Adaptive harvest management 1999 duck hunting season

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Cover art: Jim Hautman's acrylic painting of greater scaup, which was selected for the 1999 federal "duck stamp."
U. S. Fish & Wildlife Service
Adaptive
Harvest Management
1999 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
1999 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 Office
of Migratory Bird Management and the North American Waterfowl and Wetlands Office. 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), M. T. Moore (USFWS), M. C. Runge (Cornell University), and S. E. Sheaffer (New
York Cooperative Research Unit). 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, Office of Migratory Bird Management - USFWS, Arlington Square, Room 634, 4401 North Fairfax
Drive, Arlington, VA 22203.
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Annual reports on adaptive harvest management are available on the Internet at:
http://www.fws.gov/r9mbmo/reports/reports.html
TABLE OF CONTENTS
Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Mallard Stocks and Flyway Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Mallard Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Harvest Management Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Regulatory Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Optimal Harvest Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Current AHM Priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Appendix A: AHM Working Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Appendix B: Mallard Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Appendix C: Updating Model Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Appendix D: Predicting Harvest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Appendix E: Estimating the Mallard Harvest Rate for the 1998-99 Season . . . . . . . . . . . . . . . 32
Appendix F: Past Regulations and Harvest Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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EXECUTIVE SUMMARY
In 1995, the U.S. Fish and Wildlife Service embraced 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. Moreover,
adaptive harvest management (AHM) relies on an iterative cycle of monitoring, assessment, and decision making to clarify
relationships among hunting regulations, harvests, and waterfowl abundance.
To date, AHM has been based on 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 ultimate goal is to develop Flyway-specific
harvest strategies, which represent an average of optimal strategies for each mallard breeding population, weighted by
the relative contribution of each population to the respective Flyways. Geographic boundaries used to define midcontinent and
eastern mallards have been established, and mathematical models of population dynamics are available for predicting regulatory
impacts. Investigations regarding the geographic bounds and population dynamics of western mallards are ongoing.
A critical need for successful implementation of AHM is a set of regulatory alternatives that remain fixed for an extended period.
When AHM was first implemented in 1995, three regulatory alternatives characterized as liberal, moderate, and restrictive were
defined based on recent regulatory experience. The 1995 regulatory alternatives also were considered for the 1996 hunting
season. In 1997, the regulatory alternatives were modified in response to requests from the Flyway Councils. Changes included
provisions for additional hunting opportunity under the moderate and liberal alternatives, as well as the addition of a very
restrictive alternative. For the 1999 season, the USFWS is maintaining the same basic structure to the regulatory alternatives
as that in 1997 and 1998.
Harvest strategies were derived for midcontinent and eastern mallards, but they do not yet allow for Flyway-specific regulatory
choices. The strategy for midcontinent mallards was based on: (1) an objective to maximize long-term harvest and achieve a
population goal of 8.7 million; (2) regulatory alternatives that are the same as in 1998; and (3) current understanding of
regulatory impacts. Based on a breeding population size of 11.8 million mallards and 3.9 million ponds in Prairie Canada, the
optimal regulatory choice for midcontinent mallards in 1999 is the liberal alternative. The strategy for eastern mallards was
based on: (1) an objective to maximize long-term harvest; (2) regulatory alternatives that are the same as in 1998; and (3) a
“working model” of population dynamics. Based on a breeding population size of 1.1 million mallards and spring precipitation
of 8.3 inches, the optimal regulatory choice for eastern mallards in 1999 also is the liberal alternative.
Current AHM priorities include: (1) addressing communication needs related to institutional aspects of implementing AHM;
(2) restoring our ability to reliably measure the harvest rates of midcontinent mallards; (3) improving the recruitment models
for midcontinent mallards; (4) determining the appropriate number of duck stocks to include in AHM; (5) modifying the current
AHM protocol to account for eastern mallards; and (6) exploring AHM mechanisms for northern pintails.
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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 embraced 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, in terms of sustainable hunting
opportunities, can be maximized only if regulatory effects can be predicted reliably. Thus, adaptive management relies on the
iterative cycle of monitoring, assessment, and decision making described above 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) 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;
(4) 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.
Adaptive harvest management (AHM) was developed as a systematic process for dealing effectively 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 the environment 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
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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 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
Since 1995, AHM strategies have been based on the status of midcontinent mallards, which are defined as those breeding in
federal survey strata 1-18, 20-50, and 75-77, and in Minnesota, Wisconsin, and Michigan (Fig. 1). An optimal regulatory
alternative for midcontinent mallards is 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 is applied in all four Flyways,
although season lengths and bag limits are Flyway-specific.
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.
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Efforts are underway 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). Extension of the current process to account for multiple mallard stocks and
Flyway-specific regulatory choices involves:
(1) augmentation of the decision criteria to include population and environmental variables relevant to eastern and western
mallards;
(2) revision of the objective function to account for harvest management objectives for mallards outside the midcontinent
region; and
(3) modification of the decision rules to allow independent regulatory choices in the Flyways.
An optimal harvest strategy for each Flyway then can be derived, which in effect would represent an average of the optimal
strategies for each breeding stock, weighted by the relative contribution of each stock to the respective Flyways.
For the purposes of this report, 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). Western mallards currently are defined as those breeding in Washington, Oregon, and California. This
range may be extended if monitoring and assessment information becomes available for other important breeding areas of
western mallards.
MALLARD POPULATION DYNAMICS
Midcontinent Mallards
Estimates of the entire midcontinent population (as defined above) are available only since 1992. Since then, the number of
midcontinent mallards has grown by an average of 8.6 percent (SE = 1.0) per annum (Table 1).
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
a In thousands.
The dynamics of midcontinent mallards are described by four alternative models, which result from combining two mortality
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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.
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, access to hunting areas) 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 updating of model weights (Appendix C). These weights
describe the relative ability of the alternative models to mimic 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 must account for sampling error (i.e.,
partial observability) in population size and pond counts, as well as for partial 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 about harvest and environmental impacts on mallard abundance. Model weights changed
markedly in 1996, and have remained relatively stable since (Table 2). On the whole, comparisons of observed and predicted
population sizes provide some evidence of strongly density-dependent reproduction, but little indication of a compensatory
response to hunting mortality.
Year
1996 1997 1998 1999
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 error on either side of the mean estimate.
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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
Additive Strong density
dependence
0.25000 0.65479 0.53015 0.6131
1
0.6088
3
Additive Weak density
dependence
0.25000 0.34514 0.46872 0.3868
7
0.3841
6
Compensat
ory
Strong density
dependence
0.25000 0.00006 0.00112 0.0000
1
0.0000
1
Compensat
ory
Weak density
dependence
0.25000 0.00001 0.00001 0.0000
1
0.0070
0
Eastern Mallards
Midwinter counts and the Breeding Bird Survey provide evidence of exponential growth in the eastern mallard population since
the mid-1970s. This pattern of growth also is apparent in the more recent fixed-wing (strata 51-54 and 56) and northeastern
plot (New Hampshire south through Virginia) surveys (Table 3), although population growth seems to have slowed in more
recent years.
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
a In thousands.
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The population dynamics of eastern mallards were studied extensively by Sheaffer and Malecki (1996), but managers have not
yet established a set of alternative models that characterize key uncertainties about the mortality and reproductive processes.
In the interim, a “working model” has been developed to help managers understand the potential biological impacts of the current
AHM process on eastern mallards.
The working model of eastern mallards incorporates natural mortality rates that are similar to those of midcontinent mallards
and an assumption of completely additive hunting mortality. Reproductive rates are predicted based on the size of the population
and regional precipitation during March-May of the current year. The reproductive process is characterized as strongly density
dependent, predicting the highest reproductive rates during years in which population size is relatively low and spring
precipitation is high. Mathematical details of the working model for eastern mallards are provided in Appendix B.
Western Mallards
The breeding range of western mallards includes the Pacific Flyway states (including Alaska), the Yukon Territories, and British
Columbia. Although mallards breeding in southern Alberta are harvested primarily in the Pacific Flyway, estimated survival
and reproductive rates for these birds are similar to those for birds breeding in southern Saskatchewan. As a result, mallards
breeding in southern Alberta are modeled as part of the mid-continent population. Breeding pair surveys in the lower Pacific
Flyway states were initiated in 1990 and have suggested a relatively stable or slightly increasing population size in this region
during 1990-98. Highest numbers of breeding birds occur in California, Oregon, and Washington. Presently there are no
breeding pair surveys and limited banding data for mallards breeding in Alaska or British Columbia. Managers will be unable
to model the population dynamics of birds breeding in Alaska or British Columbia until operational surveys and banding
programs are established in these regions. The initial working model for western mallards will focus on identifying key
uncertainties affecting recruitment and survival of mallards breeding in the coastal states of California, Oregon, and Washington,
where there is sufficient banding, harvest, and population survey data. Estimated natural mortality rates appear to be similar
to those for mid-continent mallards. Annual reproductive rates have been estimated using female age ratios in the California
harvest during October, when the harvest is comprised primarily of stocks from the lower Pacific Flyway states. Efforts are
ongoing to identify environmental and demographic factors that affect reproductive success of western mallards. Preliminary
analyses suggest that, in contrast to eastern and mid-continent mallards, annual reproductive success is not strongly density
dependent. Environmental factors with the highest correlation to annual recruitment include spring precipitation and agriculture
parameters in California.
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, managers avoid harvest 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). 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 losses in 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.
Eastern Mallards
For the purposes of this report, the 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.
10
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 mallard harvest, expressed as a function of breeding-population
size expected in the subsequent year.
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 been unchanged since 1997 (Table 4).
Predictions of Mallard Harvest Rates
Since 1995, harvest rates associated with the AHM regulatory alternatives have been predicted using empirical estimates from
1979-84, which have been adjusted to reflect differences in season length and bag limit, and for contemporary numbers of
hunters (Table 5). These adjustments are not based on band-recovery data, but rather on estimates of hunting effort and success
from hunter surveys (Appendix D). These predictions of harvest rates have large sampling variances, and their accuracy is
uncertain. However, the estimated harvest rate during the 1998-99 season fell within the confidence limits of the prediction
(Appendix E).
Harvest rates for each of the regulatory alternatives for 1999 were predicted assuming no change in the regulatory alternatives
from 1997 and 1998. However, predicted harvest rates for 1997-99 differ from those published previously due to revised
analytical procedures, which rely on mean numbers of hunters during 1981-95, rather than on short-term trends in annual hunter
numbers. This change was made to prevent year-specificity in harvest rates predicted for a given alternative, and to better reflect
uncertainty about hunter numbers in the future.
Adult female mallards tend to be less vulnerable to harvest than adult males, while young are more vulnerable (Table 6).
Estimates of the relative vulnerability of adult females and young in the eastern mallard population tend to be higher and more
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variable than in the midcontinent population.
Table 4. Regulatory alternatives considered for the 1999 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.
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Table 5. Expected harvest rates (SE) of adult male midcontinent and eastern mallards under
different regulatory alternatives, based on mean hunter numbers during 1981-95.
Mallard
Population
Regulatory alternatives for:
Alternative 1995, 1996 1997, 1998, 1999
Midcontinent Very
restrictive
N/A 0.053 (0.011)
Restrictive 0.067 (0.014) 0.067 (0.014)
Moderate 0.089 (0.020) 0.111 (0.027)
Liberal 0.118 (0.029) 0.131 (0.032)
Eastern Very
restrictive
N/A 0.121 (0.020)
Restrictive 0.133 (0.021) 0.135 (0.022)
Moderate 0.149 (0.023) 0.163 (0.025)
Liberal 0.179 (0.028) 0.177 (0.028)
Table 6. Mean harvest vulnerability (SE) of 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
Midcontinent Mallards
The 1999 AHM strategy for midcontinent mallards was based on: (1) regulatory alternatives that are unchanged from 1997 and
1998; (2) model weights for 1999; and (3) the dual objectives to maximize long-term cumulative harvest and achieve a
population goal of 8.7 million (Table 7). This strategy provides optimal regulatory choices for midcontinent mallards assuming
that all four Flyways would use the prescribed regulation. Ultimately, regulatory choices will be Flyway-specific by accounting
for the relative contribution of the three mallard breeding populations to each Flyway. The optimal harvest strategies for the
1995-98 seasons are provided in Appendix F (Tables F-2 to F-5) so that the reader can see how the strategy for midcontinent
mallards has “evolved” over time. These strategies have been recalculated based on the most recent predictions of harvest rates
and, thus, differ slightly from those published previously.
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Table 7. Optimal regulatory choicesa for midcontinent mallards during the 1999 hunting season. This strategy
is based on regulatory alternatives unchanged from 1997 and 1998, current model weights (Table 2), 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
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.
Blank cells in Table 7 (and in other 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.
We simulated the use of the harvest strategy in Table 7 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 1.3 (SE = 0.5) million and 8.3 (SE = 0.9) million, respectively.
Based on a breeding population size of 11.8 million mallards and 3.9 million ponds in Prairie Canada, the optimal regulatory
choice for midcontinent mallards in 1999 is the liberal alternative.
Eastern Mallards
The 1999 AHM strategy for eastern mallards was based on: (1) regulatory alternatives unchanged from 1997 and 1998; (2) the
working model of population dynamics; and (3) an objective to maximize long-term cumulative harvest (Table 8). The strategy
is more liberal than that published last year, reflecting the most recent predicitions of harvest rates. Currently, this strategy
only provides optimal regulations for eastern mallards under the condition that all Flyways would use the same regulation.
Ultimately, regulatory choices will be Flyway-specific by accounting for the relative contribution of eastern and midcontinent
mallards to each Flyway.
We simulated the use of this harvest strategy with the working model of population dynamics to determine expected performance
14
characteristics. Assuming that harvest management adhered to this strategy, the annual harvest and breeding population size
would average 327 (SE = 60) thousand and 1.06 (SE = 0.13) million, respectively.
Based on a breeding population size of 1.09 million mallards and spring precipitation of 8.34 inches, the optimal regulatory
choice for eastern mallards in 1999 is the liberal alternative.
Table 8. Optimal regulatory choicesa for eastern mallards during the 1999 hunting season. This
strategy is based on regulatory alternatives unchanged from 1997 and 1998, the working model
of population dynamics, and on an objective to maximize long-term cumulative harvest.
Spring precipitationb
Mallardsc 6 7 8 9 10 11 12 13 14 15
500 VR VR VR R R M M
550 VR VR VR R M M L L L
600 VR VR R M M L L L L L
650 R M M L L L L L L L
700 M L L L L L L L L L
750 L L L L L L L L L L
800 L L L L L L L L L L
850 L L L L L L L L L L
>900 L L L L L L L L L L
a VR = very restrictive, R = restrictive, M = moderate, and L = liberal.
b March to May precipitation in the northeastern U.S., in inches.
c Estimated number of eastern mallards in the combined fixed-wing and northeastern plot surveys,
in thousands.
CURRENT AHM PRIORITIES
Communication Needs
Communication needs are becoming more complex, broader in scope, and more institutional in nature as implementation of AHM
proceeds. Because of the explicit and formal nature of the AHM process, managers are being challenged to confront long-held
beliefs about their ability to understand and influence the managed system, and about the potential of biological science to
engender policy consensus. There seem to be three problem areas in particular:
(1) Goal setting - Effective management planning and evaluation depends on agreement among stakeholders about how to
value harvest benefits, and how those benefits should be shared. These unresolved value judgements, and the lack of
effective structures for organizing debate, present a serious threat to the viability of AHM. Moreover, the lack of
information on the attitudes and preferences of the nation’s waterfowl hunters is a continuing problem in the effort to
determine appropriate harvest management objectives.
(2) Limits to system control - There are rather severe practical limits to our ability to predict, control, and measure harvests
and, therefore, significant constraints on short-term harvest yields and the learning needed to increase long-term
performance.
15
(3) Management scale - The history of waterfowl management has been characterized by persistent efforts to account for
increasingly more spatial, temporal, and organizational variability in waterfowl biology. The cost-effectiveness of this
approach is questionable; moreover, limited resources for monitoring and assessment rarely permit selection of the scale
with the highest net benefit.
It may be these institutional issues, more than any of the most difficult technical problems, that pose the greatest challenge to
the long-term success of AHM. Coping with these issues will require innovative mechanisms for producing effective dialogue,
and for handling disputes within a process that all parties regard as workable. The Service, in cooperation with the AHM
technical working group, is developing plans to address these communication issues.
Measuring Harvest Rates of Midcontinent Mallards
Since 1994 there has been a systematic effort to increase the rate at which hunters report band recoveries. This effort is expected
to greatly increase the efficiency of banding programs, but has temporarily made it difficult to estimate the harvest rate of
mallards. A study to evaluate recent changes in band-reporting rate is in the planning stage, with implementation to occur in
2000 or 2001.
As part of the planning effort, a pilot study was conducted in southern Saskatchewan in 1998 to obtain a preliminary estimate
of band-reporting rates for adult mallards. A total of 1,675 reward bands ($100.00 each) were placed on adult mallards in three
separate sites in southern Saskatchewan (Kindersley, Wynyard, Last Mt. Lake). The recovery rate for standard bands was
0.075 (S.E. 0.005), and the recovery rate for reward bands was 0.093 (S.E. 0.007). Based on these data, the estimate of the
band-reporting rate is 0.805 (S.E. 0.082), or about twice that from the late 1980's. Plans are being made to expand the area
where reward bands are used in 1999 to include the Great Lakes and other important breeding areas. This effort should permit
reasonable estimates of harvest rates for midcontinent mallards until the full band-reporting rate study is completed.
Modeling Reproduction of Midcontinent Mallards
We continue to explore predictive models that describe how the reproductive success of midcontinent mallards varies in response
to environmental conditions across the core breeding area. Recent efforts involved investigating the relationship between the
fall age ratio of female mallards and: (1) size of the mallard population; (2) number of ponds in survey strata 26-49; (3) average
latitude of ponds; (4) average longitude of ponds; (5) slope of the regression between strata-specific crop acreage and pond
abundance, and (6) selected interactions of these independent variables. The “best” model (R2 = 0.81, P < 0.01) included
negative associations with (1) and (3), and a positive relationship with (2), consistent with a priori hypotheses. The “best” six
models, as indicated by Akaike’s Information Criterion, all included a spatial component (i.e., either latitude, longitude, or their
interaction); only one included the slope variable. These results confirm that landscape attributes other than pond numbers might
be useful for predicting fall age ratios, although the appropriate environmental features, and the mechanism(s) by which those
features influence the age ratio, have not been identified. Future work will focus on methods to reliably measure recruitment
at smaller spatial scales, and on remote-sensing techniques to monitor wetland and upland habitat conditions across the Prairie
Pothole Region.
AHM Protocol for Multiple Duck Stocks
All ecological systems exhibit variability on a broad range of temporal, spatial, and organizational (or taxonomic) scales as a
function of how individuals respond to their environment. The scale at which individuals are aggregated for harvest-management
purposes is an arbitrary decision, but one that can strongly influence both the benefits and costs of management. Harvest
management systems defined at scales that account for large amounts of biological variation will produce relatively high
benefits, but also are characterized by high monitoring and assessment costs. Determining the optimal scale for harvest
management depends on the availability of explicit performance criteria (i.e., management goals, objectives, and constraints,
including how harvest should be allocated among users), and on descriptions of how biological attributes vary among different
scales.
16
We have begun to explore various temporal, spatial, and organizational scales for regulating duck harvests. We are focusing
on the appropriate number of breeding duck stocks (i.e., species and/or populations), and the spatial configuration of regulatory
decisions. Preliminary investigations of management scale suggest that harvest utility often is insensitive to the level of
aggregation, even when duck stocks are characterized by relatively large demographic differences. This lack of sensitivity is
even more pronounced in the face of poor control over stock-specific harvests. Therefore, managers potentially face a tradeoff
between relatively small gains in harvest opportunity, and the increased assessment costs and regulatory complexity associated
with fine-scale management.
AHM for Eastern and Western Mallards
Modifying the AHM protocol to account for both eastern and western mallards is perhaps the most challenging technical issue
facing duck harvest managers. Never before have we tried to consider the status of multiple mallard stocks in such a formal
way, nor have we attempted to give all Flyways the ability to choose regulations that are tied to their particular derivation of
mallards. Although progress has been significant, there are three issues in this effort that require further attention: (1)
development of population models; (2) agreement on harvest management objectives; and (3) modification of decision rules to
allow Flyway-specific regulatory choices. Population models for eastern and western mallards are still in the developmental
stage, and major sources of uncertainty have yet to be articulated or their management implications explored. Another difficulty
involves agreeing on population-specific harvest management objectives, and then combining them into one objective function
so that multiple-stock harvest strategies can be derived. Finally, we must modify the decision rules to allow for Flyway-specific
regulatory choices, but this greatly complicates the optimization process. Instead of five regulatory alternatives, we must
evaluate 54 = 625 possible decisions for each combination of population level and environmental variable for all three mallard
stocks. The USFWS and AHM working group have assigned a high priority to addressing these technical issues, and hope to
fully integrate eastern mallards and western mallards into the AHM process in 2000 and 2001, respectively.
AHM for Northern Pintails
The first step in developing an AHM process for pintails has been to develop a set of alternative models that represent the
dynamics of the continental population. Efforts to model annual recruitment for pintails have focused on four areas of
uncertainty thought to affect reproductive success: population density, population distribution, habitat conditions on the breeding
range, and habitat conditions in wintering areas. Annual productivity is indexed using age-ratios in the U.S. harvest. The initial
recruitment model for pintails predicts annual reproductive success based on the average latitude of the breeding population,
the size of the breeding population, and mean monthly precipitation (January-March) for the Prairie Pothole Region. These three
variables explain 57% of the historical variation in the index to recruitment. The average latitude of the breeding population
is a better predictor of recruitment than the May ponds index, perhaps because it is a more direct measurement of the behavioral
response of the birds to environmental conditions. Attempts to include a predictor related to wintering ground conditions have
not been successful; further work is needed to identify appropriate predictors.
Efforts to model annual survival for pintails have focused on the uncertainty about whether the harvest is an additive or
compensatory form of mortality. Analyses to estimate relationships between annual survival and factors such as population
density, reproductive success, and habitat conditions have been unsuccessful. This is primarily due to the imprecision of annual
survival rate estimates. A simple model of annual survival as a function of the kill rate, which closely follows the model for
midcontinent mallards, was developed. Additive and compensatory models were parameterized by finding a value for survival
in the absence of harvest that provided equally good fits to the historical data for both the additive and compensatory models.
At the low level of harvest typical in recent years, these two models do not differ greatly in their predictions; however, the
practical implications of the two models have not yet been determined. Finally, there is some suggestion in the immature male
survival data that some other important dynamic might be at work, but this has yet to be fully explored.
The next step in the development of an AHM process for pintails involves an exploration of the implications of these recruitment
and survival models for harvest management. If optimal harvest strategies for pintails are significantly different than those for
midcontinent mallards, managers will need to agree on acceptable mechanisms for regulating the harvests of pintails
independently of mallards.
17
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.
18
APPENDIX A: AHM Working Group
Tom Aldrich
Utah Div. of Wildlife Resources
1594 W. North Temple
Salt Lake City, UT 84116
phone: 801-538-4789
fax: 801-538-4079
e-mail: nrdwr.taldrich@state.ut.us
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
11510 American Holly Drive
Laurel, MD 20708-4017
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
19
George Haas
U.S. Fish and Wildlife Service
300 Westgate Center Drive
Hadley, MA 01035-9589
phone: 413-253-8576
fax: 413-253-8480
e-mail: george_haas@fws.gov
Jeff Haskins
U.S. Fish and Wildlife Service
P.O. Box 1306
Albuquerque, NM 87103
phone: 505-248-6827 ext 30
fax: 505-248-7885
e-mail: jeff_haskins@fws.gov
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
Arlington Square, Room 634
4401 North Fairfax Drive,
Arlington, VA 22203
phone: 703-358-1964
fax: 703-358-2217
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
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
20
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: 518-439-8083
fax: 518-478-0142
e-mail: bryan.swift@dec.mailnet.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
21
APPENDIX B: Mallard Population Models
Variable definitions:
MALM /* index to midcontinent mallard state variable */
PONDM /* index to midcontinent pond state variable */
MALE /* index to eastern mallard state variable */
PPTE /* index to eastern precipitation state variable */
RAINM /* random variable index to midcontinent precipitation */
RAINE /* random variable index to eastern precipitation */
PROP /* random variable index to proportion of midcontinent population in WI, MN, MI */
HRATEM /* index to stochastic midcontinent harvest-rate */
HRATEE /* index to stochastic eastern harvest-rate */
outcome[] /* outcome of random or stochastic variable */
cur_state[] /* current value of state variable */
nxt_state[] /* value of a state variable in the next time step */
wt1m /* model 1 weight (compensatory survival, strong density-dependent recruitment ) for
midcontinent mallards */
wt2m /* model 2 weight (compensatory survival, weak density-dependent recruitment ) for
midcontinent mallards */
wt3m /* model 3 weight (additive mortality, strong density-dependent recruitment ) for
midcontinent mallards */
wt4m /* model 4 weight (additive mortality, weak density-dependent recruitment ) for
midcontinent mallards */
nxt1m /* model 1 prediction for next state - midcontinent mallards */
nxt2m /* model 2 prediction for next state - midcontinent mallards */
nxt3m /* model 3 prediction for next state - midcontinent mallards */
nxt4m /* model 4 prediction for next state - midcontinent mallards */
Awm /* weak density-dependent fall age ratio - midcontinent mallards */
Asm /* strong density-dependent fall age ratio - midcontinent mallards */
Hafm /* harvest rate - adult females - midcontinent mallards */
Hamm /* harvest rate - adult males - midcontinent mallards */
Hyfm /* harvest rate - young females - midcontinent mallards */
Hymm /* harvest rate - young males - midcontinent mallards */
Kafm /* kill rate - adult females - midcontinent mallards */
Kamm /* kill rate - adult males - midcontinent mallards */
Kyfm /* kill rate - young females midcontinent mallards */
Kymm /* kill rate - young males - midcontinent mallards */
shafam /* hunt-season survival - adult females -additive mortality - midcontinent mallards */
shafcm /* hunt-season survival - adult females - compensatory survival - midcontinent
mallards*/
shamam /* hunt-season survival - adult males - additive mortality - midcontinent mallards */
shamcm /* hunt-season survival - adult males - compensatory survival - midcontinent mallards */
shyfam /* hunt-season survival - young females - additive mortality - midcontinent mallards */
shyfcm /* hunt-season survival - young females - compensatory surv. - midcontinent mallards */
shymam /* hunt-season survival - young males - additive mortality - midcontinent mallards */
shymcm /* hunt-season survival - young males - compensatory survival - midcontinent mallards*/
dafm=0.748 /* differential harvest vulnerability - adult females - midcontinent mallards */
dymm=1.361 /* differential harvest vulnerability - young males - midcontinent mallards */
dyfm=1.188 /* differential harvest vulnerability - young females - midcontinent mallards */
22
c=0.2 /* crippling loss rate */
s0mm=0.81 /* survival in absence of hunting - male - midcontinent mallards */
s0fm=0.64 /* survival in absence of hunting - female - midcontinent mallards */
swm=0.90 /* winter survival - midcontinent mallards */
ssmm=0.90 /* summer survival - males - midcontinent mallards */
ssfm=0.71 /* summer survival - females - midcontinent mallards */
cfm=0.36 /* compensatory threshold - females - midcontinent mallards */
cmm=0.19 /* compensatory threshold - males - midcontinent mallards */
sexm=0.55 /* proportion of males in population - midcontinent mallards */
nxt1e /* model 1 prediction for next state - eastern mallards */
nxt2e /* model 2 prediction for next state - eastern mallards */
nxt3e /* model 3 prediction for next state - eastern mallards */
nxt4e /* model 4 prediction for next state - eastern mallards */
wt1e=0.00 /* model 1 weight (compensatory survival, strong density-dependent recruitment ) for
eastern mallards */
wt2e=0.00 /* model 2 weight (compensatory survival, weak density-dependent recruitment ) for
eastern mallards */
wt3e=0.00 /* model 3 weight (additive mortality, strong density-dependent recruitment ) for eastern
mallards */
wt4e=1.00 /* model 4 weight (additive mortality, weak density-dependent recruitment ) for eastern
mallards */
BBSr /* BBS index - eastern mallards */
logits /* logit of age ratio for strong density-dependent recruitment model - eastern mallards */
Ase /* strong density-dependent age ratio - eastern mallards */
logitw /* logit of age ratio for weak density-dependent recruitment model - eastern mallards */
Awe /* weak density-dependent age ratio - eastern mallards */
Hafe /* harvest rate - adult females - eastern mallards */
Hame /* harvest rate - adult males - eastern mallards */
Hyfe /* harvest rate - young females - eastern mallards */
Hyme /* harvest rate - young males - eastern mallards */
Kafe /* kill rate - adult females - eastern mallards */
Kame /* kill rate - adult males - eastern mallards */
Kyfe /* kill rate - young females - eastern mallards */
Kyme /* kill rate - young males - eastern mallards */
shafae /* hunt-season survival - adult females - additive mortality - eastern mallards */
shafce /* hunt-season survival - adult females - compensatory survival - eastern mallards */
shamae /* hunt-season survival - adult males - additive mortality - eastern mallards */
shamce /* hunt-season survival - adult males - copmpensatory survival - eastern mallards */
shyfae /* hunt-season survival - young females - additive mortality - eastern mallards */
shyfce /* hunt-season survival - young females- compensatory survival - eastern mallards */
shymae /* hunt-season survival - young males -additive mortality - eastern mallards */
shymce /* hunt-season survival - young males - compensatory survival - eastern mallards */
dafe=0.98 /* differential vulnerability - adult females - eastern mallards */
dyme=1.45 /* differential vulnerability - young males - eastern mallards */
dyfe=1.32 /* differential vulnerability - young females - eastern mallards */
s0me=0.81 /* survival in absence of hunting - male - eastern mallards */
s0fe=0.64 /* survival in absence of hunting - female - eastern mallards */
swe=0.90 /* winter survival - eastern mallards */
ssme=0.90 /* summer survival - males - eastern mallards */
ssfe = 0.71 /* summer survival - females - eastern mallards */
cfe=0.36 /* compensatory threshold - females - eastern mallards */
23
cme=0.19 /* compensatory threshold - males - eastern mallards */
sexe=0.55 /* proportion males in May - eastern mallards */
Hunting mortality rates:
Hamm=outcome[HRATEM];
Hafm=min(1, Hamm*dafm);
Hymm=min(1, Hamm*dymm);
Hyfm=min(1, Hamm*dyfm);
Kafm=min(1, Hafm/(1-c));
Kamm=min(1, Hamm/(1-c));
Kyfm=min(1, Hyfm/(1-c));
Kymm=min(1, Hymm/(1-c));
Hame=outcome[HRATEE];
Hafe=min(1, Hame*dafe);
Hyme=min(1, Hame*dyme);
Hyfe=min(1, Hame*dyfe);
Kafe=min(1, Hafe/(1-c));
Kame=min(1, Hame/(1-c));
Kyfe=min(1, Hyfe/(1-c));
Kyme=min(1, Hyme/(1-c));
Recruitment rates:
Awm=max(0, 0.8249-(0.0547*((1-outcome[PROP])*cur_state[MALM]/1000000.0))+
(0.1130*(cur_state[PONDM]/1000000.0)));
Asm=max(0, 0.1081-(0.1128*((1-outcome[PROP])*cur_state[MALM]/1000000.0))+
(0.1460*(cur_state[PONDM]/1000000.0)));
BBSr=-2.315455 + 0.000007081*cur_state[MALE];
logits=-0.483415-0.284774*BBSr+0.121664*cur_state[PPTE];
Ase=3.0*exp(logits)/(1+exp(logits));
logitw=-0.573898-0.170945*BBSr+0.109462*cur_state[PPTE];
Awe=3.0*exp(logitw)/(1+exp(logitw));
Hunting-season survival rates:
shafam=(1-Kafm); shamam=(1-Kamm); shyfam=(1-Kyfm); shymam=(1-Kymm);
if (Kafm>cfm) shafcm=(1-Kafm)/s0fm; else shafcm=1.0;
if (Kamm>cmm) shamcm=(1-Kamm)/s0mm; else shamcm=1.0;
if (Kyfm>cfm) shyfcm=(1-Kyfm)/s0fm; else shyfcm=1.0;
if (Kymm>cmm) shymcm=(1-Kymm)/s0mm; else shymcm=1.0;
shafae=(1-Kafe); shamae=(1-Kame); shyfae=(1-Kyfe); shymae=(1-Kyme);
if (Kafe>cfe) shafce=(1-Kafe)/s0fe; else shafce=1.0;
if (Kame>cme) shamce=(1-Kame)/s0me; else shamce=1.0;
if (Kyfe>cfe) shyfce=(1-Kyfe)/s0fe; else shyfce=1.0;
if (Kyme>cme) shymce=(1-Kyme)/s0me; else shymce=1.0;
Next states:
nxt_state[PONDM=max(1, -3835087.53+0.45*cur_state[PONDM]+13695.47*outcome[RAINM]) ;
nxt1m=cur_state[MALM]*((1.-sexm)*ssfm*(shafcm+Asm*(shyfcm+shymcm)) + sexm*ssmm*shamcm)*swm;
24
nxt2m=cur_state[MALM]*((1.-sexm)*ssfm*(shafcm+Awm*(shyfcm+shymcm)) + sexm*ssmm*shamcm)*swm;
nxt3m=cur_state[MALM]*((1.-sexm)*ssfm*(shafam+Asm*(shyfam+shymam)) + sexm*ssmm*shamam)*swm;
nxt4m=cur_state[MALM]*((1.-sexm)*ssfm*(shafam+Awm*(shyfam+shymam)) + sexm*ssmm*shamam)*swm;
nxt_state[MALM]=max(0, wt1m*nxt1m+wt2m*nxt2m+wt3m*nxt3m+wt4m*nxt4m);
nxt_state[PPTE]=outcome[RAINE];
nxt1e=cur_state[MALE]*((1.-sexe)*ssfe*(shafce+Ase*(shyfce+shymce)) + sexe*ssme*shamce)*swe;
nxt2e=cur_state[MALE]*((1.-sexe)*ssfe*(shafce+Awe*(shyfce+shymce)) + sexe*ssme*shamce)*swe;
nxt3e=cur_state[MALE]*((1.-sexe)*ssfe*(shafae+Ase*(shyfae+shymae)) + sexe*ssme*shamae)*swe;
nxt4e=cur_state[MALE]*((1.-sexe)*ssfe*(shafae+Awe*(shyfae+shymae)) + sexe*ssme*shamae)*swe;
nxt_state[MALE]=max(0, wt1e*nxt1e+wt2e*nxt2e+wt3e*nxt3e+wt4e*nxt4e);
25
pt%1(model i | data) '
pt(model i) pt%1(data | model i)
jj
pt(model j) pt%1(data | model j)
, (1)
pt%1(data | model i) ' f (N data
t%1 | ˆ N (i)
t%1 ) , (2)
f (N data
t%1 | ˆ N (i)
t%1 ) - normal[E(N data
t%1 | ˆ N (i)
t%1 ), Var(N data
t%1 | ˆ N (i)
t%1 )]. (3)
ˆ N (i)
t%1
' g (i) (N data
t , P data
t , {has}t) , (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 p is the probability that model i is correct. We assume that some element of our model set is the t(model i)
“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 N data comes from the Breeding Waterfowl and Habitat Survey (May Survey), and is the predicted size of the
t%1
ˆ N (i)
t%1
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:
,
N data
t%1
&E(N data
t%1 | ˆ N (i)
t%1 )
var(N data
t%1 | ˆ N (i)
t%1 )
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 f(N data . First,
t%1 | ˆ N (i)
t%1 )
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
26
E[N data
t%1 | ˆ N (i)
t%1 ] ' ENt%1
[E(N data
t%1 |Nt%1, ˆ N (i)
t%1 )] ' E(Nt%1| ˆ N (i)
t%1 ) ' ˆ N (i)
t%1 , (5)
Var[N data
t%1 | ˆ N (i)
t%1 ] ' ENt%1
[Var(N data
t%1 |Nt%1, ˆ N (i)
t%1 )]
% VarNt%1
[E(N data
t%1 |Nt%1, ˆ N (i)
t%1 )] .
(6)
VarNt%1
[E(N data
t%1 | Nt%1, ˆ N (i)
t%1 )] ' Var [Nt%1 | ˆ N (i)
t%1 ], (7)
Vˆar[N data
t%1 | ˆ N (i)
t%1 ] ' sampling variance % Vˆar [Nt%1 | ˆ N (i)
t%1 ] , (8)
f(N boot
t ) - normal[N data
t , Var(N data
t |Nt)],
f(P boot
t ) - normal[P data
t , Var(P data
t |Nt)],
f(h boot
ast ) - normal[ ˆ hast, Var( ˆ ht|hast)].
(9)
random variable whose distribution is based on the regulatory package that is chosen in year t). In addition, N data and
t
P data are subject to error, due to partial observability of the system (i.e., sampling variation in the May Survey), but we
t
assume they are unbiased estimators. Therefore Nˆ is subject to error in predicting the actual population size, Nt+1, (i)
t%1
under model i. Based on this we derive the mean and variance of interest using conditional arguments:
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 N , , and , and to incorporate this t Pt {has}t
uncertainty we use random values from the following assumed distributions in their place:
Because we anticipate a sampling covariance between N data , and , and do not currently have an estimate of its
t P data
t
value, we make the conservative (i.e., largest Var[N data possible) assumption that the two are perfectly
t%1 | ˆ N (i)
t%1 ]
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
27
N data
t%1 |N (i)boot
t%1 -normal[N (i)boot
t%1 ,(c.v.(N data
t%1 )@N (i)boot
t%1 )2]. (10)
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 N is projected to the actual value of t Nt%1
(N (i)boot). Finally, to represent partial observability in year t+1, we generate another random number from the following
t%1
distribution:
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 N data from the May
t%1
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 N data , and incorporate it into equations (2) and (1).
t%1
28
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.
29
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
30
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
31
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 -
15,044.00
0
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 -
13,932.00
0
2,361.763 MT -887.114 1,458.939
IA -1,346.879 2,337.443 NV -2,483.897 1,369.116
KY -
15,477.00
0
2,394.393 NM -7,588.133 1,395.432
LA 41,690.000 2,543.303 OR 11,687.000 1,397.194
MI 10,232.000 2,361.763 UT 6,803.640 1,415.495
MN 61,174.000 2,635.798 WY 9,398.653 1,402.487
MS -9,207.288 2,285.436 CA_1 -3,696.948 1,385.102
MO -2,225.616 2,361.763 CA_2 -5,421.502 1,427.980
TN -6,958.016 2,361.763 CA_3 3,580.319 1,385.102
WI 27,254.000 2,361.763 CA_4 -6,475.400 1,378.069
OH_R -9,163.989 2,635.798 CA_5 29,744.000 1,385.102
32
APPENDIX E: Estimating the Mallard Harvest Rate for
the 1998-99 Season
Harvest rate estimates were obtained for banding reference areas 1-3, 5-7 and 12-14 using a single estimate from reference
area 4 (Anderson and Henny 1972), based on estimated correlations between harvest rates among reference areas. The
harvest rate estimate for reference area 4 was based on reward banding (see Current AHM Priorities). The assumed model
for the harvest rate in area j during year t was ht (j) = μj + 0jt, where μj is the area-specific mean and 0jt is correlated noise
with variance * 2 and correlations Corr(0jt,0kt) = Djk , where Djk is the correlation between the harvest rates in reference areas
j and k. These were estimated from the 36 years of available data. The strength of the correlation with area 4 harvest rates
varied substantially with a maximum value of 0.783 occurring between areas 10 and 4 and a minimum value of -0.11
(essentially 0) occurring between areas 1 and 4. We wish to predict the value of ht (j) for any value of j from the observed
value ht(4). In general, the best linear unbiased predictor (i.e., the “BLUP”) of any random variable, say Z, given another
random variable, say Y, is equivalent to the conditional expectation of Z given Y; i.e., E[Z|Y]. It can be shown that E[Z|Y] =
E[Z] + Cov(Z,Y)Var(Y)-1(Y - E[Y]).
In our case, this expression takes on the very simple form (not so, in general) E[ht (j)] = μj + Dj4(ht (4) - μ4), and it can be
shown that this is equivalent to the prediction obtained by regression of ht (j) onto ht (4). The estimate is obtained by
substituting parameter estimates into this expression (e.g., μ8j for μj ). The variance of this prediction is obtained as: Var(h8t
(j)) = * 2 (1 - D2
j4 + 1/n). Thus, we see that as the correlation with reference area 4 approaches 1, the prediction variance
approaches * 2/n, which is the variance of the mean. The prediction and associated prediction variance were computed for
each of the nine reference areas. These predictions were then weighted by the proportion of the midcontinent mallard
population in each area during spring 1998 to construct an estimate of the overall harvest rate.
The estimated harvest rate of adult male mallards in the midcontinent region during the 1998-99 hunting season was
0.10845 (SE = 0.013).
33
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.
34
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.
35
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.
36
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.
37
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.
NOTES
NOTES