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This is to certify that the

dissertation entitled

Incorporating Uncertainty Into St. Marys River Sea
Lamprey Management Through Decision Analysis

presented by

Steven Lewis Haeseker

has been accepted towards fulﬁllment
of the requirements for

Ph.D. degreein Fish. & Wildl.

 

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CM £17m

Date November 13, 2001

MS U it an Afﬁrmative Action/Eq ual Opportunity Institution 0~12771

 

 

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INCORPORATING UNCERTAINTY INTO ST. MARYS RIVER SEA LAMPREY
MANAGEMENT THROUGH DECISION ANALYSIS
By

Steven Lewis Haeseker

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Fisheries and Wildlife

2001

ABSTRACT

INCORPORATING UNCERTAINT Y INTO ST. MARYS RIVER SEA LAMPREY
MANAGEMENT THROUGH DECISION ANALYSIS

By

Steven L. Haeseker

Methods used to control St. Marys River sea lampreys (Petromyzon marinas)
include trapping adults, releasing sterilized males into the Spawning population, and
applying lampricide to kill larvae living within the strearnbed. Future control activities
will require some, or all of, these methods. In addition to the costs and logistical
challenges associated with implementing control options, considerable uncertainty exists
about lamprey population dynamics and about control option effectiveness. These
uncertainties hinder the ability of scientists to accurately forecast management option
performance and thus limit the ability of decision-makers to arrive at well-informed
decisions about which control methods to use. I use the method of decision analysis to
evaluate the performance of a set of management options for controlling the St. Marys
River sea lamprey population while explicitly accounting for uncertainty. I considered
two main sources of uncertainty: uncertainty in the stock-recruitment relationship and
uncertainty in larval distribution within the St. Marys River. To characterize the ﬁrst of
these, I developed a statistically-based, age-structured population model to estimate the
parameters of a Ricker stock-recruitment relationship for St. Marys River sea lampreys
and the uncertainty associated with these parameters. To characterize the second main
uncertainty, I developed a stochastic model that predicts the abundance of larvae at a

location in the next year based on the abundance in the current year. I applied the model

to forecast a variety of possible larval abundance maps for the lampricide treatment that
took place in 1999 to estimate the uncertainty in treatment effectiveness resulting from
spatial and temporal uncertainty in larval distribution. By combining these sources of
uncertainty into a stochastic simulation model that forecasted parasitic lamprey
abundance over time, I was able to examine the performance of a variety of management
options. I found that uncertainty in lamprey population dynamics and in treatment option
effectiveness can have large effects on the forecasts of lamprey abundance. In addition,
the relative ranking of management options depended on the performance indicator used
to evaluate them. Important tradeoffs exist between achieving management objectives
and the cost associated with achieving the management objective. Decision-makers
should evaluate option performance by considering several performance indicators.
Incorporating uncertainty into St. Marys River sea lamprey management decisions
through decision analysis should improve the quality of the decisions and increase the

probability of achieving management objectives.

ACKNOWLEDGEMENTS

This work would not have been possible without the generous funding provided
by the Great Lakes Fishery Commission, the Michigan Sea Grant College Program, a
Michigan State University Distinguished Fellowship, and a Michigan State University
Dissertation Completion Fellowship. I would also like to thank Roger Bergstedt, Mike
Fodale, Doug Cuddy, Mike Steeves, Rod McDonald, Mike Twohey, Gavin Christie, and
Lloyd Mohr for their insight, knowledge, and data. Dr. Randall Peterman provided

valuable direction on how to perform the decision analysis.

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ........................................................................................... iv
LIST OF TABLES ....................................................................................................... vii
LIST OF FIGURES .................................................................................................... viii
INTRODUCTION .......................................................................................................... 1
References ........................................................................................................... 6
CHAPTER 1
ESTIMATING UNCERTAINT Y IN THE STOCK-RECRUITMENT
RELATIONSHIP FOR ST. MARYS RIVER SEA LAMPREYS ................................. 8
Abstract ............................................................................................................... 8
Introduction ......................................................................................................... 9
Methods ............................................................................................................. l 1
Results ............................................................................................................... 23
Discussion ......................................................................................................... 25
Acknowledgements ........................................................................................... 30
References ......................................................................................................... 31
Tables ................................................................................................................ 34
Figures ............................................................................................................... 36
CHAPTER 2

QUANT IF YING SPATIAL AND TEMPORAL UNCERTAINTY IN
LARVAL LAMPREY ABUNDANCE AND ITS EFFECT ON

CHEMICAL TREATMENT SUCCESS IN THE ST. MARYS RIVER ..................... 47
Abstract ............................................................................................................. 47
Introduction ....................................................................................................... 49
Methods ............................................................................................................. 51
Results ............................................................................................................... 58
Discussion ......................................................................................................... 61
Acknowledgements ........................................................................................... 64
References ......................................................................................................... 65
Tables ................................................................................................................ 67
Figures ............................................................................................................... 69

TABLE OF CONTENTS (cont’d)

CHAPTER 3

AN APPLICATION OF DECISION ANALYSIS TO THE MANAGEMENT

OF ST. MARYS RIVER SEA LAMPREYS (PETROMYZON MARINUS) ............. 74
Abstract ............................................................................................................. 74
Introduction ....................................................................................................... 75
Description of the System and Sea Lamprey Control ....................................... 79
Conducting the Decision Analysis .................................................................... 81
Results ............................................................................................................... 97
Discussion ....................................................................................................... 101
Acknowledgements ......................................................................................... 105
References ....................................................................................................... 106
Tables .............................................................................................................. 108
Figures ............................................................................................................. 1 1 1

LIST OF TABLES

CHAPTER 1
Table A] Data used in ﬁtting the sea lamprey population model ......................... 34
Table A.2 Estimated age composition for St. Marys River sea lampreys ............. 35
Table A3 Estimated transformer age composition ................................................ 35
CHAPTER 2
Table 2.1 Algorithm for generating a single abundance map. .............................. 67
Table 2.2 Logistic GLM results. ........................................................................... 68
Table 2.3 Gamma GLM results ............................................................................. 68
CHAPTER 3
Table 3.1 A listing of the nine management options under consideration

in the decision analysis along with the speciﬁc levels and

timing associated with each control method. ...................................... 108

Table 3.2 Average net beneﬁts (over all simulations) associated with the
eight active treatment options across a range of assumed values
for the implied value, the time horizon, and the discount rate.
The rank for each treatment option is listed in superscript next
to the calculated average net beneﬁt for each set of
assumptions. . ...................................................................................... 109

Table 3.3 Proportion of simulation cases that meet Consent Agreement
implied lamprey abundance targets of 183,000 in 2006 and
114,000 in 2012 for each of the nine management options.
Also listed is the cost associated with each option. . .......................... 110

vii

LIST OF FIGURES

CHAPTER 1
Figure 1.1 Stock-recruit relationships depicting high (RI, ), moderate( Rj, ),

and low (RZ) values for the compensation ratio (R‘lRo) at a low

stock size (S*). Recruitment at low spawning stock size with
zero compensation is denoted by R0 and recruitment at high
spawning stock size (S’) is denoted by R’. ........................................ 36

Figure 1.2 MPD (modal posterior density) model estimates and observed
commercial ﬁshery catch-per-unit-effort (number of lamprey per
km of gillnet) estimates, 1967-1999. .................................................. 37

Figure 1.3 MPD (modal posterior density) model estimates and observed
spawning-phase population estimates based on mark-recapture,
1985-2000. ......................................................................................... 38

Figure 1.4 MPD (modal posterior density) model estimates and observed
parasitic-phase mark-recapture population estimates for
juvenile lampreys, 1982, 1991-1994, and 2000. ................................ 39

Figure 1.5 Residuals (observed minus predicted value) of proportions-at-
age for larval lamprey ages 2-5, 1971-1998. ..................................... 40

Figure 1.6 Frequency histogram of 12,000 MCMC samples of the larval
natural mortality rate (M). .................................................................. 41

Figure 1.7 Frequency histogram of 12,000 MCMC samples of the
proportion of the parasitic-phase population that migrates to the
St. Marys River to spawn (A). ............................................................ 42

Figure 1.8 MPD model estimates of the number of female spawners and
age-0 recruits for the St. Marys River sea lamprey population,
1968-1996. The least-squares ﬁt of a Ricker stock-recruitment
function is also plotted. ....................................................................... 43

Figure 1.9 Marginal frequency histogram of 12,000 MCMC samples of
the or (panel A), B (panel B), and c 2 (panel C) parameters
ﬁom a Ricker stock-recruitment function. ......................................... 44

Figure 1.10 Frequency histogram of 12,000 estimates of the compensation

ratio, R*/Ro, based on the MCMC samples from the joint
distribution of or and B. ...................................................................... 45

viii

Figure 1.11 Residuals (FWD minus 133,) of the I-IPD estimates of the

number of parasites (FWD) minus the estimated number of
parasites based on the least-squares stock-recruit relationship

parameters (135,) plotted over time. ...................................................... 46
CHAPTER 2
Figure 2.1 Annual means (+/- 1 SE.) of adjusted larval catches from

index sites 1-9. ...................................................................................... 69

Figure 2.2 Percent frequency histogram of adjusted larval catches from
index sites 1-9. ...................................................................................... 70

Figure 2.3 Empirical semivariogram and ﬁtted spherical semivariogram
function based on transformed larval catches from index sites
1-13. ...................................................................................................... 71

Figure 2.4 Empirical cross-semivariogram and ﬁtted spherical
semivariogram function based on transformed larval catches
from index sites 1-13. ........................................................................... 71

Figure 2.5 Observed and simulated annual distribution and abundance of
larvae in index site 6. Dot diameters are proportional to
abundance with the largest dots representing adjusted catches
of approximately 30 and the smallest dots representing zeros. ............ 72

Figure 2.6 Results of one hundred simulations of the percent of larval
population targeted versus dollars spent on Bayluscide (dots).
Also plotted is the predicted percent targeted versus dollars
spent assuming temporal stationarity in larval distribution and
abundance. ............................................................................................ 73

Figure 2.7 Results of one hundred simulations of the percent of larval
population targeted versus dollars spent on Bayluscide (dots),
holding the total abundance in non-treated areas constant. Also
plotted is the predicted percent targeted versus dollars spent
assuming temporal stationarity in larval distribution and
abundance. ............................................................................................ 74

ix

CHAPTER 3

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Trapping efﬁciency in the St. Marys River 1991-2000. The
horizontal line represents the average efﬁciency for this period.

SMRT ratio based on mark recapture versus the SMRT ratio
calculated based on egg viability assessments. The diagonal
line represents a 1:1 relationship between the two variables. ......

Percent deviation of the predicted SMRT ratio based on egg
viabilities from the SMRT ratio based on mark-recapture, over
the range observed for SMRT mark-recapture ratios. ............ ‘ .....

Decision tree representing the management options, key
uncertainties and outcomes used in the decision analysis. ..........

Deterministic forecasts of reductions in sea lampreys of two
control options that were presented to GLFC Commissioners in
1997. Vertical arrows represent the reduction in lampreys
compared to no control. ...............................................................

Forecasts of average parasitic lamprey abundance in northern
Lake Huron, incorporating all uncertainties, for decision
options 1-9. ..................................................................................

Forecasts of average parasitic lamprey abundance in northern
Lake Huron associated with using no control, and using
decision option 2, with and without uncertainty in trapping and
SMRT effectiveness. ....................................................................

Forecasts of average parasitic lamprey abundance in northern
Lake Huron associated with using no control, with full
demographic uncertainty (All models), using only the average
parameters set of demographic parameters (average
parameters), and only using the MPD set of demographic
parameters (MPD model) .............................................................

Forecasts of mean and median parasitic lamprey abundance in
northern Lake Huron associated with using no control and with
using decision option 2. ...............................................................

Box and whisker plots of the net beneﬁts across all simulations
for decision options 1-9. The dot represents the median net

beneﬁt and the box ends represent the 25th and 75th percentiles
of the distribution. ........................................................................

....... 111

....... 112

....... 112

....... 113

....... 114

....... 115

....... 116

....... 117

....... 118

....... 119

Figure 3.11

Figure 3.12

Cumulative probability of being below speciﬁed parasitic

abundance (on x-axis) in year 2012 for options 1-5 (panel A)

and Options 6-9 (panel B). The vertical arrows denote the

parasitic abundance target implied by the Consent Agreement. ........ 120

Cumulative probability of being below speciﬁed parasitic

abundance (on x-axis) in year 2030 for options 1-5 (panel A)
and options 6-9 (panel B). ................................................................. 121

xi

INTRODUCTION

Controlling the St. Marys River sea lamprey (Petromyzon marinas) population is
a major challenge facing ﬁshery managers in the Great Lakes. The sea lamprey is an
exotic ﬁsh species that has decimated ﬁsheries in the Great Lakes through parasitism, and
the St. Marys River is the largest remaining source of sea lampreys in the Great Lakes
basin (Schleen 1992, Eshenroder et a1. 1995). Because of the large number of lampreys
that are produced, mortality rates for several species (e.g., lake trout Salvelinus
namaycush and lake Whiteﬁsh Coregonus clupeaformis) have risen above target levels set
by management agencies (Sitar et al. 1999). Thus, reducing the number of sea lampreys
in northern Lakes Huron and Michigan has become a high priority for ﬁsheries managers
in the region.

Methods used to control St. Marys River sea lampreys include trapping adults,
releasing sterilized males into the spawning population, and applying lampricide to kill
larvae living within the streambed. In addition to the costs and logistical challenges
associated with implementing a lamprey control program for the St. Marys River,
considerable uncertainties exist in lamprey population dynamics and in control option
effectiveness. These uncertainties hinder the ability of scientists to accurately forecast
management option performance and thus limit the ability of decision-makers to arrive at
well-informed decisions.

Uncertainty is inherent in natural resource management decisions. Within the
ﬁeld of ﬁsheries, there is a growing trend towards incorporating uncertainty and risk in

the decision-making process (Hilbom et a1. 1993). Rosenberg and Restrepo (1994) stress

the importance of estimating and communicating uncertainty to ﬁshery managers, who
must weigh the beneﬁts, costs, and risks of various management options. They note that
while many managed ﬁsheries have incorporated elements of uncertainty and risk
analysis, the advice resulting ﬁ'om these analyses needs to be expressed to decision-
makers in an effective manner. Providing ﬁshery managers with a quantitative
evaluation of the potential consequences of alternative management actions is one of the
primary roles of stock assessment scientists (National Research Council Committee on
Fish Stock Assessment Methods 1998). Because ﬁsheries management problems
typically involve a complex system of biological and socio-economic objectives and
constraints, Lane and Stephenson (1998) contend that a conceptual change is necessary,
and ﬁsheries management needs to move toward implementing integrated decision-
making systems where uncertainty is incorporated. When uncertainty is ignored, or
accounted for in an arbitrary fashion, sub-optimal decision options may be selected,
leading to outcomes such as unnecessary losses in yields or stock collapse (Frederick and
Peterman 1994). Achieving ﬁsheries management goals is more likely when
uncertainties are acknowledged, quantiﬁed, and accounted for (Peterman et a1. 1998).
The method of decision analysis is speciﬁcally designed to quantitatively deal
with management problems in the presence of uncertainty (Raiffa and Schlaifer 1961 ,
Morgan and Henrion 1990, Clemen 1996). Decision analysis is the application of
statistical decision theory to a management decision problem, generally with the
following components: a deﬁned set of management options available to the decision-
maker, a listing of the alternative states of nature and their associated probabilities which

characterize the uncertainty in the problem, a listing of possible outcomes resulting from

the management options, and a representation of the decision-maker’s utility for the
outcomes. The relative merit of each management option is determined by the decision-
maker’s utility for the expected outcomes.

Several researchers have applied the method of decision analysis to ﬁsheries
management problems and found that uncertainty can affect both forecasts and optimal
decision-making. McAllister and Peterman (1992) used decision analysis to evaluate the
performance of experimental and status quo management strategies for pink salmon
(Oncorhynchus gorbuscha), while accounting for uncertainty in the cause of decreased
mean body weight. They concluded that the expected value of the experimental
management strategy was higher than that of the status quo under most conditions. Robb
and Peterman (1997) examined uncertainties in the stock-recruitment relationship, annual
recruitment, run timing, and catchability for a sockeye sahnon (Oncorhynchus nerka)
ﬁshery through a decision analysis. They found that the shape of the stock-recruitment
relationship had a large effect in determining the optimal management option. Hilbom et
a1. (1994) use decision analysis to examine the performance of different quotas in the
presence of uncertainty in virgin stock size. Peterman et al. (1998) summarize three case
studies where decision analysis has been applied to ﬁsheries management and
recommend that uncertainty be included in the decision making process whenever
possible to improve the quality of management decisions.

There are several reasons why decision analysis represents a valuable tool to
assist managers who wish to control St. Marys River sea lampreys. First, managing the
St. Marys River sea lamprey population is one of the most important problems '

challenging ﬁsheries managers in the Great Lakes. As a result of the high number of sea

lampreys in northern Lakes Huron and Michigan, rehabilitation of lake trout (Salvelinus
namaycush) has been difﬁcult due to the predation mortality imposed by lamprey in these
areas (Sitar et al. 1999, Sitar et a1. 1997, Eshenroder et al. 1995). Thus, the management
problem is important enough to justify a concerted effort in evaluating management
options through decision analysis. Second, major uncertainties exist regarding lamprey
population dynamics and the effectiveness of available management options. As noted
above, decision analysis is speciﬁcally designed to incorporate uncertainty into the
decision-making process. Third, economic constraints demand careful evaluation of all
lamprey management decisions in the Great Lakes. Spending too little on control of the
St. Marys may result in foregone recovery of lake trout in northern Lake Huron.
Conversely, spending too much on the St. Marys takes money away from assessment and
control needs of other lamprey populations in the Great Lakes basin. Decision makers
want and need to know the expected beneﬁts of directing funds at controlling the St.
Marys population in order to properly allocate funds available for control and assessment
throughout the basin. Fourth, several well-deﬁned management options are and will be
considered as alternatives, effectively bounding the number of options available for
consideration. Fifth, speciﬁc targets exist for lake trout recovery and sea lamprey
suppression in Lake Huron that provide indicators against which the performance of
management options can be judged. Each of these issues argues for the application of
decision analysis to the problem of managing St. Marys River sea lampreys.

In this dissertation, I applied the method of decision analysis to the problem of
controlling St. Marys River sea lampreys. I considered two main sources of uncertainty:

uncertainty in the stock-recruitment relationship and uncertainty in larval distribution

within the St. Marys River. I incorporated these two sources of uncertainty into a
stochastic simulation model that forecast parasitic lamprey abundance over time. By
using the simulation model within a decision analytic framework, I was able to examine
the performance of a set of management options for controlling the St. Marys River sea

lamprey population.

References

Clemen, RT. 1996. Making hard decisions: an introduction to decision analysis.
Duxbury Press, Paciﬁc Grove, CA.

Eshenroder, R.L., N.R. Payne, J.E. Johnson, C. Bowen H, and MP. Ebener. 1995. Lake
trout rehabilitation in Lake Huron. J oumal of Great Lakes Research 21(Supplement
l):108-127.

Frederick, SW. and RM. Peterman. 1994. Choosing ﬁsheries harvest policies: when
does uncertainty matter? Canadian Journal of Fisheries and Aquatic Sciences 52:291-
306.

Hilbom, R., E.K. Pikitch, and RC. Francis. 1993. Cm'rent trends in including risk and
uncertainty in stock assessment and harvest decisions. Canadian Journal of Fisheries and
Aquatic Sciences 50:874-880.

Hilbom, R., E.K. Pikitch, and MK. McAllister. 1994. A Bayesian estimation and
decision analysis for an age-structured model using biomass survey data. Fisheries
Research 19:17-30.

Lane, DE, and R.L. Stephenson. 1998. A framework for risk analysis in ﬁsheries
decision-making. ICES Journal of Marine Science 55:1-13.

McAllister, M.K., and RM. Peterman. 1992. Decision analysis of a large-scale ﬁshing
experiment designed to test for a genetic effect of size-selective ﬁshing on British
Columbia pink salmon (Oncorhynchus gorbuscha). Canadian Journal of Fisheries and
Aquatic Sciences 49: 1305-13 14.

Morgan, MG. and M. Henrion. 1990. Uncertainty: a guide to dealing with uncertainty
in quantitative risk and policy analysis. Cambridge University Press, Cambridge, UK.

National Research Council Committee on Fish Stock Assessment Methods. 1998.
Improving ﬁsh stock assessments. National Academy Press, Washington, DC.

Peterman, R.M., C.N. Peters, C.A. Robb, and SW. Frederick. 1998. Bayesian decision
analysis and uncertainty in ﬁsheries management. In Reinventing Fisheries Management,
Pitcher, T.J., P.J.B. Hart, and D. Pauly, editors. Kluwer Academic Publishers, The
Netherlands.

Raiffa, H. and R. Schlaifer. 1961. Applied statistical decision theory. Harvard
University, Boston, MA.

Robb, CA. and RM. Peterman. 1997. Application of Bayesian decision analysis to
management of a sockeye salmon (Oncorhynchus nerka) ﬁshery. Canadian Journal of
Fisheries and Aquatic Sciences 55:86-98.

Rosenberg, AA. and V.R. Restrepo. 1994. Uncertainty and risk evaluation in stock
assessment advice for US. marine ﬁsheries. Canadian Journal of Fisheries and Aquatic
Sciences 51 :2715-2720.

Schleen, LP. 1992. Strategy for control of sea lampreys on the St. Marys River, 1992-
95. Great Lakes Fishery Commission, Ann Arbor, Michigan.

Sitar, S.P., J .R. Bence, J .E. Johnson, and W.W. Taylor. 1997. Sea lamprey wounding
rates on lake trout in Lake Huron, 1984-1994. Michigan Academician 29:21-37.

Sitar, S.P., J .R. Bence, J .E. Johnson, MP. Ebener, W.W. Taylor. 1999. Lake trout
mortality and abundance in southern Lake Huron. North American Journal of Fisheries
Management 19:881-900.

CHAPTER 1
ESTIMATING UNCERTAINTY IN THE STOCK-RECRUITMENT RELATIONSHIP
FOR ST. MARYS RIVER SEA LAMPREYS
Abstract
For successful control of the St. Marys River sea lamprey population, one critical

uncertainty that needs to be characterized is uncertainty in the stock-recruit relationship.
In this paper I develop a statistically-based, age-structured population model to estimate
the parameters of a Ricker stock-recruit relationship for St. Marys River sea lamprey and
the associated uncertainty in these parameters. My analysis indicates that there is
considerable stock-independent variability in recruitment across a range of stock sizes,
potentially limiting the effectiveness of control options that reduce spawning population
abundance in comparison to control options that reduce the abundance of larvae. I found
evidence of negligible- to low compensation levels, which would result in nearly
proportional reductions in recruitment with spawning stock. However, I also found
evidence of strong compensation, which may reduce the effectiveness of treatment
options that attempt to reduce the size of the spawning stock. Through characterizing
these uncertainties and incorporating the results into a decision analytic framework, the
effects of compensation and stock-recruit uncertainty can be evaluated in reference to the

management options available for this system.

Introduction

Connecting Lake Superior to Lake Huron, the St. Marys River has been a primary
focus area for sea lamprey (Petromyzon marinas) management efforts since 1990. The
main reason for this focus has been the recent increase of parasitic lampreys in northern
Lakes Huron and Michigan, hindering lake trout (Salvelinus namaycush) rehabilitation
(Sitar et a1. 1999). Several studies have concluded that the St. Marys River is the primary
source of these lampreys (Johnson 1988, Schleen et al. 1992). However, because of its
large size and high ﬂow, the primary method for reducing lamprey populations (i.e., the
lampricide TFM) is infeasible for both practical and ﬁnancial reasons.

During 1998-99 the Great Lakes Fishery Commission (GLFC) initiated a program
to reduce the production of St. Marys River lampreys using a combination of adult
trapping, sterile male release, and the application of granular Bayluscide (a lampricide)
(Schleen et al., in review). The decision to treat the St. Marys River, and which treatment
option to select, was heavily based on models used to forecast the future abundance of
lamprey in Lake Huron. Although initial indications suggest that the treatment was
successful in reducing lamprey abundance (Fodale et al., in review), future control
actions will almost certainly be necessary.

Forecasting population dynamics is a common task in ﬁsheries management.
Forecasting models by deﬁnition are critically dependent upon the characterization of
system dynamics. Consequently, models that forecast lamprey abundance require the
formulation of a stock-recruit relationship, as well as other important demographic

parameters that determine sea lamprey population dynamics. Although lamprey biology

has been studied extensively, comparatively little attention has been paid to the factors
and processes that govern lamprey population dynamics.

The effectiveness of control methods that alter the number of effective spawners
(e. g., adult trapping, releasing sterilized males) depends on the reproduction and
recruitment dynamics of sea lamprey populations. Speciﬁcally, the shape of the stock-
recruitrrrent relationship and the variability around it (process error) will determine the
degree to which reductions in spawner numbers will consistently result in reductions in
recruitment. Conversely, control methods that target the larval population after year-
class strength is determined (e. g., application of larnpricides) are not affected by the
stock-recruitment relationship. For these reasons, knowledge of the stock-recruitment
relationship is key to assessing the trade-off between these control strategies.

Although estimates of recent spawner abundance for St. Marys River sea
lampreys are available, estimates of larval recruitment are not. Without estimates of
larval production, directly estimating a stock-recruit relationship was impossible.
However, six separate data sets provided information on the relative abundance and age-
composition of lamprey at various stages of their life history. This information was
combined by developing an age-structured population model describing the lamprey life
cycle and ﬁtting model parameters to the observed data using likelihood techniques.
Through this process I was able to estimate a series of historical recruitment and other
demographic parameters consistent with the observed data sets. This approach is similar
to that described by F ournier and Archibald (1982) and Methot (1989) whereby several

sources of data are incorporated into a single, statistically-based framework. To describe

10

the uncertainty in the stock-recruit relationship parameters, I utilize Bayesian methods to

obtain samples from the joint posterior density of key parameters.

Methods
Data sources

Since 1967, the Canada Department of Fisheries and Oceans (DFO) has paid Lake
Huron commercial ﬁshermen for lamprey captured while attached to host species caught
in gillnets. In addition, the Ontario Ministry of Natural Resources (OMNR) collected
effort data for Lake Huron commercial ﬁsheries, recording the ﬁsherman, location, and
km of gillnet associated with landed ﬁsh. I combined the catch and effort data to develop
a catch-per-unit-effort (CPUE) index that can be used to track the relative abundance of

parasitic-phase lamprey over time. I modeled CPUE using the relationship
CPUE... =a.ﬂ.6.xz~e”"” (1)
where a, is the effect of year t = 1967,. . .,1999, ,6,- is the effect of ﬁsherman j =
1,. . .,8, 5,. is the effect of location k = North Channel or Main Basin of Lake Huron, [1

is the overall mean CPUE and 6' t, M ~ N (0, 0'2 ). Taking the natural log of both sides
of the equation resulted in a general linear model of the form

10g, (CP UE; 13k ) = loge (a: ) +10ge (ﬂj ) + loge (5k )+ loge (1”) + £r,j,k (2)

A total of 251 observations were used to estimate the parameters of equation 2
and their associated standard errors using SAS (SAS 1997). By convention within the

SAS procedure, the last effect value for each of the effect types is set to 0 and the

11

remaining effects are scaled relative to this value. For example, loge (Cl/1999) is set to 0

and the remaining log, (69,) are scaled relative to 0. These estimates for log e (6’5I , )

were used as an annual relative abundance index for lamprey in Lake Huron. The linear
model only contained main effects because I had no a priori information suggesting
higher order (interaction) effects.

The United States Fish and Wildlife Service (U SFWS) and the DFO have
calculated mark-recapture population estimates of spawning-phase St. Marys River sea
lampreys annually since 1986. These data provide a direct estimate of the spawning
stock size within the St. Marys River. Adult lampreys migrating upstream in the St.
Marys River are collected using traps, marked, released, and a portion is subsequently
recaptured using traps. These data were used to annually estimate St. Marys River
lamprey spawning population sizes and their associated variances (Mullet et al., in
review). In addition to the Spawning-phase population estimate, data on the number of
lamprey removed by trapping, the sex ratio, and the expected reduction in successful
female spawners due to the release of sterilized males were recorded. I combined the
mark-recapture population estimates with the trapping, sex ratio, and sterile male
effectiveness data to estimate the effective number of female spawners in the St. Marys
River 1985-2000.

Mark-recapture parasitic-phase population estimates have been conducted for
Lake Huron sea lamprey intermittently since 1982. Animals were marked as newly-
metarnorphosed individuals leaving their natal streams. Two years later, adults migrating
upstream to spawn were captured using traps and investigated for marks. Bergstedt et al.

(in review) describe the methods used to estimate population sizes and their associated

12

variance. Because the animals were marked as metamorphosing larvae, the population
estimate refers to the population of metamorphosing larvae (juvenile lampreys) entering
Lake Huron (Bergstedt et al., in review). Individuals that metamorphose in the fall of
year t and winter of year t + 1 constitute the summer parasitic feeding yearclass of year t
+ 1. I will refer to these data as the parasitic-phase data set. The sea lamprey parasitic
population sizes in Lake Huron have been estimated for 1982, 1991, 1992, 1998, and
1999.

Intermittently Since 1971, sea lamprey larvae in the St. Marys River have been
sampled using Bayluscide. The chemical is applied to the water surface, and larvae
respond by swimming to the surface where they are captured using dip nets and measured
for length. I used these data to estimate an annual length-frequency distribution for years
when the survey was conducted. An analysis of data on the size-selectivity of Bayluscide
in the St. Marys River revealed no pattern in selectivity over larval length (Michael
Fodale (U SFWS), unpublished data). Therefore these distributions were assumed to be
representative of the length-frequency distributions for larvae in the river. These surveys
ended in 1989.

During 1993-1998, a deepwater electroﬁshing boat was used to sample St. Marys
River larvae. Information on the size selectivity of the deepwater electroﬁshing gear is
available (Bergstedt and Genovese 1994). Using the Bergstedt and Genoveese (1994)
gear selectivity function, I estimated annual selectivity-adjusted length-frequency
distributions for St. Marys River larval lamprey.

During 1993-96, sea lamprey larvae from the St. Marys River were aged using

statoliths, a structure analogous to otoliths in teleosts (T.B. Steeves, unpublished data). I

13

used these data and an inverse age-length key method (Hoenig and Heisey 1987) to
estimate annual age compositions from the annual length-frequency data. This method
makes the assumption that the growth pattern is constant, but recruitment can vary over
time. Because ﬂuctuations in temperature and ﬂow in the St. Marys River are buffered
by the immense volume of Lake Superior, I believe that inter-annual variation in growth
should be limited, especially relative to inter-annual variation in recruitment for a highly
fecund species such as the sea lamprey. Therefore I believe that the constant growth
pattern assumption is reasonable. By using this method, I were able to apply a recently
developed age-length key to historical data, assuming that larval growth rates in the St.
Marys River have remained similar during 1971-1998. I used this method to estimate the
annual age-compositions of larvae ages 2-5 from the Bayluscide and deepwater
electroﬁshing length-frequencies.

During 1995 and 1996, metamorphosing sea lamprey larvae ﬁom the St. Marys
River were collected and aged using statoliths (T.B. Steeves, unpublished data). I
summarized these data to estimate the age-compositions of metamorphosing sea lamprey

larvae ages 4-6 during 1995 and 1996.

Age-Structured Population Model

I constructed an age-structured population model to describe the ﬁrll lamprey life
cycle ﬁ'om age-0 recruitment through spawning, and ﬁt this model to the six data sources.
The modeled life cycle consisted of a larval (riverine) population ages 0 through 6 subject
to natural mortality constant across ages and years. A maximum larval age of 6 was used

because over 99% of the larvae aged during 1993-1996 were estimated to be 5 6 years

14

old (T.B. Steeves, unpublished data). As the model larvae age, they undergo a process of
metamorphosis from larval to parasitic form. I modeled the probability of
metamorphosis as an increasing function of age for larvae ages 4 though 6. Of the
metamorphosing larvae that were aged in 1995 and 1996, 89% were 4-6 years old (T.B.
Steeves, unpublished data). Metamorphosed larvae enter the parasitic-phase population
in Lake Huron. After 18 months in the parasitic form, a portion of the parasitic-phase
population in Lake Huron returns to the St. Marys River to reproduce.

The overall model required 39 parameters to be estimated. These included the

number of age-0 recruits from 1967 through 1996 ( N 0,, , t = 1967,] 970,...1996 ), the

initial numbers-at-age for ages 0 through 4 during 1966 ( N 3,1966 , i = 0,2,...4 ), a natural

mortality rate (M) assumed to be constant across years and ages in the larval population,
two parameters describing the probability of metamorphosis for larvae age 4 and 5, and
the proportion (A) of the parasitic-phase population in Lake Huron that migrates to the St.
Marys River during spawning. The A parameter represents a combination of two
processes: the survival ﬁ'om metamorphosis to spawner and the fraction of parasites in
Lake Huron that migrate into the St. Marys River.

Given the initial age- and year-speciﬁc abundance estimates, subsequent larval

abundances were calculated using the equation

N = Nut 1: e’” .. [1 —P(met. | age = i)] (3)

i+1,r+1
where P(met. I age = i) is a larvae’s probability of metamorphosis given that it is age i.

This formulation of the equation essentially assumes that lamprey are removed from the

larval population due to mortality and metamorphosis occurring fall through winter, and

15

maintain constant abundance over the rest of the year. The formulation of this equation is
consistent with observations on the timing of mortality and metamorphosis (Y ouson, in
review).

Although researchers have estimated the probability of larval metamorphosis as
an increasing function of length for various tributaries to the Great Lakes, there are no
estimates of these probabilities as a function of age and no estimates for the St. Marys
River. Nearly all of the metamorphosed larvae collected from the St. Marys River have
been assigned ages 4 through 6. To reﬂect this auxiliary information, I constructed the
model such that the probability of metamorphosis increases with age by assuming that the
probability was zero at age 3 and 1.0 at age 6 and estimating as parameters the increase
from age 3 to age 4 and the increase from age 4 to age 5.

The number of metamorphosed lamprey produced in a particular year was
calculated using the equation

6 .
"mm/.05.... = 2;, P(met- I age = i) * N .-,. (4)
r=
Because of the timing of the metamorphosis process relative to the operation of the
commercial ﬁshery, the metamorphosed larvae produced in the fall and early winter of
year 1 would not show up in the commercial catch until year t + 1. Similarly, parasitic
lampreys feeding in Lake Huron during the summer of year t do not spawn until the
spring and summer ofyear t + 1.

The St. Marys River is not the only river producing parasitic lamprey in Lake
Huron. At this time the river-speciﬁc contribution to Lake Huron parasitic-phase
lamprey population is largely unknown. The parasitic-phase CPUE and the parasitic-
phase mark-recapture data sets can only be used to provide information on the overall

16

abundance of lamprey in Lake Huron, not on how much the St. Marys River population
contributes to the overall population. To properly describe the overall system, I needed
an estimate of the contribution of the St. Marys River to the Lake Huron parasitic-phase
population.

The St. Marys River Assessment Plan (Bergstedt et al. 1998) estimated that the St.
Marys River produces 88% of the total parasites in Lake Huron. I used this estimate in
my model to scale the production of the St. Marys River relative to the other sources in
Lake Huron. By using this estimate I am essentially assuming that number of lamprey in
Lake Huron is a function of the amount of larval habitat, the number of spawners, and the
number of hosts available. Of these three factors, I assume that only the amount and
quality of larval habitat remains constant over time. The number of spawners and hosts is
assumed to vary over time, causing the observed variability in recruitment and parasite
densities. This assumption is supported by Young et al. (1996) who concluded that
habitat quantity and quality have remained relatively constant in the St. Marys River and
that host availability was likely a more important factor in ultimately determining parasite
abundance.

To estimate parameters, the overall model was ﬁt to the six data sets by
specifying the assumed statistical distribution of each data set and then constructing the
likelihood function. For the parasitic-phase CPUE data set, a lognonnal distribution was

assumed and the corresponding log-likelihood (ignoring constants) was

2
n, . ,.
L1 = -; 0.5 (log, [a] _ loge (at )J/a't (5)

l7

where n, is the estimated number of parasitic-phase lamprey in year t, n1999 is the

estimated number of parasitic-phase lamprey in year 1999, 63', is the relative parasitic-

phase CPUE in year t, 5', is the standard error estimate associated with each 63', . This

form for the likelihood was used because it mirrored the CPUE outputs from the linear

model used to estimate the 63', . The 5', were estimated using a separate linear model,

were treated as known values, and were not estimated within the lamprey population
model.

The parasitic-phase mark-recapture data set and the spawning-phase mark-
recapture data set were assumed to be described by lognormal distributions. The log-

likelihoods (ignoring constants) for these data sets were of the form

Li : —Z O'5[(10ge (£13: ) "' loge (x'm ”/6331 i2 (6)

where L,- is the log-likelihood for data set i, 55,3, is the empirical estimate of the

population size for data set i in year t, X23, is the model prediction of population size for

A

data set i in year t, and 0],, is the estimated standard deviation for data set i in year t.

As for the CPUE data, the 53,, were estimated in separate analyses (i.e., mark-recapture

studies). In the population model, they were treated as known values and were not
estimated during model ﬁtting.
For the Bayer survey data set, the deepwater electroﬁshing data set, and the

transforming larvae data set, a multinomial distribution was assumed to describe the

18

proportions-at-age. The corresponding log-likelihoods (ignoring constants) were of the

form

L, = 2.1.) 213:.) 108.(P.-,a,r) (7)

where L, is the log-likelihood for data set i, J” is the sample size in year t for data set

I
i, Pi,“ is the model prediction of the proportion age-a in year t, and L0,: is the

empirical estimate of the proportion age-a in year t. To prevent large sample sizes from
overwhelming the log-likelihood, a maximum effective sample size for the combined
Bayluscide and deepwater electroﬁshing data sets was determined using the iterative
method outlined in the appendix of McAllister and Ianelli (1997). If the number sampled

in a year was greater than the maximum sample size calculated, then J” was set to the

calculated maximum effective sample size. For the Bayluscide and deepwater

electroﬁshing data sets, Jmax = 30. Because the number of samples in the
metamorphosing larvae data set were only 34 in 1995 and 43 in 1996, a maximum
effective sample size was not estimated and instead the observed number of samples were
used for the J 1,, .

Combining the six log-likelihoods, the overall log-likelihood objective function

used to estimate model parameters was
L=L1+L2 +L3 +L4 +L5 +L6 (3)
I used AD Model Builder software (Otter Research Ltd. 1994) to estimate model

parameters.

19

Estimating Stock-Recruit Function Uncertainty

The primary objective of this study was to estimate the stock-recruit ﬁrnction for
St. Marys River sea lamprey and its associated uncertainty. An additional objective was
to estimate demographic parameters (M, 1, and P(met. l age = i)) and their uncertainty
for forecasting future population dynamics. To accomplish these obj ectives I adopted a
Bayesian estimation framework, placing bounded, uniform priors on each of the F
estimated parameters. The priors were bounded to avoid implausible parameter
estimates, such as a negative survival rate. I used a Monte Carlo Markov Chain (MCMC)
procedure to obtain samples from the joint posterior distribution of the estimated
parameters. These samples from the joint posterior distribution served to characterize the
uncertainty in the estimated parameters.

I assumed that lamprey recruitment was governed by a Ricker-type stock-recruit

function of the form

R = Sexp(a— ,615‘ + e) (9)
where R is the number of age-0 larvae produced, S is the number of female spawners that
produced R, a and ,6 are parameters determining the productivity and compensation,
respectively, and log(8) is distributed N (0, 0'2). Let 6 be deﬁned as a vector of the 39
parameters estimated in the population model. 0 contains the initial numbers-at-age for
1966, the number of age-O larvae from 1967-1996, M, xi, and the two parameters

describing P(met. | age = i). If I denote the information contained in the six data sets as

Z, then my primary objective is to approximate the joint posterior density

20

p(a, 5,03,01 z)

I accomplished this objective using a two-stage approach. For the ﬁrst stage I utilized the
MCMC procedure within AD Model Builder (a version of the Metropolis Hastings

algorithm) to obtain 12,000 samples from the posterior density of 9( p(67 | Z) ). I

generated a total of 12 million samples and saved every thousandth sample (to arrive at a

set of 12,000 samples) in order to reduce the degree of autocorrelation in the chain. Each
sample (19,, i = l...12,000) determines a set of stock sizes and associated number of age-0
recruits that were produced, which are completely determined by 6! I denote these stock-

recruit sets as Yg, i = 1...12,000.

The second stage consisted of obtaining samples from the joint posterior density,

Maﬁa2 | Y)

To accomplish this, ﬁrst I converted the Ricker model above into its linear form,

ln(R/S)=a—,d$'+e (10)
Then for each stock-recruit set (Y i), I drew a single sample of a,- , ﬂ,— , and 0' ,2 from the

joint posterior density p(a,- , ﬂ,- , 0' ,2 I Yr) (Gehnan et al. 1995, p. 236).

Combining the results ﬁom the two stages resulted in samples from an

approximation to the posterior density,

phaﬁﬂm)

This two-stage approach is not unique, as my approach is analogous to sampling from the

joint posterior distribution for hierarchical models (Gelman et al. 1995, p 129). I used

21

this approach to quantify the joint uncertainty in the parameters of the stock-recruit
function as well as the demographic parameters for use in forecasting models.

Trace plots (i.e., plots of the ordered, saved sample values) for the parameters
show no trends or bum-in period. However, time series analyses revealed that there was
autocorrelation among the samples up to lags of 40 saved samples (or 40,000 among the
originally-generated 12 million samples) for some of the parameters. Using the methods
described by Thiebaux and Zwiers (1984), I estimated the effective number of
independent samples among the 12,000 samples for each parameter. Based on these
methods, there were 1300 to 2000 effectively independent samples among the 12,000 for
each of the parameters.

One of my objectives was to examine the degree that compensation exhibited by
St. Marys River sea lamprey could reduce the effectiveness of the trapping and sterile
male release control options. During 1985-1990 (a time of relatively high lamprey
abundance), the estimated average number of spawning females in the St. Marys River
was ~9700. With a trapping rate of 45% and a 3:1 ratio of sterilezfertile males (both
reasonable target levels for future control, Michael Twohey (U SF WS), personal
communication), the expected number of successful female spawners would be reduced
from ~9700 to ~1100. To investigate the degree of compensation, I calculated a
compensation ratio (Figure 1.1). First I calculated the expected number of recruits (R*)
at low spawning stock size (~1100, S*=1100 in Figure 1) based on a set of stock-recruit
relationship parameters. The same parameters were also used to compute the expected
number of recruits (R’) at high stock size (S’=9700 in Figure 1). Linear interpolation

between zero and R’ deﬁnes the expected number of recruits (R0) that would be produced

22

27—f1- -

if proportional stock reductions from S’ to S* resulted in the same proportional recruit
reduction (i.e., no compensation). The compensation ratio is given by R*/Ro. For each

set of a, and ,6, (i = 1,...12,000) , I calculated the compensation ratio and deﬁned

negligible compensation as ratios less than 1, low compensation as ratios between 1 and
2, moderate compensation as ratios between 2 and 5, and high compensation as ratios

over 5.

Results

Overall, the modal estimates of CPUE, spawner abundance, parasitic abundance,
and age-composition were consistent with observed data. The modal posterior density
estimates for CPUE followed the observed pattern of low CPUE during the 1970’s,
increasing CPUE during the 1980’s, and high CPUE during the 1990’s (Figure 1.2). The
modal posterior density estimates for spawner abundance closely matched observed data,
with the possible exceptions of 1991 and 1992 (Figure 1.3). The modal posterior density
estimates of parasitic abundance were relatively close to the observed data, although
there were some departures from the observations during 1992 and 1993 (Figure 1.4).

The degree to which the model ﬁt each of the data sets is largely a reﬂection of
the number of observations in each data set and their associated variance estimates. The
CPUE data set contained 31 observations and the average coefﬁcient of variation (CV)
was 45% (Appendix Table Al). The spawner data set contained 16 observations and
average CV. was 0.6%. The parasitic data set only contained 5 observations with an
average CV. of 1.3%. Because the average CV. for the spawner data set is so low and

the number of observations is fairly high, it is not surprising that the modal estimates

23

were so close to the observed data. The objective function (equation 8) is penalized more
by parameter estimates that fail to ﬁt data sets with many observations or small
measurement error estimates. Because each data set may suggest a slightly different state
of the system, simultaneously obtaining a tight ﬁt to all of the data sets may be
impossible and suggests model error.

The modal estimates for age-composition also matched observed data (Figure 1.5,
Appendix Tables A2 and A3). I did not detect any consistent patterns in the larval age-
composition residuals over time or across ages. Because larval lampreys are difﬁcult to
age, I believe that the information contained in the age-composition data sets is rather
limited. The modal estimates of the metamorphosing larvae age-composition data were
generally poor. With only two years of data, this outcome is not surprising. However,
the metamorphosing larval age-composition data did allow for the estimation of the
parameters describing the probability of metamorphosis: 0.46 for age-4 larvae, 0.57 for
age-5 larvae, and the assumed value of 1 for age-6 larvae.

The modal estimate for the larval natural mortality rate (M) was 0.82. The
histogram in Figure 1.6 depicts the marginal posterior distribution of this parameter, with
an approximate 95% posterior probability interval of (0.73, 0.92). The modal posterior
estimate for the proportion of the parasitic population migrating to the St. Marys River
(A) was 0.045, and the histogram in Figure 1.7 depicts the marginal posterior distribution
of this parameter.

As mentioned earlier, 3. represents a combination of the survival from
metamorphosis to spawning and the fraction that migrates to the St. Marys River.

Because I was unable to estimate the survival from metamorphosis to the parasitic stage,

24

I have implicitly assumed that it was 100% and therefore my estimates for A are likely
biased low. However, survival during this period may be low, especially when small
hosts are unavailable (Young et al. 1996).

My modal estimates of historical stock sizes and the number of age-0 recruits that
were produced suggest that variation in recruitment is substantial across stock sizes

(Figure 1.8). The least squares ﬁt of the Ricker stock-recruit function to these point

estimates resulted in estimates of c“: = 9.15, ,3: 0.00018, and &2= 1.16. The marginal

11-7??‘1

posterior distributions for these parameters can be seen in Figure 1.9. There is
considerable uncertainty in each of these parameters. Marginal posterior density values
for a were generally between 7 and 10. The marginal posterior density for a" has most
of its mass between 0 and 10, suggesting that there is considerable variation in
recruitment.

A histogram of compensation ratios suggests that high compensation may be
possible (Figure 1.10). Overall, 24.5% of the ratios showed negligible compensation,
15.1% showed low compensation, 27.6% Showed moderate compensation, and 32.8%

showed high compensation.

Discussion

Understanding ﬁsh population dynamics is a difﬁcult task. Often the data
available for ﬁtting models do not come from rigorously designed population surveys and
is of questionable quality. Data sets collected ﬁ'om the same population often seem to

support different hypotheses on the state of the system or the parameters governing the

25

population’s dynamics. However, management decisions have to be made, and will be
made, regardless of the quality of the data or the sophistication of the analyses conducted.

In this paper I have attempted to gather all relevant data sources that could help
me to make inferences about St. Marys River sea lamprey population dynamics. None of
the data sets were collected with the idea that they would be used to estimate a stock-

recruit relationship. However, I combined these data sets into a single, statistically-

‘.1'

based, age-structured population model for the purpose of estimating the parameters of a

stock-recruit relationship and the associated uncertainty in these and other demographic

F.
‘1

parameters. Much of the model ﬁtting process involved balancing the information
suggested by each data set. I used estimates of the variances associated with each data

set (both empirical estimates and effective sample size estimates derived through
iteratively ﬁtting the model) to provide a basis for this balancing process. In so doing, I
was able to avoid attaching “emphasis factors” (Methot 1990), representing my degree of
belief in the individual data sets and their likelihood components. However, because
systematic errors (e. g., temporal trends in catchability) were not accounted for, the
estimated variances likely represent lower bounds for the true uncertainties in these data
sets.

The model that I developed attributed the increase in parasitic-phase abundance
over time to increased age-0 recruitment to the larval population. An alternative and
equally plausible mechanism, which I did not incorporate into my model, is that larval- or
parasitic-phase survival rates increased. Young et al. (1996) suggested that increased
survival for newly metamorphosed parasites was a better explanation for the increase in

parasitic-phase abundance than changes in the quality or quantity of larval habitat in the

26

St. Marys River. To examine this increased survival hypothesis, I compared my modal
estimates of the number of parasites (P) with the estimates of parasites based on the

least-squares ﬁt of a Ricker stock-recruit function to the modal estimates (PSR ). Figure

1.11 shows the residuals (13 minus 135,) plotted over time. This graph suggests that there

were a lower number of parasites in the 1970’s and early 1980’s than expected based on
the least-squares stock-recruit relationship parameters. The graph also suggests that the
number of parasites in 1992 and 1999 was much higher than expected based on the stock-
recruit parameters. This pattern in the residuals indicates that survival rates may have
changed over time (Peterman et a1. 1998). Unfortunately, I was unable to estimate tirne-
varying survival rates due to a lack of the appropriate data. However, my analyses
support the conclusions of Young et al. (1996) that newly metamorphosed parasitic
survival rates may have increased over time. Potential reasons for this change may be
increased prey ﬁsh abundance or biomass. Further investigations into the factors that
determine lamprey survival rates would be beneﬁcial and could improve the accuracy of
my model.

Fisheries management agencies have long recognized uncertainty, but until
recently it has generally been ignored or treated qualitatively in the decision-making .
process. The GLFC, which oversees the sea lamprey management program in the Great
Lakes, has accepted that uncertainty needs to be more formally incorporated into the
decision-making process, especially when decisions have to be made at considerable
public expense, as is the case with the St. Marys River. The decision to treat the St.
Marys River with Bayluscide in 1998 and 1999 had an associated cost of over $5 million

dollars, a substantial fraction of the total operating budget for the GLFC. Estimating a

27

single stock-recruit relationship for St. Marys River sea lamprey certainly would have
improved the understanding of this system’s population dynamics. However, it would
have overstated the certainty about the processes and demographic parameters that
govern the system dynamics. Therefore the objective of this study was to characterize
the uncertainties in the stock-recruit relationship and demographic parameters.
Examining the uncertainty in the stock-recruit function parameters yielded several
important ﬁndings. First, it is apparent that there is considerable variability in
recruitment over a range of stock sizes. The stock-recruit data set presented in Figure 1.7
represented a data set with comparatively low variability (0"2 = 1.16), but much higher
values (5 < 6'2 < 15) might be possible (Figure 1.9). Second, the amount of
compensation operating in this population may be substantial. Although the majority of
the ratios were classiﬁed as negligible, low, or moderate, the potential for high
compensation remains. If this population has low amounts of compensation, reducing the
number of spawning individuals should result in a nearly proportional reduction in the
number of recruits. But if high compensation is operating in this population then
management options that reduce the number of spawners will not result in proportional
reductions in recruitment. However, the probability of high compensation levels
suggested in this study may be overly pessimistic. Jones et al. (in review) found only
slight compensation in a meta-analysis of sea lamprey stock-recruit data from Great
Lakes tributary streams. The scale of the St. Marys River system is much larger than
those systems examined by Jones et al. (in review). If scaling issues are unimportant in
determining population compensation levels, then compensation levels should not

signiﬁcantly differ.

28

Extreme recruitment variation does not bode well for control options that aim to
reduce the number of effective spawners (e. g., trapping and releasing sterilized males).
Even if stock sizes are reduced through these control techniques, there is a high
probability that a large year-class could be produced. On the other hand, the evidence
against high compensation suggests that, on average, reducing the number of spawners
will result in a lower number of recruits. My results imply that trapping and releasing
sterilized males can be effective treatment options resulting in lower production on
average, but that strong year-classes may still result on occasion.

When the effectiveness of control options that reduce the number of effective
spawners is highly uncertain, control options that target the larval population may be
preferred. In smaller streams and rivers, TFM is still the main method for controlling sea
lamprey populations and its success has clearly been demonstrated. The uncertainty in its
effectiveness is relatively low. However, conditions in the St. Marys River make a T F M
treatment impractical, expensive, and ineffective (Shen et al., in review). Because
Bayluscide can be applied to localized areas with high larval densities, it represents an
opportunity to suppress the larval population in large systems with patchy larval
distributions like the St. Marys River. High recruitment variability at low stock sizes
may make the Bayluscide treatment option preferable in the St. Marys River compared to
trapping and sterilized male releases.

The St. Marys River will continue to pose a major challenge for the integrated
management of sea lamprey in Lake Huron. Future management decisions will be
required as to which treatment options to use and how much control is necessary to

achieve management goals. In this paper I have attempted to characterize the uncertainty

29

in St. Marys River sea lamprey population dynamics, speciﬁcally the stock-recruit
relationship. This information can serve as a valuable input to models that forecast and
evaluate the expected results of different treatment combinations, given the uncertainty
present in the system. When these models are incorporated into a formal decision
analysis, they can provide a valuable tool for decision-makers and managers alike and

should result in improved management of the system.

Acknowledgements

I would like to thank D.W. Cuddy and RB. McDonald for providing data on the
Bayluscide surveys, spawning-phase population estimates, trapping and sterile male
effectiveness, and the catches of parasites in the commercial ﬁsheries. L. Mohr provided
data on effort in the Lake Huron commercial ﬁshery. M. Fodale provided the data on the
St. Marys River deepwater electroﬁshing catches. R.A. Bergstedt provided the parasitic-

phase mark-recapture estimates.

30

References

Bergstedt, RA. and J .H. Genovese. 1994. New technique for sampling sea lamprey
larvae in deepwater habitats. North American Journal of Fisheries Management
14:449-452.

Bergstedt, RA. and eleven coauthors. 1998. St. Marys River Assessment Plan. Great
Lakes Fishery Commission, Ann Arbor, MI. 37 pp.

Bergstedt, R.A., R.B. McDonald, K.M. Mullett, G.M. Wright, W.D. Swink, and KP.
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31

 

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32

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33

Table A1 Data used in ﬁtting the sea lamprey population model

 

SE. of Variance Variance
Year Ln(CPUE) Ln(CPUE) Ln(S) of Ln(S) Ln(P) of Ln(P)
1967 -1.30 0.64
1968 -l.39 0.55
1969 -1.36 0.55
1970 -2.46 0.60
1971 -2.50 0.64
1972 -3.57 0.60
1973 I“
1974 -2.76 0.53 '
1975 -2.63 0.51 r
1976 -2.28 0.51 g. i
1977 -2.62 0.51
1978
1979 -2. 18 0.54
1980 -l .39 0.54
1981 -1 .97 0.64
1982 -2.53 0.54 12.43 0.0177
1983 -1.54 0.50
1984 -0.66 0.49
1985 -1.17 0.50 10.08 0.0007
1986 -1.15 0.50 9.73 0.0015
1987 -0.99 0.50 9.94 0.0013
1988 -1.02 0.50 9.96 0.0014
1989 -0.36 0.50 10.20 0.0013
1990 -0.84 0.50 10.05 0.0021
1991 -0.94 0.50 10.48 0.0020 13.36 0.0293
1992 -0.04 0.51 9.88 0.0020 13 .46 0.0778
1993 -0.62 0.50 10.73 0.0153 13.19 0.0206
1994 -l.l9 0.51 9.27 0.0043 13.15 0.0112
1995 -1 . 10 0.52 9.88 0.0032
1996 -0.72 0.51 10.01 0.0144
1997 -1.48 0.52 9.01 0.0147
1998 -0.66 0.52 9.92 0.0046
1999 0.00 . 9.90 0.0021
2000 10.57 0.0027 12.55 0.0174

Table A2 Estimated age composition for St. Marys River sea lampreys.

 

Year Age-2 Age-3 Age-4 Age-5 n
1971 0.34 0.47 0.18 0.01 306
1972 0.40 0.38 0.20 0.02 1 147
1978 0.92 0.08 < 0.01 < 0.01 62
1980 0.56 0.39 0.05 < 0.01 298
1981 0.76 0.22 0.02 < 0.01 2033
1982 0.30 0.64 0.06 < 0.01 44
1984 0.71 0.25 0.04 < 0.01 95
1985 0.58 0.28 0.13 0.01 779
1986 0.86 0.10 0.04 < 0.01 442
1987 0.42 0.50 0.08 < 0.01 660
1988 0.91 0.07 0.01 0.01 257
1989 0.81 0.11 0.08 < 0.01 32
1993 0.44 0.19 0.29 0.07 421
1994 0.43 0.28 0.24 0.05 898
1995 0.46 0.30 0.24 < 0.01 933
1996 0.65 0.26 0.09 < 0.01 1175
1997 0.61 0.34 < 0.01 0.06 407
1998 0.52 0.27 0.17 0.04 809

Table A3 Estimated transformer age composition.

 

Year Age-4 Age-5 Age-6 n
1995 0.53 0.38 0.09 34
1996 0.3 0.44 0.26 43

35

 

 

 

 

 

 

 

g; _
a
8
c; R H R’
G) ..
or
< R"
M /
R'L ///‘
R0 I
8* 3’

Female spawners

Figure 1.1 Stock-recruit relationships depicting high (Rm), moderate
(R*M), and low (R*,) values for the compensation ratio (R*/R0) at a low
stock size (8*). Recruitment at low spawning stock size with zero
compensation is denoted by R0 and recruitment at high spawning stock
size (S’) is denoted by R’.

36

+ Obsened
1 ‘ + Estimated

 

Parasites/Km of Gillnet

 

Figure 1.2 MPD (modal posterior density) model estimates and
observed commercial ﬁshery catch-per-unit-effort (number of
lamprey per km of gillnet) estimates, 1967-1999.

37

 

"Tl

. l .
A

50°00 ' + Observed
40000 _ + Estimated

Spawners

 

 

 

1985 1990 1995 2000
Year

Figure 1.3 MPD (modal posterior density) model estimates and
observed spawning-phase population estimates based on mark-
recapture, 1985-2000.

38

1200000 1

 

 

+ Observed
0)
E + Estimated
2- 800000 +
g
#3
8 4000007 I
3 o
_, a
0 ﬂ I I F l
1980 1985 1990 1995 2000 2005

Year

Figure 1.4 MPD (modal posterior density) model estimates and
observed parasitic-phase mark-recapture population estimates for
juvenile lampreys, 1982, 1991-1994, and 2000.

39

Residual (Obs. - Fred.)

 

 

 

 

 

 

I o Age-2 I Age-3 A Age-4 x Age-5 I
0.4 -
° 0
0.3 - . I
I
0.1 - ' ' I A A I I
I O 8 . x
0 9 x ' A x g
“a 2 “5* g"§XXI x"
-0.1 - ° ' o . o A A
0 ° A
I O I
-o.3 -
-O.4 I I '
1970 1 980 1 990 2000
Year

Figure 1.5 Residuals (observed minus predicted value) of
proportions-at-age for larval lamprey ages 2-5, 1971-1998.

40

Frequency

800 1000
I

400 600
I I

200
1

 

0

 

I
1.2

I I I
0.6 O. 8 1 .0

Natural mortality rate (M)

Figure 1.6 Frequency histogram of 12,000 MCMC samples
of the larval natural mortality rate (M).

41

Frequency

800
I

600
I

400
I

200

 

0

 

I I

l I
0.030 0.035 0.040 0.045 0.050

Proportion migrating to St. Marys (A)

Figure 1.7 Frequency histogram of 12,000 MCMC samples of
the proportion of the parasitic-phase population that migrates
to the St. Marys River to spawn (A).

42

 

Age-0 larvae (in millions)
0
O
O
0
0
6

0 4000 8000 12000
Female spawners
Figure 1.8 MPD model estimates of the number of female spawners
and age-0 recruits for the St. Marys River sea lamprey population,

1968-1996. The least-squares ﬁt of a Ricker stock-recruitment
function is also plotted.

43'

00

Frequency
4 0

 

(A)
w
6 7 8 9 10 11

 

 

a
o
a
6‘
r:
3
E g I
”- Ln
-0. 0004 -0. 0002 0.0 0.0002 0. 0004 0.0006
8 (C)
6’ 3
5
E
”t ‘ IIIIIII
o I. II.---—____.
0 5 1 15 20

Figure 1.9 Marginal frequency histogram of 12,000 MCMC '
samples of the or (panel A), [3 (panel B), and o 2 (panel C)
parameters from a Ricker stock-recruitment function.

Frequency
1000 1500 2000 2500

500
r

 

 

0

 

I
I 1 l I

0 5 10 15 20 25

Compensation Ratio

Figure 1.10 Frequency histogram of 12,000 estimates of the
compensation ratio, R*/R0, based on the MCMC samples from the joint
distribution of or and B.

45

-1.

‘2‘

Residual (Obs. - Prod.)
O

 

'3 I I v r r r
1966 1971 1976 1981 1986 1991 1996

 

Year

Figure 1.11 Residuals ( PM» minus f’sx ) of the MPD estimates
of the number of parasites ( 13ml) ) minus the estimated number of
parasites based on the least-squares stock-recruit relationship
parameters ( 13512 ) plotted over time

46

CHAPTER 2
QUANT IF YING SPATIAL AND TEMPORAL UNCERTAINTY IN LARVAL
LAWREY ABUNDAN CE AND ITS EFFECT ON CHEMICAL TREATMENT
SUCCESS IN THE ST. MARYS RIVER
Abstract
Effective chemical treatment of sea lamprey (Petromyzon marinus) larvae in the

St. Marys River requires reliable knowledge of their spatial distribution. However,
surveys to adequately characterize the distribution cannot be conducted in the same year
as the chemical treatment, thus raising questions about the accuracy of using historical
survey data to delineate treatment areas. In this project I developed stochastic models
that predict the abundance of larvae at a location in the next year based on data collected
in the current year. Recursive application of these models can allow for stochastic
forecasts at locations in Space and time, Simulating how population abundance may have
changed from the time each location was originally mapped (1993-1996) until the time
that the main treatment took place (1999). My approach to modeling was to develop two
generalized linear models (GLMs). The ﬁrst GLM model estimates the probability that
larvae would be present at a location. The second GLM model estimates parameters
deﬁning the probability distribution for larval abundance at that point, given that larvae
were present. I applied these models to forecast a variety of possible abundance maps for
the lampricide treatment that took place in 1999 to estimate the uncertainty in treatment
effectiveness resulting from spatial and temporal uncertainty in larval distribution. The
results suggest that expected treatment effectiveness did not differ from previous
predictions of treatment effectiveness at the level of treatment implemented in 1999.

However, if less area of the river had been targeted for treatment, then expected treatment

47

effectiveness would have been lower than predicted treatment effectiveness. The large
scale of the 1999 treatment may have compensated for the spatial and temporal
uncertainty by targeting the majority of the high density areas in the river, leaving few

areas with larval lamprey untreated.

48

Introduction

A combination of adult trapping, sterilized male releases, and chemical
applications is used to control the St. Marys River sea lamprey (Petromyzon marinas)
population. However, the large size and ﬂow patterns of the St. Marys River inhibit the
effective use of the lampricide 3-triﬂouromethyl-4-nitrophenol (TFM), the predominant
method for controlling sea lampreys in the Great Lakes region. In other rivers and
streams, TFM is applied at a rate that saturates the water column with a lethal dosage to
kill the majority of larvae in the riverbed. With typical TF M treatments, the spatial
distribution of larvae in the river is unimportant because it is assumed that virtually all
larvae in the stream are exposed to a lethal dose of the lampricide.

Although a substantial population exists in limited areas of the river, most of the
St. Marys riverbed area contains few larval lampreys. So TF M is not cost-effective
(Schleen et al., in review). To control the larval population in a cost-effective manner,
managers have chosen to use granular Bayluscide, an alternative lampricide that can be
applied locally to discrete patches of larval habitat rather than saturating the entire river
with TFM. With Bayluscide, managers can achieve effective population control by
targeting local areas of high larval density. However, cost-effective control of sea
lamprey larvae in the St. Marys River using Bayluscide requires a reliable knowledge of
their spatial distribution.

During 1993-1996, nearly 12,000 locations throughout the St. Marys River were
sampled using deepwater electroﬁshing to determine the distribution and abundance of
larval sea lamprey (F odale et al., in review). Based on maps generated using these data,

researchers and managers delineated areas with the highest larval densities as candidates

49

for possible treatment with Bayluscide. The larval population within each treatment area
was estimated and areas were ranked according to the number that would be killed per
dollar spent on Bayluscide. Combining these estimates resulted in a graph depicting the
proportion of the total population that would be removed as a function of the amount of
Bayluscide applied. Given this information, decision-makers chose to spend

approximately $5 million on an aggressive treatment of the St. Marys River lamprey

'3:
population that included a combination of applying Bayluscide, trapping, and releasing Ii
sterilized males into the spawning population during 1998-1999. While a small portion I

of the river was treated during pilot studies in 1998, the majority of the treatment effort
took place during 1999. Biologists predicted that 55% of the larval population would be
killed using the level of Bayluscide that was selected.

The actual distribution of larvae in the river at the time of Bayluscide treatment
affects the performance of these lampricide applications. The data used to describe larval
distribution were collected during 1993-1996. The treatment took place 2-3 years later.
This raises questions about the degree to which larval distributions are spatially and
temporally stable. If the distribution changes substantially from year to year, treatment
areas delineated from data collected in one year may substantially deviate from the
optimal treatment areas based on data collected in other years. Managers recognized that
this could be an important uncertainty, and set up a series of index stations in the St.
Marys that were sampled annually 1994-1998 to assess the degree of spatial and temporal
stability.

In this study I utilized these index site data to parameterize and develop stochastic

models that predict the abundance of larvae at a location in the next year based on data

50

collected in the current year. Recursive application of these models allows for simulating
how population abundance changed from the time each location was originally mapped
(1993-1996) until the time of the main treatment (1999). My approach to modeling was
to develop two generalized linear models (GLMs). The ﬁrst model estimates the
probability that larvae would be present at a location in the next year. The second model
estimates parameters that deﬁne the probability distribution for larval abundance at that
point in the next year, given that larvae were present. This general approach to modeling
population change over time and space has been applied quite successfully in other
contexts (Augustin et al. 1998a, Augustin et al. 1998b, Welsh et a1. 1996, Augustin et a1.
1996, Lindenmayer et a1. 1991, Osborne and Tigar 1992).

I used the index site data to parameterize the GLMs and then recursively apply the
resulting models to the areas delineated for the 1999 lampricide treatment. I applied
these models to forecast a variety of possible abundance maps for the lampricide
treatment that took place in 1999 in order to estimate the uncertainty in treatment
effectiveness resulting from the spatial and temporal uncertainty in larval distribution.
Understanding how the spatial and temporal uncertainties in larval distribution affect
lampricide effectiveness is essential for assessing the trade-offs among the management

options available for controlling St. Marys River sea lampreys.

Methods
My overall approach was to develop a stochastic simulation model that could
predict population abundance at each point given the population abundance at that point

and other information observed in the previous time step. I then applied that simulation

51

model to evaluate how well the procedure used to decide on areas for Bayluscide
treatment was likely to have performed. This was done by repeatedly simulating the
spatial patterns of larval catches in 1999 that might have arisen ﬁom those points mapped
during 1993-1996. As a preliminary step, prior to developing the stochastic simulation
model, I explored the spatial and temporal variability in the data. In part this guided my
choice of variables to include in forecasting the next year’s map of larval catches.

The data on larval distribution and catch within the St. Marys River were
collected using the boat-mounted deepwater electroﬁshing gear developed by Bergstedt
and Genovese (1994). The gear consists of a covered, electriﬁed grid connected to an
onboard suction pump. The covered grid is lowered onto the river substrate and an
electric current is initiated using a DC backpack electroﬁshing unit (Weisser and Klar
1990). The electrical ﬁeld causes larval lamprey to emerge from the substrate whereupon
they are pumped to the surface and collected in a basket, measured for length, and
enumerated. Because efﬁciency of the deepwater electroﬁshing gear varies as a function
of length, each larva was assigned an efﬁciency-adjusted catch (adjusted catch) equal to
the inverse of the capture probability calculated at its length (e.g., a larva with a capture
probability of 0.5 would be assigned an efﬁciency-adjusted catch of 2) (Bergstedt and
Genovese 1994, F odale et al., in review). At each sampled location, a differentially
corrected global positioning system was used to determine position coordinates and a
Ponar dredge was used to categorize habitat quality as preferred, acceptable, or
unacceptable. An area of approximately 2.44 m2 was sampled at each location.

During 1993-1996, biologists under the direction of the St. Marys River Task

Force (Task Force) sampled nearly 12,000 locations spanning the St. Marys River using

52

the deepwater electroﬁshing gear (F odale et al., in review). The biologists used a
combination of systematic and adaptive sampling. Systematic sample locations were 65-
140 m apart, while the adaptive samples were 35 m from systematic sample locations
where more than four animals were caught. The majority of the sampled locations had
catches of zero (94.4%).

In addition to the full river survey, the Task Force also established a set of thirteen

index sites that were sampled repeatedly during 1994-1998 (Fodale et al., in review),

manque-s my.“ I
1 AM

prior to the Bayluscide treatment. Each site consisted of a grid of 22 m x 32 In cells, with
between 48 and 244 cells per index site. Biologists sampled each cell in each index site
annually. Sites 1-4 were sampled all ﬁve years, while sites 5-9 were only sampled 1995-
1998 and sites 10-13 were only sampled 1996-1997. The index sites were established in
areas with medium to high larval densities to monitor trends in abundance and treatment
effectiveness. Figure 2.1 shows the adjusted catch (mean +/- 1 S.E.) across years for
index sites 1-9.

I used geostatistical methods to characterize the spatial and temporal variability in
the index site data. Based on initial data summaries, I observed that the variance in
adjusted catch increased with the mean and the data had a high proportion of zero values
(~7 5%) (Figure 2.2). Therefore I performed a log¢(adjusted catch + 1) transformation
(hereaﬁer referred to as transformed catch) to help stabilize the variance and better
approximate normality. To characterize the spatial correlation, I calculated an empirical,
omnidirectional semivariogram combining all ﬁve years of index site adjusted catch data.
Because preliminary analyses did not reveal anisotropy, omnidirectional semivariograms

were used (Isaaks and Srivastava 1989). To characterize the spatial and temporal

53

correlation, I calculated an empirical, omnidirectional cross-semivariogram, using a one-
year lag as the crossing variable. I used this cross-semivariogram to describe the
correlation in transformed catch between points as a function of distance at one-year lags.
I estimated the range (i.e., the distance up to which the data are spatially correlated) for
both the semivariogram and the cross-semivariogram to aid in developing the GLMs.
Using the index site data, I modeled the pointwise inter-annual changes in
adjusted catch of larval lamprey in two stages. The ﬁrst stage was a logistic GLM
whereby I estimated the probability of larvae being present at each point i as a function of

the previous year’s adjusted catch (catch,) at that point, a weighted average of the
adjusted catches at neighboring points (autocov) (Augustin et al. 1996), and the habitat

type (habtype, ). The model took the form

 

10g[1 p " J: ﬂo + ﬂlcatch, + ﬂzautocovi + ﬂ3habtypq (1)

i
where the ﬂ‘s are estimated parameters and
"i1

_ w,j catch j

autocov - J l
i - "ii (2)

is a weighted average of the adjusted catch values (catch j) among the set of "z;
neighbors within 120 m of point i. The weight applied to catch j is wg. = l/hy. , where ha.

is the Euclidean distance between points i and j. This autocovariate is intended to help
account for the ﬁne scale (< 120 m) spatial autocorrelation between points (Augustin et

al. 1996). However, it is only one of many methods available for incorporating this

54

— Venn tu'1':.- T u

information. The inﬂuence of habitat quality was incorporated through the habtype,
term where

0, if preferred

habtypei z { (3)

1, if acceptable

By deﬁnition, if a location is classiﬁed as having unsuitable habitat, then no lamprey are
present at that location.

The second stage of the model consisted of a second GLM that ﬁt a gamma
distribution to the adjusted catch data at each point i, conditional on the presence of

lamprey at point i. The model took the form

1
— = do + alcatChi (4)

#z'

where the 0's are estimated parameters, catch. is the previous year’s adjusted
catch at point i, and ‘1' is the link function for the mean (,u,) of the gamma distribution
describing the adjusted catch data, ﬁt at point i (McCullagh and Nelder 1989). The
adjusted catch data showed a high proportion of zero values and a continuous, right-
skewed distribution of positive catch values that resembled a gamma distribution or
perhaps a Poisson distribution (Figure 2). However, residual plots and histogram plots
revealed that a gamma distribution provided a better approximation to the distribution of
the positive adjusted catch data than did the Poisson distribution. The gamma
distributions were ﬁt with an assumed constant variance (shape). Because parameter
estimates for the weighted average of the adjusted catches at neighboring points

(autocov,) and habitat quality (habtypei) terms were not signiﬁcantly different from zero

55

(x2 test, p-value > 0.05), these terms were leﬁ out of the second stage model. A total of
1814 observations were available for ﬁtting the ﬁrst stage model and 474 observations
were available for ﬁtting the second stage model. I limited the data to observations
where the recorded habitat quality was the same between years at each point to avoid
incorporating habitat measurement error in the estimation process.

Using the index site data set with its known transitions, I examined the ability of
the ﬁrst stage model to correctly predict inter-annual transitions through calculation of a
matching coefﬁcient (Buckland and Elston 1993). I applied the model to each point in
the index site data set and predicted whether lamprey would be present at each point the
following year. I generated a total of 1000 stochastic realizations of the model and
calculated the mean matching coefﬁcient resulting from these simulations. As a
comparison, I also calculated the mean matching coefﬁcient for 1000 simulations where
the predicted transitions were bootstrap samples (with replacement) from a population
with the same proportion of presences and absences as the index site data set.

As mentioned earlier, the overall river sampling occurred 1993-1996 and the
treatment took place in 1998-1999. To generate a larval distribution map for 1999 I had
to recursively apply the models, starting with the most historic data. The data from each
stochastic forecast became the data set used for generating the next year’s forecast until a
map was generated for 1999 (Buckland and Elston 1993). The algorithm that I used is
described in Table 2.1. I generated a total of 100 distribution maps. Each simulated map
consisted of adjusted catches during 1999 at each of the 2042 points within the treatment
sites areas. At this point in the analysis, I switched from a spatially explicit

characterization of the population (i.e., modeling the point data) to a spatially implicit

56

characterization of the population (i.e., modeling the total abundance in each treatment
area). For each map I calculated the mean density (number of larvae/m2 ) within each of
the 46 ranked treatment areas. By multiplying the mean density of each treatment plot by
the plot area, I estimated the number of larval lampreys within each treatment plot. I then
examined the effects of selecting the ranked treatment areas (based on the original
ranking analysis performed by the Task Force) by calculating the total population that
would be targeted for each map and for each set of treatment areas.

The data used to calibrate the models came ﬁom index sites with relatively high
larval densities, similar to the treatment areas to which the models were applied. Because
the remaining portions of the river generally had much lower densities, I believed that it
would be inappropriate to apply the models to these low-density areas. Preliminary
applications of the models to these areas resulted in forecasted larval densities an order of
magnitude higher than observed densities. However, I was interested in incorporating
variability in the number of larvae outside the treatment areas in our calculations of the
percent of the total population targeted by the treatment. To do this I estimated the
coefﬁcient of variation (CV) for the mean density in the index sites between years. I
assumed that points outside the treatment area would have the same CV for mean density.

Using the empirical estimate for the mean density outside the treatment area (ﬂmm) and

the estimate of the total area outside the treatment areas, I estimated the number of larval

lampreys outside the treatment areas. I allowed this number to vary between simulations

by generating a random normal variable for the mean density (,uzmwe) for each
simulation with mean = from“, and standard deviation = CV - from“, . To calculate

treatment effectiveness (i.e., the proportion of the total population that would be removed

57

for a given amount spent on Bayluscide), I divided the cumulative population in the
ranked treatment areas by the sum of the populations within all treatment areas and the
population outside the treatment areas. All estimates reﬂect an empirical estimate for
Bayluscide efﬁciency of 75% (i.e., 75% of the larvae within an area are killed when
Bayluscide is applied to that area).

The variability in treatment effectiveness has two components: variation in the
number of larvae predicted in the treatment areas due to the stochastic nature of the
models and variation in the number of larvae predicted in the non-treated areas. To
examine the degree that these two sources affected the overall treatment effectiveness
estimates, I also calculated treatment effectiveness holding the number of larvae in the

non-treated areas constant across simulations.

Results

Although the annual means for the adjusted catch data in index sites 1-9 varied
among years, a consistent trend was not evident (Figure 2.1). A linear regression of
adjusted catch versus year did reveal a slightly negative slope, but the slope parameter
estimate was not signiﬁcantly different ﬁ'om zero (p-value = 0.63). This result suggests
that a consistent trend in abundance was not present in the index site data during 1994-
1998 and thus that parameters estimated in the model should not impose a negative trend
in abundance over time.

Calculation of the empirical semivariograms elucidated some of the spatial
autocorrelation patterns present within the data (Figures 2.3 and 2.4). Spherical models

ﬁt to the data generally had the lowest error sum of squares compared to exponential,

58

linear, or Gaussian models (Isaaks and Shirvistrana 1989). The estimate of the range for
the semivariogram (within years) was 124 m, with a sill of 0.19 and a nugget of 0.31.
The estimate of the range for the cross-semivariogram (l-year lag) was 129 m, with a sill
of 0.21 and a nugget of 0.39. The variation among points reached an asymptote of 0.50
for the serrrivariograrn and 0.60 for the cross-semivariogram. Because spatial
autocorrelation was present in the cross-semivariogram up to a distance of 129 m, we
believe that our use of a weighted average for catches within 120 m was appropriate in
our logistic model ﬁtting.

The results of the two GLMs are presented in Tables 2.2 and 2.3. For the logistic
GLM, all three variables (e.g., catch, autocov, and habtype) reduced the residual deviance
and were signiﬁcantly different from zero (38 test, p < 0.01). The highest probability of
presence at a point in the next year came when the previous year’s catch was high, the
distance-weighted average catch was high, and the habitat was classiﬁed as preferred.
For the gamma GLM, only the previous year’s catch variable was signiﬁcantly different
from zero ()52 test, p < 0.01). The estimated mean of the gamma distribution increased as
a function of the previous year’s catch.

Using the parameter estimates from the two-stage GLM and the data from index
site 6 (the largest index site), I simulated a series of abundance maps to illustrate the
observed variability in larval distribution and to demonstrate an execution of our model
(Figure 2.5). Each simulated abundance map is based on the previous year’s observed
data (i.e., the simulated map for 1996 is based on the observed data from 1995, the map
for 1997 is based on 1996 observed data, and the map for 1998 is based on 1997 observed

data) and represents a possible realization of the distribution of larvae in site 6 during the

59

speciﬁed year. Within both the observed and simulated maps, there were considerable
differences between years at individual points.

Based on the results of 1000 simulations applying the logistic model to the index
site data, the mean matching coefﬁcient was 65.1% (standard deviation = 0.95%). The
mean matching coefﬁcient for the bootstrap procedure was 60.7% (standard deviation =
1.1%). Using a t-test, I conﬁrmed that the matching coefﬁcient ﬁom the logistic model
was signiﬁcantly higher than the matching coefﬁcient from the bootstrap procedure (p <
0.01). This result does not constitute a formal test or calibration of the model because the
same data that were used to estimate model parameters were used to evaluate its
performance using the matching coefﬁcient. However, I believe that the matching
coefﬁcient does provide a relative measure of model ﬁt. Based on the matching
coefﬁcient results, the logistic model did improve predictions of the inter-annual
transitions between presences and absences.

By sequentially applying the two-stage GLM to the treatment site areas, I was
able to simulate the pointwise adjusted catch of larval lamprey in 1999, the year of the
main treatment of the St. Marys River. I generated a total of 100 maps and calculated the
efﬁciency associated with various expenditures on Bayluscide. Plotting the cumulative
proportion killed versus cumulative cost for these 100 maps, I quantiﬁed the effects of
spatial and temporal uncertainty in larval distribution on treatment effectiveness (Figure
2.6). Also plotted is the expected percent killed based on the mapping data assuming that
the historical (1993-1996) data represent the actual values in 1999 (i.e., assuming no
uncertainty in larval distribution). Our simulation results suggest that the proportion

killed is generally lower than when uncertainty is ignored. This difference is most

60

pronounced when smaller portions of the river are selected for treatment. However, as
the amount of area treated increases, which is proportional to the amount spent on
chemical treatment, this difference becomes less apparent. When the largest amount of
area is treated, the mean proportion killed for the estimates that include uncertainty is
0.57 (standard deviation = 0.04) as compared to 0.55 predicted when uncertainty is
ignored.

The total simulated variability in treatment effectiveness is due to the variability
in abundance both within the treatment areas and within the non-treated areas. Figure 2.7
shows the simulated variability in treatment effectiveness when the total abundance in the
non-treated areas is held constant. When less of the river is selected for treatment, the
variability in treatment effectiveness is similar to that shown in Figure 2.6. However,
when the largest amount of area is selected for treatment, the variability in treatment

effectiveness is less (standard deviation = 0.01).

Discussion

Effective control of sea lamprey larvae in the St. Marys River using Bayluscide
requires a reliable knowledge of their spatial distribution. However, determining the
distribution and abundance of larval lampreys immediately prior to the application of
Bayluscide in a system as large as the St. Marys River would be prohibitively expensive.
Likewise, ﬁnancial constraints make treating the entire river with lampricide impossible.
Some sort of compromise between the limitations of assessment and treatment is

necessary.

61

 

Managers and scientists involved in the management and assessment of sea
lamprey have long recognized spatial and temporal uncertainty in larval abundance
within the St. Marys and other rivers. When delineating the treatment areas in the St.
Marys, the Task Force attempted to compensate for this uncertainty by delineating low-
density buffers around the core areas of high larval density. Final treatment areas
consisted of both the high-density core areas and the buffers. While this approach was
somewhat subjective, our analysis suggests that it may have performed well.

The scale of the treatment that took place in 1999 appears to have been large
enough to counteract the effects of the spatial and temporal uncertainty. Intuitively, this
should be the case, as increasing the area treated will result in a greater proportion of the
population being killed regardless of the degree of spatial and temporal uncertainty.
Conversely, as the scale of treatment decreases, the effects of these uncertainties become
more profound, resulting in reduced effectiveness. The larval p0pulation in individual
treatment areas varies from year to year, and the optimal treatment area ranking based on
data from one year may not be optimal when data from other years is considered. This
ranking effect is most apparent when fewer areas are selected for treatment. A preferable
treatment ranking approach might be to utilize these models to rank areas for treatment
based on the frequency with which they are ranked highly.

One of the advantages of using GLMs to model changes in the spatial and
temporal distribution and abundance is that data patterns can be described without having
to model the speciﬁc processes (e. g., mortality, density dependence, emigration,

' immigration, and recruitment) that generate the patterns. Spatially explicit population

models have been used to describe changes in animal distribution and abundance, but

62

they typically require a large number of parameters to be estimated (Dunning et a1.

1995). Many of these parameters are difﬁcult or nearly impossible to estimate
empirically (Conroy ct al. 1995). In contrast, GLMs take advantage of the correlations
between covariates and the independent variable to statistically reproduce the patterns in
the data. However, the variable selection process requires special attention to avoid the
inclusion of irrelevant variables (Buckland and Elston 1993) and applications of the
model to different systems may result in poor performance. Because of the limited
number of variables available for modeling lamprey in the St. Marys River, the likelihood
that irrelevant variables were included in our model is small.

One of the shortcomings of my analysis was an inability to correctly model the
pointwise inter-annual changes in the non-treated portions of the river. The data used to
estimate the parameters of the GLMs came from index sites that have relatively high
larval densities. Data summarizing inter-annual changes in areas of low larval density
were not available. An implicit assumption of the way I modeled these non-treated areas
of the river was that mean adjusted catch in these areas did not increase or decrease over
time. Mean adjusted catch these areas may vary temporally around a mean, as I have .
modeled, or may follow a trend over time (i.e., either increasing or decreasing) with some
noise. If the latter is true, then my results (Figure 2.6) should be considered minimum
estimates of the variability in treatment effectiveness. To address this issue, index sites
need to be established in low-density areas of the river and sampled annually. These data
would provide a starting point for better describing how low-density areas behave over

time and could easily be incorporated into the model described here.

63

Related to the problem of applying the model to non-treated areas are the issues of
future assessment and treatment decision needs. An ongoing monitoring program is in
place to monitor trends in overall larval abundance within the river. However, redoing
the detailed, extensive surveys that took place during 1993-1996 would be an expensive
and difﬁcult task. To map the distribution and abundance of larvae in the river and
provide decision-makers with appropriate estimates of treatment effectiveness, a version
of our model could be applied to surveys conducted within a limited portion of the river
or to simulated maps that incorporate the reductions due to the treatment that took place
in 1999. Another alternative would be to survey a limited portion of the river and use
Gibbs sampling to estimate the abundance at non-sampled locations (Weir and Pettitt
2000, Augustin et al. 1996). Despite having samples for only 20% of the total area,
Augustin et al. (1996) was able to effectively predict presence/absence at non-sampled
locations by using Gibbs sampling. Given the rich data available in the St. Marys, this

type of approach may be feasible.

Acknowledgements
We would like to thank Mike Fodale and Roger Bergstedt for providing data and
insight regarding the sampling and assessment methods that were used on the St. Marys

River. Linda Warner developed the spatial database for use in the analyses.

References

Augustin, N.H., M.A. Mugglestone, and ST. Buckland. 1996. An autologistic model for
the spatial distribution of wildlife. Journal of Applied Ecology 33:339-347.

Augustin, N.H., D.L. Borchers, E.D. Clarke, S.T. Buckland, and M. Walsh. 1998a.
Spatiotemporal modelling for the annual egg production method of stock assessment

using generalized additive models. Canadian Journal of Fisheries and Aquatic Sciences
55:2608-2621.

Augustin, N.H., M.A. Mugglestone, and ST. Buckland. 1998b. The role of simulation
in modeling spatially correlated data. Environmetrics 9:175-196.

Bergstedt, RA. and J .H. Genovese. 1994. New technique for sampling sea lamprey
larvae in deepwater habitats. North American Journal of Fisheries Management 14:449—
452.

Buckland, S.T., and D.A. Elston. 1993. Empirical models for the spatial distribution of
wildlife. Journal of Applied Ecology 30:478-495.

Buckland, S.T., D.A. Elston, and SJ. Beaney. 1996. Predicting distributional change,
with application to bird distributions in northeast Scotland. Global Ecology and
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Conroy, M.J., Y. Cohen, F.C. James, Y.G. Matsinos, and BA. Maurer. 1995. Parameter
estimation, reliability, and model improvement for spatially explicit models of animal
populations. Ecological Applications 5(1):]7-19.

Dunning, J .B., D.J. Stewart, B.J. Danielson, B.R. Noon, T.R. Root, R.H. Larnberson, E.E.
Stevens. 1995. Spatially explicit population models: current forms and future uses.
Ecological Applications 5(1):3-1 1.

Fodale, M.F., R.A. Bergstedt, D.W. Cuddy, and J .V. Adams. In review. Planning and
executing a lampricide treatment of the St. Marys River using georeferenced data.
Submitted to the Journal of Great Lakes Research.

Isaaks, E.H., and RM. Srivastava. 1989. Applied Geostatistics. Oxford University
Press, New York, New York.

Lindenmayer, D.B., R.B. Cunningham, M.T. Tanton, H.A. Nix, and AP. Smith. 1991.
The conservation of arboreal marsupials in the montane ash forests of the Central
Highlands of Victoria, south-east Australia: III. The habitat requirements of Leadbeater’s
possum Gymnobelideus leadbeateri and models of the diversity and abundance of
arboreal marsupials. Biological Conservation 56:295-315.

65

McCullagh, P., and J .A. Nelder. 1989. Generalized Linear Models. Chapman and Hall,
New York, New York.

Schleen, L.P., G.C. Christie, J .W. Heinrich, R.A. Bergstedt, R.J. Young, T.J. Morse, D.S.

Lavis, T.D. Bills, J .E. Johnson, and MP. Ebener. In review. Control of sea lampreys in
the St. Marys River; a case study in integrated and coordinated ﬁsheries management.
Submitted to the Journal of Great Lakes Research.

Osborne, RE, and B.J. Tigar. 1992. Interpreting bird atlas data using logistic models:
an example from Lesotho, Southern Aﬁica. Journal of Applied Ecology 29:55-62.

Weir, LS, and AN. Pettitt. 2000. Binary probability maps using a hidden conditional
autoregressive Gaussian process with an application to Finnish common toad data.
Applied Statistics 49(4):473-484.

Weisser, J .W. and GT. Klar. 1990. Electric ﬁshing for sea lampreys (Petromyzon

marinas ) in the Great Lakes region of North America. Pages 59-64 in LG. Cowx, editor.

Developments in electric ﬁshing. Fishing New Books, Oxford, UK.
Welsh, A.H., R.B. Cunningham, C.F. Donnelly, and DB. Lindenmayer. 1996.

Modelling the abundance of rare species: statistical models for counts with extra zeros.
Ecological Modelling 88:297-308.

66

I.

 

Table 2.1 Algorithm for generating a single abundance map.

1. Starting with the data collected in 1993 and the logistic GLM, estimate the
probability of larval presence at each point for 1994.
2. Conduct a Bernoulli trial at each point using the probability calculated in step 1.
- If the trial results in larval presence predicted at that point, estimate the
abundance at that point by generating a random sample from a gamma
distribution with the parameters given by the gamma GLM.
3. Append the location and habitat data from 1993 to the abundance map simulated
in step 2.
4. Append the data collected in 1994 to the data from step 3.
5. Repeat steps 1—4, sequentially adding the location and habitat data that was
collected in each year, to generate abundance maps for 1995, 1996, 1997, 1998,
and 1999.

67

alarm- my: n‘r_. ‘

Table 2.2 Logistic GLM results.

 

 

 

Residual
Parameter DF Estimate S.E. Pr > x2 Deviance Residual DF
ﬂo -1.5165 0.0842 2083.97 1813
151 1 0.0686 0.0164 < 0.0001 2001.71 1812
52 1 0.3130 0.0401 < 0.0001 1941.40 1811
’63 1 -0.7624 0.1687 < 0.0001 1918.55 1810
Table 2.3 Gamma GLM results.
Residual
Parameter DF Estimate S.E. Pr > x2 Deviance Residual DF
0’0 0.2019 0.0113 354.92 473
at 1 -0.0031 0.0010 0.0063 347.46 472

68

 

 

 

é:

 

 

 

215‘
2.0 _ c)
.C
O
H
(U
0
13 _-
.93
m 1.5 r
.2.
'O
<
LO 4
(15 r

1994

Figure 2.1 Annual means (+/- 1 S.E.) of adjusted larval catches from

index sites 19

1995

69

I

1996

Year

I

1997

1998

Percent Frequency

 

85‘

 

 

.- -—
l T

12 16 20 24
Adjusted Catch

Figure 2.2 Percent ﬁ'equency histogram of adjusted larval
catches from index sites 1-9.

70

 

 

 

 

 

 

 

10.5 o ° ° 0 0
o o o 0
Y-
O
m
a O"
E
E
m
D
N
0.-
g-I
O . .
o" ob;eet1ve=0.0189
l I l I ﬂ
0 50 100 150 200
distance

Figure 2.3 Empirical semivariogram and ﬁtted spherical semivariogram
function based on transformed larval catches from index sites 1-13.

 

 

 

 

 

 

to 0 9 o o
o“ o c o
'0.—
0
Y-
o
8 84
a
N
0.-
3-
o . .
o“ objeetrve=0.0033
j l 1 l
O 50 100 150 200

distance

Figure 2.4 Empirical cross-semivariogram and ﬁtted spherical
semivariogram function based on transformed larval catches ﬁ'om index
sites 1-13.

71

Observed

 

 

0 00.0
.0'. ‘.'

0 .' .
b" 0.0. " .o... 5.0
“ .¢.Q..:... 00 0 0"

,0

1995 «.m- .
. 0- v .. 0
1 0:‘ 0.. ..

 

 

Simulated

 

 

4'.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 0 0 . ..0 .
.';:=-.-..-. --..-- 3.22.". ..
° 0 00.. 00 C . ~ . O .0.. 00 O ' ‘
*' :3? ' .- :5: :
1996 . '9‘... ' ' 0 T..o.0......0 ' ° 0 0
.0 ‘ . .. O. z 00 G .. '
I f.: ..'..°. 00... . .: .00.‘ 000..
1 0 . . 1.0 ' .
;.0?0 I 0 .. :.0
1 . 0 .g' ‘lI—t. . . 0
' "' " . . . ’ ’ 0°.- .. .. 0
1 ’:.:. o. “ 0". 2. 1o. . 0.:0..o :?° .
.ﬁ. ......~..'.: 00. .u..‘. .. 0.0. 0
1997 4 2"...” o 0 . . 4 .‘~. .‘;.. 0 0 . .
0 .. f .. 0° .. . ‘ . .0 .0 .0
+ .0. O... .... .“2 . .
0... 10.0 . .0 .
’ 0°. 0 .. ..O
0... o 0 .0 ..... :- . . . .0 ’ . .
e:.o .-.::.:O .:::o...0 . . .: ‘0
0 0.0:..0' " . '2‘! .00
d :0 .Q. ‘ . .. " ... .0 ..
1998 ,1...’ .. ”1 . ”.3. .. ,
...0 . ... . 0. 0 0 O
9 .~ .0... 0 00 0 0 0
. . o ;;:.0 J o . ‘0; ..

 

 

 

 

 

 

Figure 2.5 Observed and simulated annual distribution and abundance of
larvae in index site 6. Dot diameters are proportional to abundance with the
largest dots representing adjusted catches of approximately 30 and the
smallest dots representing zeros.

72

my- ‘   ”ﬂ-

 

Proportion killed

 

 

 

 

j I j l I
o 2 3 4
Dollars spent on Bayluscide (in millions)

Figure 2.6 Results of one hundred simulations of the percent of larval
population targeted versus dollars spent on Bayluscide (dots). Also plotted is
the predicted percent targeted versus dollars spent assuming temporal
stationarity in larval distribution and abundance.

 

Proportion killed

 

 

 

 

I I I I I
0 1 2 3 4

Dollars spent on Bayluscide (in millions)

Figure 2.7 Results of one hundred simulations of the percent of larval
population targeted versus dollars spent on Bayluscide (dots), holding the total
abundance in non-treated areas constant. Also plotted is the predicted percent
targeted versus dollars spent assuming temporal stationarity in larval
distribution and abundance.

73

CHAPTER 3
AN APPLICATION OF DECISION ANALYSIS TO THE MANAGEMENT OF
ST. MARYS RIVER SEA LAMPREYS (PETROMYZON MARINUS)
Abstract
Controlling the St. Marys River sea lamprey population is one of the most
challenging and important problems facing ﬁshery managers in the Great Lakes. In
addition to the costs and logistical challenges associated with implementing control
options, considerable uncertainties exist in lamprey population dynamics and in control
option effectiveness. These uncertainties hinder the ability of scientists to forecast
management option performance and thus limit the ability of decision-makers to arrive at
well-informed decisions. In this paper, three methods for controlling the St. Marys River
sea lamprey population are considered: trapping, releasing sterilized males, and applying
Bayluscide. Performance of the management options is measured using forecasts of
future lamprey abundance, an economic net beneﬁt statistic, and risk tolerance measures.
I found that uncertainty in lamprey population dynamics and in treatment option
effectiveness can have large effects on the forecasts of lamprey abundance. In addition,
the relative ranking of a particular management option depends on the performance
indicator used to evaluate it. Important tradeoffs exist between achieving management
objectives and the cost associated with achieving the management objective. Therefore
decision-makers will have to evaluate option performance in light of several performance
indicators. Incorporating uncertainty into St. Marys River sea lamprey management
decisions through decision analysis should improve the quality of the decisions used to

manage the system and increase the probability of achieving management objectives.

74

Introduction

The ﬁeld of statistical decision theory merges the concepts of utility and
subjective probability to quantitatively deal with management problems in the presence
of uncertainty (Raiffa and Schlaifer 1961, Morgan and Henrion 1990, Clemen 1996).
The method of decision analysis is the application of statistical decision theory to a
management decision problem, generally with the following components: a deﬁned set
of management options available to the decision-maker, a listing of the alternative states
of nature and their associated probabilities which characterize the uncertainty in the
problem, a listing of possible outcomes resulting from the management options, and a
representation of the decision-maker’s utility for the outcomes. The relative merit of
each management option is determined by the decision-maker’s utility for the expected
outcomes.

Uncertainty is inherent in natural resource management decisions. Within the
ﬁeld of ﬁsheries, there is a grong trend towards incorporating uncertainty and risk in
the decision-making process (Hilbom et al. 1993). Rosenberg and Restrepo (1994) stress
the importance of estimating and communicating uncertainty to ﬁshery managers, who
must weigh the beneﬁts, costs, and risks of various management options. They note that
while many managed ﬁsheries have incorporated elements of uncertainty and risk
analysis, the advice resulting from these analyses needs to be expressed to decision-
makers in an effective manner. Providing ﬁshery managers with a quantitative
evaluation of the potential consequences of alternative management actions is one of the
primary roles of stock assessment scientists (National Research Council Committee on

Fish Stock Assessment Methods 1998). Because ﬁsheries management problems

75

typically involve a complex system of biological and socio-economic objectives and
constraints, Lane and Stephenson (1998) contend that a conceptual change is necessary,
and ﬁsheries management needs to move toward implementing integrated decision-
making systems where uncertainty is incorporated. When uncertainty is ignored, or
accounted for in an arbitrary fashion, sub-optimal decision options may be selected,
leading to outcomes such as unnecessary losses in yields or stock collapse (Frederick and
Peterman 1994). Achieving ﬁsheries management goals is more likely when
uncertainties are acknowledged, quantiﬁed, and accounted for (Peterman et al. 1998).
Several researchers have applied the method of decision analysis to ﬁsheries
management problems and found that uncertainty can affect both forecasts and optimal
decision-making. McAllister and Peterman (1992) used decision analysis to evaluate the
performance of experimental and status quo management strategies for pink salmon
(Oncorhynchus gorbuscha), while accounting for uncertainty in the cause of decreased
mean body weight. They concluded that the expected value of the experimental
management strategy was higher than that of the status quo under most conditions. Robb
and Peterman (1997) examined uncertainties in the stock-recruitment relationship, annual
recruitment, run timing, and catchability for a sockeye salmon (Oncorhynchus nerka)
ﬁshery through a decision analysis. They found that the shape of the stock-recruitment
relationship had a large effect in determining the optimal management option. Hilbom et
al. (1994) use decision analysis to examine the performance of different quotas in the
presence of uncertainty in virgin stock size. Peterman et al. (1998) summarize three case

studies where decision analysis has been applied to ﬁsheries management and

76

 

recommend that uncertainty be included in the decision making process whenever
possible to improve the quality of management decisions.

In this paper I apply decision analysis to one of the major problems confronting
ﬁshery managers in the Great Lakes region: controlling sea lampreys (Petromyzon
marinas) originating from the St. Marys River. The sea lamprey is an exotic ﬁsh specie
that has decimated ﬁsheries in the Great Lakes through parasitism. The St. Marys River
is the primary source of sea lampreys in the Great Lakes. Because of the large number of

lampreys that are produced, mortality rates for several managed species (e. g., lake trout

and lake Whiteﬁsh) have risen above target levels. Although a substantial and effective
program exists for controlling sea lampreys in other tributaries to Great Lakes, the unique
characteristics of the St. Marys make the typical approaches for control inadequate.
Reducing the number of sea lampreys in northern Lakes Huron and Michigan is a high
priority for ﬁsheries managers in the region.

There are several reasons why decision analysis represents a valuable tool to
assist managers who wish to control St. Marys River sea lampreys. First, managing the
St. Marys River sea lamprey population is one of the most important problems
challenging ﬁsheries managers in the Great Lakes. Connecting Lake Superior to Lake
Huron, the St. Marys River has become the largest source of parasitic sea lampreys in the
Great Lakes (Schleen 1992, Eshenroder et al. 1995). The St. Marys is believed to
produce more lampreys than all other tributaries to the Great Lakes combined. As a
result of the high number of sea lampreys in northern Lakes Huron and Michigan,
rehabilitation of lake trout (Salvelinus namaycush) has been difﬁcult due to the predation

mortality imposed by lamprey in these areas (Sitar et al. 1999, Sitar et al. 1997,

77

Eshenroder et al. 1995). Thus, the management problem is important enough to justify a
concerted effort in evaluating management options through decision analysis. Second,
major uncertainties exist regarding lamprey population dynamics and the effectiveness of
available management options. As noted above, decision analysis is speciﬁcally
designed to incorporate uncertainty into the decision-making process. Third, economic
constraints demand careful evaluation of all lamprey management decisions in the Great
Lakes. Spending too little on control of the St. Marys may result in foregone recovery of
lake trout in northern Lake Huron. Conversely, spending too much on the St. Marys

takes money away from assessment and control needs of other lamprey populations in the

_‘ .mnenun I!“ ll?—

Great Lakes basin. Decision makers want and need to know the expected beneﬁts of
directing funds at controlling the St. Marys population in order to properly allocate funds
available for control and assessment throughout the basin. Fourth, several well-deﬁned
management options are and will be considered as alternatives, effectively bounding the
number of options available for consideration. Fifth, speciﬁc targets exist for lake trout
recovery and sea lamprey suppression in Lake Huron that provide indicators against
which the performance of management options can be judged. Each of these issues
argues for the application of decision analysis to the problem of managing St. Marys
River sea lampreys.

In this paper we summarize our efforts to conduct a decision analysis on the St.
Marys River sea lamprey management problem. Our summary follows the main steps in
conducting a decision analysis: (1) Identifying alternative management actions; (2)
Specifying management objectives; (3) Identifying the uncertain states of nature; (4)

Assigning probabilities to the states of nature; (5) Calculating outcomes with a simulation

78

of the ﬁshery; and (6) Using a decision analysis framework to evaluate management

options in terms of performance indicators.

Description of the System and Sea Lamprey Control

The St. Marys River is believed to be the primary source of parasitic sea lampreys
in northern Lake Huron (Schleen 1992, Eshenroder et a1. 1995). The St. Marys River is
the outflow of Lake Superior, beginning at the lake’s eastern tip. After a distance of
roughly 40 km, the river ﬂows into an area known as the North Channel of Lake Huron.
The river’s mean annual discharge is 2,140 m3/s, over 20 times the largest ﬂow
previously treated with the chemical 3-t1ifluoromethyl-4-nitrophenol (TFM), the primary
method for controlling sea lampreys in the Great Lakes. The large size and high flow of
the St. Marys rendered conventional control methods inappropriate and new approaches
needed to be developed.

Anadromous in their original range, sea lampreys have a life history adapted to
living in the Great Lakes (Lawrie 1970). In late spring-early summer, adult sea lampreys
move up Great Lakes tributaries to spawn. During spawning, eggs are deposited into
nests. Viable eggs develop into larvae, which leave the nests but remain in the streambed
for a period of 3-17 years. Once they obtain sufﬁcient size and energy reserves, they
undergo a process of metamorphosis from detrital to a parasitic feeding form. Following
metamorphosis, juvenile lampreys leave their natal stream to parasitically feed in the
Great Lakes. During this time they parasitize host species (e. g., lake trout, lake Whiteﬁsh,

salmonids) for a period of approximately 18 months, often causing death of the host.

79

Sea lamprey life history characteristics are exploited to achieve population
control in the Great Lakes. Barrier dams and weirs are control methods designed to block
lamprey access to suitable spawning locations. Trapping of adults moving upstream is
also used to help control the number of spawning individuals. Releasing chemically
sterilized males into spawning populations is a control method used to interfere with
reproductive success. However, the primary method used to control lamprey populations
in the Great Lakes is the lampricide, TFM. This liquid chemical is applied to Great
Lakes tributaries to kill larval lampreys in the streambeds and has been very effective in
controlling lamprey populations. Another lampricide, the 2-aminoethanol salt of 2’,5-
dichloro-4’-nitrosalicylanilde (Bayluscide), achieves the same effect of killing larval
lampreys, but can be applied in the form of dry granules. Because of this, Bayluscide can
be applied to small, speciﬁc areas of high larval density, whereas TFM is applied at the
whole tributary level.

The sea lamprey control program in the Great Lakes is directed and coordinated
by the Great Lakes Fishery Commission (GLF C), a bi-national governmental
organization. In 1997 the GLF C adopted a control strategy for the St. Marys River that
included trapping adults, releasing sterilized males, and applying Bayluscide to
approximately 1000 ha of high-density larval habitat in the river during 1998 and 1999.
The decision to adopt this strategy was based on deterministic models presented to the
GLF C Commissioners that forecasted the effects of various management options along
with a cost-beneﬁt analysis of those management options. An ongoing assessment

program is in place to evaluate the effects of this control strategy, but regardless of the

80

outcome, future control decisions will be necessary. The decision analysis presented here

is intended to help inform decision-makers with these future decisions.

Conducting the Decision Analysis
Identifying alternative management actions

In May 1998, we convened a meeting of scientists, agency management
personnel, and other stakeholders to explain our decision analysis approach and identify a
list of alternative management actions that could be used to control St. Marys River sea
lampreys in the future. The group identiﬁed trapping, sterile male releases, and the
application of Bayluscide as the primary actions that could be used. There was
considerable interest in examining the effects of various levels of use and timing of these
actions. The sterile male option relies on obtaining trapped animals from the St. Marys
and other Great Lakes tributaries. Because a large portion of the animals used in the
sterile male program comes from the St. Marys, the intensity of the sterile male release
technique (SMRT) is not independent of the intensity of trapping. Therefore the group
believed that different intensity levels of trapping and SMRT should be examined in
concert with each other. The group also argued that trapping (and therefore SMRT)
should be considered as a management action regardless of whether Bayluscide
applications are considered because of the relatively low costs associated with trapping
and the large capital investment of the traps currently in place. The intensity and timing
of Bayluscide applications are independent of both trapping and SMRT, and therefore
could be investigated at various intensity levels without deﬁning a relationship between

Bayluscide and the other control actions. The option of treating the river with TFM was

81

 

brieﬂy discussed, but the group concluded that this potential action was cost prohibitive

and not likely to be effective as a future control technique.

Specifying management objectives

Deﬁning the set of management objectives proved a more difﬁcult challenge for
the group. Ultimately, they agreed that the goal of controlling lamprey populations in the
Great Lakes is to provide the opportunity for a healthy, productive ﬁsh community.

Restoring lake trout in northern Lakes Huron and Michigan is a particular concern, and

I?“ IJ... [-v "

‘-

 

therefore some believed that creating self-sustaining lake trout populations should be a
management objective. However, issues of commercial and recreational ﬁshing levels,
stocking levels, and the lack of natural spawning populations all affect lake trout
restoration separate from the issue of lamprey control. As a result of discussions on this
topic, the group agreed that while restoring lake trout was indeed a management goal, it
was too broad a topic to serve as a management objective in this particular instance.
Eventually, the group agreed on an objective of reducing the number of parasitic
sea lampreys in Lake Huron. Setting this as the objective implied that reducing the
number of lampreys would aid lake trout recovery efforts in Lake Huron. The group
discussed several ways to measure achievement of the objective by various management
options. The timing and magnitude of the reductions were regarded as important
components of measuring performance, as well as the probability that a particular
management action will be successful in achieving the objective. Recent agreements
between the State of Michigan and several Native American tribes have implicitly

established target levels for future lamprey abundance in Lakes Huron and Michigan.

82

The group expressed an interest in using these targets as a measure of option
performance. Because greater reductions are generally associated with greater control
costs, some expressed an interest in incorporating economic factors such as the economic
injury level approach described by Koonce et al. (1993). We describe the speciﬁc details
of our methods for evaluating management option performance in light of the general

objective of lamprey suppression below. ' l
1

Identifying the uncertain states of nature

 

After discussing the uncertainties that could have effects on the St. Marys River
lamprey population, the group identiﬁed a set of critical uncertainties that needed to be
incorporated into the decision analysis. This exercise was important for narrowing the
large number of potential uncertainties down to a number manageable in a decision
analysis. The group identiﬁed two principal uncertainties. The ﬁrst was uncertainty in
St. Marys River lamprey population dynamics, particularly the stock-recruitment
relationship. The group recognized that little is known about lamprey demographic rates,
their productivity, and their degree of compensation (i.e., larval density-dependence).
These uncertainties limit the ability to forecast the effects of implementing different
treatment options, especially those options that affect adults attempting to spawn (e.g.,
trapping and SMRT). The second major uncertainty identiﬁed by the group was
uncertainty in larval distribution. During 1993-1996, nearly 12,000 locations in the St.
Marys River were sampled to determine the distribution of larvae. The Bayluscide
treatment that took place in 1998 and 1999 only targeted areas identiﬁed as having high

larval densities based on the historical survey. The effectiveness of any Bayluscide

83

treatment depends on the distribution of larvae in the river just prior to treatment, and any
differences due to spatial or temporal variability may reduce treatment effectiveness. The
group agreed that a characterization of this uncertainty would be critical for evaluating
the likely effects of Bayluscide treatments. To be consistent, the group agreed that
uncertainty in trapping and SMRT effectiveness should also be included in the decision
analysis. Similar to the larval distribution uncertainty, trapping and SMRT uncertainties
deal with implementation error. That is, how well will a control method actually perform
if it is selected? Thus, four uncertainties were identiﬁed to be included in the decision
analysis: demographic and stock-recruitment uncertainty, larval distribution uncertainty,

trapping uncertainty, and SMRT uncertainty.

Assigning probabilities to the states of nature

My ﬁrst analytical task was to characterize the uncertainty in d-ographics and
the stock-recruit relationship for St. Marys River sea lampreys. In doing this, my ﬁrst
objective was to develop sets of parameter values that span the range of values possibly
underlying the population dynamics of St. Marys River sea lampreys. In the terminology
of decision analysis, these sets of parameter values are known as “alternative states of
nature.” From an ecological perspective, these sets represent alternative hypotheses
about the abundance and vital rates of this sea lamprey population.

The methods describing the quantiﬁcation of the demographic and stock-recruit
uncertainty are presented in Chapter 1 of this dissertation. To summarize, I built an age-
structured population model that reﬂects the life history of St. Marys River sea lampreys,

utilizing six sources of data to ﬁt demographic parameters. These demographic

84

parameters included time series of age-0 abundance from 1967 to 1996, initial age
composition in 1966, a constant larval natural mortality rate, two parameters describing
the probability of larval metamorphosis as a function of age, and a parameter describing
the proportion of metamorphosed juvenile lampreys that survive to maturity and return to
the St. Marys River to spawn. Using a nonlinear optimization program, I obtained the
parameter estimates that maximized the objective function (i.e., the set of parameter

estimates that best ﬁt the six data sources). I refer to this set of parameters values as the

' art‘s; 1P1 . 31"
.l .L

modal posterior density (MPD) estimates.

To characterize demographic uncertainty, I used Monte Carlo Markov Chain
(MCMC) methods to obtain 1000 samples (each sample containing the full set of
demographic parameters) ﬁ'om the approximate joint posterior distribution of the
demographic parameters. The 1000 samples were independent random samples from the
autocorrelated chain of 12,000 samples described in Chapter 1. With the resulting
parameter estimates, I reconstructed 1000 stock and recruitment data sets. The stock-
recruit data sets could be determined because the age-0 recruitrnents were estimated
parameters and the stock sizes could be calculated ﬁom the recruitment time series and

the demographic parameters. I characterized the stock-recruit uncertainty by estimating

parameters of a stock-recruit function for each data set. I then obtained estimates of (1;,
Bi, and 62; (i = 1, 2, ..., 1000) for a Ricker model of the form
R=Sexp(a—ﬂ§+£) (1)

where R is the number of age-0 larvae produced, S is the number of female spawners that

produced R, and log (a) is distributed N (0, 02 ). I used the set of 1000 demographic

parameter estimates and the associated stock-recruit parameter estimates to characterize

85

the uncertainty in St. Marys River lamprey population dynamics. The MPD combined
estimates represented the most likely state of nature, while the other sets represented
alternative (possible) states of nature. Because the I-[PD estimates are biased estimates of
the expected value for each of the parameters, I also consider a set of estimates that are
the means from each distribution of parameter values (termed “average parameters”).
While the average parameters do not maintain the full covariance structure of the joint
posterior distribution, they may better approximate the central tendency of the joint
posterior than the maximum likelihood (HPD) estimates.

The second source of uncertainty to be characterized was the spatial and temporal
uncertainty in larval lamprey distribution and abundance. The principal objective in the
estimation of this uncertainty was to characterize the implementation uncertainty
associated with Bayluscide applications. In other words, how well will a Bayluscide
treatment perform if it is chosen as a management option in the future, given the spatial
and temporal variability in larval distribution and abundance?

The methods used to describe this uncertainty are presented in Chapter 2. I used
two spatially-referenced data sets containing information on larval distribution and
abundance. One data set contained the results of annual samples collected from small
index sites in the river. The other data set contained the results of an extensive, complete
sampling of the river that took place 1993-1996. Using the index site data, I estimated a
model that described the annual changes in abundance at individual sample locations
based on habitat quality at those locations, the measured abundance at those locations the
previous year, and the inverse distance-weighted average of abundance at neighboring

locations. I applied this model to the historic survey of the whole river to generate a

86

 

series of maps representing the possible distribution and abundance of larvae in 1999, the
year of the major treatment of the St. Marys with Bayluscide. I then simulated the
application of Bayluscide to the simulated maps, and calculated the proportion of larvae
that would have been killed (i.e., treatment effectiveness) for each map. The distribution
of effectiveness values for a given level of treatment characterizes the implementation
uncertainty associated with different levels of Bayluscide applications.

The simulated values of the proportion killed at the highest level of Bayluscide
treatment appeared to be normally distributed with a mean of 0.57 and a standard
deviation of 0.04. Biologists with the St. Marys River Task Force separately estimated
the proportion killed at the level of Bayluscide treatment that was selected by the
decision-makers (i.e., a large-scale treatment) to be 0.55 (Bergstedt et al. 1998). The
empirical estimate for the actual proportion killed was 0.45 (Fodale et al., in review). I
assume that the variability about the mean proportion killed at the large—scale treatment
level would be similar to the variability about the realized proportion killed. Therefore I
characterize the implementation uncertainty associated with large-scale Bayluscide
treatments to be normally distributed with a mean of 0.45 and a standard deviation of
0.04.

The third source of uncertainty to be characterized was the implementation
uncertainty associated with trapping efﬁciency. During 1991-2000 the average trapping
efﬁciency (i.e., the proportion of the spawning population removed by trapping) in the St.
Marys River was nearly 40% (Figure 3.1). Annually, efﬁciencies ranged from 20% to
54%. Based on a histogram plot of these values, they appeared to come from a uniform

distribution. Therefore I characterize the implementation errors associated with trapping

87

as unifomrly distributed from —18% to +18%. To characterize the implementation
uncertainty associated with a particular level of trapping, I take the target level and add a
randomly selected implementation error from the uniform distribution speciﬁed above.

In the decision analysis I only examine trapping levels of 40% and 70%. Because the
errors come from a uniform distribution, each alternative state of nature (i.e., combination
of target efﬁciency and implementation error) has equal probability.

The fourth source of uncertainty to be characterized was the implementation
uncertainty associated with SMRT effectiveness. Biologists with the St. Marys River
Task Force have collected data on SMRT effectiveness in the St. Marys River 1992-
2000. Complications associated with estimating the number of spawners in 1997 resulted
in an unreliable estimate of SMRT effectiveness (Mike Twohey, personal
communication). Therefore, this year’s data was excluded from my analyses. The data
include an estimate of the ratio of sterilezfertile males in the spawning population based
on mark-recapture methods. For example, a sterilezfertile ratio of 3 implies that there are
3 sterilized males for every one fertile male in the spawning population. The theoretical
proportion of viable eggs in a population where sterilized males compete with fertile
males can be estimated using the equation

V: 1
1+SFR

 

(2)

where PV is the proportion of eggs that is viable and SFR is the sterilezfertile ratio. Using
the example above, a sterilezfertile ratio of 3 would result in only 25% of the eggs being
viable.

In addition to the estimated SMRT ratio in the river, biologists with the Task

Force empirically estimated the annual PV by sampling spawning nests and comparing

88

the number of viable eggs to the total number of eggs in the nests. Even in the absence of
sterilized males, nests showed viabilities of considerably less than 100% (43.4% is the
estimate for incomplete viability in the St. Marys River nests not visited by sterile males).
After adjusting for incomplete viability, for each year I calculated the SMRT ratio that
would have resulted in the empirical estimates for PV based on the nest sampling.
Combining the two sources of data resulted in an estimated SMRT ratio based on mark-
recapture and an estimated SMRT ratio based on the egg viability sampling (Figure 3.2).
I then calculated the percent deviation of the SMRT ratio based on nest sampling from
the SMRT ratio based on mark-recapture (Figure 3.3). Essentially these values form an
estimate of the deviations from the assumed relationship (Equation 2) and form the
implementation error associated with SMRT. The mark-recapture SMRT ratio provides
an estimate of theoretical performance, while the egg viability SMRT ratio provides an
estimate of actual performance (i.e., the reduction in viability). A regression of the
percent deviation versus the mark-recapture ratio did not reveal a slope signiﬁcantly
different ﬁ'om zero, indicating that the errors did not follow a trend over the range of the
mark-recapture ratios. However, I was unable to determine the distribution of the errors.
Therefore I assumed that the observed errors deﬁne the distribution and rely on
bootstrapping to quantify the implementation uncertainty associated with a target level
for SMRT. Operationally, I would take the target ratio and multiply it by a randomly
sampled percent deviation (sampled using bootstrapping with replacement) to calculate

the PV associated with a given target for the SMRT ratio.

89

Calculating outcomes with a simulation of the ﬁshery

I used a simulation model, similar to the demographic parameter estimation
model, to forecast future abundance of parasitic sea lamprey in Lake Huron. Whereas the
parameter estimation model (Chapter 1) was utilized for reconstructing historic
population dynamics, the simulation model was used to forecast future population
dynamics in light of the uncertainties identiﬁed above and the effects of different control
options.

The simulation model shared the same age-structure characteristics and life-
history basis as the parameter estimation model. I modeled larvae ages 0-6 in the St.
Marys River with age-speciﬁc abundance determined by the previous year’s abundance
minus losses due to natural mortality and emigration (i.e., metamorphosis). Larvae ages
4-6 underwent the process of metamorphosis according to the estimated parameters that
deﬁne the probability of metamorphosis as a function of age. Juvenile (newly
metamorphosed) lampreys that leave the river in any particular year form the parasitic
population the following year, plus an addition of 30,000 parasites ﬁ'om other Lake
Huron tributaries. In the next year, a proportion of the.tota1 Lake Huron parasitic
population migrates to the St. Marys River to spawn. Trapping reduces the number of
female spawners, which produce age-0 recruits according to the parameters of the Ricker
stock-recruit relationship. The number of viable age-0 recruits .is determined by the
SMRT ratio, if SMRT is selected as a treatment option. If Bayluscide is selected as a
treatment option, then a portion (45% +/- the Bayluscide implementation error) of the

larval population is removed during the year of application.

90

Some calculations were necessary to set up the initial conditions, deﬁned as the
abundance of lampreys in the river and Lake Huron in 2000. The initial conditions were
calculated for each of the 1000 demographic parameter sets, termed “demographic
models.” Each demographic model was associated with an estimate of the larval age
composition in 1996. The numbers of age-0 recruits for 1997-2000 were determined by
the stock-recruit parameters for each demographic model and the empirical estimates of
the number of spawning lamprey present in each of those years. The stock-recruit
parameters and the forecasted number of spawners determined recruitment subsequent to
2000. In setting up the initial conditions, I incorporated the estimated effects of the
Bayluscide treatments that took place during 1998 and 1999.

As mentioned earlier, one of the 1000 demographic models represented the modal
posterior density estimates for the demographic parameters underlying sea lamprey
population dynamics. I designed the simulation model in such a way that forecasts could
be generated using only this model or using all 1000 demographic models. The

timeframe considered in the simulation model forecasts was year 2001 through 2030.

Using a decision analysis framework to evaluate management options

To evaluate the performance of management options under uncertainty, I
incorporated the four uncertainties described above into a decision analysis ﬁarnework.
Figure 3.4 offers a depiction of this framework in the form of a decision tree.
Uncertainties are represented by the circular “uncertainty nodes” while alternative
management options arise out of the square management options box. Outcomes

(deﬁned here as abundance of lamprey in Lake Huron 2001-2030) are calculated for each

91

branch arising ﬁom the uncertainty nodes and for each management option. A Monte
Carlo approach is used to obtain sample values at each of the uncertainty nodes. Values
at these nodes are sampled according to their probability distribution deﬁned above.

Based on the management options under consideration, individual uncertainties
may or may not be included in the outcome calculations. For example, to implement no
control is one potential management option. To calculate the range of possible outcomes
associated with no control, I simply calculate lamprey trajectories sampling the
demographic model uncertainty distribution. When trapping and SMRT options are
under consideration, I calculate lamprey trajectories sampling the range of trapping
uncertainty, SMRT uncertainty, and demographic model uncertainty. Likewise, all four
uncertainties are sampled for management options that consider trapping, SMRT, and
Bayluscide applications.

Measuring the performance of any management option depends critically upon
the utility decision-makers have for the various performance indicators provided.
Different performance indicators will be more meaningful to individual decision-makers
than other performance indicators. Determining which management option is “optimal”
may depend on which performance indicator is chosen. A management options that
maximizes one performance indicator may minimize other performance measures. The
decision to choose a particular management option will almost invariably involve trade-
offs in the performance measures. In this decision analysis, the management objective
was to reduce the number of parasitic sea lamprey in Lake Huron. I attempted to provide
a suite of performance indicators from which individual decision-makers could choose

indicators that they believed to be most meaningful. The indicators fall into three general

92

categories: abundance trajectories, economic cost/beneﬁt rankings, and risk-tolerance
measures.

Abundance trajectories represent trends in the abundance of parasitic sea lamprey
in Lake Huron over time. The principal type of abundance trajectory that I used was the
average parasitic abundance. To calculate this trajectory, I averaged, by year, the
individual outcomes across realizations of the uncertain states of nature for each
management option under consideration. The values in this trajectory represent how
many parasitic lampreys would be present in Lake Huron in any particular year, on
average. The second type of abundance trajectory that I used was the median parasitic
abundance. To calculate this trajectory, I estimated the median abundance, by year,
across realizations of the uncertain states of nature for each management option under
consideration. By deﬁnition, half of the outcomes associated with the management
option were above the year-speciﬁc median value and half were below. Differences exist
between the implications associated with using the median versus using the average
abundance. Individual decision-makers may prefer one over the other in evaluating
management option performance.

For sea lamprey management, there will always be a tradeoff between the beneﬁts
and costs of control. The most effective management options, in terms of suppressing the
population by the greatest amount, are also the most expensive. Following the tenets of
integrated pest management, the aim of the sea lamprey control program has been to
achieve a level of control that maximizes the difference between beneﬁts and costs. My
second type of performance indicator attempts to examine this tradeoff between beneﬁts

and costs.

93

When the GLFC Commissioners chose the control strategy for the St. Marys
River, they were supplied with graphs that depicted the relative abundance of lamprey
over time for each management option, along with its associated cost (Figure 3.5). In the
end, they chose to go with the most effective option (in terms of reducing lamprey by the
greatest amount in the shortest period of time), but it was also the most expensive option
available. I contend that this decision provides insight to the value the decision-makers
held for reducing lamprey. A simple way to look at beneﬁts is through the equation

Net beneﬁt = Outcome "' Value — Cost (3)
I assume that the decision-makers implicitly evaluated the net beneﬁts associated with
each of the different treatment options, and chose the option with the greatest net beneﬁt.
In this case, the outcome was the relative reduction in parasitic abundance and the cost
was the amount of money that the option required for implementation. The only
unknown is the value that the decision-makers held for reductions in lamprey.

One way to solve for this unknown is to examine the problem using the concept
of implied value. Because the decision-makers chose the most expensive option
presented for consideration over options that were also effective (albeit to a lesser
degree), this suggests that the decision-makers associated a high value with lamprey
reductions. By substituting different dollar amounts for the value unknown in the net
beneﬁt equation above, we can estimate the minimum dollar value where the net beneﬁt
for the most expensive treatment option exceeds the net beneﬁt for the next most
expensive treatment option. The value at which this occurs is termed the implied value.
Values higher than the implied value increase the difference in net beneﬁt between the

most expensive and next best treatment options. Therefore the implied value represents a

94

minimum value that the decision-makers held for lamprey reductions in order to justify
choosing the most expensive treatment option.

Discussions with the decision-makers revealed that they were mainly concerned
with the performance of the management options over a 15 year period (Gavin Christie,
personal communication). In addition, they preferred options that reduced lamprey faster
over options that reduced lamprey by the same amount, but over a longer period of time.
This difference in preference as a function of timing implies that the decision-makers
were employing some sort of discounting. That is, immediate beneﬁts were valued
higher than beneﬁts deferred into the future.

To calculate an implied value that the decision-makers may have used, I estimated
the net beneﬁts associated with the two most expensive treatment options considered in
1997. The more expensive option consisted of a 40% rate of trapping, a 2.8 SMRT ratio,
and removing 55% of the larvalpopulation with Bayluscide. The next most expensive
option consisted of a 40% rate of trapping and a 2.8 SMRT ratio (Figure 3.5). Using the
same graph as was presented to the decision-makers in 1997 (Figure 3.5) and my estimate
of the number of parasites present in 1997, I calculated the discounted, 15-year total
reduction in parasitic lamprey for both treatment options as compared to no control. I
used a discount rate of 6% and a 1997 parasitic abundance of 500,000 lampreys (a
reasonable number based on my model estimates). By substitution, I determined that the
implied value was $15.65 per additional lamprey removed by control. Values greater
than this amount had a higher net beneﬁt for the most expensive treatment option
(trapping, SMRT, and Bayluscide application) than the alternative treatment option

(trapping and SMRT). With an estimate of the implied value, net beneﬁts of ﬁrture

95

management options can be estimated in a similar fashion. The net beneﬁt calculation
forms my cost/beneﬁt performance indicator for the different treatment options.

The ﬁnal group of performance indicators is risk-tolerance measures. As a result
of the negations between the State of Michigan and several Native American tribes
regarding treaty-related ﬁshing rights (termed the “Consent Agreement”), there are
implied targets for reductions in lamprey abundance in Lake Huron for the years 2006
and 2012. These targets are proportional to the average abundance of lamprey in Lake
Huron during 1998-2000. My estimate for average abundance during this time period is
658,000. The Consent Agreement implies lamprey abundance targets of 183,000 in 2006
and 114,000 in 2012. As a performance indicator, for each management option I
calculate the proportion of cases where the outcomes are at or below the Consent
Agreement targets established for 2006 and 2012. These proportions represent an
estimate of the probability that the targets will be achieved by the management options
under consideration. In a related measure of risk-tolerance, I graph the probability of
being below a range of parasitic abundances in the years 2012 and 2030. These graphs
provide indications of performance based on the risk of exceeding various levels of

parasitic abundance for the management options under consideration.

Simulation details and management option notation

Within the simulation model, a standard run consisted of 100,000 30-year
forecasts for each management option. This total number is the result of 100 realizations

of the stock-recruit process error term for each of the 1,000 demographic models (100 *

96

1,000 = 100,000). Each forecast randomly samples the distributions of the
implementation uncertainties deﬁned above when applicable.

Three options are available for controlling sea lampreys from the St. Marys River:
trapping, SMRT, and Bayluscide applications. The objective of this decision analysis is
to compare different levels of use and timing for these control options. I considered nine
options and use a shorthand notation to refer to these options (Table 3.1). The notation
consists of the proportion removed by trapping followed by a “/”, the target SMRT ratio
followed by a “I”, and the proportion of the larval population removed by Bayluscide
followed by the timing of the Bayluscide applications.

Costs for the various treatment options were estimated using the data contained in
Bergstedt et al. (1998). A linear regression was ﬁt to the costs for three levels of SMRT
(0.67, 2.77, and 3.8) and was extended to predict the costs associated with SMRT ratios
of 4:1 and 7:1. The cost of a high level of trapping (70%) was based on expert opinions
(St. Marys River Decision Analysis Workshop, April 2425, 2001). To provide a relative
measure of the total costs associated with the treatment options, costs were annualized
over the thirty-year period considered in this study. When estimating net beneﬁts for the
treatment options, both the costs and beneﬁts (the reduction in the number of lamprey

compared to no control) were discounted. I used a discount rate of 6%.

Results
The results of the simulations that included all of the deﬁned uncertainties and all
of the treatment options revealed several interesting patterns (Figure 3.6). As expected,

maximum population suppression was associated with the most expensive treatment

97

option (0.7 / 7 / 0.45 every 4, 2004-2030). However, options that included trapping at
70% and a SMRT ratio of 7 performed nearly as well, even without the application of
Bayluscide (e.g., 0.7 / 7 / 0). Options that included trapping at 40% and a SMRT ratio of
4 displayed relatively poor performance, with the exception of the 0.4 / 4 / 0.45 every 4,
2004-2030 option. But even the worst option (0.4 / 4 / 0) was able to achieve suppression
levels that were, on average, 50% of the no control levels.

The results presented in Figure 3.6 incorporate the effects of all the uncertainties
that were considered in this decision analysis. When I examined the uncertainties
individually, I found that not all uncertainties affect the average number of parasites over
time. Average trajectories for treatment options where uncertainties were characterized
using symmetrical distributions (i.e., the trapping uncertainty and the Bayluscide
uncertainty) did not differ ﬁ'om average trajectories where these uncertainties were
ignored. However, uncertainty did make a difference in the average trajectories when an
asymmetric distribution was used to characterize the uncertainty (i.e., the demographic
uncertainty and the SMRT uncertainty).

Figure 3.7 shows the forecasted trajectories for no control and low levels of
trapping and SMRT, with and without uncertainty in trapping and SMRT. The
differences between those simulations with uncertainty and those without are due to the
asymmetric distribution that was used to characterize SMRT effectiveness (Frederick and
Peterman 1994). The asymmetry is due to the differences between observed and
theoretical SMRT effectives, and is a combination of process error and model error. The
process error is a result of natural variability in the underlying relationship while the

model error is a result of not having the correct model describing how SMRT ratios

98

translate into reductions in viable eggs. Of the seven estimates of the SMRT percent
deviation, ﬁve are negative (Figure 3.3). When these negative deviations are randomly
selected through bootstrapping, the predicted effectiveness of SMRT is reduced and
therefore greater lamprey abundances would be forecasted.

Figure 3.8 shows the forecasted trajectories for the no control management
option, using various descriptions of the demographic uncertainty. When I assume that
only the maximum likelihood model (the MPD model) describes the stock-recruitment
relationship and the associated demographic rates, trajectories are lower than when I
consider all models. Using the “average parameters” set is a better approximation to the
ﬁrll uncertainty (all models) results, but is biased low beyond year 2015. These effects
are due to a complex asymmetry in the parameters describing the demographic
uncertainty. Allowing for demographic uncertainty results in less optimistic forecasts of
lamprey abundance than if the MPD or average parameters models are used.

The results of the simulations are also dependent upon the statistic used to
measure central tendency. The distribution of abundance forecasts within most years
showed positive skewness. This observation is reﬂected in Figure 3.9, where the mean
and median forecasts are compared for a no control option and a 0.4 / 4 / 0 option. The
mean forecasts for the no control option approach 570,000 whereas the median forecasts
approach 270,000. Similarly, the mean of the 0.4 / 4 / 0 option forecasts is around
490,000 whereas the median forecast values lie around 130,000. With positively skewed
distributions, the mean is greater than the median. The median may be a more useful

indicator of performance than the mean for decision-makers concerned with the

99

frequency of high and low abundances, but not so much concerned with the average
abundance across simulations.

By calculating the mean net beneﬁts associated with each option, I was able to
rank the options and examine the sensitivity of this ranking to alternative assumptions of
the implied value, the time horizon, and the discount rate (Table 3.2). Option 5 was
generally ranked highest over the range of assumptions examined within the sensitivity
analysis. Option 6 was also ranked highly and was ranked highest for my approximation
of the values, time horizon, and discount rate associated with the 1997 decision by the
GLF C Commissioners. The 0.4 / 4 / 0 option ranked lowest across all assumptions.
However, there is considerable variation and overlap in the net beneﬁts across
simulations for the treatment options (Figure 3.10). While there are slight differences in
the mean net beneﬁts for each treatment option (Table 3.2), the variation in net beneﬁts
across simulations is large, making the differences in mean option performance
negligible.

In terms of achieving the Consent Agreement targets for lamprey abundance,
several of the options show a high probability of success (Table 3.3). While option 7 had
the highest chance of meeting both targets, option 6 also showed a high probability of
success. With no control, there is a 50% chance of meeting the 2006 target, but only a
18% chance of meeting the 2012 target. Option 4 had a high probability of meeting the
2006 target at relatively low cost. Interestingly, option 9 (the most expensive option) did
not perform much better (in terms of meeting the Consent Agreement goals) than option

5, at half the cost.

100

Another way to view these same results is presented in Figure 3.11. The x-axis
represents the number of parasites in 2012 while the y-axis represents the proportion of
simulation cases where the parasitic abundance was less than the values on the x-axis.
The vertical arrow marks the location of the Consent Agreement target for 2012. For
example, there is a 60% chance of less than 200,000 parasites in 2012 when the 0.4 / 4 / 0
option is selected. The y-values at the intersections of the vertical arrow with the lines
representing the treatment options are the same as those values listed in Table 3.3.
Similarly, Figure 3.12 shows the chances of exceeding parasite abundances in 2030 for
the different treatment options. Notice that over a longer time ﬁame, the performance of
a Bayluscide application in 2004 does not differ ﬁom relying only on trapping and SMRT

at the same levels.

Discussion

Over the last decade the sea lamprey control program has moved towards
incorporating greater quantitative rigor into their assessment and treatment activities.
The St. Marys River provides an excellent example of the use of quantitative methods
and assessment tools to address the problems confronting natural resource management.
When it became apparent that something needed to be done about sea lampreys in the St.
Marys River, a Task Force was formed and began the long process of identifying and
addressing the most critical knowledge gaps about the system. Through the studies that
followed, knowledge about the system grew tremendously. When the time arrived for

decision-makers to evaluate alternative control strategies, scientists and agency personnel

lOl

were well prepared to communicate their best estimates of treatment option performance
(Bergstedt et al. 1998).

This decision analysis represents the next logical step along the road of providing
decision-makers with the information necessary to make good decisions. The problem of
the St. Marys River will persist and future strategic decisions will be necessary. The
decision analysis that I have presented in this paper serves as a tool that can be used to
investigate the expected performance of alternative control strategies in the presence of
uncertainties in demographic rates and in management option implementation. Although

the GLF C did consider uncertainty when evaluating the decision options in 1997, it was

 

done subjectively (Bergstedt et al. 1998). In the presentation of each management
option, the GLF C provided a qualitative assessment of the uncertainty associated with it.
This uncertainty assessment likely inﬂuenced the decision-maker’s evaluation of the
treatment options under consideration. The decision analysis that I present forrnalizes
and quantiﬁes that uncertainty assessment. I have also presented several indicators that
can be used by decision-makers to evaluate the performance of alternative management
options. Both of these features of a decision analysis should help decision-makers to
consider the trade-offs when choosing a strategy that best meets their objectives.

One important issue to be recognized when using this decision analysis is the
feasibility of achieving the targeted levels for the control methods. In the analyses, I
assumed that the control agents would be able to achieve the speciﬁed levels and timing
of trapping, SMRT, and Bayluscide applications. Based on the recommendations and
knowledge of management personnel, I limited the range of control effort to that which

realistically could be achieved. However, up to this point in time, the control program

102

has not been able to trap 70% of the spawning population, or implement a 7:1 SMRT
ratio. Likewise, removing 45% of the larvae with Bayluscide every two years may be
difﬁcult to achieve. As the number of larvae in the river is reduced, more area may need
to be treated to achieve the target control level. Issues such as this are not reﬂected in the
analysis because I assume that the speciﬁed treatment levels and timing are achieved,
albeit with some measure of error, when evaluating option performance.

Another important issue is the limitation of the net beneﬁt calculations. The
implied value that I calculated was based on the 1997 decision. It is not the same as the
economic injury level proposed by Koonce et a1. (1993) and should not be construed as a
true economic value for lamprey suppression. It served as a ﬁrst approximation for the
values that decision-makers apparently held for comparisons among a limited set of
management options. A more comprehensive analysis of the economic values associated
with sea lamprey reductions (and lake trout restoration) is ultimately needed to properly
exarninine management option performance.

One important ﬁnding of this study was the degree of overlap in the range of
estimated net beneﬁts among treatment options. While certain treatment options may
perform better on average (in terms of maximizing the average net beneﬁt), the high
amount of overlap in net beneﬁts among decision options suggests that these differences
are small relative to their variability. The variability in net beneﬁts may itself be an
important consideration for decision-makers, however. An option that has less variability
in net beneﬁts but a lower average value may be preferable to an option with more

variability and a higher average value. Preferences regarding this variation around the

103

mean values relate to the economic risk tolerances of the decision-maker. In spite of this,
economic considerations may not strongly inﬂuence decision-maker choices.

The parameter estimates and model relationships that I used in this project
represent a summary of our current understanding of lamprey demographics and control
method effectiveness. However, the decision analysis model that I developed is a
ﬂexible one that can be re-parametcrized as new information becomes available. Each of
the uncertainties that I considered includes a combination of process error, model error,
and measurement error. Of these three types, both model error and measurement error
can be reduced by scientiﬁc investigations. If the costs and beneﬁts of these
investigations can be estimated, then this decision analysis model can be used for
prioritizing future studies aimed at reducing uncertainty using “value of information”
calculations (Morgan and Henrion 1990). This model could also be used for examining
adaptive management opportunities (Walters 1986). By altering the number of spawners
through trapping and SMRT and observing the effects on recruitment, it may be possible
to learn about the shape of the underlying stock-recruitment relationship for St. Marys
River sea lampreys (Smith and Walters 1980). However, the high variability in
recruitment for sea lamprey (see Chapter 1) may hinder learning about the stock-
recruitrnent relationship for this population (I-Iinrichsen 2001).

Similar to the ﬁndings of Robb and Peterman (1997), I found that uncertainty in
the stock-recruit relationship affected the forecasts of parasitic abundance. In addition, I
found that uncertainty in SMRT effectiveness affected forecasts. Ignoring these two
uncertainties would have resulted in more optimistic forecasts of parasitic abundance

than forecasts that ignored these uncertainties. As a general recommendation, it appears

104

that stock-recruitment uncertainty should routinely be included in the evaluations of
ﬁshery management options. When other asymmetric uncertainties affect forecasts (as
with SMRT), then these uncertainties should be incorporated as well.

The sea lamprey population of the St. Marys River will continue to present
challenges for Great Lakes ﬁshery managers. The analysis presented in this dissertation
should help managers as they consider the risks associated with alternative control
strategies. The development of these decision and simulation models represents an
incremental step forward. As knowledge about the system and the processes that govern
its dynamics advances, these models can and should be revised to reﬂect changes in
understanding. Hopeﬁrlly, as the models improve, decision-making will also improve,

and a healthier Great Lakes ecosystem will be the result.

Acknowledgements

John Netto provided valuable assistance in programming the simulation model

used to conduct the decision analysis.

105

References

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Lakes Fishery Commission, Ann Arbor, MI. 37 pp.

Clemen, RT. 1996. Making hard decisions: an introduction to decision analysis.
Duxbury Press, Paciﬁc Grove, CA.

Eshenroder, R.L., N.R. Payne, J .E. Johnson, C. Bowen' H, and M.P. Ebener. 1995. Lake
trout rehabilitation in Lake Huron. Journal of Great Lakes Research 21(Supplement
l):108-127.

Fodale, M.F., R.A. Bergstedt, D.W. Cuddy, and J .V. Adams. In review. Planning and
executing a lampricide treatment of the St. Marys River using georeferenced data.
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Frederick, SW. and RM. Peterman. 1994. Choosing ﬁsheries harvest policies: when
does uncertainty matter? Canadian Journal of Fisheries and Aquatic Sciences 52:291-
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Haeseker, S.L., M.L. Jones, and J .R. Bence. In review. Estimating uncertainty in the
stock-recruit relationship for St. Marys River sea lampreys. Submitted to the Journal of
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Haeseker, S.L. and ML. Jones. In prep. Quantifying the spatial and temporal
uncertainty in the abundance of larval lamprey and its effect on chemical treatment
success in the St. Marys River.

Hilbom, R., E.K. Pikitch, and RC. Francis. 1993. Current trends in including risk and
uncertainty in stock assessment and harvest decisions. Canadian Journal of Fisheries and
Aquatic Sciences 50:874—880.

Hilbom, R., E.K. Pikitch, and MK. McAllister. 1994. A Bayesian estimation and
decision analysis for an age-structured model using biomass survey data. Fisheries
Research 19:17-30.

Hinrichsen, R.A. 2001. High variability in spawner-recruit data hampers learning.
Canadian Journal of Fisheries and, Aquatic Sciences 58:769-776.

Koonce, J .F ., R.L. Eshenroder, G.C. Christie. 1993. An economic injury level approach
to establishing the intensity of sea lamprey control in the Great Lakes. North American
Journal of Fisheries Management 13:1-14.

Lane, DE, and R.L. Stephenson. 1998. A ﬁarnework for risk analysis in ﬁsheries
decision-making. ICES Journal of Marine Science 55:1-13.

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Lawrie, AH. 1970. The sea lamprey in the Great Lakes. Transactions of the American
Fisheries Society 99:766z775.

McAllister, M.K., and RM. Peterman. 1992. Decision analysis of a large-scale ﬁshing
experiment designed to test for a genetic effect of size-selective ﬁshing on British
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Aquatic Sciences 49:1305-1314.

Morgan, MG. and M. Henrion. 1990. Uncertainty: a guide to dealing with uncertainty
in quantitative risk and policy analysis. Cambridge University Press, Cambridge, UK.

.National Research Council Committee on Fish Stock Assessment Methods. 1998.
Improving ﬁsh stock assessments. National Academy Press, Washington, DC.

Peterman, R.M., C.N. Peters, C.A. Robb, and SW. Frederick. 1998. Bayesian decision
analysis and uncertainty in ﬁsheries management. In Reinventing Fisheries Management,
Pitcher, T.J., P.J.B. Hart, and D. Pauly, editors. Kluwer Academic Publishers, The
Netherlands.

Raiffa, H. and R. Schlaifer. 1961. Applied statistical decision theory. Harvard
University, Boston, MA.

Robb, CA. and RM. Peterman. 1997. Application of Bayesian decision analysis to
management of a sockeye salmon (Oncorhynchus nerka) ﬁshery. Canadian Journal of
Fisheries and Aquatic Sciences 55:86-98.

Rosenberg, A.A. and V.R. Restrepo. 1994. Uncertainty and risk evaluation in stock
assessment advice for US. marine ﬁsheries. Canadian Journal of Fisheries and Aquatic
Sciences 51:2715-2720.

Schleen, LP. 1992. Strategy for control of sea lampreys on the St. Marys River, 1992-
95. Great Lakes Fishery Commission, Ann Arbor, Michigan.

Sitar, S.P., J .R. Bence, J .E. Johnson, and W.W. Taylor. 1997. Sea lamprey wounding
rates on lake trout in Lake Huron, 1984-1994. Michigan Academician 29221-37.

Sitar, S.P., J.R. Bence, J .E. Johnson, M.P. Ebener, W.W. Taylor. 1999. Lake trout
mortality and abundance in southern Lake Huron. North American Journal of Fisheries
Management 19:881-900. -

Smith, A.D., and C.J. Walters. 1980. Adaptive management of stock-recruitment
systems. Canadian Journal of Fisheries and Aquatic Sciences 38:690-703.

Walters, C.J. 1986. Adaptive Management of Renewable Resources. Macmillian, New
York, New York.

107

 

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111

 

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114

 

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Net Beneﬁt ($)

Figure 3.10 Box and whisker plots of the net beneﬁts across all simulations
for decision options 1-9. The dot represents the median net beneﬁt and the
box ends represent the 25th and 75th percentiles of the distribution.

 

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Parasites In 2012

Figure 3.11 Cumulative probability of being below speciﬁed parasitic
abundance (on x-axis) in year 2012 for options 1-5 (panel A) and options 6-9
(panel B). The vertical arrows denote the parasitic abundance target implied

by the Consent Agreement.

120

Proportion of cases wlm abundance < X

Proportlon of cases with abundance < X

 

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0 - Y I I I
100000 200.000 300,000 400.000 500.000
Parasites In your 2030

Figure 3.12 Cumulative probability of being below speciﬁed parasitic
abundance (on x-axis) in year 2030 for options 1-5 (panel A) and options 6-9
(panel B).

121