Adaptive harvest management 2005 duck hunting season

              U.S. Fish & Wildlife Service
Adaptive
Harvest
Management
22000055 Huunnttiinngg SSeeaassoonn
1
Adaptive
Harvest
Management
2005 Hunting Season
PREFACE
The process of setting waterfowl hunting regulations is conducted annually in the United States (Blohm 1989).
This process involves a number of meetings where the status of waterfowl is reviewed by the agencies responsible
for setting hunting regulations. In addition, the U.S. Fish and Wildlife Service (USFWS) publishes proposed
regulations in the Federal Register to allow public comment. This document is part of a series of reports intended
to support development of harvest regulations for the 2005 hunting season. Specifically, this report is intended to
provide waterfowl managers and the public with information about the use of adaptive harvest management
(AHM) for setting duck-hunting regulations in the United States. This report provides the most current data,
analyses, and decision-making protocols. However, adaptive management is a dynamic process and some
information presented in this report will differ from that in previous reports.
ACKNOWLEDGMENTS
A working group comprised of representatives from the USFWS, the Canadian Wildlife Service (CWS), and the
four Flyway Councils (Appendix A) was established in 1992 to review the scientific basis for managing
waterfowl harvests. The working group, supported by technical experts from the waterfowl management and
research community, subsequently proposed a framework for adaptive harvest management, which was first
implemented in 1995. The USFWS expresses its gratitude to the AHM Working Group and to the many other
individuals, organizations, and agencies that have contributed to the development and implementation of AHM.
This report was prepared by the USFWS Division of Migratory Bird Management. F. Johnson and G. Boomer
were the principal authors. Individuals that provided essential information or otherwise assisted with report
preparation were D. Case (D.J. Case & Assoc.), M. Conroy (U.S. Geological Survey [USGS]), E. Cooch (Cornell
University), P. Garrettson (USFWS), W. Harvey (Maryland Dept. of Natural Resources), R. Raftovich (USFWS),
E. Reed (Canadian Wildlife Service), K. Richkus (USFWS), J. Royle (USGS), M. Runge (USGS), J. Serie,
(USFWS), S. Sheaffer (Cornell University), and K. Wilkins (USFWS). Comments regarding this document
should be sent to the Chief, Division of Migratory Bird Management - USFWS, 4401 North Fairfax Drive, MS
MSP-4107, Arlington, VA 22203.
Citation: U.S. Fish and Wildlife Service. 2005. Adaptive Harvest Management: 2005 Hunting Season. U.S. Dept. Interior,
Washington, D.C. 43pp.
Cover art: A portion of Mark Anderson’s painting of hooded mergansers (Lophodytes cucullatus ), which was chosen for
the 2005 federal Aduck stamp.@
U.S. Fish & Wildlife Service
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TABLE OF CONTENTS
Executive Summary ....................................................................................................................... 3
Background ................................................................................................................................... 4
Mallard Stocks and Flyway Management...................................................................................... 5
Mallard Population Dynamics........................................................................................................ 6
Harvest-Management Objectives ................................................................................................. 18
Regulatory Alternatives ............................................................................................................... 18
Optimal Regulatory Strategies ..................................................................................................... 22
Application of AHM Concepts to Species of Concern................................................................ 24
Literature Cited ............................................................................................................................ 35
Appendix A: AHM Working Group ........................................................................................... 37
Appendix B: Modeling Mallard Harvest Rates........................................................................... 40
This report and others regarding Adaptive Harvest Management are available online at
http://migratorybirds.fws.gov.
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EXECUTIVE SUMMARY
In 1995 the U.S. Fish and Wildlife Service (USFWS) implemented the Adaptive Harvest Management (AHM)
program for setting duck hunting regulations in the United States. The AHM approach provides a framework for
making objective decisions in the face of incomplete knowledge concerning waterfowl population dynamics and
regulatory impacts.
The original AHM protocol was based solely on the dynamics of midcontinent mallards, but efforts are being
made to account for mallards breeding eastward and westward of the midcontinent region. The challenge for
managers is to vary hunting regulations among Flyways in a manner that recognizes each Flyway=s unique
breeding-ground derivation of mallards. For the 2005 hunting season, the USFWS will continue to consider a
regulatory choice for the Atlantic Flyway that depends exclusively on the status of eastern mallards. This
arrangement continues to be considered provisional, however, until the implications of this approach are better
understood. The prescribed regulatory choice for the Mississippi, Central, and Pacific Flyways continues to
depend exclusively on the status of midcontinent mallards. Investigations of the dynamics of western mallards
(and their potential effect on regulations in the West) are continuing and the USFWS is not yet prepared to
recommend an AHM protocol for this mallard stock.
The mallard population models that are the basis for prescribing hunting regulations were revised extensively in
2002. These revised models account for an apparent positive bias in estimates of survival and reproductive rates,
and also allow for alternative hypotheses concerning the effects of harvest and the environment in regulating
population size. Model-specific weights reflect the relative confidence in alternative hypotheses, and are updated
annually using comparisons of predicted and observed population sizes. For midcontinent mallards, current
model weights favor the weakly density-dependent reproductive hypothesis (91%). Evidence for the additive-mortality
hypothesis remains equivocal (58%). For eastern mallards, current model weights favor the strongly
density-dependent reproductive hypothesis (64%). By consensus, hunting mortality is assumed to be additive in
eastern mallards.
For the 2004 hunting season, the USFWS is continuing to consider the same regulatory alternatives as last year.
The nature of the restrictive, moderate, and liberal alternatives has remained essentially unchanged since 1997,
except that extended framework dates have been offered in the moderate and liberal alternatives since 2002.
Also, at the request of the Flyway Councils in 2003 the USFWS agreed to exclude closed duck-hunting seasons
from the AHM protocol when the breeding-population size of midcontinent mallards is $5.5 million (traditional
survey area plus the Great Lakes region).
Harvest rates associated with the each of the regulatory alternatives are predicted using Bayesian statistical
methods. Essentially, the idea is to use historical information to develop initial harvest-rate predictions, to make
regulatory decisions based on those predictions, and then to observe realized harvest rates. Those observed
harvest rates, in turn, are used to update the predictions. Using this approach, predictions of harvest rates of
mallards under the regulatory alternatives have been updated based on band-reporting rate studies conducted since
1998. Estimated harvest rates from the 2002-2004 liberal hunting seasons have averaged 0.12 and 0.15 for adult
male midcontinent and eastern mallards, respectively. The estimated marginal effect of framework-date
extensions has been about 11% and 4% on mid-continent and eastern mallards, respectively.
Optimal regulatory strategies for the 2005 hunting season were calculated using: (1) harvest-management
objectives specific to each mallard stock; (2) the 2004 regulatory alternatives; and (3) current population models
and associated weights for midcontinent and eastern mallards. Based on this year=s survey results of 7.54 million
midcontinent mallards (traditional survey area plus MN, WI, and MI), 3.92 million ponds in Prairie Canada, and
1.05 million eastern mallards, the optimal regulatory choice for all four Flyways is the liberal alternative.
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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 population size, habitat conditions, and harvest levels. Data
collected from this monitoring program are analyzed each year, and proposals for duck-hunting regulations are
developed by the Flyway Councils, States, and USFWS. After extensive public review, the USFWS announces
regulatory guidelines within which States can set their hunting seasons.
In 1995, the USFWS adopted the concept of adaptive resource management (Walters 1986) for regulating duck
harvests in the United States. This approach explicitly recognizes that the consequences of hunting regulations
cannot be predicted with certainty, and provides a framework for making objective decisions in the face of that
uncertainty (Williams and Johnson 1995). Inherent in the adaptive approach is an awareness that management
performance can be maximized only if regulatory effects can be predicted reliably. Thus, adaptive management
relies on an iterative cycle of monitoring, assessment, and decision-making to clarify the relationships among
hunting regulations, harvests, and waterfowl abundance.
In regulating waterfowl harvests, managers face four fundamental sources of uncertainty (Nichols et al. 1995,
Johnson et al. 1996, Williams et al. 1996):
(1) environmental variation - the temporal and spatial variation in weather conditions and other key features
of waterfowl habitat; an example is the annual change in the number of ponds in the Prairie Pothole
Region, where water conditions influence duck reproductive success;
(2) partial controllability - the ability of managers to control harvest only within limits; the harvest resulting
from a particular set of hunting regulations cannot be predicted with certainty because of variation in
weather conditions, timing of migration, hunter effort, and other factors;
(3) partial observability - the ability to estimate key population attributes (e.g., population size, reproductive
rate, harvest) only within the precision afforded by extant monitoring programs; and
(4) structural uncertainty - an incomplete understanding of biological processes; a familiar example is the
long-standing debate about whether harvest is additive to other sources of mortality or whether
populations compensate for hunting losses through reduced natural mortality. Structural uncertainty
increases contentiousness in the decision-making process and decreases the extent to which managers can
meet long-term conservation goals.
AHM was developed as a systematic process for dealing objectively with these uncertainties. The key
components of AHM include (Johnson et al. 1993, Williams and Johnson 1995):
(1) a limited number of regulatory alternatives, which describe Flyway-specific season lengths, bag limits,
and framework dates;
(2) a set of population models describing various hypotheses about the effects of harvest and environmental
factors on waterfowl abundance;
(3) a measure of reliability (probability or "weight") for each population model; and
(4) a mathematical description of the objective(s) of harvest management (i.e., an "objective function"), by
which alternative regulatory strategies can be compared.
These components are used in a stochastic optimization procedure to derive a regulatory strategy. A regulatory
strategy specifies the optimal regulatory choice, with respect to the stated management objectives, for each
possible combination of breeding population size, environmental conditions, and model weights (Johnson et al.
1997). The setting of annual hunting regulations then involves an iterative process:
(1) each year, an optimal regulatory choice is identified based on resource and environmental conditions, and
on current model weights;
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(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 the best overall predictor of changes in population abundance. The process is optimal in the sense
that it provides the regulatory choice each year necessary to maximize management performance. It is adaptive in
the sense that the harvest strategy Aevolves@ to account for new knowledge generated by a comparison of
predicted and observed population sizes.
MALLARD STOCKS AND FLYWAY MANAGEMENT
Since its inception, AHM has focused on the population dynamics and harvest potential of mallards, especially
those breeding in midcontinent North America. Mallards constitute a large portion of the total U.S. duck harvest,
and traditionally have been a reliable indicator of the status of many other species. As management capabilities
have grown, there has been increasing interest in the ecology and management of breeding mallards that occur
outside the midcontinent region. Geographic differences in the reproduction, mortality, and migrations of mallard
stocks suggest that there may be corresponding differences in optimal levels of sport harvest. The ability to
regulate harvests of mallards originating from various breeding areas is complicated, however, by the fact that a
large degree of mixing occurs during the hunting season. The challenge for managers, then, is to vary hunting
regulations among Flyways in a manner that recognizes each Flyway=s unique breeding-ground derivation of
mallards. Of course, no Flyway receives mallards exclusively from one breeding area, and so Flyway-specific
harvest strategies ideally must account for multiple breeding stocks that are exposed to a common harvest.
The optimization procedures used in AHM can account for breeding populations of mallards beyond the
midcontinent region, and for the manner in which these ducks distribute themselves among the Flyways during
the hunting season. An optimal approach would allow for Flyway-specific regulatory strategies, which in a sense
represent for each Flyway an average of the optimal harvest strategies for each contributing breeding stock,
weighted by the relative size of each stock in the fall flight. This Ajoint optimization@ of multiple mallard stocks
requires:
(1) models of population dynamics for all recognized stocks of mallards;
(2) an objective function that accounts for harvest-management goals for all mallard stocks in the aggregate;
and
(3) decision rules allowing Flyway-specific regulatory choices.
Joint optimization of multiple stocks presents many challenges in terms of population modeling, parameter
estimation, and computation of regulatory strategies. These challenges cannot always be overcome due to
limitations in monitoring and assessment programs, and in access to sufficient computing resources. In some
cases, it may be possible to impose constraints or assumptions that simplify the problem. Although sub-optimal
by design, these constrained regulatory strategies may perform nearly as well as those that are optimal,
particularly in cases where breeding stocks differ little in their ability to support harvest, where Flyways do not
receive significant numbers of birds from more than one breeding stock, or where management outcomes are
highly uncertain.
Currently, two stocks of mallards are officially recognized for the purposes of AHM (Fig. 1). We continue to use
a constrained approach to the optimization of these stocks= harvest, whereby the Atlantic Flyway regulatory
strategy is based exclusively on the status of eastern mallards, and the regulatory strategy for the remaining
Flyways is based exclusively on the status of midcontinent mallards. This approach has been determined to
perform nearly as well as a joint-optimization approach because mixing of the two stocks during the hunting
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Fig 1. Survey areas currently assigned to the midcontinent and eastern stocks of mallards for the purposes
of AHM. Delineation of the western-mallard stock is pending further review of population monitoring
programs.
season is limited. However, the approach continues to be considered provisional until its implications are better
understood.
MALLARD POPULATION DYNAMICS
Midcontinent Mallards
Population size.--For the purposes of AHM, midcontinent mallards currently are defined as those breeding in
federal survey strata 1-18, 20-50, and 75-77 (i.e., the Atraditional@ survey area), and in Minnesota, Wisconsin, and
Michigan. Estimates of the abundance of this midcontinent population are available only since 1992 (Table 1).
Population models.-In 2002 we extensively revised the set of alternative models describing the population
dynamics of midcontinent mallards (Runge et al. 2002, USFWS 2002). Collectively, the models express
uncertainty (or disagreement) about whether harvest is an additive or compensatory form of mortality (Burnham
et al. 1984), and whether the reproductive process is weakly or strongly density-dependent (i.e., the degree to
which reproductive rates decline with increasing population size).
midcontinent
eastern
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Table 1. Estimates (N) and standard errors (SE) of mallards (in millions) in spring in the traditional survey area
(strata 1-18, 20-50, and 75-77) and the states of Minnesota, Wisconsin, and Michigan.
Traditional surveys State surveys Total
Year
N SE N SE N SE
1992 5.9761 0.2410 0.9946 0.1597 6.9706 0.2891
1993 5.7083 0.2089 0.9347 0.1457 6.6430 0.2547
1994 6.9801 0.2828 1.1505 0.1163 8.1306 0.3058
1995 8.2694 0.2875 1.1214 0.1965 9.3908 0.3482
1996 7.9413 0.2629 1.0251 0.1443 8.9664 0.2999
1997 9.9397 0.3085 1.0777 0.1445 11.0174 0.3407
1998 9.6404 0.3016 1.1224 0.1792 10.7628 0.3508
1999 10.8057 0.3445 1.0591 0.2122 11.8648 0.4046
2000 9.4702 0.2902 1.2350 0.1761 10.7052 0.3395
2001 7.9040 0.2269 0.8622 0.1086 8.7662 0.2516
2002 7.5037 0.2465 1.0820 0.1152 8.5857 0.2721
2003 7.9497 0.2673 0.8360 0.0734 8.7857 0.2772
2004 7.4253 0.2820 0.9333 0.0748 8.3586 0.2917
2005 6.7553 0.2808 0.7862 0.06503 7.5415 0.2883
All population models for midcontinent mallards share a common “balance equation” to predict changes in
breeding-population size as a function of annual survival and reproductive rates:
Nt Nt (mSt AM ( m)(St AF Rt (St JF St JM F )))
sum
M
sum
+ = + − + + 1 , 1 , , , φ φ
where:
N = breeding population size,
m = proportion of males in the breeding population,
SAM, SAF, SJF, and SJM = survival rates of adult males, adult females, young females, and young males, respectively,
R = reproductive rate, defined as the fall age ratio of females,
φ F φ
sum
M
sum
= the ratio of female (F) to male (M) summer survival, and
t = year.
We assumed that m and φ F φ
sum
M
sum are fixed and known. We also assumed, based in part on information provided by
Blohm et al. (1987), the ratio of female to male summer survival was equivalent to the ratio of annual survival
rates in the absence of harvest. Based on this assumption, we estimated φ F φ
sum
M
sum = 0.897. To estimate m we
expressed the balance equation in matrix form:
N
N
S RS
S RS
N
N
t AM
t AF
AM JM F
sum
M
sum
AF JF
t AM
t AF
+
+
⎡
⎣ ⎢
⎤
⎦ ⎥
=
+
⎡
⎣ ⎢
⎤
⎦ ⎥
⎡
⎣ ⎢
⎤
⎦ ⎥
1
1 0
,
,
,
,
φ φ
and substituted the constant ratio of summer survival and mean values of estimated annual survival and
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reproductive rates. The right eigenvector of the transition matrix is the stable sex structure that the breeding
population eventually would attain with these constant demographic rates. This eigenvector yielded an estimate
of m = 0.5246.
Using estimates of annual survival and reproductive rates, the balance equation for midcontinent mallards over-predicted
observed population sizes by 10.8% on average. The source of the bias is unknown, so we modified the
balance equation to eliminate the bias by adjusting both survival and reproductive rates:
Nt S Nt (mSt AM ( m)(St AF RRt (St JF St JM F )))
sum
M
sum
+ 1 = γ , + 1− , + γ , + , φ φ
where ( denotes the bias-correction factors for survival (S) and reproduction (R). We used a least squares
approach to estimate (S
= 0.9479 and (R = 0.8620.
Survival process.–We considered two alternative hypotheses for the relationship between annual survival and
harvest rates. For both models, we assumed that survival in the absence of harvest was the same for adults and
young of the same sex. In the model where harvest mortality is additive to natural mortality:
St sex age s sex ( K )
A
, , , t ,sex,age = − 0 1
and in the model where changes in natural mortality compensate for harvest losses (up to some threshold):
S
s if K s
t sex age K if K s
sex
C
t sex age sex
C
t sex age t sex age sex
, , C
, ,, ,
, , , , ,
=
≤ −
− > −
⎧⎨ ⎪
⎩⎪
0 0
0
1
1 1
where s0 = survival in the absence of harvest under the additive (A) or compensatory (C) model, and K = harvest
rate adjusted for crippling loss (20%, Anderson and Burnham 1976). We averaged estimates of s0 across banding
reference areas by weighting by breeding-population size. For the additive model, s0 = 0.7896 and 0.6886 for
males and females, respectively. For the compensatory model, s0 = 0.6467 and 0.5965 for males and females,
respectively. These estimates may seem counterintuitive because survival in the absence of harvest should be the
same for both models. However, estimating a common (but still sex-specific) s0 for both models leads to
alternative models that do not fit available band-recovery data equally well. More importantly, it suggests that the
greatest uncertainty about survival rates is when harvest rate is within the realm of experience. By allowing s0 to
differ between additive and compensatory models, we acknowledge that the greatest uncertainty about survival
rate is its value in the absence of harvest (i.e., where we have no experience).
Reproductive process.–Annual reproductive rates were estimated from age ratios in the harvest of females,
corrected using a constant estimate of differential vulnerability. Predictor variables were the number of ponds in
May in Prairie Canada (P, in millions) and the size of the breeding population (N, in millions). We estimated the
best-fitting linear model, and then calculated the 80% confidence ellipsoid for all model parameters. We chose
the two points on this ellipsoid with the largest and smallest values for the effect of breeding-population size, and
generated a weakly density-dependent model:
Rt = 0.7166 + 0.1083Pt − 0.0373Nt
and a strongly density-dependent model:
Rt = 1.1390 + 0.1376Pt − 0.1131Nt
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Pond dynamics.–We modeled annual variation in Canadian pond numbers as a first-order autoregressive process.
The estimated model was:
Pt + Pt t = + + 1 2.2127 0.3420 ε
where ponds are in millions and εt
is normally distributed with mean = 0 and variance = 1.2567.
Variance of prediction errors.–Using the balance equation and sub-models described above, predictions of
breeding-population size in year t+1 depend only on specification of population size, pond numbers, and harvest
rate in year t. For the period in which comparisons were possible, we compared these predictions with observed
population sizes.
We estimated the prediction-error variance by setting:
( ) ( )
( )
[ ( ) ( )] ( )
e N N
e N
N N n
t t
obs
t
pre
t
t
obs
t
pre
t
= −
= Σ − −
ln ln
~ ,
$ ln ln
then assuming
and estimating
0
1
σ 2
σ 2 2
where obs and pre are observed and predicted population sizes (in millions), respectively, and n = the number of
years being compared. We were concerned about a variance estimate that was too small, either by chance or
because the number of years in which comparisons were possible was small. Therefore, we calculated the upper
80% confidence limit for F2 based on a Chi-squared distribution for each combination of the alternative survival
and reproductive sub-models, and then averaged them. The final estimate of F2 was 0.0243, equivalent to a
coefficient of variation of about 17%.
Model implications.BThe set of alternative population models suggests that carrying capacity (average population
size in the absence of harvest) for an average number of Canadian ponds is somewhere between about 6 and 16
million mallards. The population model with additive hunting mortality and weakly density-dependent
recruitment (SaRw) leads to the most conservative harvest strategy, whereas the model with compensatory
hunting mortality and strongly density-dependent recruitment (ScRs) leads to the most liberal strategy. The other
two models (SaRs and ScRw) lead to strategies that are intermediate between these extremes. Under the models
with compensatory hunting mortality (ScRs and ScRw), the optimal strategy is to have a liberal regulation
regardless of population size or number of ponds because at harvest rates achieved under the liberal alternative,
harvest has no effect on population size. Under the strongly density-dependent model (ScRs), the density-dependence
regulates the population and keeps it within narrow bounds. Under the weakly density-dependent
model (ScRw), the density-dependence does not exert as strong a regulatory effect, and the population size
fluctuates more.
Model weights.--Model weights are calculated as Bayesian probabilities, reflecting the relative ability of the
individual alternative models to predict observed changes in population size. The Bayesian probability for each
model is a function of the model=s previous (or prior) weight and the likelihood of the observed population size
under that model. We used Bayes= theorem to calculate model weights from a comparison of predicted and
observed population sizes for the years 1996-2004, starting with equal model weights in 1995. For the purposes
of updating, we predicted breeding-population size in the traditional survey area in year t + 1, from breeding-population
size, Canadian ponds, and harvest rates in year t.
Model weights changed little until all models under-predicted the change in population size from 1998 to 1999,
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perhaps indicating there is a significant factor affecting population dynamics that is absent from all four models
(Table 2). Throughout the period of updating model weights, there has been no clear preference for either the
additive (58%) or compensatory (42%) mortality models. For most of the time frame, model weights favor the
weakly density-dependent (91%) reproductive model over the strongly density-dependent (9%) one. The reader
is cautioned, however, that models can sometimes make reliable predictions of population size for reasons having
little to do with the biological hypotheses expressed therein (Johnson et al. 2002b).
Inclusion of mallards in the Great Lakes region.--Model development originally did not include mallards
breeding in the states of Wisconsin, Minnesota, and Michigan, primarily because full data sets were not available
from these areas to allow appropriate analysis. However, mallards in the Great Lakes region have been included
in the midcontinent mallard AHM protocol since 1997 by assuming that population dynamics for these mallards
are similar to those in the traditional survey area. Based on that assumption, predictions of breeding population
size are scaled to reflect inclusion of mallards in the Great Lakes region. From 1992 through 2005, when
population estimates were available for all three states, the average proportion of the total midcontinent mallard
population that was in the Great Lakes region was 0.1143 (SD = 0.0178). We assumed a normal distribution with
these parameter values to make the conversion between the traditional survey area and total breeding-population
size.
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Table 2. Model-specific predictions and weights for midcontinent mallards (ScRs = compensatory mortality and strongly
density-dependent reproduction, ScRw = compensatory mortality and weakly density-dependent reproduction, SaRs = additive
mortality and strongly density-dependent reproduction, and SaRw = additive mortality and weakly density-dependent
reproduction). Model weights were assumed to be equal in 1995.
Model
Year Bpop(t)a Ponds(t)b Harvest
rate(t)c
ScRs ScRw SaRs SaRw
Observed
bpop(t+1)a
predicted
Bpop(t+1):
7.6740 8.0153 7.7037 8.0280
1995 8.2694 3.8925 0.1198
weight(t+1): 0.2469 0.2525 0.2482 0.2524
7.9413
predicted
Bpop(t+1):
8.0580 8.1776 8.0702 8.1841
1996 7.9413 5.0026 0.1184
weight(t+1): 0.2305 0.2666 0.2348 0.2681
9.9397
predicted
bpop(t+1):
9.0964 9.9258 9.1833 9.9768
1997 9.9397 5.0610 0.1166
weight(t+1): 0.2235 0.2722 0.2324 0.2719
9.6404
predicted
bpop(t+1):
7.4334 8.4655 7.6471 8.6474
1998 9.6404 2.5217 0.1102
weight(t+1): 0.0596 0.3801 0.0944 0.4659
10.8057
predicted
bpop(t+1):
8.5916 9.9905 8.9478 10.3308
1999 10.8057 3.8620 0.1004
weight(t+1): 0.0548 0.4007 0.0987 0.4458
9.4702
predicted
bpop(t+1):
7.3262 8.2969 7.3621 8.2718
2000 9.4702 2.4222 0.1264
weight(t+1): 0.0514 0.4033 0.0940 0.4513
7.9040
predicted
bpop(t+1):
6.9153 7.2626 7.0917 7.4301
2001 7.9040 2.7472 0.1077
weight(t+1): 0.0459 0.4035 0.0900 0.4607
7.5040
predicted
bpop(t+1):
6.1036 6.4607 6.2325 6.5766
2002 7.5040 1.4390 0.1133
weight(t+1): 0.0257 0.3928 0.0628 0.5187
7.9497
predicted
bpop(t+1):
7.3237 7.6031 7.4291 7.6983
2003 7.9497 3.5223 0.1132
weight(t+1): 0.0261 0.3956 0.0639 0.5144
7.4253
predicted
bpop(t+1):
6.5706 6.7972 6.5662 6.7794
2004 7.4253 2.5126 0.1245
weight(t+1): 0.0257 0.3960 0.0630 0.5152
6.7553
a Breeding population size (in millions) in the traditional survey area only (i.e., does not include Minnesota, Michigan, and
Wisconsin) in year t.
b Ponds (in millions) in May in Prairie Canada.
c Harvest rate of adult-male midcontinent mallards. Harvest rates for 1995 and 1996 were based on recovery rates of standard
bands, corrected for band-reporting rates. For 1997, we used the most recent estimate of the posterior mean of the harvest
rate under the 1997 liberal regulatory alternative. For 1998-2004, we used the most recent posterior estimates of the actual
harvest rates in those years.
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Eastern Mallards
Population size.--For purposes of AHM, eastern mallards are defined as those breeding in southern Ontario and
Quebec (federal survey strata 51-54 and 56) and in the northeastern U.S. (state plot surveys; Heusman and Sauer
2000) (see Fig. 1). Estimates of population size have varied from 856 thousand to 1.1 million since 1990, with
the majority of the population accounted for in the northeastern U.S. (Table 3). The reader is cautioned that these
estimates differ from those reported in the USFWS annual waterfowl trend and status reports, which include
composite estimates based on more fixed-wing strata in eastern Canada and helicopter surveys conducted by the
Canadian Wildlife Service.
Table 3. Estimates (N) and associated standard errors (SE) of mallards (in thousands) in spring in the
northeastern U.S. (state plot surveys) and eastern Canada (federal survey strata 51-54 and 56).
State surveys Federal surveys Total
Year N SE N SE N SE
1990 665.1 78.3 190.7 47.2 855.8 91.4
1991 779.2 88.3 152.8 33.7 932.0 94.5
1992 562.2 47.9 320.3 53.0 882.5 71.5
1993 683.1 49.7 292.1 48.2 975.2 69.3
1994 853.1 62.7 219.5 28.2 1072.5 68.7
1995 862.8 70.2 184.4 40.0 1047.2 80.9
1996 848.4 61.1 283.1 55.7 1131.5 82.6
1997 795.1 49.6 212.1 39.6 1007.2 63.4
1998 775.1 49.7 263.8 67.2 1038.9 83.6
1999 879.7 60.2 212.5 36.9 1092.2 70.6
2000 757.8 48.5 132.3 26.4 890.0 55.2
2001 807.5 51.4 200.2 35.6 1007.7 62.5
2002 834.1 56.2 171.3 30.0 1005.4 63.8
2003 731.8 47.0 308.3 55.4 1040.1 72.6
2004 809.1 51.8 301.5 53.3 1110.7 74.3
2005 753.6 53.6 293.4 53.1 1047.0 75.5
Population models.–We also revised the population models for eastern mallards in 2002 (Johnson et al. 2002a,
USFWS 2002). The current set of six models: (1) relies solely on federal and state waterfowl surveys (rather than
the Breeding Bird Survey) to predict reproductive rates; (2) allows for the possibility of a positive bias in
estimates of survival or reproductive rates; (3) incorporates competing hypotheses of strongly and weakly density-dependent
reproduction; and (4) assumes that hunting mortality is additive to other sources of mortality.
As with midcontinent mallards, all population models for eastern mallards share a common balance equation to
predict changes in breeding-population size as a function of annual survival and reproductive rates:
13
Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S ))
am
t
af
t
m
t
ym
t
m
t
yf
+ = ⋅ ⋅ + − ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 1 1 ψ
where:
N = breeding-population size,
p = proportion of males in the breeding population,
Sam, Saf, Sym, and Syf = survival rates of adult males, adult females, young males, and young females, respectively,
Am = ratio of young males to adult males in the harvest,
d = ratio of young male to adult male direct recovery rates,
R = the ratio of male to female summer survival, and
t = year.
In this balance equation, we assume that p, d, and R are fixed and known. The parameter R is necessary to
account for the difference in anniversary date between the breeding-population survey (May) and the survival and
reproductive rate estimates (August). This model also assumes that the sex ratio of fledged young is 1:1; hence
Am/d appears twice in the balance equation. We estimated d = 1.043 as the median ratio of young:adult male
band-recovery rates in those states from which wing receipts were obtained. We estimated R = 1.216 by
regressing through the origin estimates of male survival against female survival in the absence of harvest,
assuming that differences in natural mortality between males and females occur principally in summer. To
estimate p, we used a population projection matrix of the form:
( )
( )
M
F
S A d S
A d S S
M
F
t
t
am m ym
m yf af
t
t
+
+
+
+
⎡
⎣ ⎢
⎤
⎦ ⎥
=
+ ⋅
⋅ ⋅
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
⎡
⎣ ⎢
⎤
⎦ ⎥
1
1
1
1
0
ψ
where M and F are the relative number of males and females in the breeding populations, respectively. To
parameterize the projection matrix we used average annual survival rate and age ratio estimates, and the estimates
of d and R provided above. The right eigenvector of the projection matrix is the stable proportion of males and
females the breeding population eventually would attain in the face of constant demographic rates. This
eigenvector yielded an estimate of p = 0.544.
We also attempted to determine whether estimates of survival and reproductive rates were unbiased. We relied on
the balance equation provided above, except that we included additional parameters to correct for any bias that
might exist. Because we were unsure of the source(s) of potential bias, we alternatively assumed that any bias
resided solely in survival rates:
Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S ))
am
t
af
t
m
t
ym
t
m
t
yf
+ = ⋅ ⋅ ⋅ + − ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 1 Ω 1 ψ
(where S is the bias-correction factor for survival rates), or solely in reproductive rates:
Nt Nt ((p St ) (( p) S ) (p (A d) S ) (p (A d) S ))
am
t
af
t
m
t
ym
t
m
t
yf
+ = ⋅ ⋅ + − ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ 1 1 α α ψ
(where " is the bias-correction factor for reproductive rates). We estimated S and " by determining the values of
these parameters that minimized the sum of squared differences between observed and predicted population sizes.
Based on this analysis, S = 0.836 and " = 0.701, suggesting a positive bias in survival or reproductive rates.
However, because of the limited number of years available for comparing observed and predicted population
sizes, we also retained the balance equation that assumes estimates of survival and reproductive rates are
unbiased.
14
Survival process.–For purposes of AHM, annual survival rates must be predicted based on the specification of
regulation-specific harvest rates (and perhaps on other uncontrolled factors). Annual survival for each age (i) and
sex (j) class under a given regulatory alternative is:
( )
( ) S
h v
c t
i j j t
am i j
,
,
= ⋅ −
⋅
−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
θ 1
1
where:
S = annual survival,
θ j
= mean survival from natural causes,
ham = harvest rate of adult males, and
v = harvest vulnerability relative to adult males,
c = rate of crippling (unretrieved harvest).
This model assumes that annual variation in survival is due solely to variation in harvest rates, that relative
harvest vulnerability of the different age-sex classes is fixed and known, and that survival from natural causes is
fixed at its sample mean. We estimated θ j = 0.7307 and 0.5950 for males and females, respectively.
Reproductive process.–As with survival, annual reproductive rates must be predicted in advance of setting
regulations. We relied on the apparent relationship between breeding-population size and reproductive rates:
Rt = a ⋅ exp(b ⋅ Nt )
where Rt is the reproductive rate (i.e., At d
m
), Nt is breeding-population size in millions, and a and b are model
parameters. The least-squares parameter estimates were a = 2.508 and b = -0.875. Because of both the
importance and uncertainty of the relationship between population size and reproduction, we specified two
alternative models in which the slope (b) was fixed at the least-squares estimate ± one standard error, and in
which the intercepts (a) were subsequently re-estimated. This provided alternative hypotheses of strongly
density-dependent (a = 4.154, b = -1.377) and weakly density-dependent reproduction (a = 1.518, b = -0.373).
Variance of prediction errors.--Using the balance equations and sub-models provided above, predictions of
breeding-population size in year t+1 depend only on the specification of a regulatory alternative and on an
estimate of population size in year t. For the period in which comparisons were possible (1991-96), we were
interested in how well these predictions corresponded with observed population sizes. In making these
comparisons, we were primarily concerned with how well the bias-corrected balance equations and reproductive
and survival sub-models performed. Therefore, we relied on estimates of harvest rates rather than regulations as
model inputs.
We estimated the prediction-error variance by setting:
( ) ( )
( )
[ ( ) ( )]
e N N
e N
N N n
t t
obs
t
pre
t
t
obs
t
pre
t
= −
= Σ −
ln ln
~ ,
$ ln ln
then assuming
and estimating
0 σ 2
σ 2 2
15
where obs and pre are observed and predicted population sizes (in millions), respectively, and n = 6.
Variance estimates were similar regardless of whether we assumed that the bias was in reproductive rates or in
survival, or whether we assumed that reproduction was strongly or weakly density-dependent. Thus, we averaged
variance estimates to provide a final estimate of F2 = 0.006, which is equivalent to a coefficient of variation (CV)
of 8.0%. We were concerned, however, about the small number of years available for estimating this variance.
Therefore, we estimated an 80% confidence interval for F2 based on a Chi-squared distribution and used the upper
limit for F2 = 0.018 (i.e., CV = 14.5%) to express the additional uncertainty about the magnitude of prediction
errors attributable to potentially important environmental effects not expressed by the models.
Model implications.--Model-specific regulatory strategies based on the hypothesis of weakly density-dependent
reproduction are considerably more conservative than those based on the hypothesis of strongly density-dependent
reproduction. The three models with weakly density-dependent reproduction suggest a carrying
capacity (i.e., average population size in the absence of harvest) >2.0 million mallards, and prescribe extremely
restrictive regulations for population size <1.0 million. The three models with strongly density-dependent
reproduction suggest a carrying capacity of about 1.5 million mallards, and prescribe liberal regulations for
population sizes >300 thousand. Optimal regulatory strategies are relatively insensitive to whether models
include a bias correction or not. All model-specific regulatory strategies are Aknife-edged,@ meaning that large
differences in the optimal regulatory choice can be precipitated by only small changes in breeding-population
size. This result is at least partially due to the small differences in predicted harvest rates among the current
regulatory alternatives (see the section on Regulatory Alternatives later in this report).
Model weights.—Beginning this year, we used Bayes= theorem to calculate model weights from a comparison of
predicted and observed population sizes for the years 1996-2004. We calculated weights for the alternative
models based on an assumption of equal model weights in 1996 (the last year data was used to develop most
model components) and on estimates of year-specific harvest rates (Appendix B). There is no single model that is
clearly favored over the others at the end of the time frame, although collectively the models with strongly
density-dependent reproduction are better predictors of changes in population size as those with weak density
dependence. In addition, the change this year from the use of predicted to estimated harvest rates in the updating
of model weights led to substantial evidence of bias in extant estimates of survival and/or reproductive rates
(93%).
16
Table 4. Model-specific predictions and weights for eastern mallards (BnRw = no bias-correction and weakly density-dependent
reproduction, BnRs = no bias-correction and strongly density-dependent reproduction, BsRw = bias-corrected
survival rates and weakly density-dependent reproduction, BsRs = bias-corrected survival rates and strongly density-dependent
reproduction, BrRw = bias-corrected reproductive rates and weakly density-dependent reproduction, and BrRs =
bias-corrected reproductive rates and strongly density-dependent reproduction). Model weights were assumed to be equal in
1996.
Model
Year Bpop(t)a Harvest
rate(t)b BnRw BnRs BsRw BsRs BrRw BrRs
Observed
bpop(t+1)a
predicted
bpop(t+1): 1.2577 1.1791 1.0511 0.9854 1.0625 1.0074
1996 1.1315 0.1510
weight(t+1): 0.0565 0.1100 0.2053 0.2129 0.1996 0.2157
1.0072
predicted
bpop(t+1): 1.1197 1.1175 0.9357 0.9339 0.9428 0.9412
1997 1.0072 0.1626
weight(t+1): 0.0628 0.1232 0.1974 0.2024 0.2000 0.2142
1.0389
predicted
bpop(t+1): 1.1477 1.1267 0.9592 0.9416 0.9674 0.9527
1998 1.0389 0.1626
weight(t+1): 0.0866 0.1769 0.1842 0.1643 0.1977 0.1902
1.0922
predicted
bpop(t+1): 1.1941 1.1412 0.9980 0.9538 1.0083 0.9712
1999 1.0922 0.1626
weight(t+1): 0.0139 0.0552 0.2155 0.2409 0.2164 0.2583
0.8900
predicted
bpop(t+1): 1.0127 1.0786 0.8464 0.9014 0.8494 0.8956
2000 0.8900 0.1626
weight(t+1): 0.0229 0.0806 0.1562 0.2847 0.1623 0.2932
1.0077
predicted
bpop(t+1): 1.1201 1.1176 0.9361 0.9340 0.9432 0.9414
2001 1.0077 0.1626
weight(t+1): 0.0193 0.0689 0.1574 0.2844 0.1681 0.3018
1.0054
predicted
bpop(t+1): 1.1181 1.1169 0.9344 0.9335 0.9414 0.9406
2002 1.0054 0.1626
weight(t+1): 0.0222 0.0793 0.1523 0.2736 0.1696 0.3031
1.0401
predicted
bpop(t+1): 1.1864 1.1638 0.9915 0.9726 0.9990 0.9832
2003 1.0401 0.1471
weight(t+1): 0.0282 0.1069 0.1537 0.2427 0.1789 0.2896
1.1107
predicted
bpop(t+1): 1.2825 1.2139 1.0718 1.0145 1.0815 1.0335
2004 1.1107 0.1342
weight(t+1): 0.0101 0.0643 0.1649 0.2573 0.1894 0.3140
1.0470
a Breeding population size (in millions) in the northeastern U.S. (state plot surveys) and eastern Canada (federal survey strata
51-54 and 56) in year t.
b Harvest rate of adult-male eastern mallards. The harvest rate for 1996 was based on the recovery rates of standard bands,
corrected for band-reporting rates. For 1997-2001, we used the most recent estimate of the posterior mean of the harvest rate
under the liberal regulatory alternatives for those years. For 2002-2004, we used the most recent posterior estimates of the
actual harvest rates in those years.
Western Mallards
Substantial numbers of mallards occur in the states of the Pacific Flyway (including Alaska), British Columbia,
and the Yukon Territory during the breeding season. The distribution of these mallards during fall and winter is
centered in the Pacific Flyway (Munro and Kimball 1982). Unfortunately, data-collection programs for
understanding and monitoring the dynamics of this mallard stock are highly fragmented in both time and space.
This makes it difficult to aggregate monitoring instruments in a way that can be used to reliably model this stock=s
17
dynamics and, thus, to establish criteria for regulatory decision-making under AHM. Another complicating factor
is that federal survey strata 1-12 in Alaska and the Yukon are within the current geographic bounds of
midcontinent mallards. The AHM Working Group is continuing its investigations of western mallards and while
it is not prepared to recommend an AHM protocol at this time, progress is being made on a number of issues:
1) Population modeling – In August 2004, Drs. Herzog and Sedinger circulated a report entitled “Western
Mallard Population Model: Report to the Wildlife Management Institute and the Pacific Flyway Study
Committee.” The report represents an extensive body of work designed to help define the spatial bounds of a
western mallard stock, to estimate breeding population size, survival and recruitment rates, and to synthesize
population models that could be used for management purposes.
2) Breeding populations surveys – The development of AHM for western mallards continues to present technical
challenges that make implementation much more difficult than with either midcontinent or eastern mallards.
In particular, we remain concerned about our ability to reliably determine changes in the population size of
western mallards based on a collection of surveys conducted independently by Pacific Flyway States and the
Province of British Columbia. These surveys tend to vary in design and intensity, and in some cases lack
measures of precision (i.e., sampling error). For example, we still consider the methods for estimating
mallard abundance in British Columbia to be in the development and evaluation phase, and there are as yet
unanswered questions about how mallard abundance will be determined there on an operational basis.
Toward that end, experimental fixed-wing surveys were conducted in British Columbia this spring in a
collaborative effort by the USFWS and the Canadian Wildlife Service. Unfortunately, the terrain required
aircraft speeds and altitudes that were not conducive for surveying waterfowl, and fixed-wing surveys were
deemed an unsatisfactory (and unsafe) approach for a long-term, operational monitoring program. Plans have
now been made to evaluate the use of helicopters next spring for developing surveys that eventually could
cover the majority of key waterfowl habitats in British Columbia.
We also appreciate the cooperation of Pacific Flyway States in helping us better understand the sampling
design of their breeding-population surveys. During the past year, we were able to help the State of
California evaluate their use of helicopters to derive visibility-correction factors for their fixed-wing surveys.
Data suggested helicopter counts provided reasonable adjustments for visibility bias in fixed-wing counts, and
that the bias varied by year and survey strata. In the coming year, we hope to work collaboratively with other
Pacific Flyway States to review their mallard survey programs.
3) Harvest rates – We have been able to estimate harvest rates of adult male mallards in western breeding areas
directly from recoveries of reward bands placed on mallards prior to the hunting seasons in 2002-2004.
Generally, these rates were similar to those for mid-continent mallards, although it is not known how
representative they are of western mallards as a whole (Table 5).
Table 5. Harvest rates (h, and standard errors, se) of adult male mallards banded in states and provinces of the
Pacific Flyway as based on reward banding.
Region
Year AK BC WA OR CA
h se h se h se h se h se
2002 0.1121 0.0306 0.1504 0.0228 0.2308 0.0683 0.1173 0.0241 0.1013 0.0122
2003 0.1000 0.0391 0.1382 0.0281 0.1304 0.0502 0.1324 0.0292 0.0858 0.0135
2004 0.0968 0.0379 0.2727 0.0679 0.0853 0.0247 0.1460 0.0235
mean 0.1030 0.0057 0.1443 0.0087 0.2113 0.0517 0.1116 0.0170 0.1111 0.0221
18
HARVEST-MANAGEMENT OBJECTIVES
The basic harvest-management objective for midcontinent mallards is to maximize cumulative harvest over the
long term, which inherently requires perpetuation of a viable population. Moreover, this objective is constrained
to avoid regulations that could be expected to result in a subsequent population size below the goal of the North
American Waterfowl Management Plan (NAWMP) (Fig. 2). According to this constraint, the value of harvest
decreases proportionally as the difference between the goal and expected population size increases. This balance
of harvest and population objectives results in a regulatory strategy that is more conservative than that for
maximizing long-term harvest, but more liberal than a strategy to attain the NAWMP goal (regardless of effects
on hunting opportunity). The current objective uses a population goal of 8.8 million mallards, which is based on
8.2 million mallards in the traditional survey area (from the 1998 update of the NAWMP) and a goal of 0.6
million for the combined states of Minnesota, Wisconsin, and Michigan.
Fig. 2. The relative value of midcontinent mallard harvest, expressed as a function
of breeding-population size expected in the subsequent year.
For eastern mallards, there is no NAWMP goal or other established target for desired population size.
Accordingly, the management objective for eastern mallards is simply to maximize long-term cumulative (i.e.,
sustainable) harvest.
REGULATORY ALTERNATIVES
Evolution of Alternatives
When AHM was first implemented in 1995, three regulatory alternatives characterized as liberal, moderate, and
restrictive were defined based on regulations used during 1979-84, 1985-87, and 1988-93, respectively. These
regulatory alternatives also were considered for the 1996 hunting season. In 1997, the regulatory alternatives
were modified to include: (1) the addition of a 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. In 2002 the USFWS further modified the moderate and liberal alternatives to
include extensions of approximately one week in both the opening and closing framework dates.
Population size expected next year (millions)
0 1 2 3 4 5 6 7 8 9 10 11
0 1 2 3 4 5 6 7 8 9 10 11
Harvest value (%)
0
20
40
60
80
100
0
20
40
60
80
100
NAWMP goal
(8.8 million)
19
In 2003 the very-restrictive alternative was eliminated at the request of the Flyway Councils. Expected harvest
rates under the very-restrictive alternative did not differ significantly from those under the restrictive alternative,
and the very-restrictive alternative was expected to be prescribed for #5% of all hunting seasons. Also, at the
request of the Flyway Councils the USFWS agreed to exclude closed duck-hunting seasons from the AHM
protocol when the breeding-population size of midcontinent mallards is $5.5 million (traditional survey area plus
the Great Lakes region). Based on our assessment, closed hunting seasons do not appear to be necessary from the
perspective of sustainable harvesting when the midcontinent mallard population exceeds this level. The impact of
maintaining open seasons above this level also appears to be negligible for other midcontinent duck species
(scaup, gadwall, wigeon, green-winged teal, blue-winged teal, shoveler, pintail, redhead, and canvasbacks), as
based on population models developed by Johnson (2003). However, complete or partial season-closures for
particular species or populations could still be deemed necessary in some situations regardless of the status of
midcontinent mallards. Details of the regulatory alternatives for each Flyway are provided in Table 5.
Table 5. Regulatory alternatives for the 2005 duck-hunting season.
Flyway
Regulation Atlantica Mississippi Centralb Pacificc
Shooting hours one-half hour before sunrise to sunset
Framework dates
Restrictive Oct 1 - Jan 20 Saturday nearest Oct 1to the Sunday nearest Jan 20
Moderate and
Liberal
Saturday nearest September 24
to the last Sunday in January
Season length (days)
Restrictive 30 30 39 60
Moderate 45 45 60 86
Liberal 60 60 74 107
Bag limit (total / mallard / female mallard)
Restrictive 3 / 3 / 1 3 / 2 / 1 3 / 3 / 1 4 / 3 / 1
Moderate 6 / 4 / 2 6 / 4 / 1 6 / 5 / 1 7 / 5 / 2
Liberal 6 / 4 / 2 6 / 4 / 2 6 / 5 / 2 7 / 7 / 2
a The states of Maine, Massachusetts, Connecticut, Pennsylvania, New Jersey, Maryland, Delaware, West
Virginia, Virginia, and North Carolina are permitted to exclude Sundays, which are closed to hunting, from
their total allotment of season days.
b The High Plains Mallard Management Unit is allowed 8, 12, 23, and 23 extra days in the restrictive,
moderate, and liberal alternatives, respectively.
c The Columbia Basin Mallard Management Unit is allowed seven extra days in the restrictive, and moderate
alternatives.
Regulation-Specific Harvest Rates
Initially, harvest rates of mallards associated with each of the open-season regulatory alternatives were predicted
using harvest-rate estimates from 1979-84, which were adjusted to reflect current hunter numbers and
contemporary specifications of season lengths and bag limits. In the case of closed seasons in the U.S., we
assumed rates of harvest would be similar to those observed in Canada during 1988-93, which was a period of
restrictive regulations both in Canada and the U.S. All harvest-rate predictions were based only in part on band-recovery
data, and relied heavily on models of hunting effort and success derived from hunter surveys (USFWS
20
2002: Appendix C). As such, these predictions had large sampling variances and their accuracy was uncertain.
In 2002 we began relying on Bayesian statistical methods for improving regulation-specific predictions of harvest
rates, including predictions of the effects of framework-date extensions. Essentially, the idea is to use existing
(Aprior@) information to develop initial harvest-rate predictions (as above), to make regulatory decisions based on
those predictions, and then to observe realized harvest rates. Those observed harvest rates, in turn, are treated as
new sources of information for calculating updated (Aposterior@) predictions. Bayesian methods are attractive
because they provide a quantitative and formal, yet intuitive, approach to adaptive management.
For midcontinent mallards, we have empirical estimates of harvest rate from the recent period of liberal hunting
regulations (1998-2004). The Bayesian methods thus allow us to combine these estimates with our prior
predictions to provide updated estimates of harvest rates expected under the liberal regulatory alternative.
Moreover, in the absence of experience (so far) with the restrictive and moderate regulatory alternatives, we
reasoned that our initial predictions of harvest rates associated with those alternatives should be re-scaled based
on a comparison of predicted and observed harvest rates under the liberal regulatory alternative. In other words,
if observed harvest rates under the liberal alternative were 10% less than predicted, then we might also expect that
the mean harvest rate under the moderate alternative would be 10% less than predicted. The appropriate scaling
factors currently are based exclusively on prior beliefs about differences in mean harvest rate among regulatory
alternatives, but they will be updated once we have experience with something other than the liberal alternative.
A detailed description of the analytical framework for modeling mallard harvest rates is provided in Appendix B.
Our models of regulation-specific harvest rates also allow for the marginal effect of framework-date extensions in
the moderate and liberal alternatives. A previous analysis by the USFWS (2001a) suggested that implementation
of framework-date extensions might be expected to increase the harvest rate of midcontinent mallards by about
15%, or in absolute terms by about 0.02 (SD = 0.01) (i.e., our Aprior@ belief). Based on the observed harvest rate
during the 2002-2004 hunting seasons, the updated (Aposterior@) estimate of the marginal change in harvest rate
attributable to the framework-date extension is 0.012 (SD = 0.008). Therefore, the estimated effect of the
framework-date extension has been to increase harvest rate of midcontinent mallards by about 11% over what
would otherwise be expected in the liberal alternative. However, the reader is strongly cautioned that reliable
inference about the marginal effect of framework-date extensions ultimately depends on a rigorous experimental
design (including controls and random application of treatments).
Current predictions of harvest rates of adult-male midcontinent mallards associated with each of the regulatory
alternatives are provided in Table 6 and Fig. 3. Predictions of harvest rates for the other age-sex cohorts are based
on the historical ratios of cohort-specific harvest rates to adult-male rates (Runge et al. 2002). These ratios are
considered fixed at their long-term averages and are 1.5407, 0.7191, and 1.1175 for young males, adult females,
and young females, respectively. We continued to make the simplifying assumption that the harvest rates of
midcontinent mallards depend solely on the regulatory choice in the western three Flyways. This appears to be a
reasonable assumption given the small proportion of midcontinent mallards wintering in the Atlantic Flyway
(Munro and Kimball 1982), and harvest-rate predictions that suggest a minimal effect of Atlantic Flyway
regulations (USFWS 2000). Under this assumption, the optimal regulatory strategy for the western three Flyways
can be derived by ignoring the harvest regulations imposed in the Atlantic Flyway.
Until this year, predictions of harvest rates for eastern mallards have depended exclusively on historical (Aprior@)
information because more contemporary estimates of harvest rate were unavailable. However, we have now
begun updating the predictions of eastern-mallard harvest rates in the same fashion as that for midcontinent
mallards based on reward banding conducted in eastern Canada and the northeastern U.S. (Appendix B). Like
midcontinent mallards, harvest rates of age and sex cohorts other than adult male mallards are based on constant
rates of differential vulnerability as derived from band-recovery data. For eastern mallards, these constants are
1.153, 1.331, and 1.509 for adult females, young males, and young females, respectively (Johnson et al. 2002a).
Regulation-specific predictions of harvest rates of adult-male eastern mallards are provided in Table 7 and Fig. 4.
21
Table 6. Predictions of harvest rates of adult-male midcontinent mallards expected with
application of the 2005 regulatory alternatives in the three western Flyways.
Regulatory alternative Mean SD
Closed (U.S.) 0.0088 0.0019
Restrictive 0.0601 0.0129
Moderate 0.1121 0.0217
Liberal 0.1290 0.0219
Fig. 3. Probability distributions of harvest rates of adult male mid-continent mallards expected with
application of the 2005 regulatory alternatives in the three western Flyways.
In contrast to midcontinent mallards, framework-date extensions were expected to increase the harvest rate of
eastern mallards by only about 5% (USFWS 2001), or in absolute terms by about 0.01 (SD = 0.01) (i.e., our
Aprior@ belief). Based on the observed harvest rate during the 2002-2004 hunting seasons, the updated
(Aposterior@) estimate of the marginal change in harvest rate attributable to the framework-date extension is 0.007
(SD = 0.010). Therefore, the estimated effect of the framework-date extension has been to increase harvest rate of
eastern mallards by about 4% over what would otherwise be expected in the liberal alternative.
Harvest rate
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28
pdf(harvest rate)
0
10
20
30
200
210
Closed
Restrictive
Moderate
Liberal
22
Table 7. Predictions of harvest rates of adult-male eastern mallards expected with
application of the 2005 regulatory alternatives in the Atlantic Flyway.
Regulatory alternative Mean SD
Closed (U.S.) 0.0801 0.0233
Restrictive 0.1233 0.0392
Moderate 0.1593 0.0473
Liberal 0.1700 0.0472
Fig. 4. Probability distributions of harvest rates of adult male eastern mallards expected with
application of the 2005 regulatory alternatives in the Atlantic Flyway.
OPTIMAL REGULATORY STRATEGIES
We calculated optimal regulatory strategies using stochastic dynamic programming (Lubow 1995, Johnson and
Williams 1999). For the three western Flyways, we based this optimization on: (1) the 2005 regulatory
alternatives, including the closed-season constraint; (2) current population models and associated weights for
midcontinent mallards; and (3) the dual objectives of maximizing long-term cumulative harvest and achieving a
population goal of 8.8 million midcontinent mallards. The resulting regulatory strategy (Table 8) is similar to that
used last year.
Assuming that regulatory choices adhered to this strategy (and that current model weights accurately reflect
population dynamics), breeding-population size would be expected to average 7.34 million (SD = 1.76). Note
that prescriptions for closed seasons in this strategy represent resource conditions that are insufficient to support
Harvest rate
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28
pdf(harvest rate)
0
10
20
Closed
Restrictive
Moderate
Liberal
23
one of the current regulatory alternatives, given current harvest-management objectives and constraints.
However, closed seasons under all of these conditions are not necessarily required for long-term resource
protection, and simply reflect the NAWMP population goal and the nature of the current regulatory alternatives.
Based on an observed population size of 7.54 million midcontinent mallards (traditional surveys plus MN, MI,
and WI) and 3.92 million ponds in Prairie Canada, the optimal regulatory choice for the Pacific, Central, and
Mississippi Flyways in 2005 is the liberal alternative.
Table 8. Optimal regulatory strategya for the three western Flyways for the 2005 hunting season. This strategy is based on
current regulatory alternatives (including the closed-season constraint), on current midcontinent mallard models and weights,
and on the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.8 million mallards.
The shaded cell indicates the regulatory prescription for 2005.
Pondsc
Bpopb
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
#5.25 C C C C C C C C C C
5.50-6.25 R R R R R R R R R R
6.50 R R R R R R R R R M
6.75 R R R R R R M M M L
7.00 R R R R R M M L L L
7.25 R R R M M M L L L L
7.50 R M M M L L L L L L
7.75 M M M L L L L L L L
8.00 M M L L L L L L L L
8.25 L L L L L L L L L L
$8.5 L L L L L L L L L L
a C = closed season, R = restrictive, M = moderate, L = liberal.
b Mallard breeding population size (in millions) in the traditional survey area (survey strata 1-18, 20-50, 75-77) and Michigan,
Minnesota, and Wisconsin.
c Ponds (in millions) in Prairie Canada in May.
We calculated an optimal regulatory strategy for the Atlantic Flyway based on: (1) the 2004 regulatory
alternatives; (2) current population models and associated weights for eastern mallards; and (3) an objective to
maximize long-term cumulative harvest. The resulting strategy suggests liberal regulations for all population sizes
of record, and is characterized by a lack of intermediate regulations (Table 9). The strategy exhibits this behavior
in part because of the small differences in harvest rate among regulatory alternatives (Fig. 4).
24
Table 9. Optimal regulatory strategya for the Atlantic Flyway for the 2005 hunting season.
This strategy is based on current regulatory alternatives, on current eastern mallard
models and weights, and on an objective to maximize long-term cumulative harvest. The
shaded cell indicates the regulatory prescription for 2005.
Mallardsb Regulation
<225 C
225 R
250 R
275 M
>275 L
a C = closed season, R = restrictive, M = moderate, and L = liberal.
b Estimated number of mallards in eastern Canada (survey strata 51-54, 56) and the
northeastern U.S. (state plot surveys), in thousands.
We simulated the use of the regulatory strategy in Table 9 to determine expected performance characteristics.
Assuming that harvest management adhered to this strategy (and that current model weights accurately reflect
population dynamics), the annual breeding-population size would be expected to average 0.86 million (SD =
0.16). Based on a breeding population size of 1.05 million mallards, the optimal regulatory choice for the
Atlantic Flyway in 2005 is the liberal alternative.
Application of AHM Concepts to Species of Concern
The USFWS is striving to apply the principles and tools of AHM to improve decision-making for several species
of special concern. We here report on three such efforts in which significant progress has been made since last
year.
Pintails
We examined the harvest potential of northern pintails using a revision of the current population model in
conjunction with the “interim” harvest strategy. The revised population model accounts for overflight-bias, a new
understanding of the recruitment relationship, and recent harvest data (details of this modeling effort are available
at http://migratorybirds.fws.gov/mgmt/ahm/special-topics.htm). This model appears to be an unbiased predictor
of pintail population change, and explains a large fraction of the observed variation in annual pintail population
size. We used this model to examine harvest potential and to simulate the performance of the current harvest
strategy.
Much of the temporal variation in pintail dynamics is driven by habitat conditions on the breeding grounds, and
the average latitude of the breeding population appears to be a useful metric to capture habitat conditions.
Recruitment decreases with increasing latitude. There is evidence that the pintail population is settling, on
average, about 2.4° of latitude farther north now than it did prior to 1975, possibly as a result of changes in habitat
(Fig. A). This more northern distribution has resulted in lower reproduction, and a substantial shift in the yield
curve: a 30-45% decrease in carrying capacity, and a 40-65% decrease in sustainable harvest potential (Fig. B).
25
Fig. A. Average latitude of the center of the pintail breeding population, 1960-2004.
Fig. B. Sustainable annual harvests of pintails as a function of equilibrium population
sizes. The top curve represents harvest potential under environmental conditions
observed prior to 1975, while the bottom curve represents harvest potential after changes
in environmental conditions observed since 1975. The five horizontal reference lines show
the expected continental harvest under a series of regulatory alternatives with AHM
season lengths (letter) and pintail bag limits (number); SIS represents a restrictive pintail
season within a liberal AHM season.
26
Does the “interim” harvest strategy have the ability to respond to temporal variation in pintail dynamics? If the
“latitude” variable captures the important habitat conditions, then we have the ability to respond to annual
variation. But, the long-term shift in the latitude is more challenging, because it implies there has been a
substantial loss of harvest potential; responding to this means reducing the expectations for pintail harvest. If this
was a one-time shift in system dynamics, then we probably have suitable monitoring and regulatory control to
deal with it; if, on the other hand, the system is in a state of perpetual change, then we would need to have a better
understanding of the processes driving that change to respond properly.
If we use this model to simulate the current pintail harvest strategy, and take account of the post-1975
environmental conditions, we expect pintail season length to deviate from the AHM season length 13% of the
time, the average observed pintail breeding population to be around 3.7 million, and the average annual
continental harvest to be around 380 thousand.
Table 10. Expected performance characteristics of the “interim” pintail harvest strategy
using technical revisions of the pintail population model.
Pintail regulations
Performance metric AHM
regulation Pre-1975 Post-1975
Closed seasons 12% 12% 19%
Restrictive seasons 39% 39% 36%
Restrictive partial seasons N/A 0.1% 6%
Moderate seasons 8% 8% 6%
Liberal seasons 41% 41% 33%
Mean population size (sd), millions 6.9 (1.7) 3.7 (1.9)
Mean harvest (sd), millions 0.43 (0.19) 0.38 (0.22)
The “interim” pintail harvest strategy (without the technical revisions mentioned above) does not have good
predictive power and cannot be easily assessed. Using the revised population model for assessment, the current
pintail harvest strategy appears to be sustainable and can actually accommodate a significant amount of system
change without jeopardizing that sustainability. The tight tie to the midcontinent mallard AHM season lengths
drives a lot of the expected variation in pintail regulations. The current strategy was prescribed, however, not
derived; articulation of the underlying objectives would allow more focused assessment and improvements
Black Ducks
We examined the harvest potential of black ducks using models of population dynamics described by Conroy et
al. (2002). Our full report is available at http://migratorybirds.fws.gov/mgmt/ahm/special-topics.htm. These
models incorporate the most controversial hypotheses about reproductive and survival processes in black ducks,
and also allow for the possibility that extant estimates of reproductive and survival rates are positively biased.
Using empirically based model weights (from 1962-93) in conjunction with deterministic dynamic programming,
we derived combinations of equilibrium population size and harvest for a range of adult harvest rates. These
combinations of equilibrium population sizes and harvests can be depicted as a “yield curve,” whose shape
depends on the abundance of sympatric mallards (Fig. 5).
Because of evidence that the reproductive rate of black ducks declines with increasing numbers of mallards, the
carrying capacity (i.e., the point on the graph corresponding to 0 harvest and maximum population size) and
harvestable surplus of black ducks are smaller with higher numbers of sympatric mallards. We can account for
annual changes in both the number of black ducks and mallards with a state-dependent harvest strategy, assuming
that an unambiguous management objective can be specified. We examined the optimal harvest rates associated
27
Fig. 5. Equilibrium population sizes and harvests for black ducks for three levels of mallards (both
black ducks and mallards are in expressed in thousands in the Midwinter Index, MWI). The three
levels of mallards, in increasing order of abundance, represent the 5th percentile from 1961-2003,
the mean from 1994-2003, and the 95th percentile from 1961-2003. The North American
Waterfowl Population Management (NAWMP) is 385 thousand. These yield curves were derived
from population models provided by Conroy et al. (2002).
with three such management objectives using stochastic dynamic programming and compared them to those
estimated from reward band recoveries during 2002-2004 (Table 11).
Table 11. Observed harvest rates (h) of black ducks based on reward banding compared to those that would be optimal
under state-dependent strategies (MWI = Midwinter Index) with three different management objectives.
Observed Optimal harvest rate
Year
Black
duck
MWI
Mallard
MWI h
(se)
Maximize long-term
cumulative harvest
Maximize harvest and a
population goal of
289k
Maximize harvest and a
population goal of
385k
2002 300k 550k 0.075
(0.010) 0.13 0.12 0.04
2003 250k 300k 0.087
(0.012) 0.04 0.02 0.00
2004 250k 350k 0.080
(0.011) 0.04 0.02 0.00
The foregoing analyses do not account for an apparent temporal decline in the reproductive rate of black ducks
that cannot explained by changes in black duck and mallard abundance. The cause is unknown but may be related
to declines in the quantity and/or quality of breeding or wintering habitat or both. Whatever the cause, the
implications are profound, suggesting that carrying capacity and maximum sustainable harvest of black ducks
have decreased by 35% and 60%, respectively, in the past two decades (Fig. 6).
Since 1983, the U.S. Fish and Wildlife Service has been operating under guidance provided in an Environmental
Assessment that specified states harvesting significant numbers of black ducks achieve at least a 25% reduction in
Black duck MWI
0 100 200 300 400 500 600 700
0 100 200 300 400 500 600 700
Black duck harvest
0
20
40
60
80
100
0
20
40
60
80
100
NAWMP goal
249k mallards
395k
479k
28
harvest from 1977- 81 levels. Although this level has been achieved, black duck harvest rates have increased
recently with the return of 50-60 day duck hunting seasons associated with implementation of AHM.
Development of the recent assessment framework allows managers to account for both expected and unexpected
changes in black duck and mallard abundance, and for uncertainty in black duck population dynamics. In
addition, there is an ongoing joint effort with Canada to develop a fully adaptive framework with internationally
agreed-upon harvest management objectives and joint regulatory decision making.
Fig 6. Collapsing yield curves of black ducks as result of declining productivity. Yield curves were based on
population models provided by Conroy et al. (2002). For each period, we used the median year to represent
black duck productivity and fixed the number of mallards at their average midwinter count. The diagonal line
intersects the 10% adult harvest rate on each yield curve.
Scaup
We evaluated the harvest potential of the continental scaup (greater Aythya marila and lesser Aythya affinis
combined) population using a discrete, logistic population model and the available monitoring information
describing scaup population and harvest dynamics. Our full report is available at
http://migratorybirds.fws.gov/mgmt/ahm/special-topics.htm. We used a fully Bayesian approach to estimate
scaup population parameters and to characterize the uncertainty related to scaup harvest potential. We plotted
mean scaup equilibrium population sizes and corresponding sustainable harvests (Fig. 7) along with 95%
credibility intervals (gray shading). When observed harvests and breeding population sizes from 1994 – 2003
were added to this plot, the results suggest that harvest levels in 1997 and 1998 were significantly larger than a
maximum sustainable yield (MSY) value equal to 0.382 million. We used this estimation framework to perform a
retrospective analysis to discern changes in scaup harvest potential that may have resulted from possible large
scale system changes thought to be a factor in the scaup population decline. To perform this analysis, we
conducted two assessments using monitoring data from two different time periods: from 1961 – 1981, and from
1961 – 2001. We chose these time periods to compare population parameters based on data observed when scaup
populations were historically high (i.e., the decade of the 70’s) to population parameters based on information
collected over the time period when larges-scale system changes may have occurred (i.e., 1961 –2001). The
Black duck MWI (k)
0 100 200 300 400 500 600 700 800
Harvest (k)
0
20
40
60
80
100
0.05
0.10
0.15
0.20
2000-04
1980-84
1990-94
29
Fig. 7. Equilibrium population sizes and harvests (and 95% credibility intervals (shading) estimated for
continental scaup populations from a logistic model using a Bayesian hierarchical approach. The years
represent combinations of population sizes and harvest observed in the last decade.
retrospective analysis shows how scaup harvest potential based on an assessment conducted in 1981 is
substantially higher than the harvest potential based on an assessment conducted in 2001, indicating a decrease in
MSY from 0.600 to 0.394 million birds (Table 12).
Table 12. Estimates of model and management parameters derived from fitting a logistic population model to continental
scaup populations using a Bayesian hierarchical approach.
1961-1981 1961-2001
Parameter
mean 2.50% 97.50% mean 2.50% 97.50%
r 0.303 0.127 0.619 0.192 0.096 0.344
K 8.266 6.358 11.185 8.505 6.322 11.420
MSY 0.600 0.294 1.039 0.394 0.225 0.608
Harvest rate
at MSY 0.151 0.063 0.310 0.096 0.048 0.172
Effort at MSY 25.062 10.729 51.842 15.494 7.952 28.206
Plotting the observed harvest and population sizes on the corresponding yield curves (Fig. 8) suggests that not
only has the sustainable harvest shifted downward since 1981, but that observed scaup harvest and population
sizes have moved from the right hand shoulder to the top of the yield curve, suggesting that some of these
harvests may not be sustainable. Because scaup harvest management is predicated on the status of mallard
populations and the AHM protocol, we do not have an ability to manage the scaup harvest in relation to
population status and harvest potential. Ultimately, a state dependent harvest policy will be required to make an
informed harvest management decision in response to population changes and variation in scaup harvest potential.
30
Fig. 8. Equilibrium population sizes and harvests of scaup based on a retrospective analysis using data from
1961-1981 and from 1961-2001 and a logistic population model. The years represent combinations of
population levels and harvest observed during the 1970s and from 1994-2003.
Atlantic Population of Canada Geese
The need to identify optimal harvest policies is apparent for many waterfowl populations, and particularly for the
Atlantic Population of Canada geese (APCG) whose numbers declined significantly in the 1980’s and early
1990’s. Sport-hunting seasons for this population were closed in the U.S. from the fall of 1995 to the winter of
1999. Hunting seasons have been reinstated, but are currently at restrictive to moderate levels in the U.S.
Continuation of sport harvest for APCG and maintenance of the population within desired bounds is contingent
upon effective harvest management and monitoring programs. Effective management will need to incorporate
multiple objectives and must be accomplished with incomplete knowledge of the system and in the presence of
various types of uncertainty including environmental variation, partial system control, model uncertainty, and
partial system observability.
Adaptive management provides a useful framework for making sequential decisions in the presence of
uncertainty. AHM is currently used to set regulations for mallard harvest management, but we are not aware of
any attempts to use these decision-making techniques for any species or population of geese. Developing an
AHM protocol for APCG will require extending approaches currently used for other waterfowl to account for
fundamental differences in the demography and management of ducks and geese. To date, most applications of
adaptive management to waterfowl harvesting have relied on simple scalar population models. Such scalar
models assume all individuals in the population have the same responses to environmental stressors. By contrast,
goose populations have significant age structure as a result of relatively high survival rates and age-dependent
productivity. Previous investigations have shown that optimal harvest management of age-structured populations
is conditional on the age-structure of the population and on age-specific differences in vulnerability of harvest.
31
Adequate description of the population dynamics of geese will therefore require age-structured models; derivation
of goose harvest strategies from existing scalar models may significantly hinder management performance by
failing to take advantage of increased harvest potential and to accommodate decreased harvest potential driven by
intrinsic and extrinsic changes in the age structure.
The overall goal of this project will be to develop an AHM protocol for the U.S. sport harvest of APCG . The
specific objectives are:
1. to explore the general implications of age structure, non-equilibrium population dynamics, and population
‘momentum’ for managing the sport harvest of geese;
2. to develop a set of models describing population and harvest dynamics for geese and parameterize these
models using data specific to APCG, or to other populations comprised principally of B. canadensis
interior;
3. to identify key uncertainties in population or harvest dynamics (i.e., those to which optimal harvest
policies are sensitive); and
4. to derive adaptive policies specifying optimal state-specific harvest rates, and demonstrate the expected
performance of these policies.
For the purposes of development of this AHM application, the APCG is defined as those geese breeding on the
Ungava Peninsula. By this delineation, we assume that geese in the Atlantic population outside this area are
either few in number, similar in population dynamics to the Ungava birds, or both.
To account for heterogeneity among individuals, we developed a base model consisting of a truncated time-invariant
age-based projection model to describe the dynamics of the APCG population,
n(t+1)=An(t),
where n(t) is a vector of the abundances of the ages in the population at time t, and A is the population projection
matrix, whose ijth entry aij gives the contribution of an individual in stage j to stage i over 1 time step. The
projection interval (from t to t+1) is one year, with the census being taken in mid-June (i.e., this model has a pre-breeding
census). The life cycle diagram reflecting the transition sequence, and the corresponding projection
matrix A are shown below:
1 2
B
NB
S(1)
S P (2)
S P (2)(1- )
S P (B)
S (1- ) (NB) P
RS(0)
S P (B)(1- ) S P (NB)
,
32
A =
− − −
L
N
MMMMM
O
Q
PPPPP
0 0 0
0 0 0
0
0 1 1 1
0
1
2
2
RS
S
S P S P S P
S P S P S P
( )
( )
( ) ( ) ( )
( ) ( ) ( )
B NB
b g B b g NB b g
, nt
t
t
t
t
N
N
N
N
=
L
N
MMMM
O
Q
PPPP
( )
( )
( )
( )
1
2
B
NB
where node 1 refers to one-year-old birds, node 2 refers to two-year-old birds, node B refers to adult breeders, and
node NB refers to adult non-breeders. One immediate extension of the base model is to remove the assumption of
time-invariance, and express the parameters as time-dependent quanitites:
Pt = proportion of adult birds in population in year t which breed;
Rt = basic breeding productivity in year t (per capita);
St
(0) = annual survival rate of young from fledging in year t to the census point the next year;
St
(1) = annual survival rate of one-year-old birds in year t; etc.
The projection matrix, so extended, is equivalent to the following recursive balance equations:
N N RS t+ t t t = 1
(1) (B) (0)
N N S t+ t t = 1
(2) (1) (1)
N PN S N S N S t+ t t t t t t t = + + 1
(B) (2) (2) (B) (B) (NB) (NB)
N P N S N S N S t+ t t t t t t t = − + + 1
(NB) b1 g (2) (2) (B) (B) (NB) (NB) .
Note that we can write the number of young produced in year t as
N N R t t t
(0) = (B)
but strictly speaking, that is an intermediate variable in the model, not a state variable, because those young do not
exist on the anniversary date of the model (mid-June census point).
In our base model, we make several simplifying assumptions. First, we assume that breeding begins at age 3 in
APCG. However, while evidence from other goose populations is that breeding propensity increases with age, in
the absence of age-specific estimates for APCG, we assume that breeding propensity (P) is constant over all
breeding ages in a given year (note: in the future, we may consider relaxing this assumption, making use of
estimates from closely related species nesting at similar latitudes). Second, we assume that breeding individuals
have the same per capita breeding success (R), independent of age; while we know this is unlikely to be true,
since geese typically show age-specific differences in reproductive output until at least age 5 yr, age-specific
estimates of per capita breeding success are not available for APCG (note: in the future, we may consider relaxing
this assumption making use of estimates from closely related species nesting at similar latitudes). Third, since
goose species exhibit a monogamous breeding system, with evidence of a 50:50 sex-ratio throughout the life
cycle, we do not recognize sex structure in the model dynamics (thus, Nt
(B) is the number of adult breeders, male
and female, at time t). Fourth, we assume that the probability of a state transition (i.e., between breeder and non-
33
breeder) is random, and not Markovian, that is, that the probability of breeding, Pt in year t is not a function of
breeding state (B or NB) in year t–1. Recent evidence from snow geese and brant suggests that such transitions
are, in fact, likely to be at least first-order Markovian, but there are no data available at present for APCG, or
other populations of Canada geese.
Our objective is to make annual state-based harvest decisions, given the population objectives noted previously.
To do this requires assessment of state on a yearly basis, in time for the annual cycle by which harvest decisions
are made (note: we assume for the moment that such decisions are made annually; one area for investigation is
whether or not annual updating is optimal for goose populations).
For the APCG population, only N(B), R and z are observable annually, where N(B) is the number of breeding adults,
R is the per capita reproductive output (ratio of fledged young to breeding adults), and z is an extrinsic variable (a
function of timing of snow melt on the breeding grounds).
Note that at the time of the management decision in the United States (July), estimates for only the breeding
population size and the environmental variable(s) are available; the age-ratio isn’t estimated until later in the
summer. Thus, in year t, the directly measurable state variables are Nt
(B), zt, and Rt–1.
There are several other state variables of interest, however, namely, N(1), N(2), and N(NB). Since annual harvest
decisions need to be made based on the total population size (Ntot), which is the sum of contributions from various
non-breeding age classes as well as the number of breeding individuals, annual variation in abundance of non-breeding
individuals (N(NB), N(1), and N(2)) will need to be derived using population reconstruction techniques.
Population reconstruction involves estimation of unseen parameter values given a time series of observed
population vectors. In most cases, population reconstruction involves estimating the most likely projection
matrix, given a time series of population vectors (where number of individuals in each age class at each time is
known). However, in our case, estimates of NB, R and z only are available (not the complete population vector);
in effect, we seek to estimate some parameter values given the dynamics of other parameters in the model. Recent
extensions of Bayesian statistical methods to population reconstruction may provide an adequate solution.
Management of the APCG has, in recent years, been focused on achieving the minimum population needed to
sustain some level of sport harvest. However, there is growing concern over the potential problems caused by
overabundant goose species, and management objectives for goose species are increasingly considering
population control as an important objective.
Specification of an explicit, mathematical objective function for the APCG population will require careful
deliberation among the appropriate stakeholders. Since formal AHM is an exercise in optimization, the objective
often not only drives the outcome, but also strongly influences the development of the other components of the
decision framework (e.g., the decision variables, the projection model, etc.). As a starting point for our work in
developing an AHM application for APCG, and as a starting point for discussions about the management
objectives for this resource, we developed a candidate objective function. We propose that the management
objective needs to reflect the simultaneous problem of maximizing opportunity for harvest, while minimizing the
risk that the population will become either too large (i.e., beyond human tolerance in terms of impacts on habitat
or other species), or too small (i.e., requiring season closure for political reasons).
We believe that the critical components governing the dynamics of APCG, unlike those governing ducks, are
generally density-independent over the range of population sizes that likely characterize management objectives;
as such, harvest represents an imposed regulatory mechanism on the dynamics of the population. This requires
specification of a desired range for the population size. Let NMTP represent the maximum tolerable population size
that stakeholders would accept, given the potential for negative impacts of overabundant APCG on stakeholder
interests. Let NMin be the minimum tolerable population size, below which season closure is the only politically
viable management option. The management objective is to maintain the population in the range between the
maximum and minimum values, while simultaneously maximizing opportunity for sport harvest.
34
There is another implicit dynamic that may interact with this objective: there may be a limit to the amount of
harvest that could be induced with traditional harvest regulations. Let NMCP represent the maximum controllable
population level that could be regulated by harvest (a function of a finite number of goose hunters or hunting
effort; this is currently an unknown quantity for APCG). We think it’s most likely that NMTP < NMCP, although this
assumption won’t affect the development of any other aspect of the AHM protocol. NMCP might strongly affect
the optimal policy, however, as the policy should avoid letting the population reach an uncontrollable level,
especially if that level is also intolerable. Thus, the objective should implicitly minimize the risk of losing the
ability to control that population. Note that NMCP should be calculated from biological considerations in
conjunction with information about the limits to harvest. NMTP, however, is a purely sociological constraint.
We think this objective will hold the population as close to the maximum tolerable population size as possible
(thus, allowing the greatest harvest), while guarding against the risk of the population getting out of control.
Mathematically, these objectives can be expressed as
max ( ) ,
0
u N Ht
t
t Σ ∞
=
that is, maximizing the long-term cumulative harvest utility, where the value (utility) of harvest is decremented
relative to the bounds of the constraint (i.e., the maximum and minimum bounds). One possible form of the
utility function u is a ‘square-wave’, where utility of the harvest is 0 when the population size is above and below
NMTP and NMin, respectively. This function is shown below.
It would be valuable if the Atlantic Flyway, and other stakeholders identified by the USFWS or the Atlantic
Flyway, would begin deliberations about the specific objectives for management of the Atlantic Population of
Canada Geese. Specifically, (1) is one component of the objective to maximize long-term harvest; (2) what is the
upper tolerable level for the population size; and (3) what is the lower tolerable level for the population size? At
this point, we are defining the population size as the total population size in mid-June, during the pair surveys,
including both breeders and non-breeders; NMTP and NMin should be expressed on this scale.
Utility ( )u
1.0
0.0
Population size (N)
Nmin NMTP
35
LITERATURE CITED
Anderson, D. R., and K. P. Burnham. 1976. Population ecology of the mallard. VI. The effect of exploitation on
survival. U.S. Fish and Wildlife Service Resource Publication No. 128. 66pp.
Blohm, R. J. 1989. Introduction to harvest - understanding surveys and season setting. Proceedings of the
International Waterfowl Symposium 6:118-133.
Blohm, R. J., R. E. Reynolds, J. P. Bladen, J. D. Nichols, J. E. Hines, K. P. Pollock, and R. T. Eberhardt. 1987.
Mallard mortality rates on key breeding and wintering areas. Transactions of the North American
Wildlife and Natural Resources Conference 52:246-263.
Burnham, K. P., G. C. White, and D. R. Anderson. 1984. Estimating the effect of hunting on annual survival rates
of adult mallards. Journal of Wildlife Management 48:350-361.
Conroy, M. J., M. W. Miller, and J. E. Hines. 2002. Identification and synthetic modeling of factors affecting
American black duck populations. Wildlife Monographs 150. 64pp.
Heusman, H W, and J. R. Sauer. 2000. The northeastern states= waterfowl breeding population survey. Wildlife
Society Bulletin 28:355-364.
Johnson, F. A. 2003. Population dynamics of ducks other than mallards in midcontinent North America. Draft.
Fish and Wildlife Service, U.S. Dept. Interior, Washington, D.C. 15pp.
Johnson, F. A., J. A. Dubovsky, M. C. Runge, and D. R. Eggeman. 2002a. A revised protocol for the adaptive
harvest management of eastern mallards. Fish and Wildlife Service, U.S. Dept. Interior, Washington,
D.C. 13pp. [online] URL: http://migratorybirds.fws.gov/reports/ahm02/emal-ahm-2002.pdf.
Johnson, F. A., W. L. Kendall, and J. A. Dubovsky. 2002b. Conditions and limitations on learning in the
adaptive management of mallard harvests. Wildlife Society Bulletin 30:176-185.
Johnson, F. A., C. T. Moore, W. L. Kendall, J. A. Dubovsky, D. F. Caithamer, J. R. Kelley, Jr., and B. K.
Williams. 1997. Uncertainty and the management of mallard harvests. Journal of Wildlife Management
61:202-216.
Johnson, F. A., and B. K. Williams. 1999. Protocol and practice in the adaptive management of waterfowl
harvests. Conservation Ecology 3(1): 8. [online] URL: http://www.consecol.org/vol3/iss1/art8.
Johnson, F. A., B. K. Williams, J. D. Nichols, J. E. Hines, W. L. Kendall, G. W. Smith, and D. F. Caithamer.
1993. Developing an adaptive management strategy for harvesting waterfowl in North America.
Transactions of the North American Wildlife and Natural Resources Conference 58:565-583.
Johnson, F. A., B. K. Williams, and P. R. Schmidt. 1996. Adaptive decision-making in waterfowl harvest and
habitat management. Proceedings of the International Waterfowl Symposium 7:26-33.
Lubow, B. C. 1995. SDP: Generalized software for solving stochastic dynamic optimization problems. Wildlife
Society Bulletin 23:738-742.
Munro, R. E., and C. F. Kimball. 1982. Population ecology of the mallard. VII. Distribution and derivation of
the harvest. U.S. Fish and Wildlife Service Resource Publication 147. 127pp.
36
Nichols, J. D., F. A. Johnson, and B. K. Williams. 1995. Managing North American waterfowl in the face of
uncertainty. Annual Review of Ecology and Systematics 26:177-199.
Runge, M. C., F. A. Johnson, J. A. Dubovsky, W. L. Kendall, J. Lawrence, and J. Gammonley. 2002. A revised
protocol for the adaptive harvest management of midcontinent mallards. Fish and Wildlife Service, U.S.
Dept. Interior, Washington, D.C. 28pp. [online] URL:
http://migratorybirds.fws.gov/reports/ahm02/MCMrevise2002.pdf.
U.S. Fish and Wildlife Service. 2000. Adaptive harvest management: 2000 duck hunting season. U.S. Dept.
Interior, Washington. D.C. 43pp. [online] URL:
http://migratorybirds.fws.gov/reports/ahm00/ahm2000.pdf.
U.S. Fish and Wildlife Service. 2001. Framework-date extensions for duck hunting in the United States:
projected impacts & coping with uncertainty, U.S. Dept. Interior, Washington, D.C. 8pp. [online] URL:
http://migratorybirds.fws.gov/reports/ahm01/fwassess.pdf.
U.S. Fish and Wildlife Service. 2002. Adaptive harvest management: 2002 duck hunting season. U.S. Dept.
Interior, Washington. D.C. 34pp. [online] URL:
http://migratorybirds.fws.gov/reports/ahm02/2002-AHM-report.pdf.
Walters, C. J. 1986. Adaptive management of renewable resources. MacMillan Publ. Co., New York, N.Y.
374pp.
Williams, B. K., and F. A. Johnson. 1995. Adaptive management and the regulation of waterfowl harvests.
Wildlife Society Bulletin 23:430-436.
Williams, B. K., F. A. Johnson, and K. Wilkins. 1996. Uncertainty and the adaptive management of waterfowl
harvests. Journal of Wildlife Management 60:223-232.
37
APPENDIX A: AHM Working Group
(Note: This list includes only permanent members of the AHM Working Group. Not listed here are numerous
persons from federal and state agencies that assist the Working Group on an ad-hoc basis.)
Coordinator:
Fred Johnson
U.S. Fish & Wildlife Service
7920 NW 71st Street
Gainesville, FL 32653
phone: 352-264-3532
fax: 352-378-4956
e-mail: fred_a_johnson@fws.gov
USFWS representatives:
Bob Blohm (Region 9)
U.S. Fish and Wildlife Service
4401 N Fairfax Drive
MS MSP-4107
Arlington, VA 22203
phone: 703-358-1966
fax: 703-358-2272
e-mail: robert_blohm@fws.gov
Brad Bortner (Region 1)
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 (Region 4)
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 (contractor)
D.J. Case & Associates
607 Lincolnway West
Mishawaka, IN 46544
phone: 574-258-0100
fax: 574-258-0189
e-mail: dave@djcase.com
John Cornely (Region 6)
U.S. Fish and Wildlife Service
P.O. Box 25486, DFC
Denver, CO 80225
phone: 303-236-8155 (ext 259)
fax: 303-236-8680
e-mail: john_cornely@fws.gov
Ken Gamble (Region 9)
U.S. Fish and Wildlife Service
101 Park DeVille Drive, Suite B
Columbia, MO 65203
phone: 573-234-1473
fax: 573-234-1475
e-mail: ken_gamble@fws.gov
Diane Pence (Region 5)
U.S. Fish and Wildlife Service
300 Westgate Center Drive
Hadley, MA 01035-9589
phone: 413-253-8577
fax: 413-253-8424
e-mail: diane_pence@fws.gov
Jeff Haskins (Region 2)
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
38
Bob Leedy (Region 7)
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
Jerry Serie (Region 9)
U.S. Fish and Wildlife Service
11510 American Holly Drive
Laurel, MD 20708
phone: 301-497-5851
fax: 301-497-5885
e-mail: jerry_serie@fws.gov
Dave Sharp (Region 9)
U.S. Fish and Wildlife Service
P.O. Box 25486, DFC
Denver, CO 80225-0486
phone: 303-275-2386
fax: 303-275-2384
e-mail: dave_sharp@fws.gov
Bob Trost (Region 9)
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
Steve Wilds (Region 3)
U.S. Fish and Wildlife Service
1 Federal Drive
Ft. Snelling, MN 55111-4056
phone: 612-713-5480
fax: 612-713-5393
e-mail: steve_wilds@fws.gov
Canadian Wildlife Service representatives:
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
Eric Reed
Canadian Wildlife Service
351 St. Joseph Boulevard
Hull, QC K1A OH3, Canada
phone: 819-953-0294
fax: 819-953-6283
e-mail: eric.reed@ec.gc.ca
Flyway Council representatives:
Scott Baker (Mississippi Flyway)
Mississippi Dept. of Wildlife, Fisheries, and Parks
P.O. Box 378
Redwood, MS 39156
phone: 601-661-0294
fax: 601-364-2209
e-mail: mahannah1@aol.com
Diane Eggeman (Atlantic Flyway)
Florida Fish and Wildlife Conservation Commission
8932 Apalachee Pkwy.
Tallahassee, FL 32311
phone: 850-488-5878
fax: 850-488-5884
e-mail: diane.eggeman@fwc.state.fl.us
39
Jim Gammonley (Central Flyway)
Colorado 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
Mike Johnson (Central Flyway)
North Dakota 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
Don Kraege (Pacific Flyway)
Washington Dept. of Fish and Wildlife
600 Capital Way North
Olympia. WA 98501-1091
phone: 360-902-2509
fax: 360-902-2162
e-mail: kraegdkk@dfw.wa.gov
Dan Yparraguirre (Pacific Flyway)
California Dept. of Fish and Game
1812 Ninth Street
Sacramento, CA 95814
phone: 916-445-3685
e-mail: dyparraguirre@dfg.ca.gov
Guy Zenner (Mississippi Flyway)
Iowa Dept. of Natural Resources
1203 North Shore Drive
Clear Lake, IA 50428
phone: 515/357-3517, ext. 23
fax: 515-357-5523
e-mail: gzenner@netins.net
vacant (Atlantic Flyway)
40
APPENDIX B: Modeling Mallard Harvest Rates
We modeled harvest rates of midcontinent mallards within a Bayesian statistical framework (USFWS 2003). We
developed a set of models to predict harvest rates under each regulatory alternative as a function of the harvest
rates observed under the liberal alternative, using historical information relating harvest rates to various regulatory
alternatives. We modeled the probability of regulation-specific harvest rates (h) based on normal distributions
with the following parameterizations:
Closed:
Restrictive:
Moderate:
Liberal:
p h N
p h N
p h N
p h N
C C C
R R L R
M M L f M
L L f L
( )~ ( , )
( )~ ( , )
( )~ ( , )
( )~ ( , )
μ ν
γ μ ν
γ μ δ ν
μ δ ν
2
2
2
2
+
+
For the restrictive and moderate alternatives we introduced the parameter ( to represent the relative difference
between the harvest rate observed under the liberal alternative and the moderate or restrictive alternatives. Based
on this parameterization, we are making use of the information that has been gained (under the liberal alternative)
and are modeling harvest rates for the restrictive and moderate alternatives as a function of the mean harvest rate
observed under the liberal alternative. For the harvest-rate distributions assumed under the restrictive and
moderate regulatory packages, we specified that (R and (M are equal to the prior estimates of the predicted mean
harvest rates under the restrictive and moderate alternatives divided by the prior estimates of the predicted mean
harvest rates observed under the liberal alternative. Thus, these parameters act to scale the mean of the restrictive
and moderate distributions in relation to the mean harvest rate observed under the liberal regulatory alternative.
We also considered the marginal effect of framework-date extensions under the moderate and liberal alternatives
by including the parameter *f.
In order to update the probability distributions of harvest rates realized under each regulatory alternative, we first
needed to specify a prior probability distribution for each of the model parameters. These distributions represent
prior beliefs regarding the relationship between each regulatory alternative and the expected harvest rates. We
used a normal distribution to represent the mean and a scaled inverse-chi-square distribution to represent the
variance of the normal distribution of the likelihood. For the mean (:) of each harvest-rate distribution associated
with each regulatory alternative, we use the predicted mean harvest rates provided in USFWS (2000a:13-14),
assuming uniformity of regulatory prescriptions across flyways. We set prior values of each standard deviation
(<) equal to 20% of the mean (CV = 0.2) based on an analysis by Johnson et al. (1997). We then specified the
following prior distributions and parameter values under each regulatory package:
Closed (in U.S. only):
p N
p ScaledInv
C
C
( )~ ( . , .
)
( )~ ( , . )
μ
ν χ
0 0088 0 0018
6
6 0 0018
2
2 2 2
−
These closed-season parameter values are based on observed harvest rates in Canada during the 1988-93 seasons,
which was a period of restrictive regulations in both Canada and the United States.
For the restrictive and moderate alternatives, we specified that the standard error of the normal distribution of the
scaling parameter is based on a coefficient of variation for the mean equal to 0.3. The scale parameter of the
inverse-chi-square distribution was set equal to the standard deviation of the harvest rate mean under the
restrictive and moderate regulation alternatives (i.e., CV = 0.2).
41
Restrictive:
p N
p Scaled Inv
R
R
( )~ ( . , .
)
( )~ ( , . )
γ
ν χ
051 015
6
6 0 0133
2
2 2 2
−
Moderate:
p N
p ScaledInv
M
M
( )~ ( . , .
)
( )~ ( , . )
γ
ν χ
085 026
6
6 0 0223
2
2 2 2
−
Liberal:
p N
p ScaledInv
L
L
( )~ ( . , .
)
( )~ ( , . )
μ
ν χ
01305
0 0261
6
6 0 0261
2
2 2 2
−
The prior distribution for the marginal effect of the framework-date extension was specified as:
p( ) N( ) f δ
~ 0.02,0.012
The prior distributions were multiplied by the likelihood functions based on the seven years of data (under liberal
regulations), and the resulting posterior distributions were evaluated with Markov Chain Monte Carlo simulation.
Posterior estimates of model parameters and of annual harvest rates are provided in the following table:
Parameter Estimate SD Parameter Estimate SD
:C 0.0088 0.0007 h1998 0.1102 0.0112
<C 0.0019 0.0005 h1999 0.1004 0.0076
(R 0.5116 0.0613 h2000 0.1264 0.0099
<R 0.0129 0.0033 h2001 0.1077 0.0112
(M 0.8493 0.1057 h2002 0.1133 0.0059
<M 0.0217 0.0055 h2003 0.1132 0.0085
:L 0.1166 0.0075 h2004 0.1245 0.0111
<L 0.0219 0.0044
*f
0.0124 0.0085
42
We modeled harvest rates of eastern mallards using the same parameterizations as those for midcontinent
mallards:
Closed:
Restrictive:
Moderate:
Liberal:
p h N
p h N
p h N
p h N
C C C
R R L R
M M L f M
L L f L
( )~ ( , )
( )~ ( , )
( )~ ( , )
( )~ ( , )
μ ν
γ μ ν
γ μ δ ν
μ δ ν
2
2
2
2
+
+
We set prior values of each standard deviation (<) equal to 30% of the mean (CV = 0.3) to account for additional
variation due to changes in regulations in the other Flyways and their unpredictable effects on the harvest rates of
eastern mallards. We then specified the following prior distribution and parameter values for the liberal
regulatory alternative:
Liberal:
p N
p ScaledInv
L
L
( )~ ( . , .
)
( )~ ( , . )
μ
ν χ
01771
0 0531
6
6 0 0531
2
2 2 2
−
Moderate:
p N
p ScaledInv
M
M
( )~ ( . , .
)
( )~ ( , . )
γ
ν χ
092
028
6
6 0 0488
2
2 2 2
−
Restrictive:
p N
p ScaledInv
R
R
( )~ ( . , .
)
( )~ ( , . )
γ
ν χ
076
028
6
6 0 0406
2
2 2 2
−
Closed (in U.S. only):
p N
p ScaledInv
C
C
( )~ ( . , .
)
( )~ ( , . )
μ
ν χ
0 0800
0 0240
6
6 0 0240
2
2 2 2
−
A previous analysis suggested that the effect of the framework-date extension on eastern mallards would be of
lower magnitude and more variable than on mid-continent mallards (USFWS 2000). Therefore, we specified the
following prior distribution for the marginal effect of the framework-date extension for eastern mallards as:
p( ) N( ) f δ
~ 0.01,0.012
The prior distributions were multiplied by the likelihood functions based on the three years of data (under liberal
regulations), and the resulting posterior distributions were evaluated with Markov Chain Monte Carlo simulation.
Data-based harvest rate estimates were 0.1620 (SE = 0.0133), 0.1459 (SE = 0.0108) and 0.1322 (SE = 0.0108)
43
for 2002, 2003, and 2004, respectively. Posterior estimates of model parameters and of annual harvest rates are
provided in the following table:
Parameter Estimate SD Parameter Estimate SD
:C 0.0800 0.0100 :L 0.1626 0.0178
<C 0.0233 0.0059 <L 0.0472 0.0105
(R 0.7608 0.1137 *f
0.0070 0.0096
<R 0.0392 0.0100 h2002 0.1626 0.0128
(M 0.9242 0.1134 h2003 0.1471 0.0104
<M 0.0473 0.1200 h2004 0.1342 0.0105