ADVANCES IN THE ASSESSMENT OF THE SAGINAW BAY STOCK OF WALLEYES,
LAKE HURON AND EVALUATION OF MANAGEMENT OPTIONS
By
David G. Fielder

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Fisheries and Wildlife−Doctor of Philosophy
2014

ABSTRACT

ADVANCES IN THE ASSESSMENT OF THE SAGINAW BAY STOCK OF WALLEYES,
LAKE HURON AND EVALUATION OF MANAGEMENT OPTIONS
By
David G. Fielder

Fundamental to the careful management of fish stocks is information on mortality rates
and other dynamic functions that characterize that stock. Prior to this work, such efforts for
walleyes from the Saginaw Bay stock were conducted with a Brownie style analysis of jaw tag
reports. Unlikely assumptions and limited participation by all fisheries in their report of tags,
necessitated the elevation of stock assessment methods to state of the art methods. I developed a
statistical catch-at-age model to accomplish this and evaluated four versions including three
different treatments of natural mortality (M): a constant value, age-based M values, and timevarying M values. Deviance information criterion model selection procedures indicated that an
age-based M model version was the optimal fit of the data. I also evaluated an integrated version
that incorporated tag returns as auxiliary information for the recreational component. In this case,
model selection was based on conformity between observed and predicted data and model
convergence. The integrated version was ruled out due to poor agreement of the observed and
predicted values, and predictions of abundance that were not reflected by the fisheries. It was
concluded that the component of the population used for tagging may exhibit dynamics that
differ from the rest of the stock. Total annual mortality of walleyes was greatest for older ages in
all fisheries and ranged from 32% for age-2 fish to 39% for fish ages-10 and older. The
recreational fishery accounted for the majority of fishing mortality but the commercial trapnet

fishery in the main basin of Lake Huron and by-kill from other trapnets in the bay accounted for
proportionally greater fishing mortality of younger ages of fish. Abundance peaked in 2007 at 4
million walleyes age 2 and older but estimates indicated a previous period of high abundance in
the late 1980s, forcing the reconsideration of the past stock as depressed and dependent on
stocking. Statistical catch-at-age methods characterize the dynamics of a stock from the past up
to the present but do not project forward what the fish stock is likely to do in the future under
various management scenarios. After consulting with fishery managers, I developed a stochastic
simulation model and used it to evaluate management options for the recreational fishery in the
form of a decision analysis and a value-of-information analysis for improved estimates of by-kill
magnitude. This analysis was in light of two critical uncertain states of nature concerning the true
magnitude (catchability) of the by-kill and the future of alewives in Lake Huron, the latter being
a strong determinant of walleye recruitment. Management option evaluation indicated a greater
harvestable surplus that could be allocated. Sustainable harvest was calculated as average harvest
treating harvest in years when sustainability criteria were not met as zero. Sustainable harvest
would be maximized if recreational fishing mortality were increased 50% from recent levels.
Realizing this potential, however, would require more intensive management to ensure that
desired levels of F occurred. Choices by managers as to how to allocate surplus harvest are a
matter of policy, but concerns over maintaining predation pressure on alewives so as to suppress
any resurgence may be reasons to manage conservatively by electing to instead maintain a higher
predator abundance. The value of information analysis suggests that further research investment
in the uncertainty over by-kill catchability might be justified on the basis of producing netbenefits from the recreational fishery.

Copyright by
DAVID G FIELDER
2014
iv

ACKNOWLEDGEMENTS

Many individuals were instrumental in my successful completion of this dissertation and
the research it represents. Foremost was the assistance and support provided me by my
Committee Chair Dr. Jim Bence. His guidance and insights are greatly appreciated. I also extend
my appreciation and gratitude to my other committee members: Drs. Mike Jones, Dan Hayes,
Cheryl Murphy and Ed Roseman. Their assistance and direction was instrumental in the conduct
of this research and the development of the dissertation. Although I was unable to interact with
the staff and fellow graduate students of the Quantitative Fisheries Center as much as I would
have liked, I am also appreciative of their support and assistance, especially Associate Director
Travis Brenden. I am grateful to my employer, the Michigan Department of Natural Resources,
Fisheries Division for their cooperation, support and flexibility enabling me to pursue this
degree. Many persons from the DNR assisted me but I especially want to acknowledge Jim
Johnson, Dr. Ji He, and the other staff of the Alpena Fisheries Research Station for their support.
Lastly my deepest appreciation for the unending support of my wife Marie Fielder and our
children, who facilitated and accommodated my work and travels and were always ready with
words of encouragement. I also want to credit my parents; the late Dr. Gordon Fielder and Anita
Fielder for instilling in me a deep appreciation for education.
Funding for much of this research originated as a grant to the Michigan DNR from the
Great Lakes Fishery Commission as part of their Fisheries Research Program. Data, essential to
my research, were provided by the Michigan DNR, Great Lakes Fishery Commission, Ontario
Ministry of Natural Resources, The U.S. Geological Survey’s Great Lakes Science Center, and
the Chippewa / Ottawa Resource Authority.

v

PREFACE

The materials in chapters two and three are intended for publication in peer-reviewed
journals. I therefore would like to acknowledge my coauthors; Dr. Jim Bence for chapter 2 and
Drs. Mike Jones and Jim Bence for chapter 3. For the purpose of this dissertation, I have
generally used first person in the singular but it is understood that any publication that results
from this work may differ slightly and include plurality in language.

vi

TABLE OF CONTENTS

LIST OF TABLES…………………………………………………………………………..……ix
LIST OF FIGURES…………………….……….………………………………………………..xi
CHAPTER 1………………………………………………………………………………………1
DISSERTATION INTRODUCTION…………………………………………………….……….1
APPENDIX..……………………………………………………………….….……....…..9
CITATIONS…......…………………………………………………………..………......11
CHAPTER 2…………………………………………………………………………..…………15
INTEGRATION OF AUXILIARY INFORMATION IN STATISTICAL CATCH-AT-AGE
ANALYSIS OF THE SAGINAW BAY STOCK OF WALLEYES IN LAKE HURON…….…15
Abstract…………………………………………………………………...…….………..16
Introduction………………………………………………………………...…………….17
Methods……………………………………………………………………...……….…..20
Statistical catch-at-age model………………………………………...…….……20
Population submodel………………………………………..…………..….……20
Natural mortality options………………………………………………..……….21
Fishing mortality…………………………………………………………..….….21
Observation submodels……………………………………………………..……24
Auxiliary information………………………………………………………….....25
Model fitting……………………………………………………………………...26
Fisheries and fishery-independent data……………………………………….....30
Immigrants from Lake Erie…………………………………………….………...32
Spawning stock biomass…………………………………………………………34
Model selection…………………………………………………………..………35
Uncertainty and sensitivity analysis………………………………………..……36
Results………………………………………………………………………………..…..38
Model selection…………………………………………………………………..38
Baseline model…………………………………………………………………...39
Discussion………………………………………………………………………….…….43
APPENDIX……………………………………………………………………………….……..52
CITATIONS…………………………………………………………………………….……….69
CHAPTER 3…………………………………………………………………………….……….76
ANALYSIS OF MANAGEMENT OPTIONS AND VALUE OF INFORMATION FOR THE
SAGINAW BAY STOCK OF WALLEYES BASED ON STOCHASTIC SIMULATIONS....76
Abstract…………………………………………………………………………………..77
Introduction………………………………………………………………………………78
Methods…………………………………………………………………………………..81
Management input……………………………………………………………….81
Objectives and performance measures………………………………………….83
vii

Critical uncertainties…………………………………………………………….84
Candidate recreational harvest policies and decision analysis.………………..85
Value of by-kill information……………………………………………………..86
Simulation model and time frame……………………………………………….87
Results…………………………………………………………………………………...92
Recreational fishery performance……………………………………………….92
Value of by-kill information………………………..…………………...………..94
Discussion………………………………………………………………………………94
APPENDIX…………………………………………………………………………………….101
CITATIONS……………………………………………………………………………………110

viii

LIST OF TABLES

Table 2.1. Estimated parameters of the catch-at-age model of the Saginaw Bay stock of walleye
in Lake Huron………………….……………………………………………………………..…53
Table 2.2. Deviance information criterion (DIC) model selection of three candidate statistical
catch-at-age models for the Saginaw Bay stock of walleye in Lake Huron based on different
treatment of natural mortality (M): constant, age varying, and time varying. Included is the
effective number of parameters (pD)……………………………………………………………53
Table 2.3. Sensitivity of penalized likelihood component analysis of the Saginaw Bay stock of
walleye in Lake Huron statistical-catch-at-age model as percent change from the aged-based
natural mortality model (baseline) version. Sensitivity was tested by applying a weighting factor
lambda (  ) of either 0.5 or 2.0 to each of the 12 likelihood components in the baseline version
of the model and assessing percent change from the time series mean for four metrics of
management interest: population size (N), total annual mortality (A), spawning-stock biomass of
females (SSB), and the exploitation rate (  ) of the four fisheries. Weighting factors that resulted
in a lack of convergence are denoted by DNC (did not converge)………………………………54
Table 2.4. Sensitivity of assumptions analysis of the Saginaw Bay stock of walleye in Lake
Huron statistical-catch-at-age model from the age-based natural mortality model (baseline)
version for key assumptions: (1) An expanded (extrapolated) estimate of the magnitude of
commercial by-kill in the bay from MacMillan and Roth (2012). (2) An expanded recreational
observed catch to simulate an Ontario component. (3) Two alternative durations of residency
time for the habitation of Lake Erie (LE) walleyes in Lake Huron. (4) The ages at which Lake
Erie walleye are hypothesized to immigrate was adjusted to fully include age-3 and one-half of
age-2 fish. (5) Alternative variance ratios used in the likelihood functions for relating white-noise
or random-walk deviations of catchability to catch. (6) Treatment of recreational catchability as a
white-noise process instead of random walk. (7) Treatment of age-based natural mortality M as a
prior in the model estimation and then alternative values based on 95% confidence intervals.
Sensitivity was assessed as percent change from the time series mean for four metrics of
management interest: population size (N), total annual mortality (A), spawning stock biomass of
females (SSB), and the exploitation rate (  ) of the four fisheries. A lack of model convergence
is denoted by DNC (did not converge)…………………………………………………………..55
Table 3.1. Description of critical uncertainties (uncertain states of nature), management
scenarios, and performance measures…………………………………………………………..102
Table 3.2. Equations, symbols and descriptions of variables used in the stochastic simulation
model………………………………………………………………………………………...….103
Table 3.3. Decision analysis of eight varying degrees of fishing intensity (applied as a multiplier
to the recreational fishing mortality F) across two critical uncertainties regarding alewife futures
and true levels of commercial by-kill of walleyes resulting in a total of four combinations.
ix

Values are sustainable recreational harvest of walleyes in Saginaw Bay; mean recreational
harvest with zeros assigned to years exceeding sustainability thresholds. Maximum value is
indicated with box……………………………………………….……………………..………105
Table 3.4. Analysis of the expected value of perfect by-kill information (EVPI) and expected
value of imperfect by-kill information (EVII). High and low by-kill reference two alternative
states of nature based on the magnitude of inner Saginaw Bay state-licensed commercial by-kill
stemming from two estimates (a lower value based on an observed period and a higher value
based on an extrapolated annual basis)………………………………………………………..105

x

LIST OF FIGURES
Figure 1.1 Saginaw Bay and Lake Huron……………………………………………………….10
Figure 2.1. Saginaw Bay and Lake Huron showing geographic features of importance to the
model components. The vicinity indicated by hash marks represents the 1836 Treaty area of
Lake Huron………………………………………………………………………………………57
Figure 2.2. (A) Comparison of observed and predicted recreational walleye catch based on the
integrated model version. (B) Recreational exploitation rates between that estimated by the age
based natural mortality (baseline) version of the Saginaw Bay statistical-catch-at-age model
version and that from the Brownie model (from Fielder 2014). Error bars are +1.96 standard
errors………………………………………………………………………………………….…58
Figure 2.3. Observed and predicted values 1986 – 2011, from the M-age based (baseline) version
of the Lake Huron walleye statistical-catch-at-age model for the (A) fishery-independent survey,
(B) recreational fishery catch, (C) trapnet fishery catch, (D) lake-wide (combined Ontario and
tribal) gillnet fishery catch, and (E) the commercial by-kill…………………………………….59
Figure 2.4. Numbers of the Saginaw Bay stock of walleye in Lake Huron 1986 – 2011 age 2 and
older and the +1.96 standard errors (dashed lines) confidence interval, as predicted by the agebased natural mortality (baseline) version of the statistical-catch-at-age model………………..60
Figure 2.5. Selectivity of the three estimated fisheries scaled to 1.0 based on maximum value by
age for the age-based natural mortality (baseline) version of the Saginaw Bay stock, statisticalcatch-at-age model for Lake Huron, 1986-2011 data……………………………………………61
Figure 2.6. Time-varying catchability (q) and the +1.96 standard errors (dashed lines) confidence
interval of the (A) recreational fishery and (B) survey of the Saginaw Bay stock of walleye in
Lake Huron from the age-based natural mortality (baseline) version of statistical-catch-at-age
model 1986-2011………………………………………………………………………………..62
Figure 2.7. Total annual mortality rate (limited to age-4 and older fish), A for the Saginaw Bay
stock of walleye in Lake Huron 1986-2010 and +1.96 standard errors (dashed lines) confidence
interval, as predicted by the age-based natural mortality (baseline) version of the statisticalcatch-at-age model……………………………………………………………………………….63
Figure 2.8. Total annual mortality (A) of the Saginaw Bay stock of walleye by age (years) for
three time periods as estimated by the age based natural mortality statistical-catch-at-age model
(baseline) version………………………………………………………………………………..64
Figure 2.9. Proportion of fishing mortality (F) by age for 2011 from the age-based natural
mortality (baseline) version of the Saginaw Bay stock of walleye statistical-catch-at-age model
for walleye from the Michigan recreational fishery (Fmrc), Ontario trapnet fishery (Fotn), the

xi

lake-wide (combined Ontario and tribal) gillnet fishery (Fgln) and the by-kill stemming from the
state licensed trapnet fishery in Saginaw Bay (Fbkl)……………………………………………65
Figure 2.10. Fishing mortality (F) of age-5 walleye over time from the four fisheries represented
by the age-based natural mortality (baseline) version of the Saginaw Bay stock of walleye
statistical-catch-at-age model. Michigan recreational fishery (Fmrc), Ontario trapnet fishery
(Fotn), the lake-wide (combined Ontario and tribal) gillnet fishery (Fgln) and the by-kill
stemming from the state licensed trapnet fishery in Saginaw Bay (Fbkl)……………………….66
Figure 2.11. Recruitment of walleye to the Saginaw Bay stock as estimated by the age-based
natural mortality statistical-catch-at-age model (baseline) version based on age-2 numbers
attributed to their originating year class…………………………………………………………67
Figure 2.12. Total biomass (all ages and both sexes) and spawning-stock biomass (SSB) of
mature female walleye (in kilograms) for the Saginaw Bay stock of walleye in Lake Huron,
1986-2011 and +1.96 standard errors (dashed lines) confidence interval, as predicted by the age
based natural mortality version of the statistical-catch-at-age model……………………………68
Figure 3.1. Saginaw Bay and Lake Huron……………………………………………..………106
Figure 3.2. Decision tree reflecting alternative harvest policies for varying levels of recreational
fishing mortality (Frec depicted as low, medium, and high). In actuality, eight different levels
were evaluated………………...………………………………………………………………..107
Figure 3.3. Forecasts for different levels of recreational fishing intensity (applied as a scalar to
fishing mortality) across uncertain states of future alewife trends:
, based on means of 50 year forecasts of: total catch
(all fisheries), recreational harvest, population size, percent of years that age-3 walleye total
length exceeds management target, percent of years spawning stock biomass (SSB) drops below
the 20% of unfished threshold, and total annual mortality (A)……………………………..…108
Figure 3.4. Mean spawning stock biomass (SSB) and mean total length (TL) of age-3 walleyes
and critical thresholds of management importance across 50 year forecasts of six recreational
fishing intensity management options, across two versions of future alewife trends (without
alewife recovery; graphics A and B, and with alewife recovery; graphics C and D)…..………109

xii

CHAPTER 1

DISSERTATION INTRODUCTION

1

Of the Laurentian Great Lakes, Lake Huron is the second largest and regarded as
oligotrophic (Beeton et al. 1999). There are eutrophic regions of the lake, chief among them
being Saginaw Bay at 2,947 km2, lying entirely in the Michigan waters of the lake (Figure 1.1).
Saginaw Bay historically was the site of the second largest walleye (Sander vitreus) fishery and
population in the Great Lakes, behind only that of Lake Erie (Baldwin and Saafeld 1962;
Schneider and Leach 1979).
The history of the Saginaw Bay walleye population and its fisheries have been
characterized by three periods (Fielder and Thomas 2006). During the first period (late 1800s
~1970) there was unregulated commercial exploitation, habitat degradation, declining water
quality, and effects of invasive species. The commercial walleye fishery collapsed in the mid1940s due to recruitment failures attributed to these reasons (Schneider and Leach 1979; Keller
et al. 1987). The second period (1971 – 2002) began with the passage of water quality legislation
and the formal closure of commercial fishing for walleye in the bay. The Michigan Department
of Natural Resources (DNR) implemented a walleye fingerling stocking program in the early
1980s and a sport fishery soon emerged (Mrozinski et al. 1991; Fielder 2002, Fielder et al. 2014).
The population and fishery, however was still largely dependent on stocking and short of
recovery targets (Fielder and Baker 2004).
The most recent period in Saginaw Bay’s walleye history (2003 to present) has been the
remarkable recovery of natural reproduction and the achievement of recovery goals (Fielder and
Baker 2004; Fielder and Thomas 2014). This was brought about due to improved reproductive
success which in turn was attributed to the decline of alewives (Alosa pseudoharengus) in Lake
Huron (Fielder et al. 2007). Alewives are predators and competitors on newly hatched Percid fry
(Kohler and Ney 1980; Brandt et al. 1987; Brooking et al. 1998) and use Saginaw Bay as

2

spawning and nursery grounds (Organ et al. 1979). Walleye recovery targets were formally met
in 2009.
Food web changes and trophic interactions are believed to have precipitated the near
extirpation of alewives and decline in rainbow smelt (Osmerus mordax) in Lake Huron in the
early 2000s (Riley et al. 2008; Riley and Roseman 2013). This is hypothesized to have been due
to a combination of sequestering of productivity into the nearshore zone of the lake (Hecky et al.
2004; Cha et al. 2011), a decline in offshore productivity affecting phytoplankton and
zooplankton as well as macroinvertebrates (Nalepa et al. 2007, 2009; Barbiero et al. 2009, 2011),
and heavy predation from the lake’s predators (Bence 2008, He et al. in press). The decline of
alewives led to a severe contraction of the popular Chinook salmon (Oncorhynchus tshawytscha)
fishery which drove a corresponding decline in recreational fishing effort in the Michigan waters
of the lake (Michigan DNR, unpublished data). Walleye have emerged as an integral component
in the remaining recreational fishery in Lake Huron (Fielder et al. 2014).
Saginaw Bay walleye have long been hypothesized to contribute to the main basin
walleye population by some degree of migration (Hile 1954). A study of Saginaw Bay mortality
and exploitation via jaw tagging further confirmed that at least some walleyes migrate
throughout the Michigan waters of the lake (Fielder 2014). Genetic studies indicated that the
Tittabawassee River (a tributary to the Saginaw River; Figure 1.1) spawning genotype was a
substantial component of the commercial walleye fishery operating in Ontario’s southern main
basin waters (McParland et al. 1996). More recently, a telemetry study of Saginaw Bay walleye
movement has confirmed that not only are walleyes making such a migration, but as many as
half the adult fish emigrate as early as June after spawning in the bay’s tributaries (T. Hayden,
U.S. Geological Survey, Great Lakes Science Center, personal communication).

3

Large-scale emigration of walleye from Saginaw Bay to the main basin of the lake
underscores the relevance of this species in understanding the whole lake fish community,
including efforts to manage for a balance between predators and prey (He et al. in press). Such
movement also contributes to spatial complexity of walleye exploitation across fishery types and
jurisdictions. These revelations in combination with the recovery of the stock herald an elevated
need for improved understanding of stock dynamics, exploitation, and sustainability. This
information is needed to inform management for the future as a now recovered walleye stock
will play an increasing important role in Lake Huron.
At the outset of this research, there were several unknowns and misconceptions. The
awareness around the full extent of the movement made by Saginaw Bay walleyes was only
revealed concurrently with this analysis. One of the benefits of stock assessment modeling is that
it forces one to think explicitly about what processes shape the stock including the sources of
mortality (Hilborn and Mangel 1997). Previously little regard was given to how Saginaw Bay
walleyes might be exploited outside of the bay or how other fisheries may act in concert to
constitute a collective source of mortality. These included recreational harvest outside of the bay,
commercial harvest by Ontario fisheries and tribal commercial harvest (permitted as a by-catch)
in the 1836 Treaty Waters of northern Lake Huron (Figure 1.1). An additional source of
mortality not previously considered was that of by-kill (mortality of by-catch) by state of
Michigan licensed commercial fisheries still operating in inner Saginaw Bay (permitted for other
species aside from walleye). That assessment, performed by MacMillan and Roth (2012), added
to the scope of Saginaw Bay walleye exploitation.
The stock assessment work undertaken in this research was not the first for Saginaw Bay
walleyes. The Michigan DNR had sought to characterize the dynamics of mortality and

4

exploitation for more than three decades using a Brownie-style analysis of tag recoveries (Fielder
2014). That work provided estimates of exploitation rate, total mortality rate, fishing mortality
rate, and natural mortality rate. These metrics served as the basis for evaluating the status and
sustainability of the walleye fishery for the Michigan DNR. That study, however, was almost
exclusively limited to the recreational fishery as the other fisheries typically would not report the
jaw tags. Consequently, the rates estimated were specific to only one component of several
shaping that stock. This limitation was not fully apparent until those findings could be contrasted
with the statistical catch-at-age (SCA) stock assessment performed in Chapter 2.
The motivation for developing a SCA model for Saginaw Bay walleye was driven by
another study entitled “Quantifying new top-down influences on the rapidly changing food web
in the main basin of Lake Huron.” This work sought to evaluate certain predator/prey questions
for the lake via a bioenergetics approach, which necessitated estimates of predator populations
by age (He et al. in press). The Brownie-style analysis employed by the Michigan DNR could
generate estimates of abundance but was not age structured. Aside from accommodating the
bioenergetics model, I wished to elevate the stock assessment for Saginaw Bay walleyes to state
of the art methods, so as to provide a stronger basis for management. SCA methods more
thoroughly accommodate multiple fisheries than the earlier Brownie-style tag recovery analysis.
SCA analysis posed its own unique challenges, however, in that estimates of natural mortality
are not typically generated as a product of the analysis (Quinn and Deriso 1999) in contrast to
tagging studies. A goal of my SCA analysis was to evaluate alternative estimates and sources of
natural mortality information.
A further element of SCA modeling is the ability to integrate a greater variety of
information (Punt et al. 2001) to improve model fit, resulting in more realistic parameter

5

estimates. The ability to potentially combine the walleye tag returns as auxiliary information to
help inform the SCA model fitting process could result in a hybrid model. The intent of this
analysis was not just improved estimates but a strategy to guide stock assessment and inform
management in the future.
An additional challenging aspect of this work was a need to account for the role of
walleyes that immigrate to Lake Huron from Lake Erie on the Saginaw Bay population
assessment. Because I derive the model based on the concepts of maximum penalized likelihood
estimation by parameter fitting to match observed and predicted values such as harvest, I also
had to account for harvest of Lake Erie stock fish in Lake Huron, so to be able to predict the
harvest correctly without overestimating harvest from the Saginaw Bay stock.. This meant
developing a means to predict the number of Lake Erie immigrants, so their contribution to the
Lake Huron harvest could be accounted for. Lake Erie immigrants were initially believed to
account for overall greater walleye abundances in the 1980s (i.e. more Lake Erie walleyes in
Lake Huron than Saginaw Bay walleyes). This analysis provides the ability to evaluate that
hypothesis.
In spite of the insights gained from the SCA analysis reported in Chapter 2, key
uncertainties persist. The original estimate of by-kill reported by MacMillan and Roth (2012)
was based on an observed period of May – August and a much larger extrapolated year-round
value was also reported. Abundance estimates from the SCA model proved sensitive to which
estimate was used. This uncertainty in the stock assessment model could translate into important
uncertainty about the performance of future management strategies. Another key uncertainty
governing the future of the stock is how alewives will trend. Because the food web changes in
Lake Huron since the early 2000s were so unprecedented, it is uncertain whether alewives will

6

remain scarce or will recover to previous levels of abundance. Alewife population recovery in
Saginaw Bay may be tied to maintenance of predation pressure by walleyes. The future of
alewives will likely be a critical determinant of future walleye recruitment and the whether the
stock will sustain its recovery (Fielder et al. 2007).
The SCA model seeks to describe the status of the stock “today” based on how it
performed in the past. Projecting the status of the stock out into the future is an entirely different
matter but one closely tied to the stock assessment work. Ultimately fishery managers want to be
able to design management strategies to sustain and optimally utilize the resource in the future.
Because there are a number of uncertainties, I could not predict a single outcome for any given
management strategy. Instead I sought to embrace uncertainty through use of decision analysis
(Peterman and Anderson 1999) and a value of information analysis (Clemen and Reilly 2001). In
Chapter 3 I describe the development of a stochastic simulation model based on the parameters
developed in the SCA model. This is the basis to then conduct a decision analysis on
management options for the future of the recreational fishery; asking the question; given the
probability of the various combinations of potential states of nature represented by the
uncertainties, which management option maximizes our goal while minimizing any exceedance
of our limits of sustainability? A form of this decision analysis is an examination of the value of
information. This analysis asks the question; is further investment in research justified to reduced
or eliminate an uncertainty? That is the question asked about if improved understanding of the
by-kill magnitude was warranted. These are the sort of practical management questions that can
be tackled by the process of decision analysis accomplished through simulation modeling.
Management of the Saginaw Bay stock of walleye is transitioning from a recovery effort
to one of maintenance and possibly allocation. The cessation of stocking in 2006 means that the

7

principal future role of fishery managers will be to choose harvest policies and regulations as the
primary means to meet future management objectives, balancing between meeting the harvest
goals of the fisheries yet ensuring sustainability. The question of sustainability is elusive,
however. Thresholds can be defined and strategies evaluated within the simulation model,
however, the exact role of walleye as a predator on alewives in the lake is a harder question to
isolate. Harvest reduces the walleye stock predictably but what is the threshold needed to ensure
sufficient predation pressure on alewives to minimize the risk of a recovery? These questions are
also explored but prove difficult since the full answer lies in the combined suite of predators in
the lake, something beyond the scope of this work.
As is often the case with these types of analyses, as our understanding grows and
questions are answered, new questions emerge exposing additional uncertainties. This becomes
the context for further research and adaptive management options. The context for this research
needs to be contemplated based partly on what the understanding of the stock was at the outset of
the work. This work advances our understanding by more explicitly characterizing the status of
the Saginaw Bay walleye stock and providing tools for crafting future management direction.
Considerable discoveries have been made, and misconceptions revealed if not answered. Equally
important is the recognition of new critical uncertainties that will confront future management
and research. The ultimate utility of this work will depend on fishery managers and their
willingness to advance management of this resource based on these findings.

8

APPENDIX

9

Figure 1.1 Saginaw Bay and Lake Huron.

10

CITATIONS

11

CITATIONS

Baldwin, N.S., and R.W. Saafeld. 1962. Commercial fish production in the Great Lakes 18671960. Great Lakes Fisheries Commission, Technical Report No. 3 (and 1970 supplement
covering the years 1961-1968), Ann Arbor.
Barbiero, R. P., M. Balcer, D. C. Rockwell, and M. L. Tuchman. 2009. Recent shifts in the
crustacean zooplankton community of Lake Huron. Canadian Journal of Fisheries and
Aquatic Sciences, 66:816-828.
Barbiero, R. P., B. M. Lesht, and G. J. Warren. 2011. Evidence for bottom-up control of recent
shifts in the pelagic food web of Lake Huron. Journal of Great Lakes Research, 37:78-85.
Beeton, A. M., C. E. Sellinger, and D. F. Reid. 1999. An introduction to the Laurentian Great
Lakes Ecosystem, Pages 3-54 in W. W. Taylor, and C. P. Ferreri [editors] Great Lakes
fisheries policy and management a binational perspective. Michigan State University
Press, East Lansing.
Bence, J.R. N. E. Dobiesz, C.P. Madenjian, R. Argyle, R. Barbiero, J.N. Bowlby, R.M.
Claramunt, R. O’Gorman, and T. Schaner. 2008. Top-down effects of open-water
Salmonine predators in the Great Lakes. Michigan State University, Quantitative
Fisheries Center Technical Report T2008-07.
Brandt, S. B., D. M. Mason, D. B. MacNeill, T. Coates, and J. E. Gannon. 1987. Predation by
alewives on larvae of yellow perch in Lake Ontario. Transactions of the American
Fisheries Society 116:641-645.
Brooking, T. E., L. G. Rudstam, M. H. Olson, and A. J. VanDeValk. 1998. Size dependent
alewife predation on larval walleyes in laboratory experiments. North American Journal
of Fisheries Management 18:960-965.
Cha, Y., C. A. Stow, T. F. Nalepa, and K. H. Reckhow. 2011. Do Invasive Mussels Restrict
Offshore Phosphorus Transport in Lake Huron? Environmental Science and Technology
45:7226–7231.
Clemen, R. T., T. Reilly. 2001. Making hard decisions with decision tools. Second Edition.
Duxbury Thomson Learning, Pacific Grove, CA. ISBN: 0-534-36597-3.
Fielder, D. G. 2002. Sources of Walleye Recruitment in Saginaw Bay, Lake Huron. North
American Journal of Fisheries Management, 22:1032-1040.
Fielder, D. G. 2014. Mortality, exploitation, movement, and stock size of Saginaw Bay walleyes,
1981-2011; 31 years of tag return analysis. Michigan Department of Natural Resources,
Fisheries Report 04. Lansing.

12

Fielder, D. G., and J. P. Baker. 2004. Strategy and options for completing the recovery of
walleye in Saginaw Bay, Lake Huron. Michigan Department of Natural Resources,
Fisheries Special Report No. 29. Ann Arbor.
Fielder, D. G., T. L. Kolb, T. M. Goniea, D. L. Wesander, and K. S. Schrouder. 2014. Fisheries
of Saginaw Bay, Lake Huron 1986-2010. Michigan Department of Natural Resources,
Fisheries Report 02. Lansing.
Fielder, D. G., J. S. Schaeffer, and M. V. Thomas. 2007. Environmental and ecological
conditions surrounding the production of large year classes of walleye (Sander vitreus) in
Saginaw Bay, Lake Huron. Journal of Great Lakes Research 33:118-132.
Fielder, D. G., and M. V. Thomas. 2006. Fish Population Dynamics of Saginaw Bay, Lake
Huron 1998 – 2004. Michigan Department of Natural Resources, Fisheries Research
Report. No. 2083. Ann Arbor.
Fielder, D. G., and M. V. Thomas, 2014. Status and Trends of the Fish Community of Saginaw
Bay, Lake Huron 2005–2011. Michigan Department of Natural Resources, Fisheries
Report 03. Lansing.
He, J. X., J. R. Bence, C. P. Madenjian, S. A. Pothoven, N. E. Dobiesz, S. C. Riley, D. G.
Fielder, J. E. Johnson, M. P. Ebener, A. R. Cottrill, L. C. Mohr, and S. R. Koproski. In
press. Using a system of time-varying models to quantify piscivory patterns during the
rapid food-web changes in the main basin of Lake Huron. Canadian Journal of Fisheries
and Aquatic Sciences
Hecky R.E., Smith R.E.H., Barton D.R., Guildford S.J., Taylor W.D., Charlton M.N. and Howell
T. 2004. The nearshore phosphorus shunt: a consequence of ecosystem engineering by
dreissenids in the Laurentian Great Lakes. Canadian Journal of Fisheries and Aquatic
Science, 61, 1285–1293.
Hilborn, R. and M. Mangel. 1997. The ecological detective confronting models with data.
Monographs in Population Biology; 28. Princeton University Press.
Hile, R. 1954. Fluctuations in growth and year-class strength of the walleye in Saginaw Bay.
Fishery Bulletin 91. U. S. Fish and Wildlife Service. Washington D. C.
Keller, M., J. C. Schneider, L. E. Mrozinski, R. C. Haas, and J. R. Weber. 1987. History, status,
and management of fishes in Saginaw Bay, Lake Huron, 1981-1986. Michigan
Department of Natural Resources, Fisheries Technical Report 87-2, Ann Arbor.
Kohler, C. C., and J. J. Ney. 1980. Piscivory in a landlocked alewife (Alosa pseudoharengus)
population. Canadian Journal of Fisheries and Aquatic Sciences 37:1314-1317.
MacMillan, E.A., and B.M. Roth. 2012. By-catch in the Saginaw Bay, Lake Huron commercial
trap net fishery. Journal of Great Lakes Research 38:353-361.
13

McParland, T. L., M. M. Ferguson, and A. P. Liskauskas. 1996. Genetic Population Structure
and Mixed-Stock Analysis of Walleye in the Lake Erie–Lake Huron Corridor using
Allozyme and Mitochondrial DNA Markers. Transactions of the American Fisheries
Society. 128: 1055-1067.
Mrozinski, L. E., J. C. Schneider, R. C. Haas, and R. E. Shepherd. 1991. Rehabilitation of
walleye in Saginaw Bay, Lake Huron. Pages 63-84 in P. J. Colby, C. A. Lewis, and R. L.
Eshenroder, editors. Status of walleye in the Great Lakes: case studies prepared for the
1989 workshop. Great Lakes Fishery Commission, Special Publication 91-1, Ann Arbor.
Nalepa, T. F., D. L. Fanslow, and G. A. Lang. 2009. Transformation of the offshore benthic
community in Lake Michigan: recent shift from the native amphipod Diporeia spp. To
the invasive mussel Dreissena rostriformis bugensis. Freshwater Biology, 54:466-479.
Nalepa, T. F., D. L. Fanslow, S. A. Pothoven, A. J. Foley III, and G. A. Lang. 2007. Long-term
trends in benthic macroinvertebrate populations in Lake Huron over the past four
decades. Journal of Great Lakes Research, 33:421-436.
Organ, W. L., G. L. Towns, M. O. Walter, R. B. Pelletier, and D. A. Riege. 1979. Past and
presently known spawning grounds of fishes in the Michigan coastal waters of the Great
Lakes. Michigan Department of Natural Resources, Technical Report 79-1, Ann Arbor.
Peterman, R. M., and C. N. Anderson. 1999. Decision analysis: a method for taking uncertainties
into account in risk-based decision making. Human and Ecological Risk Assessment
5:231-244.
Punt, A. E., D. C. Smith, R. B. Thomson, M. Haddon, X. He, and J. Lyle. 2001. Stock
assessment of the blue grenadier Macruronus novaezelandiae resource off south-eastern
Australia. Marine and Freshwater Research, 52:701-717.
Quinn, T. J. II. and R. B. Deriso. 1999. Quantitative Fish Dynamics. Oxford University Press,
Oxford.
Riley, S. C., and E. F. Roseman. 2013. Status of the offshore demersal fish community. Pages
21-28 in S. C. Riley (Editor) The state of Lake Huron in 2010. Great Lakes Fishery
Commission Special Publication 13-01.
Riley, S. C., E. F. Roseman, S. J. Nichols, T. P. O’Brien, C. S. Kiley, and J. S. Schaeffer. 2008.
Deepwater demersal fish community collapse in Lake Huron. Transactions of the
American Fisheries Society, 137:1879-1890.
Schneider, J. C., and J. H. Leach. 1979. Walleye stocks in the Great Lakes, 1800-1975:
fluctuations and possible causes. Great Lakes Fishery Commission, Technical Report
No. 31. Ann Arbor.

14

CHAPTER 2

INTEGRATION OF AUXILIARY INFORMATION IN STATISTICAL CATCH-AT-AGE
ANALYSIS OF THE SAGINAW BAY STOCK OF WALLEYES IN LAKE HURON

15

Abstract

Estimates of mortality rates and abundance for the Saginaw Bay stock of walleye Sander
vitreus has traditionally been performed with an analysis of tag returns using a Brownie style
analysis. An estimation approach that more fully accounted for sources of exploitation in
addition to the recreational fishery in Saginaw Bay and inclusive of the rest of Lake Huron was
needed. I developed a statistical catch-at-age model to accomplish this and evaluated four
versions including three different treatments of natural mortality (M): a constant value, age-based
M values, and time-varying M values. Deviance information criterion model selection procedures
indicated that an age-based M model version was optimal. I also evaluated an integrated version
that incorporated tag returns as auxiliary information for the recreational component. In this case,
model selection was based on conformity between observed and predicted data and model
convergence. The integrated version was ruled out due to poor agreement of the observed and
predicted values, and predictions of abundance that were not reflected by the fisheries. It was
concluded that the component of the population used for tagging may exhibit dynamics that
differ from the rest of the stock. Total annual mortality of walleyes was greatest for older ages in
all fisheries and ranged from 32% for age-2 fish to 39% for fish ages-10 and older. The
recreational fishery accounted for the majority of fishing mortality but the commercial trapnet
fishery in the main basin of Lake Huron and by-kill from other trapnets in the bay accounted for
proportionally greater fishing mortality of younger ages of fish. Abundance peaked in 2007 at 4
million walleyes age 2 and older but estimates indicated a previous period of high abundance in
the late 1980s, forcing the reconsideration of the past stock as depressed and dependent on
stocking.

16

Introduction

Statistical catch-at-age (SCA) methods are widely regarded as the state of the art
approach to stock assessment (NRC 1998; Quinn and Deriso 1999; Maunder 2003; Hilborn
2012) and are a form of “integrated analysis” (Quinn and Deriso 1999; Punt et al. 2001).
Statistical catch-at-age models are age-structured and describe changes in abundance of
individual cohorts over time (Megrey 1989). Such models include a population submodel from
which estimates of abundance and mortality rates are derived. In addition, observation
submodels describe the fisheries and fishery-independent assessment (Fournier and Archibald
1982; Methot 1990, 2000). The modern day approach to model fitting is a likelihood based
approach (Methot 1990, 2000), sometimes in a Bayesian context (Maunder 2003), from which
the model parameters are adjusted so that a match to observed data (and prior information) is
achieved as measured by an objective function. Statistical catch-at-age methods in combination
with the likelihood approach to model fitting are hailed for their flexibility in utilizing a variety
of data to characterize complex dynamics of fish stocks and their fisheries (Magnusson and
Hilborn 2007; Butterworth and Rademeyer 2008; Methot 2009; Hilborn 2012).
Walleye Sander vitreus from Saginaw Bay in Lake Huron is a stock with complex
dynamics that requires a flexible approach to assessment. Walleyes are an important native
predator in Lake Huron (Roth et al. 2013) and the Saginaw Bay stock is the largest source of
recruits (Schneider and Leach 1979). Saginaw Bay is a relatively shallow, coolwater embayment
of about 2,947 km2, that lies entirely in the Michigan waters of Lake Huron (Figure 2.1). The
Saginaw Bay walleye fishery dates to the late 1800s and historically was the second largest in the
Great Lakes, behind that of Lake Erie (Baldwin and Saalfeld 1962; Schneider 1977). Fielder and

17

Thomas (2006) characterized the history of walleye in Saginaw Bay as having three phases. The
first period, characterized by unbridled commercial exploitation, severe habitat degradation, and
effects of invasive species, ended with collapse of the fishery in the mid-1940s (Schneider 1977;
Schneider and Leach 1977, 1979). After the passage of water quality legislation in the 1970s,
closure of the commercial fishery, and the initiation of a fingerling stocking program, a
recreational fishery emerged (Fielder et al. 2014) but it was believed to have remained dependent
on stocking (Fielder 2002; Fielder and Thomas 2006). Most recently a period of recovery has
occurred and is attributed to the disappearance of the invasive alewife Alosa pseudoharengus
and substantial decline in rainbow smelt Osmerus mordax (Riley and Roseman 2013), which
were predators on newly hatched percids (Fielder et al. 2007).
Since the early 1970s, the Michigan Department of Natural Resources (DNR) has
recognized the importance of stock assessment information for Saginaw Bay walleye.
Investments in assessment included analysis of trends in abundance, recruitment, and growth
rates since 1971 (Fielder and Thomas 2014), creel surveys to document extraction since 1986
(Fielder et al. 2014) and an analysis of mortality and exploitation rates based on a tagging
program conducted since 1981 (Fielder 2014).
The spatial extent of the Saginaw Bay stock of walleye, however, reaches beyond the
confines of the bay (Hile 1954). Return of jaw tags has been limited to the recreational fishery
(Fielder 2014) but has indicated considerable out-migration from Saginaw Bay proper to much
of the rest of Lake Huron. Genetic studies have also documented that Tittabawassee River
genotypes of walleye (a spawning tributary within Saginaw Bay) have been found to comprise as
much as 9% of the harvest of walleyes in commercial fisheries operating in the southern Ontario
waters of the lake even before the recovery (McParland et al. 1996). More recently, an acoustic

18

telemetry study has indicated that as many as half of the adult walleyes are emigrating from
Saginaw Bay during the open water months (T. Hayden, U.S. Geological Survey, personal
communication). Other fisheries thought to be likely exploiting Saginaw Bay walleyes outside
the bay includes the recreational fishery in the nearshore waters of Lake Huron and a tribal
gillnet fishery in the area of northern Lake Huron defined by the 1836 Treaty (Figure 2.1) where
walleyes are retained as by-catch. By-kill of walleyes also occurs in the state-licensed
commercial fishery that operates in the bay and constitutes another source of mortality
(MacMillan and Roth 2012).
Compounding the challenges of stock assessment for Saginaw Bay walleye is the effect
of immigration of walleyes from Lake Erie. Some walleyes from the central and western basins
of Lake Erie are documented to inhabit portions of Lake Huron seasonally (Wolfert 1963;
Ferguson and Derksen 1971; Thomas and Haas 2005; Wang et al. 2007). A mixed-stock analysis
of walleye from commercial fisheries on the Ontario portion of southern Lake Huron indicated
that walleyes from western Lake Erie contributed as much as 67-72% of the total commercial
catch in 1994 and 1995 (McParland et al. 1996). Tag returns from Lake Huron of walleyes
tagged in Lake Erie average about 1% but have been as great as 2.6% in some years (Lake Erie
Walleye Task Group, Great Lakes Fishery Commission, unpublished data). These are
proportions of the entire exploitable walleye population in central and western Lake Erie, which
regularly number in the tens of millions. This outside source of walleyes must also be accounted
for in a comprehensive stock assessment for Saginaw Bay walleye.
The objective of this study was to use SCA methods to more fully characterize population
dynamics of the Saginaw Bay stock of walleye. I sought to develop a model that accounted for
four fisheries (Michigan recreational fishery, Ontario trapnet fishery, Ontario and tribal gillnet

19

fisheries, and the effects of commercial by-kill within the bay) and adjusted for immigration of
Lake Erie walleyes and their contribution to each of the fisheries. I also evaluated whether
existing tag return data could be incorporated as auxiliary information to yield a superior
integrated model.
Methods

The analytical approach to this analysis was to (1) develop an SCA model for the
Saginaw Bay stock of walleye in Lake Huron, including adjustments for migrants from Lake
Erie, (2) evaluate three options of natural mortality and settle on a final “baseline” model
version, and lastly (3) develop an integrated model version incorporating tag returns as auxiliary
information and determine the optimal model (baseline versus the integrated version).
Statistical catch-at-age model. -- The SCA analysis was conducted on annual time steps
from 1986 through 2011. The modeling began with 1986 because recreational harvest estimates
were not consistently available before that year. Ages of walleye modeled were 2 through 13,
and age 13 fish included an aggregate of all walleyes age 13 and over. These ages were selected
because younger walleyes (< age 2) are not typically recruited to most of the fisheries. I chose
age 13 to aggregate because the Saginaw Bay stock of walleye has exhibited considerable
longevity at certain points in its history. There was no attempt to quantify aging error from any
of the fisheries or survey.
Population submodel.-- Predicted walleye numbers (N) at age a+1 at the start of year y+1
were derived by the population equation:
Na  1, y  1  Na , ye

 Z a,, y

 Na , ye

20

 ( Ma , y  Fa , y )

[1]

where Z is the total instantaneous mortality rate for the corresponding age and year; Z is the sum
of instantaneous natural mortality, M, and the total instantaneous fishing mortality, F, for age a
and year y.
Natural mortality options.-- Instantaneous natural mortality is not typically estimable
from within a SCA model and instead is supplied (Quinn and Deriso 1999). Three alternative
assumptions regarding natural mortality were used in alternative assessment models compared in
this analysis, and in each case treating values for M as known. First, I used time-varying (annual)
estimates (My) derived from the analysis of the tagging operation conducted annually by the
Michigan DNR (Fielder 2014). Second, I used age-based estimates of Ma borrowed from the
walleye tagging assessment in the neighboring western basin Lake Erie (Vandergoot and
Brendon 2014). Those values were 0.335 for ages 2-4 and 0.152 for ages 5+. Lastly I used a
value for M=0.23 that was constant over years and ages derived from the Pauly (1980) equation,
taking into account Von Bertlanaffy growth parameters and temperature data. Growth
parameters of L and K were obtained from survey data for Saginaw Bay walleye (Fielder et al.
2000; Fielder and Thomas 2006, 2014) based on all available data from the initiation of the
survey in 1989 through 2011, and temperature was derived from mean annual air temperature
data obtained from Midland-Bay City-Saginaw (MBS) airport.
Fishing mortality.—All model variations evaluated included the same treatment of
fishing mortality. For the recreational, commercial trapnet, and commercial gillnet fisheries,
instantaneous fishing mortality was treated as separable by age and year such that
Fa, y  s a qE y  y

21

[2]

where sa is the age-specific selectivity of the fishery, q is the catchability of the fishery, Ey is the
year specific fishing effort, and  y represents the catchability deviations (process error). These
elements, aside from the effort, are estimated as parameters (or calculated from estimated
parameters) in the model. Selectivity for each fishery was freely estimated out to age 10, and fish
older than age 10 were assumed to have the same selectivity as age-10 fish. Depending on the
fishery, the catchability deviations were either based on a white-noise (trapnet and gillnet
commercial fisheries) or a random walk (recreational fishery) model. In the case of white noise,

 y are assumed to come from a lognormal distribution. For the random walk,  y are modeled as

 y   y 1 y 1

y  1986

 y  1 y  1986

[3]

and  y (random walk deviations) are lognormal. In the case of white noise, the catchability
deviations allow for variation around a mean catchability but the model fit is penalized when  y
deviates from that mean. In this case the  y are estimated parameters. In the case of the random
walk, the random walk deviations,  y , are what is penalized in the model fit for changes in
catchability from one year to the next, and these rather than the  y are the estimated parameters .
I used a random walk to model  y for the recreational fishery because of the colonization of
Saginaw Bay by dreissenid mussels in 1992 – 1993 midway through the time series (Nalepa et
al. 1995). Dreissenid mussels were documented to increase water clarity and are theorized to
have affected catchability in some fisheries including that in Saginaw Bay (Fielder et al. 2000).
The use of random walks is an omnibus approach to estimating time-varying parameters when
they may not come from a distribution with a consistent mean (Wilberg and Bence 2006;
Wilberg et al. 2010). I presumed that catchability would be more stable for the commercial
22

trapnet and gillnet fisheries, which are prosecuted outside Saginaw Bay, and hence used the
white noise model in those cases.
Instantaneous fishing mortality stemming from the commercial by-catch (constituting bykill) in Saginaw Bay could not be derived as it was for the other fisheries because only one
estimate of by-kill was available (for 2010). However, this source of extraction is believed to be
large (MacMillan and Roth 2012) and could not be ignored. To calculate F for this source of
extraction, the Newton-Raphson method (Quinn and Deriso 1999; Haddon 2001) was employed
to find a catchability value so that the 2010 by-kill was matched. Year-specific F values were
then determined as the product of catchability and annual commercial effort in the bay (Michigan
DNR, unpublished data), which is functionally equivalent to equation 2 with an assumption of
constant selectivity (1.0) across ages. The resulting annual values for F derived for the
commercial by-kill in Saginaw Bay were then included in population calculations (as a
component of the total instantaneous F value). While the inclusion of by-kill influences fit to the
data, it was not directly part of the objective function given that there was only one by-kill
observation that was exactly matched. MacMillan and Roth (2012) offered two estimates of bykill for Saginaw Bay commercial fisheries: a lesser value of 23,500 walleyes represented their
observed period of May-August and an extrapolated larger value of 102,000 walleyes was given
for the entire year. The larger expanded value represented a set of assumptions that were
regarded as tenuous so I used the lower value from the observed period in my model fitting.
Using my final model (after model selection), I evaluated sensitivity to the by-kill by refitting the
model with the larger expanded value. I did not directly incorporate other sources of discard
mortality. However, as part of my sensitivity analysis I did evaluate how estimates were
influenced when a higher than reported recreational harvest occurred.

23

Observation submodels.-- Predicted catch from each of the four fishery components was
treated the same across the model variations and was derived from the Baranov Catch equation
(Quinn and Deriso 1999):

Z


Fa , y
Ca, y 
N a , y 1  e a , y

Z a, y







[4]

where the predicted catch at age ‘a’ in year ‘y’ is the product of the ratio of fishing mortality to
total mortality and the numbers dying each year.


The predicted survey catch-per-unit-of-effort (CPUE) ( I ) was derived as:
8.5
 Z a, y

surv surv
I a , y  q y s a N a , y e 12

[5]

where qysurv is survey catchability and sasurv is age specific survey selectivity. The term e



8.5
Z
12 a, y

sets the corresponding population size for that age and year to the time of year the survey is
conducted. As with the fisheries, the survey selectivity was fixed for ages 11-13 to that estimated
for age 10. The qysurv was modeled as a random walk, similarly to the catchability deviations for
the recreational fishery (equation 3), for the same reasons (as this survey is conducted in
Saginaw Bay) but starting in 1989 rather than 1986 when this survey was initiated. The
treatment of survey CPUE was the same across the model variations.
Other parameters include initial abundance at age of walleyes (N2 to N13) in 1986 (first
year of modeling) and initial numbers at age 2 for each year of the model (recruitment) (Table
2.1). Because of the use of random walk and white noise penalties in the objective function,
however, fewer than the total 181 parameters represented in Table 2.1 were freely estimated.

24

Auxiliary information.-- Each year during the time series, about 3,000 walleyes were jawtagged and released during the annual spawning run (late March or early April) at Dow Dam on
the Tittabawassee River (Figure 2.1). Tag returns came from the recreational fishery, and tags
were rarely or never reported from the other fisheries even when encountered. I developed a
version of SCA that included the fit to the recreational tag return data as a component in the
objective function, and I evaluated this integrated model version. The number reported by
anglers in any given year ‘j’ after initial release year ‘r’ is a function of annual survivals S i , for i
in years r to j-1, and the recovery rate in the recreational fishery f j . Thus, the probability of a
tag return being observed in year j in the recreational fishery given the fish was tagged in year r
is

 j 1

p r , j    S i  f j
 i r


[6]

where  is a tag retention rate described by Fielder (2014) and originally obtained from
Vandergoot et al. (2012). In conventional Brownie et al. (1985) tag-return analysis, the values of

S i and f j are parameters estimated solely based on the tagging data, but in my integrated model
their values are obtained as functions of parameters that are already estimated as part of the
population model. I assumed that equation 6 applied only to walleyes ages 4 and above, given
that fish younger than that age are not typically tagged as part of that survey (Fielder 2014).
During model fitting, estimates of S i were calculated as the ratio of modeled numbers of age-5+
fish alive in year i+1 to the number of age-4+ fish alive in year i. The f j values were calculated
as the exploitation rate for year j (estimate of harvest of fish age 4 and older based on equation 4
divided by estimate of N for fish age 4 and older at start of year j). Equation 6 is not age

25

structured. While methods exist for age structuring Brownie style analyses, that has not been the
approach by Michigan DNR, as often the number of aged specimens was insufficient to organize
an age-based tag-return analysis.
Model fitting.-- I used highest posterior density (HPD) estimation to obtain point
estimates of parameters and quantities estimated from parameters. This Bayesian approach and
its application in fisheries have evolved from a maximum likelihood approach (Fournier and
Archibald 1982; Methot 1990; Aldrich 1997) to account for process errors (Schnute 1994; Linton
and Bence 2008). Using this approach, parameters are estimated based on the minimization of
an objective function, which is the sum of the negative log likelihood for the data and the
negative log prior densities for the parameters. The separate negative log-likelihood components
for each type of data predicted by each of the submodels were summed, yielding the joint
negative log likelihood term. In addition, for parameters allowing for catchability variations,
there was a negative log prior term associated with each fishery or the survey based on the
assumed prior distribution for either the  y (white noise) or  y (random walk) deviation for that
fishery or survey. These prior terms were summed to obtain the joint negative log prior density.
I assumed bounded uniform priors for other parameters, and thus prior densities were constant
within the bounds so were not included in the objective function. The objective function was
minimized using AD Model Builder version 10.0 (AD Model Builder Project 2011; Fournier et
al. 2012). The prior components can be viewed as penalties for deviations from the values
deemed most likely a priori, and HPD estimation is also referred to as penalized likelihood
estimation.

26

For each of the observed fishery catches (recreational, Ontario trapnet, lake-wide gillnet),
each of the four sets of fishery or survey catchability deviations (  y for white noise or  y
random walks), and for the survey CPUE, a lognormal distribution was assumed

1  ln X i , y  ln Xˆ i , y 
Li  IC  n ln  i   


2 y 
i



2

[7]

where IC is an ignorable constant that was not included in my calculations. Each data source is


designated by i, sample size is denoted by n, and  i is the standard deviation that applies to each
quantity X i , y . In the case of fishery catch and survey CPUE, Xˆ i , y represents a model prediction
of observed catch or CPUE and X i , y represents the observed number of fish caught. In the cases
of the components for white noise catchability, X i , y represents the catchability deviations (  y ),
and for the random walk components, X i , y represents random-walk deviations (  y ). In both of
these cases Xˆ i , y has an assumed value of 1. Thus there are 8 lognormal components (Li): three
for the different fitted fishery catches (all fisheries except the by-kill from state-licensed
trapnets), one for the CPUE from the survey, two for random-walk deviations (for the survey and
recreational fishery), and two for white-noise catchability deviations (Ontario trapnet fishery and
lake-wide gillnet fishery).


The values of  i play a role in all eight lognormal components. Unfortunately, we know

from the theory of penalized likelihood that not all of these  i can be estimated during model

fitting (Linton and Bence 2008). My approach was to estimate the  i associated with the

27


observed data (fishery catch or CPUE), so four of these were estimated. The  i for the white
noise or random-walk deviations were then calculated based on an assumed ratio of their

variance (  i2 ) relative to the variance for catch or CPUE from the same fishery or survey.

We set the ratios for both the survey and recreational fishery catchability random walk
variance to 0.85, hypothesizing that the interannual changes in catchability, as a proportion of the
current value, would be of lesser magnitude than the observation error associated with catch (as a
proportion of the true or expected value), but nearly as large. I set the ratios for the commercial
fishery effort deviation variances to 1.0 for the lake-wide gillnet fishery and to 0.25 for the
Ontario trapnet fishery. For the gillnet fishery I had no specific reason to expect actual changes
in catchability, but I assumed that estimation error variance for effort would be at least as large
as for the actual landings, as effort reporting is not a focus. For the Ontario trapnet fishery I
hypothesized that variance in effort estimates would be substantially less than those of harvest,
as they are not subject to all of the errors that beset harvest (e.g., weighing, converting from
weight to numbers), and the Ontario effort reporting system is well developed. I acknowledge
that these ratios, while based on my best judgment are somewhat arbitrary. Consequently I
explore the influence of the ratios in the sensitivity analysis.
The age-composition data (proportions at age) from each fishery and the survey were
based on age samples assumed to arise from a multinomial distribution, leading to the likelihood
equation


Li   N i , y  pi , y ,a ln pi , y ,a
y

a

28

[8]

where N i , y is the sample size for the number of specimens aged but was capped at an effective
sample size of 200. This cap limited the influence of the component in the objective function and
prevented this likelihood component from being over weighted for a specific year and data
source (Fournier and Archibald 1982). The cap of 200 is consistent with the examination of
residuals, which suggested markedly better fits were not obtained for year and data sources when
age-composition sample size exceeded the cap. The observed proportion of age ‘a’ in year ‘y’
for each source of age composition data was denoted as pi , y ,a and the corresponding predicted

value as pi , y ,a is the predicted proportion.

The likelihood component for the auxiliary information (tag return comparison) was
based on the multinomial distribution as





L13     Rr , j ln p r , j   URr ln 1   p r , j 
r 
j



 j

[9]

where p r , j is the probability of the tag recovery from the rth tagging year in the jth recovery year,

URr is the number of tags not recovered from the original lot tagged in year r, and Rr , j is the
number of tags recovered from tagging year r in recovery year ‘j’; Rr , j was the adjusted value of
the actual number of tags reported expanded by the year–specific correction factor for
nonreporting (Fielder 2014).
Two model versions were developed. The baseline version omitted the tag return
component from the objective function, and the integrated version included the tag return
evaluation in the objective function. The baseline version was the SCA model chosen from the

29

three candidate models (based on the natural mortality options) as a result of the model selection
criteria. Because the predicted tag returns were generated with existing parameters, the total
parameter set (181) is the same for the baseline and integrated model versions. Comparison of
the two versions is addressed in following sections on sensitivity analysis and model selection.
Fisheries and fishery-independent data.-- Estimates of recreational harvest and effort
were available for most of the Michigan waters of Lake Huron since 1986. In this context,
harvest refers to actual number retained and does not account for any discarded that die due to
hooking mortality. While I do not adjust for such mortality, I do evaluate the consequences of
underreported recreational harvest in the sensitivity analyses. Because the Saginaw Bay stock of
walleye is the subject population, decisions had to be made as to what estimates of harvest to
include. Although there is undoubtedly some local natural reproduction outside the bay, the
decision was made to assume that it was negligible and the majority of Michigan’s main-basin
walleye harvest could be credited to the Saginaw Bay stock because of the seasonal movement
from the bay.
While walleyes are sought by recreational anglers in Ontario, recreational harvest is not
regularly estimated there. Based on the bathymetry of Lake Huron, it was rationalized that
Saginaw Bay walleye would not likely reach Ontario waters past the abyssal area of Lake Huron
north of the Sixth Fathom Scarp, effectively limiting them to the Ontario waters in the
southernmost portion of the main basin from Pt. Clark, to Sarnia, Ontario. For the purpose of this
SCA analysis, the recreational fishery there was regarded as negligible, but as indicated above
the influence of higher-than-reported recreational harvest is evaluated as an assumption in the
sensitivity analysis.

30

Recreational walleye fishery harvest estimates were obtained from the Michigan DNR’s
statewide annual creel survey study and direct reporting for charter boat operations. The creel
survey follows the methods of Schneider (2000). Michigan DNR creel survey estimates were
obtained from Fielder et al. (2014) and as unpublished data. Ages of walleyes were obtained
from annual biological samples of the recreational fishery throughout the survey season. Ages
were estimated from hard structures: scales for early years and spines since 2009. Numbers of
fish aged each year averaged 521. Proportions at age for the recreational fishery were derived
from those annual biological samples.
Commercial harvest numbers are calculated based on yield-reporting programs and the
average weight of harvested fish each year for each fishery, based on biological sampling.
Walleye is a highly valued species and I assumed that commercial discard was negligible. The
provincially licensed commercial trapnet fishery occurs in the Ontario southern basin waters of
Lake Huron from Pt. Clark to Sarnia. Most of the effort is reported to occur in the most southwesterly area around Sarnia. While all reported harvest was included as the observed harvest for
this fishery, I only used effort targeted on walleyes (number of trapnet lifts). Ages of walleyes
were obtained from samples of the trapnet harvest and estimated from hard structures (scales or
spines), and numbers averaged 527 per year. Proportions at age for the trapnet fishery were
derived from the sample.
The commercial gillnet fishery exists in two regions of Lake Huron thought to include
Saginaw Bay stock of walleye. There is a Provincially licensed gillnet fishery targeting walleye
and other species in the southern main basin of the Ontario waters of the lake from Pt. Clark to
Sarnia concurrent with the trapnet fishery. The second portion of the gillnet fishery is a tribal
fishery authorized under the 1836 Treaty and 2000 Consent Decree (USA v. State of Michigan

31

2000). That fishery exists from the Straits of Mackinaw east to De Tour Passage (Figure 2.1)
excluding the embayments of the Les Cheneaux Islands. That fishery is similar to the Ontario
gillnet fishery and nets comprise mesh sizes from 114 to 140 mm stretch measure. Tribal harvest
is permitted as a retention of the by-caught walleyes. Annual effort was recorded as the
cumulative km of nets fished. These data were obtained from the Chippewa-Ottawa Resource
Authority and Ontario Ministry of Natural Resources. Age distributions were also similar
allowing them to be combined with sample sizes of around 100 per year. All SCA model
variations combined the two gillnet fisheries as a single fishery in the estimation.
The fishery independent survey is a gillnet-based assessment operation, using variablemesh gear, conducted annually each September by the Michigan DNR since 1989 (Fielder et al.
2000; Fielder and Thomas 2006, 2014). Walleyes as young as age 1 were vulnerable to this gear
in all years (Fielder and Thomas 2006). Age distributions were obtained from ages of hard
structures (scales in early years, spines since about 1997) numbering about 500 each year. All the
walleyes in the catch were aged.
Immigrants from Lake Erie.-- Each fishery and survey component had the potential for
augmentation by migratory Lake Erie walleyes and needed some level of adjustment. In each
case, the predicted fishery catch or survey CPUE compared with the observed value in the
objective function was the sum of Saginaw Bay stock of walleye and Lake Erie walleye.
Immigrant walleyes from Lake Erie were given by

nyE

 N yE Py ,aTy  y Ca

32

[10]

where

n yE

is the number of walleyes at age a in year y in Lake Huron that results from Lake Erie

migrants, N yE is the total walleye population in the central and western basins of Lake Erie in a
given year, Py ,a is the year-specific age distribution of the Lake Erie walleye population, T y is the
year-specific proportion of walleyes migrating to Lake Huron as based on the jaw tag returns.
Those values are reported by the Lake Erie Walleye Task Group (LEWTG) of the Great Lakes
Fishery Commission (Thomas et al. 2011). The proportion of Lake Erie walleyes migrating to
Lake Huron based on jaw tags reported from Lake Erie fish ( T y ) was further adjusted based on a
running 3-year average (the average of the previous two years and the reported value for the
current year became the new value for the “current” year). A year-specific correction factor for
nonreporting of tags from the recreational fishery in Lake Huron was incorporated as  y
(Fielder 2014). The contribution (expressed as a proportion) of each age that is thought to make
the migration from Lake Erie is denoted as C a . This value is 1.0 for ages 5 and older but is
reduced to 0.5 for age 4 and 0.0 for ages 2-3. The reduction for younger fish is based on
information reported by Wolfert (1963), Ferguson and Derksen (1971), and Wang et al. (2007),
who observed that younger Lake Erie walleyes were less prone to large migrations.
The estimates generated by the Lake Erie SCA model are specific to the western- and
central-basin stocks in that lake (Thomas et al. 2011). The same model also estimates proportions
at age, however those are aggregated at age 7+. To adjust the Lake Erie estimated age;
distribution out to age 13 to conform to that used in this (Saginaw Bay) model, the aggregated
age-7+ fraction was distributed across ages out to age 13 based on assuming a 50% survival rate
of walleyes for each subsequent age beginning with age 6.

33

Unlike the estimation process for the catch of the Saginaw Bay stock of walleye, fishing
mortality F was not estimated directly for Lake Erie walleye but rather as a fraction of the F for
Saginaw Bay fish for each of the fisheries. The Lake Erie F within Lake Huron was set at half
(0.5) of the corresponding fishery F for each of the fisheries operating in Lake Huron including
the by-kill in the Saginaw Bay commercial fishery. This was based on the belief that once in
Lake Huron, Lake Erie walleyes would be subject to the same fishing mortality rate as the
Saginaw Bay stock but for approximately half the year. The one exception is for the gillnet
fishery, which was spatially split between southern Lake Huron and northern Lake Huron.
Rationalizing that Lake Erie walleyes will not substantially migrate as far north as northern Lake
Huron, only half of the gillnet fishery was set to exploit fish from Lake Erie, and thus the
multiplier for the gillnet F was 0.25. Values for natural mortality of Lake Erie immigrants were
similarly adjusted by these fractions. The catch attributable to Lake Erie fish was derived from
the Baranov Catch Equation (equation 4) applying the fishing mortality rates described above,
and

n yE .
The survey CPUE was adjusted for Lake Erie contributions by using equation 5 (for the

predicted survey CPUE) but applying the formulation to numbers and total mortality rate specific
to Lake Erie fish. This approach used the same estimated survey selectivity and catchability as
estimated by the model for Saginaw Bay fish. As with the fisheries, the predicted values used in
the calculation of the likelihood component was the sum of the two predicted CPUE values.
Spawning stock biomass.-- The mean weight at age and by sex was available for the time
series from the walleyes caught in the Michigan DNR survey (Fielder and Thomas 2014).
Similarly the survey provided a matrix of maturity (expressed as proportions) of females by year

34

and age. Spawning stock biomass at age and year was derived as half the total biomass (to reflect
females only) and the product of the maturity matrix.
Model selection.-- Deviance information criterion (DIC) was used to select among the
models of the three candidate treatments of natural mortality M. Deviance information criterion
is a Bayesian approach to selecting among models, analogous to Akaike's Information Criterion
used when fitting models by maximum likelihood (Spiegelhalter et al. 2002). Because DIC is
limited to models involving the same data, this technique could not be used to select between
models including or excluding the tag return data. The "best" model is the one with the lowest
DIC value. Wilberg and Bence (2008) found DIC to work well as a model selection technique
for SCA analysis.
Deviance information criterion model selection is based on Markov Chain Monte Carlo
(MCMC) analysis which is implemented in AD Model Builder based on the Metropolis-Hastings
algorithm and begins by first obtaining the parameter values based on the HPD and the
associated asymptotic variance-covariance matrix (Wilberg and Bence 2008; AD Model Builder
Project 2011). When calculating DIC, I used half of the variance of the individual deviance
values method for estimating the effective number of parameters (Spiegelhalter et al. 1998;
Gelman et al. 2004) and HPD point estimates.
In my MCMC chain I implemented 1,000,000 steps and saved every 200th step to thin the
chain and save on computing time, which resulted in 5,000 values. The first 1,000 were
discarded as a burn-in period and DIC analysis was based on the remaining 4,000 values. To
determine whether the length of the burn-in period was long enough to ensure convergence, I
visually inspected trace plots the MCMC chains (of the objective function) and evaluated the
burn-in point with stabilization in the plot (Gelman et al. 2004).

35

The model that I selected using DIC was then deemed the “baseline” model and
contrasted with the model integrated with tag returns as auxiliary information. The question was
whether the added influence of tag returns in the evaluation of the model objective function
would result in an overall improved estimation. Selection between these two candidate models
(with and without tag returns) was based on model fit between observed and predicted values,
and particularly evaluation of whether there were systematic patterns to residuals.
Uncertainty and sensitivity analysis.-- Sensitivity analysis was only applied to the
baseline model selected by DIC. Analysis of the sensitivity of the model to the various likelihood
components was based on performance metrics of management interest. Performance metrics
included spawning stock biomass of females (SSB) and total annual mortality (A). Further
performance metrics included the exploitation rates of each of the four fisheries modeled but
limited to age-4 and older walleyes to limit comparison to age-groups fully selected in each
fishery type. Lastly the annual population sizes were compared as a measure of sensitivity to
each model component. All metrics were evaluated on time series means as percent of the
optimal model. Minimum estimates of 95% confidence intervals were based on +1.96 of the
asymptotic posterior standard deviation.
Analysis of sensitivity was conducted by applying a weighting factor lambda (  ) of
either 0.5 or 2.0 to each of the likelihood and prior components. This served to either
deemphasize or over emphasize the effect of that component in order to examine the effect on
the aforementioned performance metrics. Lambda was left at 1.0 for all other likelihood
components so as to evaluate each component’s sensitivity singularly. In some cases, the model
would not converge.

36

Starting values for parameters were varied typically as half and twice the selected values
to test for robustness of model convergence. Failure to converge or large departures of model
predictions from observed values would increase concerns that the final "converged" estimates
obtained from the starting values might not be at the global minimum for the objective function.
Starting values were adjusted until values that met this criterion were identified.
The SCA model was structured by having to make certain assumptions about walleye
stock structure and fishery dynamics. These assumptions constitute a source of uncertainty. To
evaluate the significance of these assumptions, performance metrics were also examined by
increasing and decreasing assumed values. These assumptions included the following: (1) The
duration of Lake Erie walleyes inhabiting Lake Huron. This was set in the model by assigning
mortality rates that were one-half of those of the Saginaw Bay fish, but was trialed here at +/20% implying 7.2-month and 4.8-month durations as opposed to a six-month duration. (2) The
proportion of fish of each age of Lake Erie walleyes migrating to Lake Huron. Fish of ages 2, 3,
and 4 were trialed at 50, 100, and 100%, respectively, in their propensity to migrate to Lake
Huron instead of none for ages 2 and 3 and 50% for age 4. (3) The assumption of Ontario’s
recreational fishery harvest being nil is improbable as is the assumption of no catch-and-release
mortality, so elevating the recreational catch by an additional 10% of the present observed values
was evaluated for effect on the performance metrics. (4) The larger expanded value of by-kill
reported by MacMillan and Roth (2012) was evaluated. (5) I evaluated the sensitivity to variance
ratios by choosing alternative values. Recognizing that a variance ratio of 1.0 implied the same
variance for the effort deviations or catchability random walk as the catch or CPUE, I chose
alternative values to test for sensitivity by selecting values of similar proportions above or below
1.0. (6) I evaluated the sensitivity of model predictions to the assumption of time-varying

37

catchability for the recreational fishery. Preliminary analysis suggested that catchability did vary
but not in anticipated patterns. To explore the influence of allowing expected catchability to drift
over time for the recreational fishery, I refit the model with a white-noise treatment of the
variability of recreational catchability, which assumes catchability varies about a constant mean.
(7) Lastly, I evaluated the treatment of the age-based M as fully known. This was done two
ways: first I treated Mage as a prior of its own and incorporated in the objective function based on
a lognormal distribution using equation 7. For that purpose  was approximated at one-quarter of
the maximum range of the confidence intervals for the age-based M values from Vandergoot and
Brenden (2014). Secondly I evaluated model sensitivity to alternative set values of Mage using the
upper and lower 95% confidence intervals.
Standardized residual values between observed and predicted for the baseline model were
examined. Residuals were based on the proportions at age for the survey CPUE and each fishery
catch with the residuals computed as the observed minus predicted and standardized as the
quotient of the difference and the predicted standard deviation. A uniform scatter for each about
zero was interpreted to mean that no systematic pattern existed.

Results

Model selection-- Application of the DIC model selection procedures indicated that the
age-varying natural mortality best reflected the observed data. In spite of this candidate model
having the greatest effective number of parameters, analysis indicated it had the lowest overall
DIC value (Table 2.2). Consequently, it was the age-varying natural mortality model version that
then constituted the baseline model version for integration with tag returns as an alternative
model version for further evaluation. The integrated model resulted in a fit of predicted and
38

observed values that compared favorably with the baseline model except that the predicted
recreational catch was over estimated relative to the observed (Figure 2.2A). The mean (over
years) recreational catch predicted by the integrated model was more than twice the mean catch
predicted by the baseline model. In contrast the baseline model predictions of recreational catch
were within 1% of the observed values.
The integrated version of the model generated predicted tag returns by utilizing the
annual recreational exploitation and survival rates to generate the tag return probabilities.
Exploitation rates differed between the Brownie model and the baseline model since 2004
(Figure 2.2B). The elevated exploitation rate depicted by the observed tag returns probably
resulted in an inflation of the recreational catch causing the departure from the observed. The
result was an elevated (72% greater) population estimate. The integrated version also estimated
that the population had declined in the last two years of the time series (2010 and 2011) to record
lows, which were not consistent with the observed fishery trends. From this, and the systematic
overestimation of observed recreational harvest by the integrated model, I concluded that the
inclusion of the tag return auxiliary information did not result in a superior model and I retained
the baseline version as my selected model.
Baseline model-- The baseline model achieved an overall good fit of the observed data
sets (fishery extractions and fishery-independent survey; Figure 2.3). An examination of
standardized residual proportions at age for the survey and each fishery generally revealed no
consistent pattern. There was a slight preponderance of positive residuals over negative values,
especially for older ages across fisheries. This may have been a result of aging error or over
estimation of certain terminal age-groups.

39

Abundance of the walleye stock was estimated by the baseline SCA model to have
declined steadily since 1988 and then increased beginning in 2005. The stock peaked at about 4
million walleyes (age 2 and older) in 2007 before declining again and leveling off at about 2.4
million by the end of the time series (Figure 2.4). Uncertainty in the abundance was greatest in
recent years likely reflecting the model's estimation uncertainty over year-classes not yet
depleted, a characteristic common to SCA fits (Figure 2.4).
Selectivities steadily increased for older ages of walleyes through age 10 in the lake-wide
gillnet and recreational fisheries (at which point selectivities were assumed to remain at the age10 value and were not estimated; Figure 2.5). In contrast, the highest selectivity for the trapnet
fishery was ages 3 and 4. Minimum mesh sizes in the gillnet fishery and minimum length limits
in the recreational fishery may have contributed to the low selectivity of age-2 and age-3 in those
fisheries.
Estimated catchability of the recreational fishery and survey varied considerably over the
series (Figure 2.6). The survey catchability reflected a greater catchability before dreissenid
mussel colonization (about 1993), as I hypothesized would occur due to changes in water clarity.
However, catchability of walleyes in the recreational fishery reflected a pattern that was more
characteristic of recent population trends or perhaps the disappearance of alewives.
Total annual mortality ranged from a low of 32% in 1991 to a high of 53% in 1986
(Figure 2.7). Uncertainty about total mortality was greatest for 1986 and was likely associated
with the need to estimate an initial population consisting of some cohorts that were reflected in
only a few years of data. Uncertainty in estimates of total mortality increased slightly beginning
in 2001. Total annual mortality rates generally followed a pattern that suggested, to some degree,
mortality was an inverse function of abundance of fish (Figures 2.4, 2.7).

40

Estimated age-specific total annual mortality of walleyes showed a consistent qualitative
pattern over time, in which a peak occurred at age 4 and a decline to a substantially lower value
for age 5, followed by a gradual increase to the levels experienced by age-10 and older fish
(Figure 2.8). The total annual mortality reached for fish of ages 10 and older, however, decreased
over the time series. The age specific patterns reflect selectivity and natural mortality rates, and
the sharp decrease from age 4 to age 5 corresponds to the lower M for age-5 and older walleyes
and a decrease in selectivity for the trapnet fishery.
Recreational fishing mortality (Fmrc) of walleyes was greatest in proportion of all sources
of combined fishing mortality for all ages and was nearly 80% of the total by age 7 and older
(Figure 2.9). Fishing mortality increased with walleye age for the recreational and gillnet
fisheries, likely reflecting increasing selectivity as a function of length limits and mesh sizes.
This was in contrast with the Ontario trapnet fishery, where fishing mortality was highest on the
younger age groups and then declined with age. Recreational fishing mortality increased over the
time series, especially after 2000, where age 5 was used as an indicator (Figure 2.10). Fishing
mortality declined or was steady for the other fisheries over the same time series.
Estimated recruitment of walleyes at age 2 clearly indicates the resurgence in
reproductive success beginning with the 2003 year class (Figure 2.11). Total biomass and
spawning-stock biomass of the Saginaw Bay stock followed a similar trajectory (Figure 2.12).
Biomass metrics were much higher earlier in the time series, peaking in 1989, and then exhibited
a decline until the recent resurgence beginning in 2006. Uncertainty for the estimates was
generally less for this time series relative to other estimated metrics but also increased for the
three most recent years.

41

The baseline SCA model exhibited high sensitivity to weighting for the CPUE of the
survey and gillnet harvest, but not the recreational fishery harvest (Table 2.3). The baseline
model was somewhat sensitive to weighting of the age structure of the trapnet and gillnet
fisheries. Aside from those cases, there was generally little departure on a percentage basis from
the baseline version for most metrics as component weightings were changed. Collectively,
altering weighting factors resulted in the model failing to converge just three times.
Although by-kill estimation was not directly part of the objective function, the value used
to derive the catchability for application to commercial effort of past years did result in
estimation differences from the baseline model (Table 2.4). The higher extrapolated (year-round)
estimate of by-kill in 2010 (102,000 walleyes) increased the population estimate by 36% and
SSB by 26% (Table 2.4). Total annual mortality was not affected on average, but corresponding
exploitation rates in the fisheries were reduced by about 20% each except for the by-kill
exploitation rate, which increased 202%. Increasing recreational fishery harvest from the
baseline model by 10% resulted in a 5-6% increase in population and SSB estimates and a
similar magnitude decline in exploitation rates except for the recreational fishery (Table 2.4).
The influence of the duration of Lake Erie walleye habitation in Lake Huron had only
minor effects on model estimates when increased from 6 months to 7.2 months (Table 2.4). A
lack of convergence prevented evaluation of a shorter duration. Similarly, the full inclusion of
age-3 and one-half of the age-2 walleyes in the immigration had only a minor effect on the
estimates of the baseline model (Table 2.4).
The baseline (age-based M) model proved reasonably robust to the assumptions of
variance ratios as shifting the weight of the variance had only minor effects on the metrics of

42

management interest (Table 2.4). This suggests that, while somewhat arbitrary, the choice of
variance ratios did not have a profound effect on the overall estimation.
Model estimates were sensitive to the assumption of time-varying catchability in the
recreational fishery (Table 2.4). Application of an alternative white noise treatment of q resulted
in a 22% increase in total population size on average with most of the departure occurring in the
last seven years of the time series. In order to fit the model with a white noise treatment of q
deviations, the recreational fishery variance ratio had to be increased to 1.10.
Treating the age based M value as a prior in the estimation process had little effect on
model performance in which there was a <1% departure from the baseline for all metrics and an
increase in the standard deviation by just 0.1% over the baseline. It did increase uncertainty
slightly for some other metrics and added to the parameter load. Trialing age based M at the
upper and lower 95% confidence limits did result in a modest effect on model metrics (Table
2.4): generally a 10-20% effect with greater population estimate resulting from the upper limit
value, and a lower population from the lower limit value.

Discussion

The SCA model presented in this study allows the explicit incorporation of multiple
fisheries in the stock assessment and overcomes a weakness of the tag-recapture assessment
methods that have been used in the past. By incorporating all the fisheries exploiting this stock,
more plausible estimates of overall mortality, and consequently fishing mortality were obtained.
Although the recreational fishery accounted for the majority of the total fishing mortality (Figure
2.9), the omission of the other fisheries clearly underestimated overall fishing mortality,
especially for younger ages of walleyes. This was most apparent in the annual estimates of
43

natural mortality derived from the Brownie data (as the difference between Z and F). This
method of M estimation passes any bias in F to M and in the instance of the Brownie model
resulted in an over estimation of M. That greater value almost certainly reflected the fishing
mortality of the trapnet, lake-wide gillnet, and commercial by-kill fisheries. The difficulty in
obtaining tag returns from commercial fisheries is widespread and methodologies for addressing
this have had only limited success especially in improved estimates of M (Eveson et al. 2007).
The elevated M values of the time-varying M model version resulted in the over estimation of the
population. That version of the SCA model likely estimated the greater abundance so as to fulfill
the observed fisheries and the supplied high annual M values. While an annual estimate of M
might be desirable for use in a SCA model, in this case model selection procedures favored the
age-based natural mortality expression.

Natural mortality rates are generally difficult to assess in a fish population and are often
assigned an assumed set value for population modeling (Quinn and Deriso 1999; Nate et al.
2011). I treated M as known and assumed age-specific natural mortality borrowed from Lake
Erie because this proved to fit the data better than other expressions of M (constant and timevarying), but I was unable to fully evaluate the accuracy of these values. The DIC selection of
the age-varying natural mortality model version may be a reflection that natural mortality of the
Saginaw Bay stock does vary by age. It doesn’t necessarily mean that those are the most correct
values. Misspecification of M can have considerable influence over model estimates (Clark
1999; Deroba and Schueller 2013). Deroba and Schueller (2013) evaluated reliance of a constant
M value on stock assessments through simulations and found it to bias estimates of SSB and F
by as much as +/-100% of the estimate for both long- and short-lived species. The degree of bias
of a misspecified natural mortality may depend on how M was treated in the model and what the
44

true values were. Deroba and Schueller (2013) reported that parameter bias was generally less
when time-varying M values are misapplied than when misspecified as age varying or as a
constant, and as such, bias affected estimates of SSB and recruitment.
The difficultly remains in obtaining reliable estimates of time- or age-varying M. Quinn
and Deriso (1999) favored using tag-based studies to derive M, but in this instance, it failed to
result in an accurate estimate of M because of the lack of representation of the competing
fisheries. More stock assessment efforts are turning to estimating M within a SCA analysis,
although typically this requires an informative prior value (Wang and Liu 2006; Lee et al. 2011;
Maunder and Wong 2011; Vincent 2013). I evaluated this in the sensitivity analysis and found
that when the specified values for M were treated as medians of a prior lognormal distribution,
this did not materially affect the model point estimates. There was some increase in uncertainty
but this was modest. In this instance, I concluded that this does not strengthen the model. I
believe that there is more uncertainty associated with transferring estimates of M from Lake Erie
to Lake Huron than was captured as estimation error for the Lake Erie estimates. I tested the
impact of specifying M at the upper and lower bounds of the Lake Erie estimates and this did
have a moderate effect on the abundance and exploitation estimates, suggesting that this is a
consequential model term, especially if it were to turn out that natural mortality of walleyes in
Lake Huron was substantially different than in Lake Erie. Future improvements to the model
may be better realized by development of stock-specific, age-based M values for Saginaw Bay
walleyes based on a tagging study following the methods of Vandergoot and Brenden (2014).
The time series mean estimate of total annual mortality for Saginaw Bay walleye was
37%, which was the same as that from the Brownie model (Fielder 2014) for the same years
suggesting that both models are in agreement for total mortality. Brownie model-based estimates

45

of total annual mortality ranged more over time than those generated by SCA, but on average
both estimates agree suggesting Brownie estimation (from tagging analysis) could accurately
estimate total mortality, but it overestimated M because of the lack of representation from
fisheries other than the recreational.
The trapnet by-kill decrease over time resulted from the large decreases in inner Saginaw
Bay trapnet fishing effort (Fielder et al. 2014) and the assumption that this source of mortality
was directly proportional to the inner Saginaw Bay trapnet fishing effort. Only the recreational
fishing mortality increased substantially in recent years, a response to the stock recovery and
increases in vulnerability as confirmed by trends in catchability (Figure 2.6). Generally
recreational fishing effort has declined or been stable in recent years (Fielder et al. 2014) so the
increasing recreational F must be a function of the time-varying catchability.
The initial peak abundance of Saginaw Bay walleyes in 1988 is evident in the population
estimate (Figure 2.4) as well as the biomass trends (Figure 2.12). Previously, when indices of
abundance were noted to be high for those early years it was dismissed as an artifact of Lake Erie
immigration and or increased catchability due to changes in water clarity stemming from
dreissenid mussel colonization in 1993 (Fielder et al. 2000; Fielder and Thomas 2006). My
version of the SCA model generated these estimates of abundance and biomass specific to the
Saginaw Bay stock, effectively factoring out any effect of Lake Erie fish. Similarly, the
utilization of a random-walk procedure for application of changing catchability was intended to
allow for any change in catchability stemming from dreissenid colonization and appears to have
achieved that for the survey CPUE (Figure 2.6). In spite of these efforts, the model is still
projecting high abundance at a time when generally the walleye population was still regarded as
unrecovered. Alternatively, if my assumption of time-varying catchability in the recreational

46

fishery is mistaken and the white-noise depiction is more accurate, then the abundance in recent
years is greater, suggesting that abundance in the early years was not similar to the more recent
recovery.
Operating on the premise that recreational fishery catchability was in fact time varying,
then this analysis forces one to consider that abundance of the Saginaw Bay walleye stock most
likely was genuinely substantial in those early years of the time series. While it was believed that
the fishery was dependent on stocking until 2003 (Fielder 2002; Fielder and Thomas 2014), the
use of oxytetracycline marking to identify hatchery fish in stocking evaluations was not available
until 1997 (Fielder 2002). Consequently it is difficult to rule out a surge in reproductive success
early in the time series. While the abundance estimate in 1988 rivaled the peak of the more
recent recovery (2007), earlier biomass was much greater in both total and spawning stock than
during the more recent resurgence (Figure 2.12). This suggests that the high abundance in 1988
was not merely due to large numbers of young fish but was comprised of mature fish and older
or fast growing fish. Recruitment estimated by the SCA model at age 2 indicated year-class
strength in 1984-1986 that was on the same magnitude of some recent years (Figure 2.11). The
downward decline of abundance in the interceding years is consistent with the lower recruitment
of the same years. Overall, the phenomenon illustrates that gains in abundance can be lost if
recruitment fails to remain strong, at least periodically. The more recent decline in recruitment
does not bode well for the on-going recovery of the Saginaw Bay walleye stock. Fielder et al.
(2007) forecasted that recruitment should remain strong as long as alewives remain scarce, but
Fielder and Thomas (2014) suggested that recent lower recruitment may be reflecting density
dependence in the stock/recruitment relationship.

47

Failure of the attempt to strengthen the baseline SCA model by integrating auxiliary
information in the form of tag returns raises the question as to why these data did not have the
intended benefit. I followed the recommendations of Maunder (1998) and Maunder and Punt
(2013) for the optimal methods of incorporating tag return as estimating the probabilities within
an SCA model for the corresponding likelihood component, but this still resulted in a poor fit.
Customarily if the addition of information reduces the fit of the model, it suggests conflicting or
contradictory dynamics (Richards 1991; Schnute and Hilborn 1993; Haddon 2001).
Homogeneity of capture probability is a fundamental assumption in tag return analysis (Brownie
et al. 1985) and the tag-based method equivalent to the assumption of single-stock estimation
within SCA. This raises the question of how representative the Tittabawassee River spawning
run of walleyes (used as the tagging source) is of the rest of the bay’s population. There are other
reproductive sources of walleye within the bay (Fielder 2002). The operational premise in the
management of the bay’s walleye fishery to date is: (1) That the combined reproductive sources
operate with a single dynamic (i.e. as a single population). (2) That the Tittabawassee River is
the single largest source and would be most reflective of the bay’s walleye population. Recent
attempts to better understand the mix of reproductive sources within Saginaw Bay, based on
otolith microchemistry methods, have suggested that the Tittabawassee River may, in fact, not be
the single greatest reproductive source (B. Murry, Central Michigan University, personal
communication).
New information on movement dynamics of Tittabawassee River walleye from a
telemetry study has suggested that fish are exposing themselves to certain fisheries and not
others, by virtue of movement choices each year (T. Hayden, USGS, personal communication).
Variable spatial structure of fish stocks complicates stock assessment and can be difficult to

48

account for (Goethel et al. 2011). Accounting for movement of fish, usually from source
locations to harvest or ‘sink’ localities in stock assessment typically involves the incorporation of
a movement matrix (Quinn et al. 1990; Quinn and Deriso 1999; Bence et al. 2011) and is
regarded to be among some of the challenges of the future of stock assessment modeling in
fisheries (Quinn 2003). Such movement can have implications for management of the stocks
(Wilberg et al. 2008), and is a challenge to the assessment of other species in the Great Lakes as
well (Nalepa et al. 2005), and for walleye in other locations (Thomas et al. 2011). Some
investigators have resorted to entirely different modeling approaches to compensate for the
mixing of stocks (Michielsens et al. 2006; Molton et al. 2012). In this instance I concluded that
attempting to use auxiliary information from one specific breeding source in the SCA objective
function exposed the differences in dynamics and confounded the model’s estimation. Fielder
(2014) has recognized this limitation and has recommended a diversification of source spawning
runs for the future continuation of that tagging study. Better stock definition for walleye is
needed in Lake Huron, with the definition not only determined by genetic or microchemistry
type analyses but by differences in population metrics such as mortality, exploitation patterns,
and movement.
Structure of the Saginaw Bay baseline SCA model exhibited sensitivity to only certain
components of the joint likelihood function (among data sources). The failure to result in an
iterative convergence in just three sensitivity trials suggests that the model structure is
reasonably robust. The baseline SCA model is complex, however, given the four fisheries. It is
difficult to say how the 181 parameters are interrelated during the model fitting process. Detailed
analysis of parameter correlation may disentangle those relationships, but it is apparent from the
sensitivity analysis that the model is generally consistent with its convergence. Future work on

49

this model might explore whether similar estimates and confidence could be obtained with a
simpler model or achieve greater resiliency. Key assumptions in this baseline model appear to
include the by-kill values used with the larger extrapolated year-round values that resulted in a
larger Saginaw Bay walleye population estimate. From this, it might be concluded that the
estimates of the baseline are conservative in that relaxed assumptions would tend to lead to a
larger estimated population.
The development of an SCA model to describe the Saginaw Bay stock of walleye in Lake
Huron is a significant advancement in the assessment of this population. Not only has it offered
age-based estimates of mortality rates and abundance, but it has also helped shed light on
deficiencies of the Brownie model and biases that affect those estimates. The integrated version
did not result in an improved model in this instance, but if the improvements in the Brownie
model suggested by Fielder (2014) are achieved, future integration may still be possible.
Statistical catch-at-age models are often used as the basis for forecasting models that allow
evaluation of alternative fishery management strategies. Estimates like those generated from this
SCA analysis would be essential in developing such a model. Management choices for walleye
in most of the Michigan waters of Lake Huron has been, to date, primarily as a fixed rule (e.g.,
length limit, bag limit, and season closure in the spawning rivers) for the recreational fishery, and
a yield allowance for by-catch in the tribal gillnet fishery in the northern portion of the lake. A
simulation model based on the availability of estimated dynamics including recruitment,
abundance and mortality rates would assist managers to design harvest regulations that are more
state-based, and even address allocation if different jurisdictions ever feel that is necessary.
Ultimately, the utility of this assessment model and its estimates will depend on fishery

50

manager’s willingness to advance management to keep pace with the information stemming from
the assessment of this stock.

51

APPENDIX

52

Table 2.1. Estimated parameters of the catch-at-age model of the Saginaw Bay stock of walleye
in Lake Huron.
Parameter
Sa
sasurv

Description
Age-specific selectivity for each of the
fisheries
Age specific selectivity for the survey

q

Catchability coefficient for each of the
fisheries (or starting value for recreational
fishery random walk)

qsurv

Random walk time varying catchability for
the survey
Catchability deviations (white noise
process error term) for the trapnet and
gillnet fisheries or random walk deviations
for recreational fishery and survey


Na,1986

Initial numbers for the beginning year of
1986

N2,y

Initial numbers of age-2 walleye for each
modeled year (recruitment)
Observation standard deviations for each
fishery & survey except the state-licensed
by-kill.



Table 2.2. Deviance information criterion (DIC) model selection of three candidate statistical
catch-at-age models for the Saginaw Bay stock of walleye in Lake Huron based on different
treatment of natural mortality (M): constant, age varying, and time varying. Included is the
effective number of parameters (pD).
Model
Constant M
Age-varying M
Time-varying M

DIC value
60208.4
60142.2
60340.8

Delta from min
66.2
0.0
198.7

53

‘Best’ model
No
Yes
no

pD
160
180
149

Table 2.3. Sensitivity of penalized likelihood component analysis of the Saginaw Bay stock of
walleye in Lake Huron statistical-catch-at-age model as percent change from the aged-based
natural mortality model (baseline) version. Sensitivity was tested by applying a weighting factor
lambda (  ) of either 0.5 or 2.0 to each of the 12 likelihood components in the baseline version
of the model and assessing percent change from the time series mean for four metrics of
management interest: population size (N), total annual mortality (A), spawning-stock biomass of
females (SSB), and the exploitation rate (  ) of the four fisheries. Weighting factors that resulted
in a lack of convergence are denoted by DNC (did not converge).
Weighting
factor

Recreational

Trapnet

A

SSB



0.42

-0.50

2.66

0.63

0.23

1.99

1 =2.0
0.51
Recreational random walk
 2 =0.5
0.12

-0.32

0.75

-1.77

-0.94

-0.73

-1.34

0.28

0.22

0.29

-0.16

-0.22

1.84

 2 =2.0
0.68
Recreational age structure
3 =0.5
-0.28

-1.01

0.24

-1.75

-0.64

-0.61

-5.78

-0.06

-0.85

1.55

2.63

1.43

-3.73

0.34

2.22

-1.00

1.36

14.23

251.98

-57.56

240.57

128.19

43.02

0.09

-0.16

-0.13

-0.14

-0.19

N

Recreational harvest
1 =0.5
0.02

3 =2.0

-0.89
1.93
Survey catch-per-unit-of-effort
 4 =0.5
289.27
39.29
 4 =2.0
0.09
-0.04
Survey random walk
5 =0.5
-0.18
0.17
5 =2.0
DNC
DNC
Survey age structure
6 =0.5
1.47
-1.01
6 =2.0
4.53
6.47
Trapnet harvest
7 =0.5
-0.23
0.33
7 =2.0
11.80
5.63
Trapnet effort
8 =0.5
DNC
DNC
8 =2.0
0.53
-0.32
Trapnet age structure
9 =0.5
0.94
-1.15
9 =2.0
625.12
11.49
Gillnet harvest
10 =0.5
417.59
44.18
10 =2.0
DNC
DNC
Gillnet effort
11 =0.5
-0.81
1.03
11 =2.0
1.50
-1.42
Gillnet age structure
12 =0.5
-0.02
-0.01
12 =2.0
411.51
628.02



Gillnet



By-kill



-0.09

0.36

0.15

0.18

1.09

DNC

DNC

DNC

DNC

DNC

2.26

-4.74

-3.55

-6.73

0.35

-0.72

26.18

4.79

6.77

17.84

-0.01

0.23

-0.05

0.54

1.39

8.54

-9.97

-5.68

95.48

11.29

DNC

DNC

DNC

DNC

DNC

0.25

-0.13

-0.46

-0.94

-1.17

-0.53

0.48

-0.13

3.89

-6.87

567.72

-85.23

-85.23

693.50

-73.49

361.30

-68.16

234.87

216.26

-57.33

DNC

DNC

DNC

DNC

DNC

-0.12

0.81

0.50

3.27

5.69

0.40

-1.38

-0.99

-2.99

-7.58

-0.19

0.01

-0.11

0.05

-2.46

375.61

-12.94

283.44

-50.40

411.51

54

Table 2.4. Sensitivity of assumptions analysis of the Saginaw Bay stock of walleye in Lake
Huron statistical-catch-at-age model from the age-based natural mortality model (baseline)
version for key assumptions: (1) An expanded (extrapolated) estimate of the magnitude of
commercial by-kill in the bay from MacMillan and Roth (2012). (2) An expanded recreational
observed catch to simulate an Ontario component. (3) Two alternative durations of residency
time for the habitation of Lake Erie (LE) walleyes in Lake Huron. (4) The ages at which Lake
Erie walleye are hypothesized to immigrate was adjusted to fully include age-3 and one-half of
age-2 fish. (5) Alternative variance ratios used in the likelihood functions for relating white-noise
or random-walk deviations of catchability to catch. (6) Treatment of recreational catchability as a
white-noise process instead of random walk. (7) Treatment of age-based natural mortality M as a
prior in the model estimation and then alternative values based on 95% confidence intervals.
Sensitivity was assessed as percent change from the time series mean for four metrics of
management interest: population size (N), total annual mortality (A), spawning stock biomass of
females (SSB), and the exploitation rate (  ) of the four fisheries. A lack of model convergence
is denoted by DNC (did not converge).
Model
version
N
By-kill magnitude
Extrapolated
36.18
value

Recreational
SSB

-0.03

26.34

-19.36

-20.14

-19.68

201.83

5.15

3.63

-5.63

-5.86

-4.27

-0.80
DNC

-0.34
DNC

0.01
DNC

0.02
DNC

0.11
DNC

-1.68

6.10

-1.90

-4.50

1.26

-3.15

-0.02

-0.18

0.31

0.19

0.15

-0.24

0.15

0.19

0.10

-1.23

0.03

0.72

-0.06
0.08

0.00
-0.02

-0.01
0.03

0.00
0.02

-0.69
0.84

-0.36
0.56

-0.05

0.01

-0.09

-0.02

-0.02

-0.28

12.21

-19.34

-15.91

-17.95

-47.00

Duration of LE habitation
0.60 year
-0.75
-0.19
0.40 year
DNC
DNC

Variance ratios
Recreational
1.15
-0.08
Trapnet
1.25
0.16
Gillnet
1.25
0.04
0.75
-0.06
Survey
1.15
0.04

Catchability deviations
Recreational
22.22
-10.07
white noise

55



By-kill

A

Expanded recreational fishery
10%
5.75
-0.05
increase

LE immigration
Includes
-2.89
ages 2 and 3

Trapnet



Gillnet





Table 2.4 (cont’d)

Age-based M values
Mage as a
0.45
-0.03
prior
Mage 95%
upper limit
Mage 95%
lower limit

0.01

-0.16

-0.40

-0.04

0.05

21.93

3.22

12.83

-8.76

-9.81

-9.19

-15.74

-15.51

-3.17

-9.94

7.93

8.77

8.01

15.99

56

Figure 2.1. Saginaw Bay and Lake Huron showing geographic features of importance to the
model components. The vicinity indicated by hash marks represents the 1836 Treaty area of
Lake Huron.

57

Figure 2.2. (A) Comparison of observed and predicted recreational walleye catch based on the
integrated model version. (B) Recreational exploitation rates between that estimated by the age
based natural mortality (baseline) version of the Saginaw Bay statistical-catch-at-age model
version and that from the Brownie model (from Fielder 2014). Error bars are +1.96 standard
errors.

58

Figure 2.3. Observed and predicted values 1986 – 2011, from the M-age based (baseline) version
of the Lake Huron walleye statistical-catch-at-age model for the (A) fishery-independent survey,
(B) recreational fishery catch, (C) trapnet fishery catch, (D) lake-wide (combined Ontario and
tribal) gillnet fishery catch, and (E) the commercial by-kill.

59

Figure 2.4. Numbers of the Saginaw Bay stock of walleye in Lake Huron 1986 – 2011 age 2 and
older and the +1.96 standard errors (dashed lines) confidence interval, as predicted by the agebased natural mortality (baseline) version of the statistical-catch-at-age model.

60

Figure 2.5. Selectivity of the three estimated fisheries scaled to 1.0 based on maximum value by
age for the age-based natural mortality (baseline) version of the Saginaw Bay stock, statisticalcatch-at-age model for Lake Huron, 1986-2011 data.

61

Figure 2.6. Time-varying catchability (q) and the +1.96 standard errors (dashed lines) confidence
interval of the (A) recreational fishery and (B) survey of the Saginaw Bay stock of walleye in
Lake Huron from the age-based natural mortality (baseline) version of statistical-catch-at-age
model 1986-2011.

62

Figure 2.7. Total annual mortality rate (limited to age-4 and older fish), A for the Saginaw Bay
stock of walleye in Lake Huron 1986-2010 and +1.96 standard errors (dashed lines) confidence
interval, as predicted by the age-based natural mortality (baseline) version of the statisticalcatch-at-age model.

63

Figure 2.8. Total annual mortality (A) of the Saginaw Bay stock of walleye by age (years) for
three time periods as estimated by the age based natural mortality statistical-catch-at-age model
(baseline) version.

64

Figure 2.9. Proportion of fishing mortality (F) by age for 2011 from the age-based natural
mortality (baseline) version of the Saginaw Bay stock of walleye statistical-catch-at-age model
for walleye from the Michigan recreational fishery (Fmrc), Ontario trapnet fishery (Fotn), the
lake-wide (combined Ontario and tribal) gillnet fishery (Fgln) and the by-kill stemming from the
state licensed trapnet fishery in Saginaw Bay (Fbkl).

65

Figure 2.10. Fishing mortality (F) of age-5 walleye over time from the four fisheries represented
by the age-based natural mortality (baseline) version of the Saginaw Bay stock of walleye
statistical-catch-at-age model. Michigan recreational fishery (Fmrc), Ontario trapnet fishery
(Fotn), the lake-wide (combined Ontario and tribal) gillnet fishery (Fgln) and the by-kill
stemming from the state licensed trapnet fishery in Saginaw Bay (Fbkl).

66

Figure 2.11. Recruitment of walleye to the Saginaw Bay stock as estimated by the age-based
natural mortality statistical-catch-at-age model (baseline) version based on age-2 numbers
attributed to their originating year class.

67

Figure 2.12. Total biomass (all ages and both sexes) and spawning-stock biomass (SSB) of
mature female walleye (in kilograms) for the Saginaw Bay stock of walleye in Lake Huron,
1986-2011 and +1.96 standard errors (dashed lines) confidence interval, as predicted by the age
based natural mortality version of the statistical-catch-at-age model.

68

CITATIONS

69

CITATIONS

AD Model Builder Project. 2011. An Introduction to AD Model Builder Version 9.0.0 for use in
Nonlinear Modeling and Statistics. Available: http://admbproject.googlecode.com/files/admb.pdf (July 2013).
Aldrich, J. 1997. R. A. Fisher and the making of maximum likelihood 1912-1922. Statistical
Science 12:162-176.
Baldwin, N. S., and R. W. Saalfeld. 1962. Commercial fish production in the Great Lakes
1867–1960. Great Lakes Fisheries Commission, Technical Report 3 (1970 supplement).
Bence, J., S. Sitar, and M. Ebener. 2011. Stock Assessment Models. Pages 5-16 in: D.C.
Caroffino and S.J. Lenart, editors. Statistical catch-at-age models used to describe the
status of lean lake trout populations, a report prepared by the 1836-Treaty ceded waters
of Lakes Michigan, Huron, and Superior at the inception of the 2000 consent decree.
Available:
http://www.michigan.gov/documents/dnr/LakeTroutLongReport_353000_7.pdf. (July
2014).
Brownie, C., D., R. Anderson, K. P. Burnham, and D. S. Robson. 1985. Statistical inference
from band recovery data: a handbook. U. S. Fish and Wildlife Service, Resource
Publication 156.
Butterworth, D. S., and Rademeyer, R. A. 2008. Statistical catch-at-age analysis vs. ADAPTVPA: the case of Gulf of Maine cod. ICES Journal of Marine Science 65: 1717–1732.
Clark, W. G. 1999. Effects of an erroneous natural mortality rate on a simple age-structured
stock assessment. Canadian Journal of Fisheries and Aquatic Sciences 56:1721-1731.
Deroba, J. J., and A. M. Schueller. 2013. Performance of stock assessments with misspecified
age- and time-varying natural mortality. Fisheries Research 146:27-40.
Eveson, J. P., T. Polacheck, and G. M. Laslett. 2007. Incorporating fishery observer data into an
integrated catch-at-age and multiyear tagging model for estimating mortality rates and
abundance. U.S. National Marine Fisheries Service Bulletin 105:493-508.
Ferguson, R.G., and A. J. Derksen. 1971. Migrations of adult and juvenile walleyes (Stizostedion
vitreum vitreum) in southern Lake Huron, Lake St. Clair, Lake Erie, and connecting
waters. Journal of the Fisheries Research Board of Canada 28:1133-1142.
Fielder, D. G. 2002. Sources of Walleye Recruitment in Saginaw Bay, Lake Huron. North
American Journal of Fisheries Management 22:1032-1040.

70

Fielder, D. G. 2014. Mortality, exploitation, movement, and stock size of Saginaw Bay walleyes,
1981-2011; 31 years of tag return analysis. Michigan Department of Natural Resources,
Fisheries Report 04. Lansing.
Fielder, D. G., J. E. Johnson, J. R. Weber, M. V. Thomas, and R. C. Haas. 2000. Fish population
survey of Saginaw Bay, Lake Huron, 1989-97. Michigan Department of Natural
Resources, Fisheries Research Report, 2052. Ann Arbor.
Fielder, D. G., T. L. Kolb, T. M. Goniea, D. L. Wesander, and K. S. Schrouder. 2014. Fisheries
of Saginaw Bay, Lake Huron 1986-2010. Michigan Department of Natural Resources,
Fisheries Report 02. Lansing.
Fielder, D. G., J. S. Schaeffer, and M. V. Thomas. 2007. Environmental and ecological
conditions surrounding the production of large year classes of walleye (Sander vitreus) in
Saginaw Bay, Lake Huron. Journal of Great Lakes Research 33:118-132.
Fielder, D. G., and M. V. Thomas. 2006. Fish Population Dynamics of Saginaw Bay, Lake
Huron 1998 – 2004. Michigan Department of Natural Resources, Fisheries Research
Report. No. 2083. Ann Arbor.
Fielder, D. G., and M. V. Thomas. 2014. Status and Trends of the Fish Community of Saginaw
Bay, Lake Huron 2005–2011. Michigan Department of Natural Resources, Fisheries
Report 03. Lansing.
Fournier, D., and C. P. Archibald. 1982. A general theory for analyzing catch at age data.
Canadian Journal of Fisheries and Aquatic Sciences 39:1195-1207.
Fournier, D. A., H. J. Skaug, J. Ancheta, J. Ianelli, A. Magnusson, M. N. Maunder, A. Nielsen
and J. Sibert 2012. AD Model Builder: using automatic differentiation for statistical
inference of highly parameterized complex nonlinear models. Optimization Methods and
Software 27:233-249.
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B., 2004. Bayesian Data Analysis, 2nd edition.
Chapman and Hall, Boca Raton, Florida.
Goethel, D. R., T. J. Quinn II, and S. X. Cadrin. 2011. Incorporating spatial structure in stock
assessment: movement modeling in marine fish population dynamics. Reviews in
Fisheries Science 19:119-136.
Haddon, M. 2001. Modelling and quantitative methods in fisheries. CRC Press. New York.
Hilborn, R. 2012. The evolution of quantitative marine fisheries management 1985-2010.
Natural Resource Modeling 25:122-144.
Hile, R. 1954. Fluctuations in growth and year-class strength of the walleye in Saginaw Bay. U.
S. Fish and Wildlife Service Fishery Bulletin 91.
71

Lee, H.H., L., M. N. Maunder, K. R. Piner, and R. D. Methot. 2011. Estimating natural mortality
within a fisheries stock assessment model: an evaluation using simulation analysis based
on twelve stock assessments. Fisheries Research 109:89-94.
Linton, B. C., and J. R. Bence. 2008. Evaluating methods for estimating process and observation
error variances in statistical catch-at-age analysis. Fisheries Research 94:26-35.
MacMillan, E.A., and B.M. Roth. 2012. By-catch in the Saginaw Bay, Lake Huron commercial
trap net fishery. Journal of Great Lakes Research 38:353-361.
Magnusson, A., and R. Hilborn. 2007. What makes fisheries data informative? Fish and Fisheries
8:337-358.
Maunder, M. N. 1998. Integration of Tagging and Population Dynamics Models in Fisheries
Stock Assessment. Doctoral dissertation. University of Washington, Seattle.
Maunder, M. N. 2003. Paradigm shifts in fisheries stock assessment: from integrated analysis to
Bayesian analysis and back again. Natural Resource Modeling 16:465-475.
Maunder, M. N., and A. E. Punt. 2013. A review of integrated analysis in fisheries stock
assessment. Fisheries Research 142: 61-74.
Maunder, M.N. and Wong, R.A. 2011. Approaches for estimating natural mortality: Application
to summer flounder (Paralichthys dentatus) in the U.S. mid-Atlantic. Fisheries Research
111: 92– 99.
McParland, T. L., M. M. Ferguson, and A. P. Liskauskas. 1996. Genetic Population Structure
and Mixed-Stock Analysis of Walleye in the Lake Erie–Lake Huron Corridor using
Allozyme and Mitochondrial DNA Markers. Transactions of the American Fisheries
Society 128: 1055-1067.
Megrey, B. A. 1989. Review and comparison of age-structured stock assessment models from
theoretical and applied points of view. Pages 8-48 in E. F. Edwards and B. A. Megrey,
editors. Mathematical analysis of fish stock dynamics. American Fisheries Society,
Symposium 6, Bethesda Maryland.
Methot, R. D. 1990. Synthesis model: an adaptable framework for analysis of diverse stock
assessment data. International North Pacific Fisheries Commission Bulletin 50:259-277.
Methot, R.D. 2000. Technical Description of the Stock Synthesis Assessment Program. NOAA
Technical Memorandum NMFS-NWFSC-43.
Methot, R. D. 2009. Stock assessment: operational models in support of fisheries management.
Pages 137-165 in Beamish R. J., Rothschild B. J., editors. The Future of Fishery Science
in North America. Springer, Fish and Fisheries Series 31, Seattle.
72

Michielsens, C. G. H., M. K. McAllister, S. Kuikka, T. Pakarinen, L. Karlsson, A.
Romankkaniemi, I. Pera, and S. Mantyniemi. 2006. A Bayesian state-space markrecapture model to estimate exploitation rates in mixed-stock fisheries. Canadian Journal
of Fisheries and Aquatic Sciences 63:321-334.
Molton, K. J., T. O. Brenden, and J. R. Bence. 2012. Control rule performance for intermixing
lake whitefish populations in the1836 Treaty waters of the Great Lakes: a simulationbased evaluation. Journal of Great Lakes Research 38:686-698.
Nalepa, T.F., Mohr, L.C., Henderson, B.A., Madenjian, C.P., and Schneeberger, P.J. 2005. Lake
whitefish and Diporeia spp. in the Great Lakes: an overview. Great Lakes Fishery
Commission Technical Report 66:3-20.
Nalepa, T. F., J. A. Wojick, D. L. Fanslow, and G. A. Lang. 1995. Initial colonization of the
zebra mussel (Dreissena polymorpha) in Saginaw Bay, Lake Huron: population recruitment,
density, and size structure. Journal of Great Lakes Research 21:417-434.
Nate, N. A., M. J. Hansen, L. G. Rudstam, R. L. Knight, and S. P. Newman. 2011. Population
and community dynamics of walleye. Pages 321-374 in B. A. Barton, editor. Biology,
management, and culture of walleye and sauger. American Fisheries Society, Bethesda,
Maryland.
NRC (National Research Council). 1998. Review of the northeast fishery stock assessments.
National Academy Press, Washington, D.C.
Pauly, D. 1980. On the interrelationships between natural mortality, growth parameters, and
mean environmental temperature in 175 fish stocks. Journal du Conseil, Conseil
International pour 1'Exploration de la Mer 39:175-192.
Punt, A. E., D. C. Smith, R. B. Thomson, M. Haddon, X. He, and J. Lyle. 2001. Stock
assessment of the blue grenadier Macruronus novaezelandiae resource off south-eastern
Australia. Marine and Freshwater Research 52:701-717.
Quinn , T. J. II, 2003. Ruminations on the development and future of population dynamics
models in fisheries. Natural Resource Modeling 16:341-392.
Quinn, T. J. II. and R. B. Deriso. 1999. Quantitative Fish Dynamics. Oxford University Press,
Oxford, New York.
Quinn, T. J. II, R. B. Deriso, and P. R. Neal. 1990. Migratory catch-age analysis. Canadian
Journal of Fisheries and Aquatic Sciences 47: 2315-2327.
Richards, L. J. 1991. Use of contradictory data sources in stock assessments. Fisheries Research
11:225-238.

73

Riley, S. C., and E. F. Roseman. 2013. Status of the offshore demersal fish community. Pages
21-35 in S. C. Riley, editor. The state of Lake Huron in 2010. Great Lakes Fishery
Commission Special Publication 13-01.
Roth, B. M., N. E. Mandrak, T. R. Hrabik, G. G. Sass, and J. Peters. 2013. Fishes and decapod
crustaceans of the Great Lakes Basin. Pages 105-135 in W. W. Taylor, A. J. Lynch, and
N. J. Leonard, editors. Great Lakes Fisheries Policy & Management. 2nd edition.
Michigan State University Press, East Lansing.
Schneider, J. C. 1977. History of the walleye fisheries of Saginaw Bay, Lake Huron. Michigan
Department of Natural Resources, Fisheries Research Report 1850, Ann Arbor.
Schneider, J. C. 2000. Manual of fisheries survey methods II: with periodic updates. Michigan
Department of Natural Resources, Fisheries Special Report No. 25. Ann Arbor.
Schneider, J. C., and J. H. Leach. 1977. Walleye (Stizostedion vitreum vitreum) fluctuations in
the Great Lakes and possible causes, 1800-1975. Journal of Fisheries Research Board of
Canada 34:1878-1889.
Schneider, J. C., and J. H. Leach. 1979. Walleye stocks in the Great Lakes, 1800–1975:
fluctuations and possible causes. Great Lakes Fishery Commission, Technical Report 31.
Schnute, J.T. 1994. A general framework for developing sequential fisheries models. Canadian
Journal of Fisheries and Aquatic Sciences 51:1676–1688.
Schnute, J. and R. Hilborn. 1993. Analysis of contradictory data sources in fish stock assessment.
Canadian Journal of Fisheries and Aquatic Science 50:1916-1923.
Spiegelhalter, D. J., N. Best, and B. P. Carlin. 1998. Bayesian deviance, the effective number of
parameters, and the comparison of arbitrarily complex models, University of Minnesota,
Division of Biostatistics, Research Report 98-009, Minneapolis.
Spiegelhalter, D.J., N. G. Best, B. P. Carlin, and A. van der Linde. 2002. Bayesian measures of
model complexity and fit. Journal of the Royal Statistical Society Series B 64:583–639.
Thomas, M. V., D. Einhouse, K. Kayle, M. Turner, C. Vandergoot, M. Belore, A. Cook, K. Ho,
T. MacDougall, K. Soper, Y. Zaho, and C. Murray. 2011. Report for 2010 by the Lake
Erie Walleye Task Group, Lake Erie Committee, Great Lakes Fishery Commission.
Available:
http://www.glfc.org/lakecom/lec/WTG_docs/annual_reports/WTG_report_2011.pdf
(January 2014).
Thomas, M. V., and R. C. Haas. 2005. Status of Yellow Perch and Walleye in Michigan Waters
of Lake Erie, 1999–2003. Michigan Department of Natural Resources, Fisheries
Research Report 2082, Ann Arbor.

74

USA v. State of Michigan. 2000. 1836 Great Lakes Treaty Waters consent decree manual. U. S.
District Court, Western District of Michigan Southern Division, 2:37 CV26.
Vandergoot, C. S. and T. O. Brenden. 2014. Spatially varying population demographics and
fishery characteristics of Lake Erie walleyes inferred from a long-term tag recovery
study. Transactions of the American Fisheries Society 143:188-204.
Vandergoot, C. S., T. O. Brenden, M. V. Thomas, D. W. Einhouse, H. A. Cook, and M. W.
Turner. 2012. Estimation of tag shedding and reporting rates for Lake Erie jaw tagged
walleyes. North American Journal of Fisheries Management 32:211-223.
Vincent, M. T. 2013. Estimability of Natural Mortality within a Statistical Catch-at-Age Model:
A Framework and Simulation Study based on the Gulf of Mexico Red Snapper. Master’s
Thesis. Fisheries and Wildlife Department, Virginia Polytechnic Institute and State
University, Blacksburg.
Wang, Y., and Q. Liu. 2006. Estimation of natural mortality using statistical analysis of fisheries
catch-at-age data. Fisheries Research 78:342-351.
Wang, H., E . Rutherford, A Cook, D. W. Einhouse, R. C. Haas, T. B. Johnson, R Kenyon, B.
Locke, M. W. Turner. 2007. Movement of walleye in Lakes Erie and St. Clair inferred
from tag recapture and fisheries data. Transactions of the American Fisheries Society
136:539–551.
Wilberg, M. J., and J. R. Bence. 2006. Performance of time-varying catchability estimators in
statistical catch-at-age analysis. Canadian Journal of Fisheries and Aquatic Sciences.
93:2275-2285.
Wilberg, M. J., and J. R. Bence. 2008. Performance of deviance information criterion model
selection in statistical catch-at-age analysis. Fisheries Research 93:212-221.
Wilberg, M. J., B. J. Irwin, M. L. Jones, and J. R. Bence. 2008. Effects of source-sink dynamics
on harvest policy performance for yellow perch in southern Lake Michigan. Fisheries
Research 94:282-289.
Wilberg, M. J., J. T. Thorson, B. C. Linton, and J. Berkson. 2010. Incorporating time-varying
catchability into population dynamic stock assessment models. Reviews in Fisheries
Science 18:7-24.
Wolfert, D. R. 1963. The movement of walleyes tagged as yearlings in Lake Erie. Transactions
of the American Fisheries Society 92:414-420.

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CHAPTER 3

ANALYSIS OF MANAGEMENT OPTIONS AND VALUE OF INFORMATION FOR THE
SAGINAW BAY STOCK OF WALLEYES BASED ON STOCHASTIC SIMULATIONS

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Abstract

Saginaw Bay walleyes contribute to fisheries throughout Lake Huron, including a
recreational fishery and by-kill stemming from the state-licensed commercial fishery in the bay.
Two critical uncertain states of nature exist concerning the true magnitude (catchability) of the
by-kill and the future of alewives in Lake Huron, the latter being a strong determinant of walleye
recruitment. After consulting with fishery managers, a stochastic simulation model was
developed and used to evaluate management options for the recreational fishery in the form of a
decision analysis and the value of information for improved estimates of by-kill magnitude.
Management option evaluation indicated a greater harvestable surplus that could be allocated.
Sustainable harvest was average harvest treating harvest in years when sustainability criteria
were not met as zero. Sustainable harvest would be maximized if recreational fishing mortality
were increased 50% from recent levels. Realizing this potential, however, would require more
intensive management to ensure that desired levels of F occurred. Choices by managers as to
how to allocate surplus harvest are a matter of policy, but concerns over maintaining predation
pressure on alewives so as to suppress any resurgence may be reasons to manage conservatively
by electing to instead maintain a higher predator abundance. The value of information analysis
revealed that further research investment in the uncertainty over by-kill catchability could
provide net benefits from the recreational fishery.

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Introduction

The Saginaw Bay (Figure 3.1) stock of walleye is the single largest source of walleye in
Lake Huron (Schneider and Leach 1977; 1979) and historically sustained the largest walleye
fishery in the Great Lakes outside of Lake Erie (Baldwin and Saalfeld 1962). The stock declined
and its fisheries collapsed in the mid-20th century due to a series of year class failures, which
were attributed to habitat destruction, poor water quality, and effects of invasive species
(Schneider 1977; Schneider and Leach 1977; 1979). After improving conditions and initiation of
a fingerling stocking program in the early 1980s (Fielder 2002), a recreational fishery emerged
(Fielder et al. 2014). Full recovery and the discontinuation of stocking came in 2006 (Fielder and
Thomas 2006; 2014); the recovery was attributed to the decline of alewives in Lake Huron which
were predators and competitors on newly hatched walleye fry (Fielder at al. 2007; Riley et al.
2008; Dunlop et al. 2010).
Early walleye fisheries in Saginaw Bay were almost exclusively commercial (Baldwin
and Saalfeld 1962; Schneider 1977). The recreational fishery that emerged in the 1980s was all
that remained in the bay after formal closure of commercial walleye fisheries in the early 1970s
(Schneider 1977; Fielder et al. 2014). Commercial fishing continued in Saginaw Bay for other
species. Modern day management efforts such as habitat improvement and stocking focused on
promoting the recovery of the walleye population (Fielder and Baker 2004). Aside from those
efforts, management of the stock was based principally on statewide harvest regulations on the
recreational fishery including a five fish per day bag limit, a 381 mm minimum length limit, and
a spring spawning closure in tributaries to the open water.

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Today, several fisheries, both in and outside Saginaw Bay are believed to exploit this
stock of walleye including commercial trapnet and gillnet extractions in Ontario’s southern Lake
Huron waters, a tribal gillnet fishery in northern Lake Huron permitted as a by-catch under the
1836 Treaty 2000 consent decree, recreational fisheries both in and outside the bay, and lastly a
by-kill (by-catch mortality) of walleyes in Saginaw Bay from the remaining commercial fishery.
The exploitation outside the bay stems from migration of some walleyes, initially confirmed by a
tagging program that began in 1981(Fielder 2014). Those angler-reported tag returns amounted
to 9% being reported from outside the bay over the years. More recently, a telemetry study of
walleye movement indicated that about half of the adults make a seasonal emigration from the
bay to points throughout Lake Huron (T. Hayden USGS, Personal communication). Fielder and
Bence (2014) assessed the Saginaw Bay stock of walleyes through the use of statistical-catch-atage (SCA) analysis. That analysis concluded that the dynamics of the stock were affected by
each of these fisheries.
The commercial by-kill of walleyes in Saginaw Bay was quantified by MacMillan and
Roth (2012) who generated two estimates; a lesser value of 21,500 for their observation period of
May through August 2010, and a larger estimate of 102,000 extrapolated to the entire year.
Analysis by Fielder and Bence (2014) found the estimates generated by their SCA model to be
sensitive to the choice of the by-kill estimate (observed seasonal or extrapolated year-round)
used to parameterize the model. Additional research has been proposed to refine the estimates of
commercial walleye by-kill in Saginaw Bay (B. Roth Michigan State University, personal
communication).
With the transition from an emphasis on stock recovery to the next stage of management,
fishery managers today on Lake Huron are seeking to define new management goals and

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objectives for the Saginaw Bay walleye population and fisheries. The walleye fishery in Lake
Erie is managed through a combined assessment and allocation process under the guidance of the
Great Lakes Fishery Commission. Goals there have been to fairly allocate resources, and fully
utilize the resource but within the boundaries of sustainability (GLFC 2005). Part of the
motivation to update management practices on Saginaw Bay is to reflect new information about
the extent and complexity of exploitation of the stock. In spite of this new information,
considerable uncertainty remains making management decision making and research planning
difficult.
Several sources of uncertainty about the dynamics of this stock persist. In this analysis
we concern ourselves principally with two; what is the true amount of walleye by-kill occurring
in Saginaw Bay’s state-licensed commercial fishery and what will alewives do in the future? If
alewives remain scarce, walleye are expected to continue to sustain strong year classes but
should alewives recover, walleye year classes are expected to become weak and reduced again
(Fielder et al. 2007; 2010). There is a lack of consensus on the future of alewives in the lake
(Riley et al. 2008). Accounting for uncertainty in fishery management does not necessarily result
unless specifically sought. Williams (1997) describes natural resource management in the face of
uncertainty and argues that too often, decision makers ignore uncertainty. Wise fishery
management will seek to consider uncertainty for making decisions (Walters 1986; Lane and
Stephenson 1998; Jones and Bence 2009) and, where appropriate, seek opportunities to reduce
critical uncertainties.
Not all uncertainties, however, are of sufficient magnitude or significance to warrant
investment in their elimination (Hansen and Jones 2008). Analysis of the value of information is
one form of decision analysis. The expected value of perfect information (EVPI) is an estimate

80

of the expected increases in benefits or decrease in expected losses, if we could eliminate a
source of uncertainty (Peterman and Peters 1998). The concept of EVPI is closely tied to those of
adaptive management where one seeks to reduce uncertainty so as to further management of a
resource (Walters 1986). Simply put, analysis of EVPI can be thought to answer the question
"does the elimination of uncertainty justify the cost of doing an experiment or paying for new
information?" (Clemen and Reilly 2001).
My objective with this analysis was to offer insights to managers on the development of
new management goals and objectives for this stock, specifically with regards to the extent to
which the recreational fishery may exploit walleyes within limits of sustainability. Secondly, I
address whether management would benefit from refined estimates of state-licensed commercial
by-kill in Saginaw Bay by conducting a value of information analysis, thereby determining the
utility of reducing one element of uncertainty about Saginaw Bay walleye management. To
accomplish the examination of management options, and to enable the EVPI analysis, I develop
a stochastic (system) model and use it to simulate the dynamics of the stock, its exploitation and
test the population and fishery response across the range of uncertainties and management
scenarios. My intent is to better frame the range of management options available and the
tradeoffs that will face future management.
Methods

Management input.--I framed the decision analysis by first meeting with State of
Michigan fishery managers from the Department of Natural Resources (DNR) whose jurisdiction
included the recreational fisheries within Michigan waters of Lake Huron and the by-kill within
the state-licensed commercial fishery. Fielder and Bence (2014) found these sources to

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collectively constitute the majority of total fishing mortality exerted on the stock. Interaction
with fishery managers was based on one meeting devoted to discussing management objectives
and the current state of knowledge, and uncertainty, concerning Saginaw Bay walleye stocks and
fisheries, and also periodic participation in the Michigan DNR’s Lake Huron Basin Team
meetings, which included ongoing discussions of walleye management and information needs.
Between February 2013 and February 2014, researchers and managers met three times to explore
these issues.
Among the topics discussed was the existing state of management. Participants were in
agreement that emphasis over the previous three decades was on stock recovery and that
management has relied on statewide recreational harvest regulations. Considerable investment
had been made in research and assessment, and while substantial knowledge had been amassed,
many uncertainties remained, especially in light of the growing awareness of the complexity of
the lake wide nature of the stock, its movement, and exploitation. Additional discussions were
focused on identifying uncertainties that managers felt were necessary to reduce, to make
management decisions. Performance measures and critical thresholds were discussed that helped
frame a basis for simulation modeling. As development of the simulation model progressed,
discussions included feedback on the model structure and performance as a basis to reflect the
primary considerations of walleye management.
Aside from the state-wide recreational fishery regulations, the only state-based
management rule that existed was a pledge that managers would annually revisit the decision to
suspend walleye stocking and may reinstate it if alewives ever became abundant again. Alewife
abundance was annually estimated by the U.S. Geological Survey’s (USGS) Great Lakes Science
Center (GLSC)’s bottom trawling and hydroacoustic survey. Analyses have suggested that a

82

threshold of alewife density existed at 20 age-1 and older alewife/ha such that densities less than
that allowed for recruitment of walleye (Fielder et al. 2007; Fielder and Thomas 2014).
Managers maintained that walleye fingerling stocking likely would be resumed if and when
alewife density ever increased above this threshold in the future. This suggested one obvious
management option to include in the simulation model.
Objectives and performance measures.-- Managers were in agreement that stock
sustainability was a primary objective, as was the desire to manage for the greatest recreational
harvest (in numbers) possible. Knowing that walleye management in Lake Erie has led to a
liberalization of recreational harvest regulations over state-wide rules in many years, managers
wanted to determine the greatest recreational harvest that could be sustained with an expectation
that harvest regulations could then be crafted to achieve or allow for that.
On the other hand, managers were also concerned that a liberal harvest policies or
allocation of walleye may lead to a reduced stock. To capture the idea of sustainability being a
function of spawning stock, we identified the ratio of mature female spawning stock biomass
(SSB) of walleyes relative to the unfished SSB (β0) as a metric representing a threshold of
management concern. Specifically the proportion of years across multiple simulations for any
given management scenario that dropped below 20% β0 was designated as a performance
measure to address sustainability with respect to the risk of recruitment overfishing. The 20% β0
threshold was chosen because it was consistent with the inflection point in the stock-recruitment
relationship such that stock sizes less than that risked recruitment-overfishing.
Criteria for detection of walleye recovery from the degraded state was defined by Fielder
and Baker (2004) and offered some additional basis for defining performance measures for
future management. The principal benchmark for defining recovery was a mean total length (TL)

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of age-3 walleye (sexes combined) at or below 110% the state average rate at the time of survey
capture which was 425mm (for a September collection). The benchmark was based on the belief
that mean TL at age could serve as a surrogate for growth rate, which is affected by density.
When size at age is above this threshold (i.e., relative high growth rates), the system should be
able to support higher densities of walleye. Walleye mean TL at age 3 slowed, falling below this
threshold for the third consecutive year in 2009 (Fielder and Thomas 2014) and was the primary
basis for declaring the stock recovered.
With the above objectives and considerations in mind, specific performance measures
were identified. Mean TL at age-3, SSB ratio (the proportion of years SSB fell below 20% of
β0), and recreational harvest were therefor selected as the performance measures (Table 3.1). The
decision analysis and value of information analysis required the combination of these measures
into one overall objective function. For this purpose we used the average recreational harvest
(over years for a simulation). Harvest in years when one or both thresholds of sustainability (TL
of walleye >425mm, SSB below 20% β0) were violated was set to zero in the calculation of the
average. From here on I refer to this function as “sustainable harvest”.
Critical uncertainties.-- Managers emphasized that they remain uncertain about future
trends of alewives in Lake Huron (Table 3.1). Although alewives have not recovered after 10
years post collapse, it remained unclear whether their continuing scarcity was due to lower
productivity (bottom up) or suppression via predation (top down). Recognizing that alewives
were a strong determinant of walleye recruitment (Fielder et al. 2007), their future was included
as a critical uncertainty. After reflection, the managers and researchers concluded that a
reasonable characterization of expert opinion would be that on-going suppression of alewives
was three times more likely than their resurgence. I examined the sensitivity of my conclusions

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to this choice of probabilities by repeating the analysis at a lesser 10% likelihood of alewife
recovery.
The second critical uncertainty was whether the true value of commercial by-kill was best
reflected by the lower value estimated for the period when by-kill was directly observed by
MacMillan and Roth (2012) or the larger value that reflects an extrapolation to the remainder of
the year (Table 3.1). Most managers were skeptical that the true value was as great as the
extrapolated value because the measurements were limited primarily to one fisher and because of
the potential for seasonal differences outside the warmer months. However, they also conceded
that the true value had to be greater than that which was reported for the observation period
alone. Not having any clear basis for assigning probabilities beyond that rationale, it was agreed
to treat the two as equally likely. The assumed magnitude of by-kill directly affected the estimate
of by-kill catchability from Fielder and Bence (2014). In turn, the differing estimates of by-kill
catchability affected most other population metrics and parameters estimated by the SCA model.
Thus, when evaluating the effect of the two alternative by-kill hypotheses, the simulation model
had to be reparameterized with values from the SCA model fit to the appropriate by-kill value.
We developed a decision tree to graphically depict the potential decision framework
(Figure 3.2). Other uncertainties also existed such as the future of recruitment, fishing effort,
natural mortality, and catchability in the other fisheries aside from the by-kill. These
uncertainties, however, were either accounted for by incorporating stochasticity about their
relationships in the simulation model (e.g. stock/recruitment function and fishery catchability) or
were treated as constants (effort and natural mortality).
Candidate recreational harvest policies and decision analysis.-- Recognizing that the
management agency principally had the most influence over the recreational fishery, analysis

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was limited to those extractions. In the simulations I evaluated a range of recreational fishing
mortality rates. In particular, I varied recreational fishing mortality from 10% to 400% of recent
levels here after referred to as fishing intensity and applied as scalars (multipliers) to the current
fishing mortality rate (Table 3.1) and represented by the notation Fvalue such that F1.5 means
1.5 times the current fishing mortality. Managers were not ready to identify any new state-based
management options for the recreational fishery but did want to answer the objective of learning
the range of population response to varying amounts of recreational harvest with an emphasis on
the upper limit to exploitation that would still keep the frequency of SSB falling below 20% of β0
rare, and a mean TL of age-3 walleyes < 425 mm.
The sustainable harvest for each level of fishing intensity was analyzed as a decision
analysis according to Figure 3.2. Decision analysis principally followed the methods of
Peterman and Anderson (1999). This decision analysis was premised on the uncertain states of
nature regarding by-kill catchability and future alewife trends and their probabilities described in
Table 3.1. The combined probability of occurrence of these alternative states of nature was the
product of their individual probabilities (Table 3.1). Thus to calculate the average sustainable
harvest for a given level of fishing intensity, a weighted average was calculated based on the
simulation results for each of the four uncertain states of nature and using the probabilities of
those states as weights.
Value of by-kill information.-- The calculation of EVPI followed the methods of Clemen
and Reilly (2001) and also used sustainable harvest as the objective function. The management
options evaluated were the same as previously described for the recreational fishing mortality
intensities (Table 3.1). The expected value after knowing the true model (elimination of
uncertainty) was the average of the maximum performance (maximum recreational harvest)

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obtained under conditions of either high or low by-kill. Maximizing separately for each by-kill
condition was premised on the idea that recreational F could be optimized for the given by-kill
level if this were known. The EVPI was the difference between that value, and maximum value
of the “best” management option when the same recreational F had to be applied regardless of
by-kill because the level of by-kill was unknown. In this case, that was the greatest sustainable
recreational harvest across management options (the ranges of fishing intensities simulated)
within the four uncertain states of nature.
The expected value of imperfect information (EVII) is an extension that attempts to
further account for the uncertainty in the ability of experiments to successfully reduce or
eliminate the uncertainty about the states of nature. We regarded the probability of a new study
of walleye by-kill to successfully achieve its objectives as 95% likely based on the standard
usage of a significance level of 0.05 for a type-I error in any statistical tests stemming from
further by-kill research.
This approach to value of information is premised on the two estimates of by-kill. In
reality the true value may be an intermediate level and as such the dichotomous choice is really a
simplification. Thus my two estimates of EVPI and EVII may be optimistic, recognizing that an
intermediate value would likely yield intermediate outcomes.
Simulation model and time frame.-- For each of the four uncertain states of nature, 250
simulations were conducted for each fishing intensity scenario (including the by-kill scenarios
needed for the value of information analysis). Thus there were 1,000 simulations for each
fishing scenario. Each simulation was done over a 50 year time-horizon, reflecting a desire that
outcomes not be dominated by initial conditions. I chose to base decision analysis as well as
scenario performance on the entire 50 year time span recognizing that the result would then be a

87

reflection of both transient dynamics as the system moved toward a stationary set of results and
long term performance. When calculating the average sustainable harvest, the results for each
uncertain state of nature were weighted by the assumed probability for that state. All model
coding was performed in AD Model Builder (Fournier et al. 2012; ADMB 2013).
The principal state variable modeled was number of walleyes by year at age represented
by the exponential population equation (equation 2.1 in Table 3.2). The model was formulated
and parameterized based on the fitted SCA model from Fielder and Bence (2014) and
represented ages 2 – 13+. The gillnet fisheries were estimated as a single fishery in that original
analysis, and continue to share the same catchability and selectivity here in the simulation model
but their effort is broken out. Scenarios held effort constant at the average value observed in the
various fisheries since walleye recovery was achieved (based on values from 2004 – 2011).
Stochasticity was incorporated in each of the fisheries as an error term about catchability
‘q’ either as a white noise process error variation or random walk deviations (for the recreational
fishery) for log-scale q. To ensure, however, that recreational catchability did not trend to
extremes from the random walk process over long scenario durations, I limited the product of the
recreational q and  y from equation T2.3 to no more than twice or no less than ½ the starting q
value. I did not have an estimate of the variance for the by-kill catchability, so I applied the
estimated variance from the Ontario trapnet fishery to this quantity, because these two fisheries
use similar gear types.
Recruitment was incorporated two ways, from natural reproduction and from stocking.
The stock-recruitment (S/R) relationship was based on a Ricker model (Ricker 1975) and
followed the methods of Fielder et al. (2007), which previously analyzed the S/R relationship of
Saginaw Bay walleyes. That work concluded that the abundance of alewives was the single best

88

determinant of walleye recruitment for this stock as alewives have been demonstrated to be a
formidable predator and competitor on newly hatched percid larvae.
I wanted a S/R function that reflected the sensitivity to alewife abundance as
demonstrated by Fielder et al. (2007), but one that also was responsive to changes in stock
density in simulations. Therefore, I developed a new S/R function for wild recruits that used both
stock (represented as number of eggs produced in 100,000 increments) and density of alewives
in Lake Huron following a multivariate Ricker S/R function as described by Chen and Irvine
(2001) and Haddon (2001).
To accommodate scenarios where stocking would resume because alewives again became
abundant, a separate recruitment function for stocked fish was needed. The stocked fish
recruitment model predicted age-2 stocked recruits from spring fingerling stocking numbers two
years earlier, following a Ricker model. The model was estimated using data from 1997 – 2005
although 2003 was omitted as an outlier. That year was a transitional year in recovery with
unusual dynamics (Fielder and Thomas 2006). The model also included a coefficient for the
effect of the abundance of wild recruits (equation T2.6) due to observations of a negative
relationship between survival of stocked fish and wild fish recruitment from the same year (R2
=0.48), consistent with the general concept of stocking success being inversely related to natural
reproduction (Laarman 1978; Li et al. 1996).
Total recruitment in any given year in the simulation model was then the sum of wild
recruitment R Wy and hatchery recruitment R yH , if any. Total recruitment was then incorporated in
the population equation (equation T2.1) as the starting numbers of walleyes (age-2) for each
cohort. The simulated recruitment for both wild and hatchery recruitment included a stochastic
component based on the observed process error in the S/R models.

89

Because alewife density was an important determinant of wild recruitment, and because
some management rules about walleye stocking were defined on alewife abundance, alewife
trends had to be part of the simulation model. I developed a separate alewife population surplus
production sub-model (Hilborn and Walters 1992) based on a logistic function requiring two
parameters; the finite rate of increase or R (based on the intrinsic rate of increase) and the
carrying capacity k (Table 3.2, equation T2.7).
To accommodate alewife trends as a critical uncertainty, I incorporated two alternatives
futures for alewives in Lake Huron: (1) alewives return to high levels of the 1970s and 1980s
over a 25 year period, providing that predation is not inhibitory; and (2) alewives remain scarce
at the levels observed between 2004 – 2011. Both alternatives used the aforementioned logistic
function, both with different finite rates of increase (R); for alternative 2, R was simply set to
zero. I estimated k from the USGS GLSC alewife bottom trawl data, using the highest value
observed over the time series (600/ha). I then estimated a value of R (for alternative 1) that
would result in the alewife population increasing to k over 25 years.
Walleyes regularly prey on alewives when present (Fielder and Thomas 2006; Schaeffer
1994) so alewife numbers were further adjusted based on walleye consumption. Walleye
consumption was modeled based on a Type II functional response of prey to predation (equation
T2.8, Holling 1959). Parameters include the number of days per year that alewives are exposed
to walleye predation (270), the area of available alewife habitat in Lake Huron (1,522,618 ha),
which is necessary to convert alewife abundance to density, the handling time (one alewife per
day), and the attack rate. I estimated the attack rate parameter by comparing stomach contents
(consumption) to estimated alewife density, and assuming the other functional response
parameters were known. Total walleye consumption of alewives was subtracted from the

90

population predicted from the logistic population equation (T2.7) for the same year and the
resulting number of alewives was used for alewife abundance in the walleye S/R function for
wild recruits (equation T2.5). Although alewives are treated as an uncertainty in this analysis, the
feedback of walleye consumption also afforded some degree of performance measure in that
walleye abundance, as a function of management, had some degree of influence over alewife
trends.
A pivotal value in the calculation of alewife abundance was the area of alewife habitat in
Lake Huron used to return numbers of alewives consumed to a density basis. My value of habitat
area was derived by first limiting alewife habitat to the main basin of the lake and then
subtracting the near shore area based on the belief that warmer near shores waters would not
constitute age-1 and older alewife habitat and also subtracted the mid lake area (about 1.4 million
ha) based on the description of Eshenroder and Burnham-Curtis (1999) that the mid-lake main
basin is likely largely devoid of pelagic fishes. This process left us with the conservative estimate
of 1,522,618 ha of adult alewife habitat in the main basin of Lake Huron. I note that these
calculations treat the trawl swept area values for alewife as reflective of actual alewife densities
(see He et al. in press, for further discussion of this assumption).
To predict fall mean walleye TL at age-3 in the simulations, I quantified the relationship
between walleye abundance (all ages combined) and observed September mean TL for age-3
following the methods of Shuter and Koonce (1977). I regressed the log of the difference
between the mean TL at age 3 and the largest observed length for that age, against the log of the
3-year running average of estimated population abundance (age-2+) for the first three years of
the cohort. Length data were taken from the annual Saginaw Bay Fish Community Survey
(Fielder and Thomas 2014) for 2003-2011 and the population values were from the SCA model

91

projection (ages combined) from Fielder and Bence (2014) for the same years. Data were also
available for years 1989-2002, but were not included in the analysis because they did not
represent the growth dynamics of the recovered walleye population. The resulting relationship
exhibited an R2 of 0.70 and was significant at (F test) P=0.005. The predicted mean TL at age-3
was the observed maximum less the predicted value according to equation T2.10.

Results
Recreational fishery performance.-- The greatest sustainable harvest was achieved at
recreational F1.5 (approximately 50% increase in the recreational fishing mortality) (Table 3.3).
Although sustainability was taken into account, this level of fishery expansion (F1.5), still
exceeded the sustainability thresholds in some years. The proportion of years (with alewives
remaining scarce), for example, that SSB fell below 20% β0 was 8% for F1.5, while only 1% at
F1.0 (Figure 3.3E). Similarly, the proportion of years that the mean TL of age-3 walleyes
exceeded the 425mm threshold rose from 21% (F1.0) to 26% (F1.5) (Figure 3.3D). Proportional
increases were even greater for scenarios that accounted for the recovery of alewives (Figure
3.3). By defining a composite performance metric, the decision analysis accounted for these
trade-offs to yield an overall preferred option in light of the uncertainty and stochasticity,
suggesting that these slight increases in risk were an acceptable trade off up to F1.5.
When alewives remained scarce, the mean SSB and mean TL of age-3 walleyes
(averaged within years across the 250 simulations) fell within their thresholds of sustainability
up to a fishing intensity as great as F3.0 for SSB and F 2.0 for mean TL of age-3 walleyes
(Figures 3.4A and B). These means, however, conceal the range of performance, and exceedance
of the sustainability criteria in some years even at lower fishing intensities and thus the lesser
92

value of just F1.5 indicated by the decision analysis as the preferred option. The performance of
the recreational fishery, however, varied considerably with the uncertainty of future alewife
trends (Figures 3.3 and 3.4). In every instance, the hypothetical recovery and presence of
alewives reduced the sustainable intensity of the recreational fishery. To maintain the mean SSB
above 20% β0, fishing intensity would have to be reduced to at least F0.5 (Figure 3.4C) and no
degree of reduction in fishing intensity would prevent the mean TL age-3 from exceeding the
target (Figure 3.4D). In spite of this, repeating the decision analysis at a lower probability of
alewife recovery (10% likelihood instead of 25% likelihood) did not substantively change the
outcome.
Predictably, the walleye population size decreased with increasing fishing intensity
across both possible alewife futures (Figure 3.3C). Average recreational harvest itself continued
to increase up to F3.0 but gains in harvest beyond F1.5 were marginal, with F1.5 as 86% of the
maximum. The corresponding upper limit of total annual mortality (A) for an F1.5 was 0.41
(Figure 3.3F). By contrast, in the presence of alewives, average SSB fell below 20% β0 for F1.0
(Figure 3.4C) corresponding to a total annual mortality of about 0.35 (Figure 3.3F). Of the two
performance measures of sustainability, the ratio of SSB to unfished SSB appears to be
somewhat more forgiving to the effect of harvest than the mean TL at age-3 threshold. The
growth rate-based criterion reaches its threshold slightly before the SSB-based criteria does. For
example, average SSB was at 20% β0 at F3.0 (Figure 3.4A), but mean TL at age-3 exceeded its
threshold value at this F (Figure 3.4B). Generally the two metrics agree, however. The
sustainable maximum fishing intensity of F1.5 from the decision analysis approximately
corresponds to an instantaneous recreational fishing mortality F of about 0.13 of the fully
selected ages.

93

Value of by-kill information.-- The analysis of perfect information indicated a positive
gain to the fishery in terms of sustainable harvest from knowing the true by-kill magnitude
(Table 4). Because the units for these values was the sustainable recreational harvest, derived by
assigning zero to years when the sustainability criteria was exceeded, the magnitude of the value
of the perfect information is not a representation of the actual gains in the numbers of walleyes
that would be harvested as a result from eliminating this uncertainty. As a proportion of the total
expected value of the best decision, however, we see that the EVPI amounts to about 5.5% as
much. One might conclude then that the value of information is equivalent to about 5.5% of
value of the recreational fishery. Analysis of imperfect information is intended to provide a
similar valuation but incorporating a degree of uncertainty over the study’s ability to fully deliver
on the necessary findings. There is no particular reason to hypothesize that an expanded study of
by-kill in Lake Huron would not fulfill its objectives but hypothesizing a potential 5% error rate
in any statistical analysis, the EVII reduces the benefit to 0.3% of the recreational fishery value
(Table 4). Testing sensitivity of the value of information to a lesser probability of alewife
recovery (10% likelihood verses 25% likelihood of recovery) did reduce the EVPI from 5.5% of
the fishery value to 4.2%.
Discussion
Nearly all metrics and performance measures, (validated by the outcome of the decision
analysis), indicated that the Saginaw Bay walleye population is capable of sustaining greater
recreational harvest. The choice of whether to adopt management options that allow for
increased harvest within limits of sustainability is partly a consideration of how conservative
managers want to be. Walleye recovery on Lake Erie led to a management practice of annually
determining the harvestable surplus and making allocation choices amongst the various fisheries
94

and jurisdictions (GLFC 2005). In many years, this constituted more liberal recreational harvest
regulations than the customary statewide regulations enforced by Michigan DNR. The
importance of the Lake Erie approach is that the regulations were regularly revisited and adjusted
based on updated management information and model projections. A decision to manage for a
Saginaw Bay walleye fishery closer to the limits of sustainability would necessitate regular stock
assessments and the willingness to modify harvest regulations as needed to stay within the limits
of sustainability.
Michigan DNR law enforcement officers report that harvest beyond the daily bag limit of
five walleyes per angler is a common violation. “Over bagging” or double bagging (making two
fishing trips in one day) is an enforcement problem when angler catch rates are high as in
Saginaw Bay at certain times of the year. This suggests that increased recreational allocation in
the form of an increased daily bag limit may result in greater utilization of the walleye
population and help alleviate the temptation to violate daily bag limits. This has been Michigan
DNR’s management tool of choice for increased harvest of walleyes in Lake Erie’s recreational
fishery. In 2013, 32% of open water (April – October) angler parties of four or less anglers on
Saginaw Bay reached their collective party limit (Michigan DNR unpublished data).
While not detailed in this analysis, it was clear from the simulations that population
effects from one fishery in turn affected the other walleye fisheries around the lake. The impacts
of increased allocation to recreational fisheries in Michigan on the other walleye fisheries was
not part of the decision analysis. Certainly, however, any increased mortality of walleyes (via
increased recreational exploitation) will reduce the overall population and cause some degree of
contraction in the other fisheries unless their fishing effort increased. Similarly, increased harvest
from those fisheries would also have impacts on the Michigan fisheries. This underscores the

95

need to begin to coordinate management of all the fisheries that exploit the Saginaw Bay stock of
walleyes in Lake Huron, if necessary by working under the aegis of the Great Lakes Fishery
Commission as is done in Lake Erie (GLFC 2005).
Increased exploitation of walleye as a management option has consequences beyond
sustainability questions that need to be considered. The management implications of the ongoing uncertainty regarding alewife futures means that managers may be wise to go slow in their
allocation of additional surplus walleye as their role as a predator is still not fully understood.
Similarly, if increased harvest is opted for by fishery managers, how to allocate that across
fisheries is largely a matter of policy. In spite of the recovery of the walleye population,
recreational fishing effort in Saginaw Bay has been steadily declining (Fielder et al. 2014).
Fielder et al. (2014) offers two hypotheses for this, first that fishing effort in the bay is driven
more by availability of yellow perch (Perca flavescens) than by walleye, and yellow perch are in
severe decline. Secondly, as walleye angler catch rates increase, trip length declines as anglers
reach either their daily bag limits or a satisfactory catch sooner.
The true magnitude of the commercial by-kill was one critical uncertainty identified at
the outset of this analysis. Fielder et al. (2014) estimated that the value of the recreational fishery
in terms of economic activity generated was $33 million per year between 2008 and 2010. As a
follow up to MacMillan and Roth (2012), an expanded study of walleye commercial by-kill in
Lake Huron was proposed, estimated to cost $496,000 (B. Roth, Michigan State University,
personal communication). Amortized over the 50-year simulation period, the study cost would be
$9,920/year, This is well below EVPI of 5.5% ($1,815,000) of the annual recreational fishery
value, of $33 million total (Fielder et al. 2014). The lesser utility based on the EVII still
constitutes a $99,000/year value easily exceeding the expected cost of the study. However

96

trialing probabilities of a successful study of by-kill at less than 95% results in a loss of positive
value to the reduction of this uncertainty. This suggests that the benefit of further by-kill research
is at its margins of justification should the study’s probability of success be questionable in
anyway. One possibility is that the actual by-kill is not close to the two alternatives we
considered, but rather at a more intermediate level. This would reduce the benefits associated
with improving the estimated level of by-kill. Nevertheless, from the value of information
analysis, I conclude that further research to eliminate the uncertainty over the by-kill catchability
magnitude is justified. In addition to potential fishery gains from optimizing management in light
of this added information, the Saginaw Bay walleye SCA model would also be improved by
better by-kill information, and this would benefit all other uses of those estimates.
The uncertainty over alewife futures is another matter. While not the subject of its own
value of information analysis, due to the complexity or inability to reduce that uncertainty, it
unquestionably is a profound driver of all the simulations. Fielder et al. (2007) aptly concluded
that alewife effects dwarfs all other determinants of walleye recruitment in Saginaw Bay. For the
wild walleye S/R function, I included stock size as I wanted that feedback from population trends
but any stock size effect seems to be vastly overshadowed by trends with alewives. Of the two
sustainability based performance measures, the maintenance of the SSB at or above 20% β0
might be less consequential.
The decision and value of information analysis was not substantially sensitive to the
probability of alewife recovery. The lower tested probability of 10% likelihood (versus 25%
likelihood of alewife recovery) did not alter the optimal management option choice. In fact, it
appears from Table 3.3 that if the true probability of alewife recovery was nil, the optimal
management option would remain unchanged from the 1.5 recreational fishing intensity. Thus

97

lingering uncertainty over the future of alewives does not greatly affect the benefits from
management options I explored. The lower, 10% probability of alewife recovery did, reduce the
EVPI over by-kill catchability from 5.5% of the fishery value to 4.2%. This suggests that as
alewife recovery likelihood decreases, the benefit of further by-kill research becomes less. The
low sensitivity of the probability of alewife recovery and the profound effects if alewives do
recover as indicated in Figures 3.3 and 3.4, are not contradictory. The probability of recovery is a
variable in testing management options and a reduction to 10% probability or less was not
consequential, but the effects of alewives, if they in fact do recovery, is a function of their
influence on recruitment of walleye, which is profound.
Herein I explored options where recreational fishing mortality rates were fixed over time,
based on the assumed level of knowledge regarding an alewife recovery. In reality a recovery of
alewife would probably not take managers by surprise, given that alewives would likely need to
build at least two year classes (thus giving two years advanced notice) before having deleterious
effects on walleye recruitment. Results presented here suggests that under conditions when a
recovery does occur, substantially lower fishing mortality rates are necessitated to maximize
benefits. It is possible that an adaptive policy where harvest rates are reduced (and stocking of
walleye is initiated) when a recovery of alewife begins would provide benefits and be more
practical than determining in advance whether alewife will recover or not.
Managers wanted to be able to evaluate the tradeoffs between management options based
partly on the effects of walleye predation on alewives, and how that in turn modulated the effect
of alewife on walleye recruitment. While these interactions were incorporated in the model and
consequently the decision and value of information analysis, I do not feel as if I fully captured
that complex dynamic. The abundance of alewives in the model was a function of the expansion

98

of alewife densities across the total alewife habitat of the lake. I rationalized a minimal value (1.5
million ha) in an attempt to maximize The model’s sensitivity to this dynamic while others (He
et al. in press) have used a much greater estimate of alewife habitat area of 3.2 million ha. I
recognize that the ability of alewives to be suppressed by predation is a function of the suite of
all predators in Lake Huron. He et al. (in press) estimated that walleyes in Lake Huron
constituted about 10% of the collective consumption demand on available prey forms in the main
basin since 2003. Very possibly alewives cannot recover in the face of predation across all
predators but such an analysis was beyond the scope of this study.
Fielder and Bence (2014) in their analysis of walleye stock dynamics in Lake Huron,
included the effects of walleye immigration from Lake Erie and their contribution to the overall
fisheries in Lake Huron. Some proportion of adult walleyes has been documented to make such a
migration (Thomas and Haas 2005, Wang et al. 2007). As part of their analysis, Fielder and
Bence (2014) estimated that from 1986-2011 Lake Erie walleyes averaged 8% as much as
Saginaw Bay fish but only 2.8% since 2004. While at times these numbers were considerable, in
recent years jaw tag returns from Lake Erie tagged fish have trailed off to zero (Lake Erie
Walleye Task Group of the Great Lakes Fishery Commission, unpublished data; Thomas et al.
2011). The simulation analysis was limited to the scope of the Saginaw Bay walleye stock. I
elected not to attempt to additionally account for the effects of Lake Erie walleyes. If there were
to be greater future contributions by Lake Erie fish, likely the realized fishery response would be
slightly greater than the predictions in the analysis reflecting the supplementary catch. Predatory
effects of Lake Erie walleyes on alewives might be another effect but given the difficultly of
documenting any limiting effect on alewives by Saginaw Bay fish alone, I hypothesize that Lake
Erie predation effects would be negligible.

99

While this analysis may be a step forward in the evolution of management tools for the
Saginaw Bay stock of walleyes in Lake Huron, further advancement will necessitate more
deliberate objective setting by fishery managers. An essential element to making decisions is the
formulation of management objectives (Peterman and Peters 1998). Without defined
management objectives, it is difficult or impossible for management strategy evaluations or
decision analysis to effectively proceed. Collaborations for this analysis were merely a first step
of the sort of efforts required. Further deliberation, and clear declarations of goals and objectives
in consultation with stakeholders, will be needed before further evaluation of options can be
generated. Quantitative stock assessment, management option evaluation, and decision analysis
can only proceed so far in the absence of these goals and objectives.

100

APPENDIX

101

Table 3.1. Description of critical uncertainties (uncertain states of nature), management
scenarios, and performance measures.
Component
Uncertainties

Description

By-kill low

State-licensed by-kill catchability is correctly depicted
by 21,500 value for May-August observed period.
Assigned probability of 0.5

By-kill high

State-licensed by-kill catchability is correctly depicted
by 102,000 extrapolated value for entire year. Assigned
probability of 0.5

Alewives recover

Alewives in Lake Huron follow a logistic population
trend with a finite rate of increase ‘R’ of 1.5. Assigned
probability of 0.25

Alewives remain scarce

Alewives in Lake Huron don’t recovery (follow a
logistic population trend with a finite rate of increase ‘R’
of 0). Assigned probability of 0.75

Harvest policies
Scalars (multipliers) of Frecreational
intensity

0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0

x the current F

Performance measures

Sustainable harvest

Average recreational harvest with zeros applied for
years when other performance measure thresholds are
exceeded. This measure to be maximized.

% years SSB < 20% β0

Percentage of years with SSB below 20% unfished
level (β0)

% years mean TL (age-3) > 425mm

Percentage of years that mean total length of age 3
walleyes exceeds 110% state average growth rate
(recovery index)

102

Table 3.2. Equations, symbols and descriptions of variables used in the stochastic simulation
model.
Equation
Number Symbol
Description
Index variables
Na  1, y  1  Na, ye

 Z a,, y

 Na, ye

 (M a  Fa , y )

T2.1

F  f a, y  s a qE y  y

T2.2

 y   y 1 y 1 for recreational fishery only

T2.3

Fa , y   F  f a , y

 R Wy
ln 
 Sy

R
ln 
 Sy


H
y

T2.4


  ln( W )   s S y   A Ay  ln( Wy )



  ln( H )   P Py   W R Wy  ln( yH )



T2.5

T2.6

y

a
Age
State and control variables
N
Number of walleye
Z
Instantaneous total mortality
M
Instantaneous natural
mortality
Instantaneous fishing
F f 
mortality for each individual
fishery
Instantaneous total fishing
F
mortality
R Wy

Yearly wild recruitment

R yH

Yearly hatchery recruitment

Sy

Yearly stock size (100K egg
increments)

Ay
Ay 


Ay 1  Ay  RA y 1 
k 


T2.7

Py
Cony

 åA y 

 N (270)
 1  åhA y  y


Con y 
1522618

T2.8

Ca,y
L3,y

Ca, y 

Fa , y
Z a, y

Z

N a , y 1  e a , y







9 1.62134774N y  y  2

L3, y  505  2.7824786610 e

T2.9

T2.10

103

Year

N y  y 2

Yearly density of alewives
Yearly number of fingerlings
planted
Yearly consumption of
alewives by walleyes,
expressed on a density unit
Catch (harvest) from each of
the fisheries
Yearly mean total length age3 walleyes
Three prior year runningaverage number of walleye

Structural parameters
Age specific selectivity for
sa
each fishery
q
Catchability of each fishery
Year specific fishing effort
Ey
for each fishery

Table 3.2 (cont’d)
13

SSB y   Mat aWa N a , y 1
a2

2

T2.11

Mata
Wa

Maturity by age

Weight at age
Ricker stock-wild recruitment
βs
parameter for stock size
Ricker stock-wild recruitment
βA
parameter for alewife density
Ricker stock-hatchery
βP
recruitment parameter for
number planted
Ricker stock-hatchery
βW
recruitment parameter for
wild recruitment
Ricker stock-wild recruitment
W
parameter
Ricker-hatchery recruitment
H
parameter
Finite rate of increase for
R
alewives
Carrying capacity for
k
alewives
Attack rate by walleyes on
å
alewives
Alewife handling time by
h
walleyes
Distributional parameters and
associated stochastic errors
Yearly wild recruitment
 Wy
deviation
H
Yearly hatchery recruitment
y
deviation
Catchability deviation for
each value of q (from a
random walk process for the
y
recreational fishery or as
white noise drawn from a
normal distribution for all
others)
Yearly deviation from
random-walk process for
y
recreational fishery
catchability

104

Table 3.3. Decision analysis of eight varying degrees of fishing intensity (applied as a multiplier
to the recreational fishing mortality F) across two critical uncertainties regarding alewife futures
and true levels of commercial by-kill of walleyes resulting in a total of four combinations.
Values are sustainable recreational harvest of walleyes in Saginaw Bay; mean recreational
harvest with zeros assigned to years exceeding sustainability thresholds. Maximum value is
indicated with box.

Recreational F
multiplier
Probability
0.1
0.5
1.0
1.5
2.0
2.5
3.0
4.0

Low by-kill
rate &
alewives
scarce
0.375
4896
23253
45173
52922
49799
45056
40763
31487

Low by-kill
rate &
alewives
recover
0.125
18335
53722
50609
46431
35397
28858
23755
16601

High by-kill
rate &
alewives
scarce
0.375
3120
15420
30317
43355
45952
41847
38144
29737

High by-kill
rate &
alewives
recover
0.125
12817
46192
43422
41435
36124
27480
21324
12728

Expected
value
6900
26992
40063
47087
44847
39631
35225
26625

Table 3.4. Analysis of the expected value of perfect by-kill information (EVPI) and expected
value of imperfect by-kill information (EVII). High and low by-kill reference two alternative
states of nature based on the magnitude of inner Saginaw Bay state-licensed commercial by-kill
stemming from two estimates (a lower value based on an observed period and a higher value
based on an extrapolated annual basis).
Metric
Expected value of “best” decision (maximum value from
Table 3.3)
EVPI Calculation
Value if we know high by-kill model was true
Value if we know low by-kill model was true
Expected value after knowing true by-kill model
EVPI
% EVPI of total fishery
EVII Calculation
Value if we know high by-kill model was true
Value if we know low by-kill model was true
Expected value after knowing true by-kill model
EVII
% EVII of total fishery

105

Value
47087

49697- 47087

47212- 47087

46072
53322
49697
2610
5.5%

43768
50656
47212
125
0.3%

Figure 3.1. Saginaw Bay and Lake Huron.

106

Figure 3.2. Decision tree reflecting alternative harvest policies for varying levels of recreational
fishing mortality (Frec depicted as low, medium, and high). In actuality, eight different levels
were evaluated.

107

500,000

500,000

A

Recreational harvest

Total harvest

400,000
300,000

200,000
100,000

300,000
200,000
100,000
0

0

0.1

0.5

1.0

1.5

2.0
2.5
F scalar

3.0

4.0

5.0

6.0

0.1

C

100

% years mean TL age-3
exceeeds target

Population size (N)

4,000,000

B

400,000

3,000,000
2,000,000
1,000,000

0.5

1.0

1.5

2.0 2.5
F scalar

3.0

4.0

5.0

6.0

0.5

1.0

1.5

2.0
2.5
F scalar

3.0

4.0

5.0

6.0

0.5

1.0

1.5

2.0 2.5
F scalar

3.0

4.0

5.0

6.0

D

80
60
40
20

0

0.5

1.0

1.5

2.0 2.5
F scalar

3.0

4.0

5.0

0

6.0

0.1

100

0.7

E

80

Total Annual Mortlity (A)

% years SSB below 20% unfished

0.1

60
40
20
0
0.1

0.5

1.0

1.5

2.0
2.5
F scalar

3.0

4.0

5.0

6.0

0.6

F

0.5
0.4
0.3
0.2

0.1
0.1

Figure 3.3. Forecasts for different levels of recreational fishing intensity (applied as a scalar to fishing mortality) across uncertain
states of future alewife trends:
, based on means of 50 year forecasts of: total catch (all
fisheries), recreational harvest, population size, percent of years that age-3 walleye total length exceeds management target, percent of
years spawning stock biomass (SSB) drops below the 20% of unfished threshold, and total annual mortality (A).

108

1,400,000

A

1,200,000

Mean SSB

1,000,000

F0.1
F0.5

800,000

F1.0
F1.5

600,000

F2.0
F3.0

400,000

F4.0
200,000

20% unfished SSB

0
0

470

20

30

40

50

Year

450

Mean TL age-3 (mm)

10

B
F0.1

430

F0.5
F1.0

410

F1.5

F2.0

390

F3.0
F4.0

370

Target not to exceed

350
0

10

20

30

40

50

Year
600

C

1,200,000

500
400

800,000
300
600,000
200

400,000

100

200,000
0

F0.5
F1.0

F1.5
F2.0
F3.0
F4.0

20% unfished SSB

0

0

10

20

30

40

50

Year

510

600

D

490

Mean TL age-3 (mm)

F0.1

Alewife

500

470
400

450
430

300

410

200

390
100

370

350

0
0

10

20

30

40

Alewife density (no./ha)

Mean SSB

1,000,000

Alewife density (no./ha)

1,400,000

F0.1

F0.5
F1.0
F1.5
F2.0

F3.0
F4.0
Target not to exceed

Alewife

50

Year

Figure 3.4. Mean spawning stock biomass (SSB) and mean total length (TL) of age-3 walleyes
and critical thresholds of management importance across 50 year forecasts of six recreational
fishing intensity management options, across two versions of future alewife trends (without
alewife recovery; graphics A and B, and with alewife recovery; graphics C and D).
109

CITATIONS

110

CITATIONS

AD Model Builder-Project.org (2013). An Introduction to AD Model Builder Version 9.0.0 for
use in Nonlinear Modeling and Statistics. http://admbproject.googlecode.com/files/admb.pdf Accessed July, 2013. 185 p.
Baldwin, N. S., and R. W. Saalfeld. 1962. Commercial fish production in the Great Lakes
1867–1960. Great Lakes Fisheries Commission, Technical Report 3 (and 1970
supplement covering the years 1961–1968), Ann Arbor.
Chen, D.G., and Irvine, J.R. 2001. A semiparametric model to examine stock–recruitment
relationships incorporating environmental data. Canadian Journal of Fisheries and
Aquatic Sciences 58: 1178–1186.
Clemen, R. T., T. Reilly. 2001. Making hard decisions with decision tools. Second Edition.
Duxbury Thomson Learning, Pacific Grove, CA. ISBN: 0-534-36597-3.
Dunlop, E. S., S. W. Milne, and M. S. Ridgeway. 2010. Temporal trends in the numbers and
characteristics of Lake Huron fish schools between 2000 and 2004. Journal of Great
Lakes Research, 36:74-85.
Eshenroder, R.L., and Burnham-Curtis, M.K. 1999. Species succession and sustainability of the
Great Lakes fish community. In Great Lakes fishery policy and management: a binational
perspective. Edited by W.W. Taylor and C.P. Ferreri. Michigan State University Press,
East Lansing, Mich. pp. 145–184.
Fielder, D. G. 2002. Sources of Walleye Recruitment in Saginaw Bay, Lake Huron. North
American Journal of Fisheries Management, 22:1032-1040.
Fielder, D. G. 2014. Mortality, exploitation, movement, and stock size of Saginaw Bay walleyes,
1981-2011; 31 years of tag return analysis. Michigan Department of Natural Resources,
Fisheries Report 04. Lansing.
Fielder, D. G., and J. P. Baker. 2004. Strategies and options for the recovery of walleye in
Saginaw Bay, Lake Huron. Michigan Department of Natural Resources, Fisheries Special
Report 29. Ann Arbor.
Fielder, D. G., and J. R. Bence. 2014. Integration of auxiliary information in statistical catch-atage analysis of the Saginaw Bay stock of walleye in Lake Huron. North American
Journal of Fisheries Management 34:970-987.
Fielder, D. G., T. L. Kolb, T. M. Goniea, D. L. Wesander, and K. S. Schrouder. 2014. Fisheries
of Saginaw Bay, Lake Huron 1986-2010. Michigan Department of Natural Resources,
Fisheries Report 02. Lansing.

111

Fielder, D. G., A. P. Liskauskas, D. J. A. Gonder, L. C. Mohr, and M. V. Thomas. 2010. Status
of walleye in Lake Huron. Pages 71-90 In Status of walleye in the Great Lakes:
proceedings of the 2006 Symposium. Edited by E. Roseman, P. Kocovsky, and C.
Vandergoot. Great Lakes Fishery Commission Technical Report 69.
Fielder, D. G., J. S. Schaeffer, and M. V. Thomas. 2007. Environmental and ecological
conditions surrounding the production of large year classes of walleye (Sander vitreus) in
Saginaw Bay, Lake Huron. Journal of Great Lakes Research 33:118-132.
Fielder, D. G., and M. V. Thomas. 2006. Fish Population Dynamics of Saginaw Bay, Lake
Huron 1998 – 2004. Michigan Department of Natural Resources, Fisheries Research
Report. No. 2083. Ann Arbor.
Fielder, D. G., and M. V. Thomas. 2014. Status and Trends of the Fish Community of Saginaw
Bay, Lake Huron 2005–2011. Michigan Department of Natural Resources, Fisheries
Report 03. Lansing.
Fournier, D. A., H. J. Skaug, J. Ancheta, J. Ianelli, A. Magnusson, M. N. Maunder, A. Nielsen
& J. Sibert 2012. AD Model Builder: using automatic differentiation for statistical
inference of highly parameterized complex nonlinear models. Optimization Methods and
Software, 27:233-249.
Great Lakes Fishery Commission (GLFC). 2005. Lake Erie walleye management plan. Lake Erie
Committee, GLFC. http://www.glfc.org/lakecom/lec/WTG_docs/other_reports_and_docs
/wmp20051207.pdf Accessed May, 2014.
Haddon, M. 2001. Modelling and quantitative methods in fisheries. Chapman & Hall/CRC 406p.
Hansen, G. J. A., and M. L. Jones. 2008. The value of information in fishery management.
Fisheries 33:340-348.
He, J. X., J. R. Bence, C. P. Madenjian, S. A. Pothoven, N. E. Dobiesz, S. C. Riley, D. G.
Fielder, J. E. Johnson, M. P. Ebener, A. R. Cottrill, L. C. Mohr, and S. R. Koproski. In
press. Using a system of time-varying models to quantify piscivory patterns during the
rapid food-web changes in the main basin of Lake Huron. Canadian Journal of Fisheries
and Aquatic Sciences.
Hilborn, R., & Walters, C. J. 1992. Quantitative Fisheries Stock Assessment: Choice, Dynamics
and Uncertainty. Springer.
Holling, C. S. 1959. Some characteristics of simple types of predation and parasitism. Canadian
Entomology. 91:385-398.

112

Jones, M. L., and J. R. Bence. 2009. Uncertainty and fishery management in the North American
Great Lakes: lessons from applications of decision analysis. American Fisheries Society
Symposium 70:1-23.
Laarman, P. W. 1978. Case histories of stocking walleyes in inland lakes, impoundments and the
Great Lakes-100 years with walleyes. American Fisheries Society Special Publication II:
254-260.
Lane, D. E., and R. L. Stephenson. 1998. A framework for risk analysis in fisheries decisionmaking. ICES Journal of Marine Science55:1-13.
Li, J., Cohen, Y., Schupp, D. H., & Adelman, I. R. 1996. Effects of walleye stocking on
population abundance and fish size. North American Journal of Fisheries Management,
16, 830-839.
MacMillan, E.A., B.M. Roth. 2012. By-catch in the Saginaw Bay, Lake Huron commercial trap
net fishery. Journal of Great Lakes Research 38:353-361.
Peterman, R. M., and C. N. Anderson. 1999. Decision analysis: a method for taking uncertainties
into account in risk-based decision making. Human and Ecological Risk Assessment
5:231-244.
Peterman, R. M., & Peters, C. N. 1998. Decision analysis: taking uncertainties into account in
forest resource management. Statistical methods for adaptive management studies. Land
Management Handbook 42: 105-128.
Ricker, W. E. 1975. Computation and interpretation of biological statistics of fish populations.
Fisheries Research Board of Canada Bulletin 191.
Riley, S. C., E. F. Roseman, S. J. Nichols, T. P. O’Brien, C. S. Kiley, and J. S. Schaeffer. 2008.
Deepwater demersal fish community collapse in Lake Huron. Transactions of the
American Fisheries Society, 137:1879-1890.
Schaeffer, J. S. 1994. Mechanisms regulating growth and food consumption of Walleyes and
yellow perch in Saginaw Bay, Lake Huron. Ph.D. dissertation, University of Michigan,
Ann Arbor, Michigan, USA.
Schneider, J. C. 1977. History of the walleye fisheries of Saginaw Bay, Lake Huron. Michigan
Department of Natural Resources, Fisheries Research Report 1850, Ann Arbor.
Schneider, J. C., and J. H. Leach. 1977. Walleye (Stizostedion vitreum vitreum) fluctuations in
the Great Lakes and possible causes, 1800-1975. Journal of Fisheries Research Board of
Canada 34:1878-1889.
Schneider, J. C., and J. H. Leach. 1979. Walleye stocks in the Great Lakes, 1800–1975:
fluctuations and possible causes. Great Lakes Fishery Commission, Technical Report 31.
113

Shuter, B. J., and J. F. Koonce. 1977. A dynamic model of the western Lake Erie walleye
(Stizostedion vitreum vitreum) population. Journal of the Fisheries Research Board of
Canada 34:1972-1982.
Thomas, M. V., D. Einhouse, K. Kayle, M. Turner, C. Vandergoot, M. Belore, A. Cook, K. Ho,
T. MacDougall, K. Soper, Y. Zaho, and C. Murray. 2011. Report for 2010 by the Lake
Erie Walleye Task Group, Lake Erie Committee, Great Lakes Fishery Commission.
Available:
http://www.glfc.org/lakecom/lec/WTG_docs/annual_reports/WTG_report_2011.pdf
(January 2014).
Thomas, M. V., and R. C. Haas. 2005. Status of Yellow Perch and Walleye in Michigan Waters
of Lake Erie, 1999–2003. Michigan Department of Natural Resources, Fisheries
Research Report 2082. Ann Arbor.
Walters, C. J. 1986. Adaptive management of renewable resources. Macmillan, New York.
Wang, H., E . Rutherford, A Cook, D. W. Einhouse, R. C. Haas, T. B. Johnson, R Kenyon, B.
Locke, M. W. Turner. 2007. Movement of walleye in Lakes Erie and St. Clair inferred
from tag recapture and fisheries data. Transactions of the American Fisheries Society
136:539–551.
Williams, B. K. 1997. Approaches to the management of waterfowl under uncertainty. Wildlife
Society Bulletin 25:714-720.

114