244
phone: 916-653-8709
fax: 916-653-1019
e-mail: dyparrag@hq.dfg.ca.gov
25
APPENDIX B: Mallard Population Models
State and random variable definitions:
MBPOP /* state variable index - midcontinent population */
PONDS /* state variable index - Canadian ponds */
EBPOP /* state variable index - eastern population */
RRES /* random variable index - recruitment residuals for eastern population */
SSFVAR /* random variable index - female summer survival - eastern population */
SSFRES /* random variable index - residuals for female summer survival - eastern population */
PPT /* random variable index - Canadian Prairie precipitation */
PROP /* random variable index - proportion of midcontinent population in Lake States */
MRATE /* random variable index - harvest rate - midcontinent adult males */
ERATE /* random variable index - harvest rate - eastern adult males */
outcome[] /* outcome of random variable */
cur_state[] /* current value of state variable */
nxt_state[] /* value of state variable in next time step */
Eastern mallard parameters:
wtr1b1s1 /* weight for model r1b1s1 - neg. exp. reproduction, log. BBS, constant survival */
wtr1b1s2 /* weight for model r1b1s2 - neg. exp. reprod., log. BBS, density-dependent survival */
wtr1b2s1 /* weight for model r1b2s1 - neg. exp. reprod., exp. BBS, constant survival */
wtr1b2s2 /* weight for model r1b2s2 - neg. exp. reprod., exp. BBS, density-dependent survival */
wtr2b1s1 /* weight for model r2b1s1 - logistic reprod., log. BBS, constant survival */
wtr2b1s2 /* weight for model r2b1s2 - logistic reprod., log. BBS, density-dependent survival */
wtr2b2s1 /* weight for model r2b2s1 - logistic reprod., exp. BBS, constant survival */
wtr2b2s2 /* weight for model r2b2s2 - logistic reprod., exp. BBS, density-dependent survival */
nxr1b1s1 /* model-specific prediction of population size in next time step */
nxr1b1s2 /* model-specific prediction of population size in next time step */
nxr1b2s1 /* model-specific prediction of population size in next time step */
nxr1b2s2 /* model-specific prediction of population size in next time step */
nxr2b1s1 /* model-specific prediction of population size in next time step */
nxr2b1s2 /* model-specific prediction of population size in next time step */
nxr2b2s1 /* model-specific prediction of population size in next time step */
nxr2b2s2 /* model-specific prediction of population size in next time step */
Ne /* breeding population size */
bbsb1 /* BBS index - logarithmic model */
bbsb2 /* BBS index - exponential[max] model */
ar1b1 /* male age ratio - neg. exp. reproduction - log. BBS */
ar1b2 /* male age ratio - neg. exp. reproduction - exponential[max] BBS */
ar2b1 /* male age ratio - logistic reproduction - log. BBS */
ar2b2 /* male age ratio - logistic reproduction - exponential[max] BBS */
hafe /* harvest rate - adult females */
hame /* harvest rate - adult males */
hyfe /* harvest rate - young females */
hyme /* harvest rate - young males */
26
kafe /* kill rate - adult females */
kame /* kill rate - adult males */
kyfe /* kill rate - young females */
kyme /* kill rate - young males */
dafe=1.19 /* differential vulnerability - adult females */
dyme=1.47 /* differential vulnerability - young males */
dyfe=1.62 /* differential vulnerability - yong females */
ce=0.2 /* crippling loss rate */
swe=0.90 /* winter survival */
ssme=0.81 /* summer survival - males */
ssfvar /* summer survival - females - random variation */
ssfmb1 /* summer survival - females - density dependent survival - log. BBS */
ssfmb2 /* summer survival - females - density dependent survival - exponential[max] BBS */
sexe=0.55 /* proportion males in May */
Eastern mallard dynamics:
hame = outcome[ERATE];
hafe = min(1.0, hame*dafe);
hyme = min(1.0, hame*dyme);
hyfe = min(1.0, hame*dyfe);
kafe = min(1.0, hafe/(1-ce) );
kame = min(1.0, hame/(1-ce) );
kyfe = min(1.0, hyfe/(1-ce) );
kyme = min(1.0, hyme/(1-ce) );
bbsb1 = 0.6185*exp(1.3534*cur_state[EBPOP]/1000000);
bbsb2 = 11292.7921*(1-exp(-0.0002*cur_state[EBPOP]/1000000));
ar1b1 = max(0.0, (1.7330*exp(-0.2036*bbsb1))+outcome[RRES]);
ar1b2 = max(0.0, (1.7330*exp(-0.2036*bbsb2))+outcome[RRES]);
ar2b1 = max(0.0, (1.5027/(1+exp(-(bbsb1-2.8608)/-0.6490)))+outcome[RRES]);
ar2b2 = max(0.0, (1.5027/(1+exp(-(bbsb2-2.8608)/-0.6490)))+outcome[RRES]);
ssfvar = outcome[SSFVAR];
ssfmb1 =
exp(1.6746-0.5422*bbsb1+outcome[SSFRES])/(1+exp(1.6746-0.5422*bbsb1+outcome[SSFRES]));
ssfmb2 =
exp(1.6746-0.5422*bbsb2+outcome[SSFRES])/(1+exp(1.6746-0.5422*bbsb2+outcome[SSFRES]));
Ne = cur_state[EBPOP];
nxr1b1s1=max(0.0, Ne*((1-sexe)*ssfvar*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar1b1*(1-kyfe)*swe + (sexe)*ssme*ar1b1*(1-kyme)*swe));
nxr1b1s2=max(0.0, Ne*((1-sexe)*ssfmb1*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar1b1*(1-kyfe)*swe + (sexe)*ssme*ar1b1*(1-kyme)*swe));
nxr1b2s1=max(0.0, Ne*((1-sexe)*ssfvar*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar1b2*(1-kyfe)*swe + (sexe)*ssme*ar1b2*(1-kyme)*swe));
nxr1b2s2=max(0.0, Ne*((1-sexe)*ssfmb2*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar1b2*(1-kyfe)*swe + (sexe)*ssme*ar1b2*(1-kyme)*swe));
nxr2b1s1=max(0.0, Ne*((1-sexe)*ssfvar*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar2b1*(1-kyfe)*swe + (sexe)*ssme*ar2b1*(1-kyme)*swe));
nxr2b1s2=max(0.0, Ne*((1-sexe)*ssfmb1*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
27
(sexe)*ssme*ar2b1*(1-kyfe)*swe + (sexe)*ssme*ar2b1*(1-kyme)*swe));
nxr2b2s1=max(0.0, Ne*((1-sexe)*ssfvar*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar2b2*(1-kyfe)*swe + (sexe)*ssme*ar2b2*(1-kyme)*swe));
nxr2b2s2=max(0.0, Ne*((1-sexe)*ssfmb2*(1-kafe)*swe + sexe*ssme*(1-kame)*swe +
(sexe)*ssme*ar2b2*(1-kyfe)*swe + (sexe)*ssme*ar2b2*(1-kyme)*swe));
nxt_state[EBPOP] =
max(0.0, nxr1b1s1*wtr1b1s1+nxr1b1s2*wtr1b1s2+nxr1b2s1*wtr1b2s1+nxr1b2s2*wtr1b2s2+
nxr2b1s1*wtr2b1s1+nxr2b1s2*wtr2b1s2+nxr2b2s1*wtr2b2s1+nxr2b2s2*wtr2b2s2);
Midcontinent mallard parameters:
wt1 /* model 1 weight - compensatory mortality, strong density-dependent reproduction
(ScRs) */
wt2 /* model 2 weight - compensatory mortality, weak density-dependent reprod. (ScRw) */
wt3 /* model 3 weight - additive mortality, strong density-dependent reprod. (SaRs) */
wt4 /* model 4 weight - additive mortality - weak density-dependent reprod. (SaRw) */
nxt1 /* model-specific prediction of population size in next time step */
nxt2 /* model-specific prediction of population size in next time step */
nxt3 /* model-specific prediction of population size in next time step */
nxt4 /* model-specific prediction of population size in next time step */
Nm /* breeding population size */
P /* May ponds in Prairie Canada */
Ai /* fall age ratio - females - weakly density-dependent reproduction */
Ad /* fall age ratio - females - strongly density-dependent reproduction */
Hafm /* harvest rate - adult females */
Hamm /* harvest rate - adult males */
Hyfm /* harvest rate - young females */
Hymm /* harvest rate - young males */
Kafm /* kill rate - adult females */
Kamm /* kill rate - adult males */
Kyfm /* kill rate - young females */
Kymm /* kill rate - young males */
shafim /* hunt season survival - adult females - additive mortality */
shafdm /* hunt season survival - adult females - compensatory mortality */
shamim /* hunt season survival - adult males - additive mortality */
shamdm /* hunt season survival - adult males - compensatory mortality */
shyfim /* hunt season survival - young females - additive mortality */
shyfdm /* hunt season survival - young females - compensatory mortality */
shymim /* hunt season survival - young males - additive mortality */
shymdm /* hunt season survival - young males - compensatory mortality */
dafm=0.748 /* differential vulnerability - adult females */
dymm=1.361 /* differential vulnerability - young males */
dyfm=1.188 /* differential vulnerability - young females */
cm=0.2 /* crippling loss rate */
s0m=0.81 /* survival in absence of hunting - male */
s0f=0.64 /* survival in absence of hunting - female */
swm=0.90 /* winter survival */
ssmm=0.90 /* summer survival - males */
28
ssfm=0.71 /* summer survival - females */
ctf=0.36 /* compensatory threshold - females */
ctm=0.19 /* compensatory threshold - males */
sexm=0.55 /* proportion of males in May */
Midcontinent mallard dynamics:
Nm = cur_state[MBPOP];
P = cur_state[PONDS];
Hamm = outcome[MRATE];
Hafm = min(1.0, Hamm*dafm);
Hymm = min(1.0, Hamm*dymm);
Hyfm = min(1.0, Hamm*dyfm);
Kafm = min(1.0, Hafm/(1-cm) );
Kamm = min(1.0, Hamm/(1-cm) );
Kyfm = min(1.0, Hyfm/(1-cm) );
Kymm = min(1.0, Hymm/(1-cm) );
Ai = max(0.0, 0.8249-(0.0547*((1-outcome[PROP])*Nm/1000000.0))+(0.1130*(P/1000000.0)));
Ad = max(0.0, 1.1081-(0.1128*((1-outcome[PROP])*Nm/1000000.0))+(0.1460*(P/1000000.0)));
shafim=(1-Kafm); shamim=(1-Kamm); shyfim=(1-Kyfm); shymim=(1-Kymm);
if (Kafm>ctf) shafdm=(1-Kafm)/s0f; else shafdm=1.0;
if (Kamm>ctm) shamdm=(1-Kamm)/s0m; else shamdm=1.0;
if (Kyfm>ctf) shyfdm=(1-Kyfm)/s0f; else shyfdm=1.0;
if (Kymm>ctm) shymdm=(1-Kymm)/s0m; else shymdm=1.0;
nxt_state[PONDS] = max(1.0, -3835087.53+0.45*P+13695.47*outcome[PPT]) ;
nxt1=Nm*((1.-sexm)*ssfm*(shafdm+Ad*(shyfdm+shymdm)) + sexm*ssmm*shamdm)*swm;
nxt2=Nm*((1.-sexm)*ssfm*(shafdm+Ai*(shyfdm+shymdm)) + sexm*ssmm*shamdm)*swm;
nxt3=Nm*((1.-sexm)*ssfm*(shafim+Ad*(shyfim+shymim)) + sexm*ssmm*shamim)*swm;
nxt4=Nm*((1.-sexm)*ssfm*(shafim+Ai*(shyfim+shymim)) + sexm*ssmm*shamim)*swm;
nxt_state[MBPOP] = max(0.0, wt1*nxt1+wt2*nxt2+wt3*nxt3+wt4*nxt4);
29
(2)
(3)
(4)
APPENDIX C: Updating of Model Weights
Adaptive harvest management prescribes regulations for midcontinent mallards based on passive adaptive optimization
using weighted models of population and harvest dynamics (Johnson et al. 1997). We update model weights (or
probabilities) based on how predictions from each of the four population models compare to the observed breeding
population in year t+1. This posterior updating of model probabilities is based on a version of Bayes Theorem:
where is the probability that model i is correct. We assume that some element of our model set is the
“correct” model for the system, and remains the correct model throughout. Equation (1), then, tracks the probability that
each of the candidate models is the correct one through time. The state of the system in year t+1 consists of breeding
population size (Nt+1) and number of ponds (Pt+1). Under our current approach, information on ponds in year t+1 is not
informative with respect to updating model probabilities in year t, because all four candidate models predict the same
number of ponds every year. We can rewrite the likelihood above as:
where comes from the Breeding Waterfowl and Habitat Survey (May Survey), and is the predicted size of the
population based on model i.
A formal approach involves modeling the conditional likelihood in (2) as a normal distribution:
This form is intuitively appealing, because the value of the likelihood for the observed population size will depend on:
,
which includes the difference between the observed population size and that predicted by model i, and the variance in the
observed state of the system one would expect under model I.
Next, we must address the estimation of the mean and variance of . First,
where g(i) is a model-specific description of population dynamics and {has}t is the set of age- and sex-specific harvest rates
in year t. All of the models we are considering are stochastic, allowing for partial controllability of the system (i.e., hast is
a random variable whose distribution is based on the regulatory package that is chosen in year t). In addition, and
are subject to error, due to partial observability of the system (i.e., sampling variation in the May Survey), but we
assume they are unbiased estimators. Therefore is subject to error in predicting the actual population size, Nt+1,
under model i. Based on this we derive the mean and variance of interest using conditional arguments:
30
(5)
(6)
(7)
(8)
(9)
We estimate the first term in equation (6) with the sampling variance from the May Survey in year t+1. The second term
can be simplified to:
Therefore (6) can be reexpressed as:
The variance in the second term of (8) is derived from the sources of uncertainty inherent in the function in (4): partial
observability of the state of the system, and partial controllability of harvest, in year t.
We use parametric bootstrapping for approximating the likelihood in (2) without assuming a distributional form. It also
precludes the need to derive an explicit estimate of the variance in (8). Instead we assume distributional forms for more
basic quantities.
We simulate the transition from the state of the system in year t, to the state of the system in year t+1, under each model,
described by g in (4). We acknowledge uncertainty about the values of , , and , and to incorporate this
uncertainty we use random values from the following assumed distributions in their place:
Because we anticipate a sampling covariance between , and , and do not currently have an estimate of its
value, we make the conservative (i.e., largest possible) assumption that the two are perfectly correlated.
Practically speaking, this implies that the simulation of these two random variables will be based on the same draw from a
standard normal distribution.
Because we update the model probabilities after direct recovery rates are available from the hunting season in year t, we
use estimates and sampling variances of realized harvest rates (recovery rates, adjusted for reporting rate) in the updating
process whenever possible. Because there is no sampling covariance between estimates of harvest rate for the four age-sex
classes, we generate an independent normal random variate for each.
For each model, in each repetition of the simulation, the generated value of is projected to the actual value of
( ). Finally, to represent partial observability in year t+1, we generate another random number from the following
distribution:
31
(10)
We base the variance of the model-dependent distribution in (10) on the estimated coefficient of variation from the May
Survey, instead of its variance, because experience has shown that the standard error is proportional to population size.
This process produces an observed population size in year t+1 for each repetition of the simulation. By repeating the
process a large number of times we produce an empirical distribution to compare against the realized from the May
Survey. We use 10,000 iterations and then use smoothing techniques to estimate a likelihood function. Finally, we
determine the likelihood value for model i based on , and incorporate it into equations (2) and (1).
32
APPENDIX D: 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 D-1), and the “hunter” model which
predicts the number of successful duck hunters (Table D-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; the Atlantic Flyway portion without compensatory days; the Mississippi Flyway; the low plains
portion of Central Flyway; the High Plains Mallard Management Unit in the Central Flyway; the Columbia Basin Mallard
Management Unit in the Pacific Flyway; and the remainder of the Pacific Flyway excluding Alaska. 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.
33
Table D-1. Parameter estimates by management area for models of seasonal harvest per successful hunter.
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
34
Table D-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
35
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
36
APPENDIX E: Estimating the Mallard Harvest Rate for
the 1999-00 Season
We estimated the overall harvest rate of adult male mallards in the midcontinent region using harvest-rate estimates for
reference areas 2, 4 and 5 (Anderson and Henny1972) that were derived from reward banding. Harvest rates in the
unsampled banding reference areas (1, 3, 6-7 and 12-14) was treated as missing, and conventional data augmentation (or
multiple imputation) techniques were employed (Schafer, J. L. 1997. Analysis of incomplete multivariate data.
Chapman and Hall, London. 430pp.).
The model under which estimates were produced assumes that the vector of harvest rates for the 10 reference areas is a
multivariate normal random variable with some unknown mean vector and variance-covariance matrix. The variance-covariance
matrix describes the correlations between harvest rate among the reference areas. Nominally, the harvest rate
for a given reference area is correlated with the harvest rate in the other reference areas, and it is this aspect which
facilitates estimation of “unobserved” harvest rates from those which are estimated from data. The mean vector and
variance-covariance matrix in the model were estimated from 36 years of historic data.
Estimates and their variances were computed for each of the seven unsampled reference areas. These predictions were then
weighted by the proportion of the midcontinent mallard population in each area during spring 1999 to construct an
estimate of the overall harvest rate. The estimated harvest rate of adult male mallards in the midcontinent region during
the 1999-00 hunting season was 0.098 (SE = 0.0097), which is consistent with the liberal regulatory alternative.
37
Appendix F: Past Regulations and Harvest Strategies
Table F-1. Regulatory alternatives considered for the 1995 and 1996 duck-hunting seasons.
Flyway
Regulation Atlantic Mississippi Centrala Pacificb
Shooting hours one-half hour before sunrise to sunset for all Flyways
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.
38
Table F-2. Optimal regulatory choicesa for midcontinent mallards during the 1995 hunting
season. This strategy is based on the regulatory alternatives for 1995, equal weights for four
alternative models of population dynamics, and the dual objectives of maximizing long-term
cumulative harvest and achieving a population goal of 8.7 million.
Pondsb
Mallardsc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
4.5 M M M L L L L L L L
5.0 L L L L L L L L L L
5.5 L L L L L L L L L L
6.0 L L L L L L L L L L
6.5 L L L L L L L L L L
7.0 L L L L L L L L L L
7.5 L L L L L L L L L L
8.0 L L L L L L L L L L
8.5 L L L L L L L L L L
9.0 L L L L L L L L L L
9.5 L L L L L L L L L L
10.0 L L L L L L L L L L
10.5 L L L L L L L L L L
11.0 L L L L L L L L L L
a 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.
39
Table F-3. Optimal regulatory choicesa for midcontinent mallards during the 1996 hunting
season. This strategy is based on the regulatory alternatives and model weights for 1996, and
the dual objectives of maximizing long-term cumulative harvest and achieving a population goal
of 8.7 million.
Pondsb
Mallardsc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
4.5
5.0 R R R
5.5 R R R R M M
6.0 R R R R R R M M L L
6.5 R R R M M M L L L L
7.0 M M M L L L L L L L
7.5 M L L L L L L L L L
8.0 L L L L L L L L L L
8.5 L L L L L L L L L L
9.0 L L L L L L L L L L
9.5 L L L L L L L L L L
10.0 L L L L L L L L L L
10.5 L L L L L L L L L L
11.0 L L L L L L L L L L
a 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.
40
Table F-4. Optimal regulatory choicesa for midcontinent mallards during the 1997 hunting
season. This strategy is based on regulatory alternatives and model weights for 1997, and on
the dual objectives of maximizing long-term cumulative harvest and achieving a population goal
of 8.7 million.
Pondsb
Mallardsc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
4.5
5.0
5.5 VR VR VR
6.0 VR VR VR VR VR R R R
6.5 VR VR VR VR R R R M M M
7.0 R R R R R M M M L L
7.5 R R M M M M L L L L
8.0 M M M M L L L L L L
8.5 M M L L L L L L L L
9.0 L L L L L L L L L L
9.5 L L L L L L L L L L
10.0 L L L L L L L L L L
10.5 L L L L L L L L L L
11.0 L L L L L L L L L L
a 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 mid-continent mallards during May, in millions.
41
Table F-5. Optimal regulatory choicesa for midcontinent mallards during the 1998 hunting
season. This strategy is based on regulatory alternatives and model weights for 1998, and on
the dual objectives of maximizing long-term cumulative harvest and achieving a population goal
of 8.7 million.
Pondsb
Mallardsc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
4.5
5.0 VR
5.5 VR VR VR R
6.0 VR VR VR VR VR R R R M
6.5 VR VR VR R R R M M M L
7.0 R R R R M M M L L L
7.5 R M M M M L L L L L
8.0 M M M L L L L L L L
8.5 M L L L L L L L L L
9.0 L L L L L L L L L L
9.5 L L L L L L L L L L
10.0 L L L L L L L L L L
10.5 L L L L L L L L L L
11.0 L L L L L L L L L L
a 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 mid-continent mallards during May, in millions.
42
Table F-6. Optimal regulatory choicesa for midcontinent mallards during the 1999 hunting
season. This strategy is based on regulatory alternatives and model weights for 1999, and on
the dual objectives of maximizing long-term cumulative harvest and achieving a population goal
of 8.7 million.
Pondsb
Mallardsc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
<5.0
5.0 VR
5.5 VR VR VR R
6.0 VR VR VR VR VR R R R M
6.5 VR VR VR R R R M M M L
7.0 R R R R M M M L L L
7.5 R M M M M L L L L L
8.0 M M M L L L L L L L
8.5 M L L L L L L L L L
9.0 L L L L L L L L L L
9.5 L L L L L L L L L L
10.0 L L L L L L L L L L
10.5 L L L L L L L L L L
11.0 L L L L L L L L L L
a 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.