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A MODEL OF SEA LAMPREY FEEDING WITH IMPLICATIONS
FOR LAKE TROUT DYNAMICS IN LAKE HURON

By

Michael A. Rutter

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

2004

ABSTRACT

A MODEL OF SEA LAMPREY FEEDING WITH IMPLICATIONS
FOR LAKE TROUT DYNAMICS IN LAKE HURON

By
Michael A. Rutter

The dynamics of sea lamprey (Petromyzon marinas) and lake trout (Salvelinus
namaycush) in Lake Huron were investigated and a population model for lake trout in the
main basin of Lake Huron was parameterized. I also examined the implications of
uncertainty in these dynamics on the outcomes of sea lamprey control programs and lake
trout stocking strategies using a stochastic model.

To better estimate wounding rates on hosts of sea lamprey in the Great Lakes, I
developed a method to fit a logistic model for the mean number of wounds per host as a
function of host length. Wounds per fish were assumed to follow a Poisson distribution,
and a number of alternative models were considered using maximum likelihood
techniques. Parameters were allowed to vary spatially and temporally, and my results
suggest that the asymptote, or wounding rate on the largest lake trout, varied over years
and among lake regions. In addition, the inflection point (where wounding rates
increased most rapidly toward the asymptote) varied among regions, shifting toward
smaller lake trout lengths further north in Lake Huron.

Using a statistical catch-at-age model as a framework, I parameterized a
population model for lake trout in the main basin of Lake Huron. Natural, commercial L
fishing, and recreational fishing mortality were estimated separately for each of the three
regions. Sea lamprey-induced mortality was determined by a multi-species Type II

functional response model parameterized across three regions in the main basin of Lake

Huron. The functional response parameters were estimated during the model fitting
process using observed wounding rates on lake trout as one important measure of
goodness of fit. The effective search rate of sea lamprey, a key component of the
functional response model, was modeled as a logistic function of prey length and was
shown to vary spatially across the regions of Lake Huron. I used the functional response
model to estimate sea lamprey-induced mortality as a function of sea lamprey density,
lake trout density, and the density of alternative prey. Sea lamprey-induced mortality
was the largest component of annual lake trout mortality rates in central and southern
Lake Huron, while commercial fishing was the largest component of lake trout annual
mortality in the north.

To assess the impacts of sea lamprey control and lake trout restoration programs, I
used a stochastic model to forecast future lake trout population dynamics. This model
was based on the catch-at-age model, and incorporated the parameterized type II
functional response. A variety of possible lake trout and sea lamprey management
scenarios were forecast for Lake Huron. A Markov Chain Monte Carlo analysis of the
catch-at-age model accounted for uncertainty in model parameters and I allowed for
stochastic temporal variation in model inputs such as sea lamprey abundance and lake
trout stocking when assessing the uncertainty of the forecasts. In northern Lake Huron,
reducing sea lamprey abundance 90% and increasing lake trout stocking 200% reduced
sea lamprey-induced mortality on large lake trout 85%. Reduced commercial fishing
mortality rates in northern Lake Huron will allow sea lamprey-induced morality rates to
decease as lake trout density increases when lamprey populations are reduced 90%,

indicating that sea lamprey are partially saturated with prey hosts under these conditions.

ACKNOWLEDGMENTS

I would like to thank my advisor, James Bence, as well as my committee
members Michael Jones, Robert Tempelman, and Habib Salehi for their assistance and
guidance in completing this dissertation. I would also like to thank Gavin Christie, Great
Lakes Fishery Commission, and Mark Ebener, Chippewa Ottawa Resource Authority, for
the data they supplied and the insight they provided. The Michigan Department of
Natural Resources, especially Shawn Sitar, Aaron Woldt, and, Jim Johnson, US Fish and
Wildlife Service, and Ontario Ministry of Natural Resources, especially Rob Young, all
supplied important data for this project. I would also like to thank my friends from the
Fisheries and Wildlife department for their support during my graduate career. Finally, I
would like to thank for family for their undying support during my entire college career,
especially my wife Natalie.

Financial support for this project was provided by the Great Lakes Fishery

Commission.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES ........................................................................................................... ix
CHAPTER ONE

AN IMPROVED METHOD TO ESTIMATE SEA LAMPREY
WOUNDING RATE ON HOSTS WITH APPLICATION TO
LAKE TROUT IN LAKE HURON .................................................................................... 1

CHAPTER TWO
PARAMETER ESTIMATION OF A FUNCTIONAL RESPONSE MODEL
TO IMPROVE ESTIMATES OF SEA LAMPREY-INDUCED MORTALITY ON

LAKE TROUT IN LAKE HURON .................................................................................. 14
Introduction ............................................................................................................ 14
Methods .................................................................................................................. l9

Observed data and annual inputs ............................................................... 20

Population model ....................................................................................... 23

Submodels for predicting observed quantities ........................................... 28

Fitting the catch-at-age model ................................................................... 30

Results .................................................................................................................... 35

Discussion .............................................................................................................. 41
CHAPTER THREE

PROJECTING THE OUTCOMES OF LAKE TROUT RESTORATION
EFFORTS IN LAKE HURON: INCORPORATING A SEA
LAMPREY FUNCTIONAL RESPONSE AND ACCOUNTING

FOR UNCERTAINTY ...................................................................................................... 86
Introduction ............................................................................................................ 86
Methods .................................................................................................................. 91

Population model ....................................................................................... 92
Simulations ................................................................................................ 95
Comparisons among models ...................................................................... 97
Scenarios varying lake trout stocking ...................................................... 100
Sea lamprey saturation in northern Lake Huron ...................................... 100
Lake trout damage goals .......................................................................... 101
Marginal distributions for parameters of interest ..................................... 101
Results .................................................................................................................. 103
Comparisons among models .................................................................... 103
Scenarios varying lake trout stocking ...................................................... 105
Sea lamprey saturation in northern Lake Huron ...................................... 108
Lake trout damage goals .......................................................................... 108
Discussion ............................................................................................................ l l l

REFERENCES ................................................................................................................ l 50

vi

LIST OF TABLES

Table 1: Years in which data sources are available for the commercial and

recreational harvest in Lake Huron for each region ........................................................... 46
Table 2: Model equations describing the catch-at-age model .......................................... 47
Table 3: Likelihood equations used to fit the catch-at-age model .................................... 49
Table 4: List of variables and parameters used in the model ............................................ 50

Table 5: Area of lake trout habitat (<40 fathoms) in km2 for each region
of Lake Huron (Ebener 1998). ........................................................................................... 53

Table 6: Variability (0,3)“, ) of data for log-normal likelihood

components (assumed constant across all year) estimated from the data .......................... 5 3

Table 7: Yearly variability (0”,, ) of the log-normal likelihood

component of survey CPUE based on mixed-effect linear model analysis.
An entry of zero indicates no data for that year ................................................................. 54

Table 8: Predicted average number of attacks per lamprey (1997-1998)
for assumed handling times of 6.8, 11, and 20 days .......................................................... 54

Table 9: Predicted average sea lamprey-induced mortality rates (1997-
1998) for age 4 and age 10 lake trout for assumed handling times of 6.8,
11, and 20 days .................................................................................................................. 55

Table 10: Estimated background natural mortality rates for age 2+ lake
trout for each region of Lake Huron .................................................................................. 55

Table 11: Estimated parameters for the effective search rate for sea
lamprey (eq. 12). 95% Bayesian prediction interval based on MCMC
analysis ............................................................................................................................... 55

Table 12: Area of lake trout habitat (<40 fathoms) in square km for each
region of Lake Huron (Ebener 1998) ............................................................................... 116

Table 13: The average number of yearling equivalents recruiting in each

region of Lake Huron for models I and II. This is based on the average

number of yearling equivalents stocked from 1984-1998, and takes into

account an assumed probability fish stocked in one region will move to

another immediately after stocking (see chapter two) ..................................................... 116

vii

Table 14: Assumed abundances (numbers of fish) assumed constant in the

simulation models based on the average values estimated from 1984-1998

(see chapter two). For sea lamprey these represent the baseline values,

which were adjusted by stated percentages for different scenarios ................................. 116

Table 15: Means of the normal distribution used to randomly generated

simulation model inputs for each year in model III. Chinook salmon,

Whitefish, sea lamprey and yearling equivalents are numbers as defined in

Table 14. Fishing intensity represents the instantaneous fishing mortality

rate on a fully selected age . The mean and coefficient of variation (CV)

are based on the values estimated from 1984-1998 (see chapter two).

Coefficient of variation is reported for each quantity because the CV was

held constant when sea lamprey abundance and stocked yearling

equivalents were changed from baseline values .............................................................. 117

Table 16: Descriptions of management scenarios simulated in terms of
changes in sea lamprey abundance and lake trout stocking ............................................. 118

Table 17: Probability of reducing sea lamprey-induced mortality on age 9

lake trout in northern Lake Huron below 0.10 based on marginal

distribution determined under model 111 for each management scenario

described in Table 16 ....................................................................................................... 119

Table 18: Probability of reducing sea lamprey-induced mortality on age 9

lake trout in northern Lake Huron below 0.05 based on marginal

distribution determined under model 111 for each management scenario

described in Table 16 ....................................................................................................... 119

viii

LIST OF FIGURES

Figure 1: Map of Lake Huron with model regions based on the statistical
districts listed (Smith et al. 1961). Northern Lake Huron consists of MH-l
and portions of OH-l. Central Lake Huron consists of MH-Z, portions of
OH-l, and OH-2. Southern Lake Huron consists of MH-3, MH-4, MH-S,
MH-6, OH-3, OH-4, and OH-5. Dotted line indicates boundary between
Canada and United States. .............................................................................

Figure 2: Example of a double logistic selectivity curve estimated by the
model for commercial fishery selectivity and spring survey selectivity ........

Figure 3: Total commercial harvest (in 10005 of lake trout) for Lake
Huron comparing observed harvest to harvest predicted by the model for
each region of Lake Huron for 1984-1998. Note the difference in scale for
each region .....................................................................................................

Figure 4: Total recreational harvest (in 10005 of lake trout) for Lake
Huron comparing observed harvest to harvest predicted by the model for
each region of Lake Huron for 1985-1998. Note the difference in scale for
each region .....................................................................................................

Figure 5: Commercial effort for Lake Huron in 10008 of meters of large-
mesh gill-net set per year comparing observed effort to effort predicted by
the model for each region of Lake Huron for 1984-1998. Note the
difference in scale for each region .................................................................

Figure 6: Recreational effort for Lake Huron in 10003 of angling hours
comparing observed effort to effort predicted by the model for each region
of Lake Huron for 1985-1998. Note the difference in scale for each region

Figure 7: Average age at harvest in the commercial fishery for Lake
Huron comparing observed average age to average age predicted by the
model for each region of Lake Huron for 1984-1998 ....................................

Figure 8: Average age at harvest in the recreational fishery for Lake
Huron comparing observed average age to average age predicted by the
model for each region of Lake Huron for 1985-1998 ....................................

Figure 9: Average age in the spring graded-mesh gill-net survey for Lake
Huron comparing observed average age to average age predicted by the
model for each region of Lake Huron for 1984-1998 ....................................

ix

.................... 56

.................... 57

.................... 58

.................... 6O

.................... 62

.................... 64

.................... 66

.................... 68

.................... 7O

Figure 10: Observed asymptotic wounding rates from Rutter and Bence
(2003) compared to the asymptotic wounding rates predicted by the model
for each region of Lake Huron for 1984-1998 ................................................................... 72

Figure 11: Observed wounding rates from Rutter and Bence (2003) for
1987 in each region of Lake Huron compared to wounding rates predicted
by the model for 1987. Note the difference in scale for each region ................................ 74

Figure 12: Total number of lake trout (in 10003) age 1 to age 15 predicted
by the model for the three regions of Lake Huron from 1984-1998 .................................. 76

Figure 13: Predicted spawning stock biomass (biomass of age 7+ female
lake trout assuming 50% gender ratio) in 10008 of kg of lake trout in three
regions of Lake Huron from 1984-1998 ............................................................................ 77

Figure 14: Predicted average instantaneous mortality rates due to

commercial fishing, recreational fishing, and sea lamprey on age 5-10 lake

trout in each region of Lake Huron from 1984-1998. Note the difference

in scale for each region ...................................................................................................... 78

Figure 15: Predicted average sea lamprey-induced mortality rates on age

5-10 lake trout for each region of Lake Huron from 1984-1998. The 95%

Bayesian prediction interval of the marginal distribution is shown, with

the median of the marginal indicated by the square .......................................................... 80

Figure 16: Predicted average total (natural, sea lamprey, commercial and
recreational fishing) annual mortality on age 5-10 lake trout for each
region of Lake Huron from 1984—1998 .............................................................................. 82

Figure 17: Estimated effective search rate (eq. 12) for central Lake Huron
as a function of lake trout length with the IMSL effective search (Grieg et
al. 1992) given for reference .............................................................................................. 83

Figure 18: Sea lamprey induced-mortality rates as a function of the
percent increase in lake trout population based on 1998 levels of sea
lamprey, lake trout and alternative prey ............................................................................ 84

Figure 19: Number of age 2+ lake trout in northern, central, and southern

Lake Huron for model I. The 95% Bayesian prediction interval of the

marginal distribution of age 2+ lake trout is shown, with the median of the

marginal indicated by the square. The point estimate of age 2+ lake trout

based on the joint posterior modal estimates from chapter two is indicated

by the gray circle. For information on scenarios, see Table 16. Note the

difference in scale between regions ................................................................................. 120

Figure 20: Density of age 2+ lake trout in northern, central, and southern

Lake Huron for model I. The 95% Bayesian prediction interval of the

marginal distribution for age 2+ lake trout density is shown, with the

median of the marginal indicated by the square. The point estimate of the

age 2+ lake trout density based on the joint posterior modal estimates from

chapter two is indicated by the gray circle. For information on scenarios,

see Table 16 ..................................................................................................................... 121

Figure 21: Number of age 7+ lake trout in northern, central, and southern

Lake Huron for model I. The 95% Bayesian prediction interval of the

marginal distribution of age 7+ lake trout is shown, with the median of the

marginal indicated by the square. The point estimate of age 7+ lake trout

based on the joint posterior modal estimates from chapter two is indicated

by the gray circle. For information on scenarios, see Table 16. Note the

difference in scale between regions ................................................................................. 122

Figure 22: Density of age 7+ lake trout in northern, central, and southern

Lake Huron for model I. The 95% Bayesian prediction interval of the

marginal distribution for age 7+ lake trout density is shown, with the

median of the marginal indicated by the square. The point estimate of the

age 7+ lake trout density based on the joint posterior modal estimates from

chapter two is indicated by the gray circle. For information on scenarios,

see Table 16 ..................................................................................................................... 123

Figure 23: Sea lamprey-induced mortality on age 5 lake trout in northern,

central, and southern Lake Huron for model I. The 95% Bayesian

prediction interval of the marginal distribution of mortality is shown, with

the median of the marginal indicated by the square. The point estimate of

mortality based on the joint posterior modal estimates from chapter two is

indicated by the gray circle. For information on scenarios, see Table 16.

Note the difference in scale between regions .................................................................. 124

Figure 24: Sea lamprey-induced mortality on lake trout for the youngest

age lake trout with mean length exceeding 700 mm in northern (age 9),

central (age 9), and southern (age 8) Lake Huron for model I. The 95%

Bayesian prediction interval of the marginal distribution of mortality is

shown, with the median of the marginal indicated by the square. The

point estimate of mortality based on the joint posterior modal estimates

from chapter two is indicated by the gray circle. For information on

scenarios, see Table 16. Note the difference in scale between regions ........................... 125

Figure 25: Number of age 2+ lake trout in northern, central, and southern
Lake Huron for three different models. The 95% Bayesian prediction
interval of the marginal distribution of age 2+ lake trout is shown, with the
median of the marginal indicated. The models with an adjusting [3 (model
II), a fixed [3 (model I), and the stochastic simulations with an adjusting [3

xi

(model 111) are shown. For information on scenarios, see Table 16. Note
the difference in scale between regions ........................................................................... 126

Figure 26: Density of age 2+ lake trout in northern, central, and southern

Lake Huron for three different models. The 95% Bayesian prediction

interval of the marginal distribution for age 2+ lake trout density is shown,

with the median of the marginal indicated. The models with an adjusting

B (model II), a fixed B (model I), and the stochastic simulations with an

adjusting B (model III) are shown. For information on scenarios, see

Table 16 ........................................................................................................................... 128

Figure 27: Number of age 7+ lake trout in northern, central, and southern

Lake Huron for three different models. The 95% Bayesian prediction

interval of the marginal distribution of age 7+ lake trout is shown, with the

median of the marginal indicated. The models with an adjusting B (model

11), a fixed B (model I), and the stochastic simulations with an adjusting B

(model III) are shown. For information on scenarios, see Table 16. Note

the difference in scale between regions ........................................................................... 130

Figure 28: Density of age 7+ lake trout in northern, central, and southern

Lake Huron for three different models. The 95% Bayesian prediction

interval of the marginal distribution for age 7+ lake trout density is shown,

with the median of the marginal indicated. The models with an adjusting

B (model 11), a fixed B (model I), and the stochastic simulations with an

adjusting B (model HI) are shown. For information on scenarios, see

Table 16 ........................................................................................................................... 132

Figure 29: Sea lamprey-induced mortality on age 5 lake trout in northern,

central, and southern Lake Huron for three different models. The 95%

Bayesian prediction interval of the marginal distribution of mortality is

shown, with the median of the marginal indicated. The models with an

adjusting B (model II), a fixed B (model I), and the stochastic simulations

with an adjusting B (model III) are shown. For information on scenarios,

see Table 16. Note the difference in scale between regions ............................................ 134

Figure 30: Sea lamprey-induced mortality for the youngest age lake trout

with mean length exceeding 700 mm in northern (age 9), central (age 9),

and southern (age 8) Lake Huron for three different models. The 95%

Bayesian prediction interval of the marginal distribution of mortality is

shown, with the median of the marginal indicated. The models with an

adjusting B (model II), a fixed B (model I), and the stochastic simulations

with an adjusting B (model III) are shown. For information on scenarios,

see Table 16. Note the difference in scale between regions ............................................ 136

Figure 31: Number of age 2+ lake trout in northern, central, and southern
Lake Huron for model 111 under a series of proposed lake trout

xii

management scenarios. The 95% Bayesian prediction interval of the

marginal distribution of age 2+ lake trout is shown, with the median of the

marginal indicated by the square. For information on scenarios, see Table

16. Note the difference in scale between regions ............................................................ 138

Figure 32: Density of age 2+ lake trout in northern, central, and southern

Lake Huron for model 111 under a series of proposed lake trout

management scenarios. The 95% Bayesian prediction interval of the

marginal distribution for age 2+ lake trout density is shown, with the

median of the marginal indicated by the square. For information on

scenarios, see Table 16 .................................................................................................... 139

Figure 33: Number of age 7+ lake trout in northern, central, and southern

Lake Huron for model 111 under a series of proposed lake trout

management scenarios. The 95% Bayesian prediction interval of the

marginal distribution of age 7+ lake trout is shown, with the median of the

marginal indicated by the square. For information on scenarios, see Table

16. Note the difference in scale between regions ............................................................ 140

Figure 34: Density of age 7+ lake trout in northern, central, and southern

Lake Huron for model 111 under a series of proposed lake trout

management scenarios The 95% Bayesian prediction interval of the

marginal distribution for age 7+ lake trout density is shown, with the

median of the marginal indicated by the square. For information on

scenarios, see Table 16 .................................................................................................... 141

Figure 35: Sea lamprey-induced mortality on age 5 lake trout in northern,

central, and southern Lake Huron for model 111 under a series of proposed

lake trout management scenarios. The 95% Bayesian prediction interval

of the marginal distribution of mortality is shown, with the median of the

marginal indicated by the square. For information on scenarios, see Table

16. Note the difference in scale between regions ............................................................ 142

Figure 36: Sea lamprey-induced mortality for the youngest age lake trout

with mean length exceeding 700 mm in northern (age 9), central (age 9),

and southern (age 8) Lake Huron for model 111 under a series of proposed

lake trout management scenarios. The 95% Bayesian prediction interval

of the marginal distribution of mortality is shown, with the median of the

marginal indicated by the square. For information on scenarios, see Table

16. Note the difference in scale between regions. ........................................................... 143

Figure 37: Number of age 2+ lake trout in northern Lake Huron for model
III under a series of proposed lake trout management scenarios.

Commercial fishing rates are reduced 56%. The 95% Bayesian prediction
interval of the marginal distribution of age 2+ lake trout is shown, with the

xiii

median of the marginal indicated by the square. For information on
scenarios, see Table 16 .................................................................................................... 145

Figure 38: Density of age 2+ lake trout in northern Lake Huron for model

III under a series of proposed lake trout management scenarios.

Commercial fishing rates are reduced 56%. The 95% Bayesian prediction

interval of the marginal distribution for age 2+ lake trout density is shown,

with the median of the marginal indicated by the square. For information

on scenarios, see Table 16 ............................................................................................... 145

Figure 39: Number of age 7+ lake trout in northern Lake Huron for model

111 under a series of proposed lake trout management scenarios.

Commercial fishing rates are reduced 56%. The 95% Bayesian prediction

interval of the marginal distribution of age 7+ lake trout is shown, with the

median of the marginal indicated by the square. For information on

scenarios, see Table 16 .................................................................................................... 146

Figure 40: Density of age 7+ lake trout in northern Lake Huron for model

111 under a series of proposed lake trout management scenarios.

Commercial fishing rates are reduced 56%. The 95% Bayesian prediction

interval of the marginal distribution for age 7+ lake trout density is shown,

with the median of the marginal indicated by the square. For information

on scenarios, see Table 16 ............................................................................................... 146

Figure 41: Sea lamprey-induced mortality on age 5 lake trout in northern

Lake Huron for model 111 under a series of proposed lake trout

management scenarios. Commercial fishing rates are reduced 56%. The

95% Bayesian prediction interval of the marginal distribution of mortality

is shown, with the median of the marginal indicated by the square. For

information on scenarios, see Table 16 ............................................................................ 147

Figure 42: Sea lamprey-induced mortality on age—9 lake trout in northern

Lake Huron for model 111 under a series of proposed lake trout

management scenarios. Commercial fishing rates are reduced 56%. The

95% Bayesian prediction interval of the marginal distribution of mortality

is shown, with the median of the marginal indicated by the square. For

information on scenarios, see Table 16 ............................................................................ 147

Figure 43: Sea lamprey-induced mortality on age 9 lake trout in northern

Lake Huron for model 111 as the number of lake trout stocked is increased.

Sea lamprey populations are based on a 90% reduction in the average

value observed in Lake Huron from 1984-1998 .............................................................. 148

Figure 44: Sea lamprey-induced mortality on age 9 lake trout in northern
Lake Huron for model III as the number of lake trout stocked is increased.

xiv

Sea lamprey populations are based on the average value observed in Lake
Huron from 1984-1998 .................................................................................................... 148

Figure 45: Age specific sea lamprey-induced mortality rates for lake trout

in northern Lake Huron under scenario C (90% reduction in sea lamprey)

and scenario F (90% reduction in sea lamprey and a 200% increase in

stocking) using model 111. The median of the marginal distribution of

mortality is shown ............................................................................................................ 149

Figure 46: Age specific sea lamprey-induced mortality rates for lake trout

in southern Lake Huron under scenario C (90% reduction in sea lamprey)

and scenario F (90% reduction in sea lamprey and a 200% increase in

stocking) using model 111. The median of the marginal distribution of

mortality is shown ............................................................................................................ 149

XV

CHAPTER 1

Rutter, MA. and Bence, J.R.. 2003. An improved method to estimate sea lamprey
wounding rate on hosts with application to lake trout in Lake Huron. J. Great
Lakes Res. 29 (Supplement 1): 320-331.

J. Greai lakes Res. 29 (Supplement 1):320-331
Internet. Assoc. Great Lakes Res., 2003

An Improved Method to Estimate Sea Lamprey Wounding Rate
on Hosts with Application to Lake 'Iinut in Lake Huron

MiehadARntter‘andJameslfienoe

Department of Fisheries and Wildlife
Michigan State University
13 Natural Resources Building
East Lansing, Michigan 48824-1222

ABSTRACT. To better estimate wounding rates on hosts of sea lamprey (Petromyzon marinas) in the
Great Lakes. methods were developed to fit a logistic model for the mean number of wounds per host as a
function of host length. These methods were applied to the number of wounds (the sum of type 11-] to A-III
marks on hosts collected in spring) on individual lake trout (Salvelinus namaycush) collected by surveys
and by a commercial fishery in three regions of U.S. waters of Lake Huron from 1984 to 2000. Wounds
per fish were assumed to follow a Poisson distribution, and a number of models were examined using
maximum likelihood techniques. Parameters were allowed to vary spatially and temporally, and Markov
Chain Monte Carlo techniques were used to evaluate uncertainty in parameter values. By using data for
individual hosts and modeling the efl’ect of host length as a continuous fitnction. this method makes more
complete use of available data. increases precision, and removes biases in comparison with widely used
approaches for estimating wounding rates. In this application, the asymptote, or wounding rate on the
largest lake trout, varied over years and among lake regions. In addition, the inflection point (where
wounding rates increased most rapidly toward the asymptote) varied among regions, shitting toward
smaller lake trout lengths further north in Lake Huron. This change in shape suggests some complexity in
the sea lamprey-lake trout interaction. For a 500-mm lake trout, a host size observed in all areas, esti-

mated wounding rates were highest in the north.

INDEX WORDS: Lake trout, sea lamprey, wounding, Markov Chain Monte Carlo, Lake Huron.

INTRODUCTION

Wounds on host fish that survive sea lamprey
(Petromyzon marinas) attacks provide an important
source of information 'to fishery managers and sci-
entists in the Great Lakes. Although attacks by sea
lamprey often lead to host death, many fish survive
and bear marks resulting from the attacks (King
1980). “Wounds” are marks made by lamprey on
hosts that heal to “scars” within a period of a year.
There is evidence that mortality experienced by the
host population can be positively correlated with
frequency of wounds (Bence et al. 2003), and the
mean number of wounds per fish is used to assess
the success of the sea lamprey management pro-
gram (Adams et al. 2003), estimate host mortality

 

Corresponding area. Email: mar360peuedn
Cmaddless:1‘he8choolof8eience.PennStateErie,TheBehrend
College. Suic- Road, Erie. PA 16562-0203

rates (Eshenroder et al. 1995, Sitar et al. 1999), and
to calibrate sea lamprey-host models that provide
guidance to the sea lamprey management program
(Koonce et al. 1993, Larson et al. 2003). The host
species emphasized in all these applications is lake
trout (Salvelinus namaycush). ,

The incidence of and characteristics of sea 1am-
prey marks vary seasonally (Spangler et al. 1980,
Jacobson 1989) and with host size (Wrgley 1959,
Pycha and King 1975, Sitar 1996). Most parasitic-
phase sea lamprey leave streams and enter the Great
Lakes during fall and spring but do not begin to
grow rapidly until the summer after they enter the
lake. Most potentially lethal attacks on larger hosts
such as lake trout or lake Whitefish (Coregonus cla-
peafqrmis) are by rapidly growing sea lamprey in
the autumn (Spangler et al. 1980, Bergstedt and
Schneider 1988, Bence et al. 2003) following one
summer’s growth. The King (1980) system for refer-

Estimating Sea lamprey Wounding Rates 321

ring to marks was used. In this system, type A
marks result from attacks where the skin has been
broken and stages I to IV represent progressive
stages of healing. For hosts collected and observed
in the autumn, A-I marks unquestionably are the re-
sult of attacks by the currently feeding cohort of sea
lamprey, whereas marks in later stages of healing
may represent a mix of attacks by the current and
prior year’s cohort (Eshenroder and Koonce 1984,
Schneider et al. 1996, Ebener et al. 2003). When ob-
served in the spring, larger type A-I, A-II, and A-III
marks are most likely the result of the cohort that
was responsible for lethal attacks the previous au-
tumn (Eshenroder and Koonce 1984). Virtually all
A-I to A-III marks observed on lake trout in the
upper Great Lakes in the spring are larger marks
(Mark Ebener, Chippewa-Ottawa Resource Author-
ity, and James Johnson, Michigan Department of
Natural Resources, personal communication). Thus,
for lake trout, A-I marks in the autumn and the sum
of A-I to A-III marks in the spring provide two ways
to assess wounding on lake trout for the same cohort
of sea lamprey. The frequency of wounds generally
increases with host size (Wigley 1959, Pycha and
King 1975, Sitar 1996). Examination of patterns re-
ported in the literature, observations from laboratory
experiments (Farmer and Beamish 1973, Swink
1991), and unpublished data from Lakes Michigan,
Huron, and Superior all suggest a general pattern
where the mean wounds per fish increases with fish
size up to an asymptote (Bence et al. 2003).

Current practice for recording and reporting sea
lamprey marks in the Great Lakes generally follows
recommendations made by Eshenroder and Koonce
(1984). They recommended reporting marks of the
year (what are called wounds) on lake trout from ei-
ther autumn or spring surveys. They assumed that
A-I to A-III marks would qualify as marks of the
year at both times of year, but fall A-III marks in
Lake Ontario were found to be better correlated
with A-l marks from the previous year (Schneider
et al. 1996). More recently Ebener et al. (2003) rec-
ommend only including A-I marks when fish are
observed in the autumn. Eshenroder and Koonce
(1984) also recommended that marks be summa-
rized as the mean number of wounds per lake trout
(wounding rate) for each of several broad length
categories or bins (432 to 533 mm, 534 to 635 mm,
636 to 737 mm, > 737 mm).

The common practice for estimating mean
wounds has several substantial limitations and does
not make full use of available data. First, it treats
each year and region independently, and thus makes

no use of any regularity among regions or years.
Second, it does not take advantage of the fact that
the expected number of wounds is related to host
length in a predictable way. Finally, the use of
broad host length bins can produce biases that will
vary depending upon the length-composition of
hosts within each bin. Because sample sizes are
often small for the largest length bin of hosts, esti-
mated wounding rates for such bins often vary
wildly from year to year.

In this paper, an approach for estimating mean
wounds per fish was developed by fitting a model
based on a logistic function to observed wounding
data, and applying this approach to spring wound-
ing data for lake trout in Lake Huron. The logistic
equation allows wounding rates to increase as a
continuous function of host length, eventually lev-
eling off at an asymptotic rate for large hosts. This
model is only applicable to wounds, since scars can
accumulate over the life of the host and the number
of scars per fish is not expected to reach an upper
asymptote with increasing fish size. The basic idea
was to encapsulate the relationship between wound-
ing rate and host size through a limited number of
parameters, while allowing for wounding rates that
vary from region to region and year to year.

This approach differs from conventional methods
by making use of individual observations on the
number of wounds and the size of each fish, and
makes the reasonable assumption that the distribu-
tion of wounds among fish of a given length is
Poisson (Bence et al. 2003). This approach makes
more complete use of the available information
than conventional approaches, provides wounding
rates for any length of host, and circumvents some
of the problems associated with small sample sizes
and bias by assuming a functional form for the rela-
tionship between wounds per host and host length.

METHODS

peritoneum

Data used in the analysis came from spring lake
trout surveys and sampling of tribal commercial
harvest during the spring (April through June) in
US. waters of the main basin of Lake Huron during
1984 to 2000. Marking data were recorded using
the protocol developed by King (1980), and A-1, A-
II, and A-III marks were considered to be wounds.
Earlier marking data exist, but were not used be-
cause they were not recorded using the King (1980)
protocol. Only data collected in the spring were an-
alyzed, as this is the one time period consistently

322 Rutterand Bence

sampled for lake trout in all regions of U.S. waters
of the main basin of Lake Huron. In addition,
spring wounding was of special interest because es-
timated spring wounding rates have been trans-
formed into estimated per capita host mortality,
based on an assumed direct proportionality that de-
pends on the lethality of an attack (Eshenroder er
al. 1995, Sitar et al. 1999, Bence et al. 2003). Sur-
veys were conducted using gill nets by both the
Michigan Department of Natural Resources
(MDNR) and the Chippewa-Ottawa Resource Au-
thority (CORA), and were done during April to
June of each year. The MDNR gill nets were graded
mesh in 125-cm intervals ranging from 5 to 15 cm,
typically 3,000 m in length (Merena et al. 1981,
Johnson and VanAmberg 1995). Survey sites were
fixed stations in U.S. waters of Lake Huron. Survey

data were collected near Drummond Island in

northern Lake Huron by CORA using similar meth-
ods. Commercial samples came from sampling by
CORA of a tribal fishery in the northern part of the
lake during April through June, and fish were col-
lected by trap nets, small-mesh (6.35 cm and 7.6
cm) gillnets, and large-mesh (11.4 cm and larger)
gillnets (Mark Ebener, CORA, Sault Ste. Maire,
MI, personal communication).

Data were pooled spatially into three regions:
north, central, and south (Fig. 1). Only fish greater
in total length (TL) than 430 mm were considered
in this study, which is consistent with current meth-
ods and low wounding rates on smaller lake trout.
Only 0.5% of sampled lake trout 430 mm and
smaller had wounds. Wounding data were available
for 18,284 individual fish greater than 430 mm, TL,

in length.

Models for Observed Wounding Rates

The logistic function is a simple model that al-
lows wounding rates to increase gradually with host
length, eventually reaching an asymptote, in the
same way actual wounding rates typically do (see
Introduction). This three parameter function:

. 0 .
WIFE-my (1)

describes the average wounding rate per fish, it. ob-
served on a fish of length l. The parameter 6 de-
scribes the average wounding rate for large fish (the
asymptotic wounding rate), while a and B affect the
steepness and position of the curve, respectively
(Fig. 2). The length at which lake trout wounding

 

 

Michigan \
Southern

Ontario

FIG. 1. Lake Huron with the regions used in
this study demarcated. The north, central and
south regions correspond to regions used in lake
trout assessments (Sitar et al. 1999). Following
the statistical districts devised by Smith et al.
(1961), U.S. waters of the north region corre-
sponds to MH-I, U.S. waters of the central region
corresponds to MH-Z and U.S. waters of the
southern region corresponds to MH-J, MH-4 and
MH-S combined The international boundary is
denoted by a dashed line.

reaches 50% of the asymptote is given by Band is
the inflection point. The rate at which the wounding
rate approaches the asymptote while passing
through the inflection point is determined by a,
with the slope at the inflection point being (re/4. As
described below, a range of model variants were ex-
plored in which the parameters of the logistic func-
tion were allowed to vary among regions and years.

The model was fit to the observed data using
maximum likelihood techniques. The observed
number of wounds for a lake trout of a given length
was assumed to follow a Poisson distribution
(Bence et al. 2003). The mean of the Poisson varies
as a function of length, and is described by the 10-
gistic wounding model (eq. 1). Parameters were es-
timated by maximizing the log-likelihood:

no,a,m=2(w,,,,1ntnl-¢v.) (2)
51

Estimating Sea lamprey Wounding Rates 323

 

    

 

o .-i” of.

no no no sso soo no no no no no no no 910
rant-minim

FIG. 2. An example of a logistic function relat-
ing expected or average wounds per fish (wound-

ing rate) to total fish length, with the asymptote
indicated by 0 and the inflection point indicated
by l1

7 fir T

where n is the number of fish sampled, wow is the
observed number of wounds on fish i, and w,- is the
predicted number of wounds for fish i from equa-
tion 1. The maximum likelihood approach allows
the use of data available for each individual fish, in-
cluding length and number of observed wounds. Pa-
rameter estimation was done using the optimization
software Admodel Builder (Otter Research 2000).
Akakie’s Information Criteria (AIC) (Akaike 1973)
was used to determine which model variant pro-
vided the best fit to the data.

It was originally hypothesized that the shape of
the logistic curve would be constant, and that only
the asymptote (9) potentially varied temporally and
spatially. This premise was initially used because
foraging models where both sea lamprey size selec-
tion and the lethality of their attacks are constant
produce this pattern (Bence et al. 2003). Following
this approach, year-to-year (1984 to 2000) and re-
gion-to-region (north, central and south) differences
in wounding rates can be estimated through differ-
ences in the asymptote parameter. Within this class
of models, the simplest case where a common as-
ymptote was shared among all regions and years
was fit first (Table 1, model A). Models where the
asymptote varied among regions but not years
(model B), among years but not regions (model C),
as an additive sum of region and year effects
(model D), and freely among regions and years
(i.e., with 51 values of 9 representing each combi-
nation of year and region, model B) were then con-
sidered. -

The shape of the logistic model was explored

next to determine if it varied among regions. It was
expected that changes in the logistic shape would
largely reflect changes in sea lamprey behavioral
responses to host populations (Bence et al. 2003),
and would thus change gradually in response to
gradual changes in the host community. This con-
trasts with the expectation for the overall level of
wounding, which was expected to reflect both large
year to year variations in sea lamprey abundance
(Morse et al. 2003, Young et al. 2003) and possible
variations in how wounds were recorded (Ebener et
al. 2003). Regional rather than temporal effects
were focused on first because over the study period
spatial differences in the lake trout stocks appear
more substantial than temporal ones (Eshenroder et
al. 1995, Sitar 1996, Sitar et al. 1999, Wilberg et al.
2002). The set of models considered allowed a or B
to vary among regions with the other shape parame-
ter fixed (Table 1, models F and G, respectively),
and for both shape parameters to vary among re-
gions at the same time (model H). This exploration
of regional variation in the logistic shape was
viewed as building upon the previous analysis of
variation in the asymptote, so the approach was to
add these variants in the shape parameters to the
best model obtained when the shape parameters
were assumed constant. In addition, these three
treatments of spatial variationin a and B were
added to each of the four other treatments for spa-
tial and temporal variation in 6 (as in models A
through D), to check that none of these alternatives
had a better fit (lower AIC).

Next, the shape parameters a and B were allowed
to vary temporally. Initial attempts to fit models
that allowed these parameters to vary freely from
year to year were unsuccessful, either because the
resulting AIC was large or because not all parame-
ters were estimable (convergence to a solution was
not achieved). Therefore, an approach was imple-
mented that reflected the premise that temporal
changes in the shape parameters would result from
responses to changes in host populations, and that
these changes would occur gradually over time.
Gradual temporal changes in a and B were esti-
mated by modeling them using a random walk
model (Gudmundsson 1998). A single set of ran-
dom walk variations were applied to all regions
(Table 1, models I and J) and independent random
walks for each region were also considered (models
K and L). In the former case, the random walk ef-
fects were added to regional differences, if any (es-
sentially additive year and region effects), using
both the best model with fixed shape parameters

324

Rutter and Bence

TABLE 1. Description of models used in estimating sea lamprey wounding rates on lake trout. Sab-
scripts N, C, and S refer to the northern (MR—I), central (MB-2) and southern (MH-JMHJ, and M36)
regions (Fig. 1) respefliyely. Subscripts 1984, . . . , 2000 refer to the year lake trout were collected.

 

 

 

 

 

Model 0 a and B Random effects
A Constant across years and regions Constant across years and regions none
3 Region effects (0N,0c, 03), constant Constant across years and regions none

across years '
C Year effects (01934, . . . , 0mg) Constant across years and regions none
D Additive region and year effects Constant across years and regions none
(0Ns%s%sol984s°s°som) .
B Asymptote for each year and region Constant across years and regions none
(9N.|984.9c.1934. - - - . 932000)
F Asymptote for each year and region a: Region efi'ects, constant across years; none
, B: constant across regions and years
G Asymptote for each year and region a: Constant across regions and years; none
B: Region effects, constant across years
H Asymptote for each year and region Region effects for both a. and B, none
both constant across years
I Asymptote for each year and region a: One random walk across years for none
all regions. B: region effects.
constant across years.
I Asymptote for each year and region a: Constant across years and regions. none
B: region effects and one random walk
across years for all regions.
K Asymptote for each year and region a: Independent random walks across none
years for each region. B: region effects
and constant across years.
L Asymptote for each year and region a: Constant across years and regions. none
B: independent random walks across years
for each region
M Asymptote for each year and region a: Constant across regions and years; site by year
B: Region effects, constant across years
N Asymptote for each year and region a: Constant across regions and years; .. net set

B: Region effects, constant across years

 

and the best model that allowed regional differ-
ences in the shape parameters. In the latter case, the
separate random walks were added to the best
model with a constant shape, as this way of using
the random walk for the shape parameters already
incorporates regional effects, temporal efi‘ects, and
interactions. When considering a and B separately,
the random walk year effects did not improve
model fit, so variants where both a and B varied
temporally at the same time were not considered.
All the previous models treat each observed lake
trout within a given year and region as an indepen-
dent observation. In reality, multiple lake trout are
collected from within a given year and region at
each fixed site, and within a fixed site multiple lake

trout are taken from individual sets of gill nets. It is
conceivable that lake trout taken together either at
the level of a site or at the level of an individual net
set would have different average wounding rates
than one might expect on average for the region and
year. Ignoring such “random-cluster effects” if they
exist would lead to underestimating uncertainty as-
sociated with the parameter estimates. This possi-
bility was explored by adding to the best model
without random effects (model G, Table 1) random
effects in 0, first for site by year combinations and
then for individual net sets (models M and N, re-
spectively). Mixed models, that combine fixed and
random effects, are increasingly being used in fish-
eries to address such “cluster sampling” and corre-

Estimating Sea Lamprey Wounding Rates 325

lations among observations (Conover et al. 1997,
Sitar et al. 1999, Bergstedt et al. 2003).

For the overall best model (model G, table 1),
overdispersion was tested for using the T-test de-
scribed by Dean and Lawless (1989). Substantial
overdispersion could compromise the use of AIC
for model choice or otherwise influence conclu-
sions. The assumption in the model that the ob-
served wounding frequency follows a Poisson
distribution means that the variance of the observed
wounds is equal to the mean wounding rate. If
wounding rates are overdispersed, then the variance
would be greater than assumed, and there would be
more fish with multiple wounds than is expected for
the Poisson distribution.

Assessment of Uncertainty

A Bayesian approach was adopted to assess uncer-
tainty through posterior distributions for the parame-
ters. Given the nonlinear model, correlation among
parameters, and small sample sizes for some year
and region combinations, more familiar asymptotic
standard errors would provide a less reliable assess-
ment of uncertainty. The joint posterior distribution
of the parameters was esn'mated using Markov Chain
Monte Carlo (MCMC) methods (Brooks 1998). It is
possible to calculate a posterior distribution for any
quantity of interest that can be calculated as a func-
tion of the parameters. This was done for the mean
number of wounds on a 500-mm fish as a basis for
comparison among areas. This length was chosen as
it represents fish that are observed commonly in all
three regions of Lake Huron.

Assuring that calculated posterior distributions are
valid is not an automatic nor simple task (Cowles
and Carlin 1996), and care was taken in this area. To
ensure that the posterior distribution was a proper
one, all parameters were bounded during estimation,
which effectively gave each parameter a uniform
prior distribution between the bounds (set to be well
above or below what was viewed as possible values
for the parameters). Thus, the priors were only
weakly informative, as a priori all values within the
bounds were considered equally likely. To ensure the
MCMC chains converged to the posterior distribu-
tions, a number of methods were used, including vi-
sual inspection of trace plots. The effective sample
size, or the number of independent samples con-
tained within the correlated MCMC chain, for each
parameter of interest was also calculated. This was
done using the methods outlined in Thiebaux and
Zwiers (1984), estimating the antocorrelation func-

tion for lags up to 150 steps and insuring the antocor-
relation function converged to zero. The length of the
MCMC chain was chosen with a goal of achieving
an effective sample size of at least 1,000. Diagnos-
tics of the MCMC chains showed that convergence
was satisfactory, and that efiective sample sizes were

greater than 1,000 for all parameters.
RESULTS
. Model Selection and Evaluation

Model G was selected as the best model (Tables 1
and 2). This model allowed the average wounding
rate on large fish (0) to vary freely among years and
regions, and allowed for regional differences in B
(the lake trout length at which 50% of the asymp-
totic rate was reached). Among models with fixed
shape parameters, model B, which allowed 0 to vary
freely over years and regions (Table 1), had a sub-
stantially lower AIC than the alternatives and was
selected as best (Table 2). When possible regional
differences in the shape parameter were considered,
model G was selected, which added a regional effect
to B. Models that include regional effects on a
(which would alter the rate at which wounding rate
approached the asymptote), either alone or in com-
bination with regional effects for B, had higher AIC
values (Table 2). The attempts to include annual ef-
fects on the shape parameters (a and B) led to very
small year effects and no improvement in goodness-
of-fit. None of the mixed effects were significant
when added to model G, thus there was no evidence
that lake trout sampled together tended to have a
common wounding rate that difiered from the over-
all value for the region and year.

In general, comparison of model predictions with
observed wounding data indicated the model fit

TABLE 2. Ahahie’s Information Criteria (AIC)
values for wounding models A through H (Table
I). The lower the AIC value, the better the model
fit the data.

E

 

 

Number of Akakie's
Model Parameters Information Criteria
A 3 16674
B 5 16538
C 19 16560
D 23 16383
E 53 16326
F 55 16291
G 55 16283
ll 57 16286

 

326 Rutter andBence

 

 

 

0.6 » e
i 0.5 r
80.0 [
$0.3 l
0.2
0.1
0 t 9 - - . . l - . . . O .
430 no 010 550 590 630 610 710 150 190 030 m
Totalbnmhlnnln.

FIG. 3. An example of wounds per lake trout as
a function of lake trout length (southern Lake
Huron in 1985). Solid circles represent observed
data grouped by 20-mm length bins for visual
clarity, and the solid line is the model estimated
mean wounds per lake trout.

well (Fig. 3). In graphically comparing observed
and predicted values, the data were summarized
into 20-mrn length bins, but even so, large devia-
tions between observed and predicted wounding
rates for large fish were typical because few lake
trout were observed for the larger length bins.

The t-test for overdispersion using the entire data
set suggests that there was overdispersion (t ‘= 9.02,
p—value < 0.01), These results imply that the vari-
ance of the distribution of wounds is greater than
the Poisson model predicts. The year and area with
the greatest level of overdispersion was northern
Lake Huron in 1995. Closer examination of the data
for this extreme case showed that the model failed
to predict accurately the number of fish with three
wounds, per fish or greater (Fig. 4), indicating a
larger variance than assumed by the Poisson distrib-

ution. For model G, the deviance was determined to ,

be 1.08, indicating that no adjustments to the AIC
values in Table 2 are needed. (McCullagh and

Nelder 1989). This, together with the fact that very _

few fish are causing the overdispersion, suggests
that overdispersion had little influence on parame-
ter estimates and model choice, although this could
be explored further through simulation.

PatternslnEstlmatedWonndlng
ParametersandRates

The estimated asymptotic wounding rates (0)
fluctuated significantly from year to year (Table 3).
While there is a fairly large amount of temporal

300
l

250 .

N

O

O
arrests...

Ftsh
H
U!
0
I.
sailfish

3‘
-.,1."- s

at ~+

:"J’t‘fst

p
O
O

.1

05‘3",
l

at
O
a
J.

- “.25.”

 

 

Wourds per fish

FIG. 4. Expected and observed number of
wounds per fish on lake trout for northern Lake
Huron in 1995.

variation in the estimates for each region, central
Lake Huron had the highest estimated asymptotic
wounding rates. Yearly fluctuations in 0 did not
track across regions (Table 3). For example, asymp-
totic wounding rates were low during 1990 and
1991 in northern Lake Huron, whereas they were
high at the same time in central Lake Huron. The
lake trout length'at which 50% of the asymptotic
rate was reached (B) was smallest in the north and
largest in southern Lake Huron (Table 4).

A consequence of the fact that B varies spatially
is that asymptotic wounding rates cannot be inter-
preted in a simple way. This is evident when con-
sidering the fact that estimates of 0 for northern
Lake Huron tended to be less than the estimates for
cental Lake Huron (Table 3), a result that at first ap-
pears to contradict observations that sea lamprey at-
tacks are greatest in the northern portions of Lake
Huron (Rakoczy and Rogers 1991a, 1991b;
Rakoczy 1992; Rakoczy and vaoda 1993, 1994;
Sitar 1996). However, as the inflection point B oc-
curred at progressively lower lengths when moving
from the southern to northern region, the resulting
rates of sea lamprey wounding for lake trout of the
same length were greater in the north than in either
central or southern Lake Huron. For example, the
expected wounding rate was higher in northern
Lake Huron than in central Lake Huron in 1992 for
the size range that includes most of the fish sam—
pled, even though 0 was estimated to be lower in
the north (Fig. 5).

To compare wounding rates across regions and

Estimating Sea Lamprey Wounding Rates 327

TABLE 3. Maximum likelihood estimates and 95%
probability intervals ofasymptotic wounding rate, 0,
for northern (MH-I), central (ME-2), and southern
(MH-SMH-d, and MH-S) Lake Huron fiorn 1984 to

 

 

 

2000. Results based on model G (Table 1).
Region Year 0 MCMC 95% Prob. Interval
North 1984 0.235 [0.2260385]
1985 0.306 [0.2650592]
1986 0.338 [0.3130598]
1987 0.524 [0.4890921]
1988 0.269 [0.2140587]
1989 0.167 [0.1020512]
1990 ‘ 0.064 [0.0330259]
1991 0.089 [0.0550266]
1992 0.361 [0.2970755]
1993 0.327 [0.2860598]
1994 0.333 [0.2820582]
1995 0.547 [0.5160909]
1996 0.390 [0.3630682] ‘
1997 0.580 [0.5380915]
1998 0.523 [0.4930859]
1999 0.181 [0.1550313]
2000 0.278 [0.2360485]
Central 1984 0.816 [0720,1200]
' 1985 0.955 [0.878,].367]
1986 0.527 [0.4600808]
1987 0.277 [0.2260453]
1988 0.217 [0.1 170547]
1989 0.470 [0.4060724]
1990 0.666 [0487,1221]
1991 0.259 [0.2100408]
1992 0.424 [0.3690620]
1993 0.450 [0.3610779]
1994 0.443 [0.3860665]
1995 0.866 [0767,1291]
1996 0.337 [0.2950514]
1997 0.323 [0.2590540]
1998 0.561 [0.5290812]
1999 0.386 [0.3560572]
2000 0.497 [0.4670708]
South 1984 0.272 [0.2490330]
1985 0.448 [0.4250528]
1986 0.238 [0.2130304]
1987 0.400 [0.3670492]
1988 0.255 [0.2270321]
1989 0.273 [0.2420348]
1990 0.456 [0.4210551]
1991 0.323 [0.2700421]
1992 0.301 [0.2640374]
1993 0.521 [0.4580652]
1994 0.432 [0.3850539]
1995 0.355 [0.3020461]
1996 0.524 [0.4650668]
1997 0.486 [0.4300627]
1998 0.450 [0.3930596]
1999 0.361 [0.3280466]
2000 0.313 [0.2850400]

 

TABLE 4. Maximum likelihood estimates of the
shape parameters. Results based on model G
(Table 1). Subscripts N, C, S refer to northern,
central and southern regions (Fig. 1) respectively.

 

 

 

Parameter MIE Estimate
(1 0.0231
BN 528.825
BC 576.088
BS 595.897

 

years, given the differences in inflection points of the
logistic function for different regions, the estimated
sea lamprey wounding rate for 500-mm lake trout
was focused on. Wounds per 500-mm lake trout are
highest in the north and lowest in the south (Fig. 6).
Following the patterns seen in 0, temporal fluctua-
tionsinthewoundsperfishdidnottrackacrossre—
gions. Due to the variability in year-to-year
wounding rates, there did not appear to be a trend of
increasing or decreasing wounding rates on 500-mm
laketroutoverthepastfifteen years inanyofthere-
gions.

DISCUSSION

The model based on the logistic function was
successfully fit to wounding data from Lake Huron,
thereby estimating how sea lamprey wounding rates
on lake trout varied among regions, years, and as a
function of lake trout size. In this application, sepa-

 

 

 

‘1

430 470 510 550 590 630 670 71:
Lemthinmm.

FIG. 5. Predicted wounds on lake trout in north-
ern and central Lake Huron in 1992. The solid
line extends to a length below which 95% of the
observed lake trout lengths occurred, and the
dashed line continues the plot for larger lake trout
MS.

328

n

1996

“f“

1984

,l

2000

 

H14”

1988 1992
Year

Northern Lake Huron

O O
o e

t-I . N

01 N 01

l
Ml

1996

O
H

24*,

1984

h;

2060

Wounds per fish

[ll M

1992
Year

 

 

1988'

Central Lake Huron

 

a‘o‘oo‘44“““‘5
2000

 

1 992 1996

Year

1964 1988

Southern Lake Huron

FIG 6. Maximum likelihood estimates of
wounds per fish for a 500-mm lake trout in north-
ern (MH-I), central (MH-Z), and southern (MH-
3,MH-4, and Mil-5) Lake Huron from 1984 to
2000. Solid line indicates 95% probability interval
determined by MCMC methods.

10

Rutter and Bence

rate wounding rates for large lake trout (0) for each
region and year combination needed to be esti-
mated, but the shape of the logistic function only
varied through region-specific inflection points (B).

The initial expectation was that the wounding
rate on large lake trout (0) would vary among re-
gions and years, but it was expected that the shape
of the logistic function would be fairly constant.
Large temporal and regional variations in parasitic-
phase sea lamprey density (Young et al. 2003) are
one reason large temporal variations in wounding
rates were expected. Although wounding rates var-
ied spatially and temporally, the temporal corre-
spondence between wounding rates and
independent estimates of sea lamprey abundance
(Young et al. 2003) is not a tight one. For example,
the large number of sea lamprey estimated to be
present in Lake Huron in 1993 was not followed by
an increase in wounding rates on lake trout sampled
in the spring of 1994.

Variations in the density of host populations can
also cause variations in wounds per fish, as there is
an upper limit to the total number of attacks a sea
lamprey can make in a year (Koonce et al. 1993,
Bence et al. 2003). Wounding rates can also vary if
the procedures followed when recording wounds
vary over time or geographically (Ebener et al.
2003). In addition, any other factor that influences
either the feeding activity of sea lamprey or the
healing of the resulting marks could cause varia-
tions in observed wounding rates.

The shift of the logistic function to the right
along the lake trout length axis when going from
north to south in Lake Huron (as indicated by in-
creasing value of B) can not be as easily explained.
The shape was originally expected to remain con-
stant because foraging models based on a type-2
functional response predict that the relative magni-
tude of attack rates on different categories of hosts
should not change as host or parasitic-phase sea
lamprey densities change (Bence er al. 2003). For
differences in procedures for recording marks to
produce such an artifactual pattern, the method by
which wounds are recorded would need to change
differently as a function of host size in the different
regions. This seems very unlikely, especially since
all the data for the central and southern region were
collected by the same personnel. One possible ex-
planation stems from the fact that in the northern
part of the main basin of Lake Huron large lake
trout have been at low abundance. Given the low
abundance of large lake trout, sea lamprey might
direct more attacks toward smaller fish, causing the

Estimating Sea lamprey Wounding Rates

inflection point to shift towards a smaller host size
in the north. Such a broadening of diets to include
lower ranked prey when high quality prey are
scarce is a qualitative prediction of optimal diet the-
ory (Pulliam 1974) and has been observed in many
other systems (Dill 1983, Sih and Christensen
2001). The failure to see temporal variations in the
shape of the logistic function is not inconsistent
with this explanation, because temporal changes in
lake trout densities and size compositions during
the 19803 and 19903 have been less pronounced
than the regional differences (Bshenroder et al.
1995). This hypothesis to explain regional differ-
ences in shape of the logistic function needs to be
explored further along with any other potential ex—
planations for the pattern.

The approach suggested herein avoids biases
caused by grouping hosts into arbitrary length bins
and better accounts for the efi'ects of host size and
other sources of variation. Previous efforts to com-
pare annual wounding rates in Lake Huron have
been challenged by low sample sizes for many re-
gion and size-bin combinations and differences in
size compositions within size bins between regions
(Sitar er al. 1997). These problems were encoun-
tered even though Sitar et al. (1997) attempted to
share information among regions and years by esti-
mating wounding rates of lake trout in Lake Huron
using a general linear model that included effects
for length bin, region, year, and associated interac-
tions. They attributed some of the patterns in their
results to exactly the kind of bias this approach
avoids. For example, they estimated lower mean
wounds per fish in MH-l (north) than in Mil-2
(central) for the 534 to 635 mm bin, and attributed
this to the fact that most of the lake trout in this bin
in MH-l had lengths below the midpoint of the bin.
It is instructive to compare the 95% probability in-
terval for wounding rate on SOO-mm fish in central
Lake Huron in 1994 (0.046,0.089) with a 95% con-
fidence interval (0, 0.215) for the wounding rate for
the 432 to 533 mm size bin of lake trout reported by
Sitar (1996) based on these same data. The proba-
bility interval is a Bayesian version of a confidence
interval and should approximate a confidence inter-
val given the diffuse nature of the priors. While the
development of a comprehensive model that allows
information to be shared among years and regions
is certainly partly responsible for the tighter inter-
val, it is clear that the ability of the model to ac-
count for effects of host size within the broader size
categories used in the conventional approach pro-
vides a huge advantage.

329

The approach of Jacobson (1989) is related to the
approach described herein in that he modeled either
the proportion wounded or wounding rate for a
monthly collection of fish as continuous function of
time of year, treating time of year in much the same
way as fish size was dealt with here. Therefore, in
his model be accounted for seasonal changes using
a restricted size range of fish, while for the ap-
proach applied herein effects of size were ac-
counted for using data collected from a restricted

- set of dates. In a situation where wounding data are

available from a number of times during the year, it
might be possible to improve on both approaches
by modeling the joint effects of host size and sea-
son. Eshenroder and Koonce (1984) also discussed
the possibility of summarizing wounding in the
form of parameters of linear regressions of wound-
ing rates versus host size. This analysis can be
viewed as the further exploration and refinement of

, the approach they called for.

11

The lack of significant random effects in this ap-
plication indicates that sea lamprey wounding rates
on lake-trout were homogenous within each region
for a given year. Although the actual process of sea
lamprey attacks on lake trout almost certainly
varies spatially within regions, because both the sea
lamprey and lake trout move extensively (Bergstedt
and Seeleye 1995, Ebener 1998) there appears to be
sufficient mixing to eliminate strong correlations
that could potentially be induced by the sampling
scheme. Thus, while the amount of wounding varies
from year to year and region to region, this analysis
indicates that the estimated rates can be applied to
different locations or samples within a region.

In conclusion, this approach can be strongly rec-
ommended to be broadly applied as a method for
summarizing wounding data. As experience with
this approach is gained, it is suggested that this
technique should replace the reporting of mean
wounding rates for broad size bins, which is cur-
rently the accepted standard in the Great Lakes (Es-
henroder and Koonce 1984). Some movement in
this direction has already occurred, as lake trout
wounding rates used in lake trout assessment mod-
els for Michigan's waters of Lakes Huron and Su-
perior are already estimated using this approach
(Bence and Ebener 2002), and wounding rates esti-
mated through this approach have been used to de-
sign how to assess the success of sea lamprey
control on the St. Marys River (Adams et al. 2003).
As this approach is extended into other areas, and
applied to different host species and/or ranges of
years, the assumed logistic relationship and how the

330 Rutter and Bence

parameters of logistic function vary regionally and
temporally will need to continue to .be checked. Ex-
amination of many data sets with this approach may
reveal new insights regarding sea lamprey parasite-
host interactions.

‘ ACKNOWLEDGMENTS

.We thank those who provided the funding, data,
and advice that made this study possible. Funding
was provided by the Fisheries Division of the
Michigan Department of Natural Resources (par-
tially through support funds administered by the
Department of the Interior, U.S. Fish and Wildlife
Service, Federal Aid in Sport Fish Restoration), and
by the Great Lakes Fishery Commission. Access to
data were provided by the Michigan Department of
Natural Resources, Chippewa-Ottawa Resource Au-
thority, Gavin Christie, Mark Ebener, James John-
son, Shawn Sitar, and Aaron WoldL Gavin Christie,
Mark Ebener, James Johnson, Mike Jones, Shawn
Sitar, and Robert Tempelman provided advice as
the research progressed. Larry Jacobson, Roger
Bergstedt, and two anonymous reviewers provided
very useful comments on an earlier draft of this
manuscript.

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_, Bence, J.R., Taylor, W.W., and Johnson, J.E.
1997. Sea lamprey Wounding Rates on Lake Trout in
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Submitted: 20 December 2000
Accepted: 6 October 2002
Editorial handling: Roger A. Bergstedt

l3

CHAPTER 2

PARAMETER ESTIMATION OF A FUNCTIONAL RESPONSE MODEL
TO IMPROVE ESTIMATES OF SEA LAMPREY-INDUCED MORTALITY

ON LAKE TROUT IN LAKE HURON

Introduction

Prior to the 19505, the lake trout (Salvelinus namaycush) fishery in the Great
Lakes represented one of the largest freshwater fisheries in the world. Annual
commercial yield varied over time and among the Great Lakes, and on Lake Huron
ranged from 1.3 to 3.0 million kg from 1912 through 1940 (Baldwin et al. 2002).
This peak period was quickly followed by a drastic decrease in yield to the point that
lake trout were considered commercially extinct in Lake Huron in the 19508
(Bshenroder et al. 1995).

Before the collapse of lake trout, they were an important top level predator in
the Great Lakes’ ecosystem. The rapid decline in lake trout abundance created large
perturbations throughout the ecosystem and heightened the detrimental influence of
other nonindigenous species. For example, the lack of abundant top level piscivores
may have contributed to the spread and population explosion of alewife (Alosa
pseudoharengus) in the Great Lakes (Smith 1970). Alewife in turn have had negative
effects on other native species (Smith 1970).

The cause of this decline in lake trout sparked a debate in the 19503 that has
continued well into the 19908. Possible explanations for the drastic decline in lake
trout include intense overfishing (Bshenroder et al. 1995), predation by the
nonindigenous parasitic sea lamprey (Petromyzon marinas) (Coble et al. 1990), or the

combination of overfishing and the sea lamprey predation (Hansen 1999). Soon after

14

the collapse of the lake trout fishery, efforts began to reestablish self-sustaining lake
trout populations. Lake trout restoration remains a major goal of fishery management
on the Great Lakes. Ongoing restoration efforts include extensive stocking of lake
trout and control of fishing effort and sea lamprey populations.

While the role of sea lamprey in the collapse of lake trout populations has
been debated, there remains little doubt that sea lamprey are now a major impediment
to lake trout restoration efforts (Bence et al. 2003; Morse et al. 2003). After the
modifications to the Welland Canal in 1919, sea lamprey began colonizing the Upper
Great Lakes and were established by 1947. Sea lamprey predation in the Great Lakes
is the result of “parasitic" juvenile stage sea lamprey that have migrated from rearing
habitats in streams into the lakes where they generally spend up to 18 months feeding
before maturing and returning to Great Lakes tributaries to spawn. These juveniles
preferentially attack large fish, attach to and gnaw through the prey’s skin and feed on
body fluids. These attacks frequently result in the death of the fish that is attacked.
Sea lamprey negatively impact species other than lake trout, including burbot (Lota
Iota) and lake Whitefish (Coregonus clupeaformis).

In 1958, the Great Lakes Fishery Commission (GLFC) began efforts to control
sea lamprey populations in the Great Lakes. Control methods have included both
chemical and biological approaches (see Morse et al. 2003 for details), and the results
of sea lamprey control have been somewhat successful in terms of lake trout
restoration. Self-sustaining populations of lake trout have been restored in Lake
Superior, and reduced sea lamprey abundance has increased the amount of lake trout
spawning stock produced due to stocking in other areas of the Great Lakes. While
stocked lake trout have contributed to a resurgence in recreational fishing throughout

Lake Huron, self sustaining populations have not been observed in Lake Huron

15

outside of an isolated embayment of Georgian Bay (Reid et al. 2001). However, there
has been evidence of natural reproduction of lake trout in the main basin at the Six
Fathom Bank Refuge since 1992 (Ebener 1998).

Currently, the GLF C takes the lead in planning sea lamprey management, with
facilities and support provided by the U.S. Fish and Wildlife Service and the
Department of Fisheries and Oceans, Canada. The approach, referred to as Integrated
Management of Sea Lamprey (IMSL) (Greig et al. 1992), is based on the philosophy
of integrated pest management (Sawyer 1980). The GLFC has worked with state and
provincial fishery management agencies to develop lake trout restoration plans and
lake trout mortality limits, with the aim of reestablishing self-sustaining lake trout
populations. In the upper Great Lakes, current estimates of sea lamprey-induced
mortality are based primarily upon Observed wounds per lake trout of fish sampled in
the spring of each year (Bshenroder et al. 1995, Sitar et al. 1999, Rutter and Bence
2003). In order to predict future levels of sea lamprey-induced mortality, another key
component of IMSL, a sea lamprey/lake trout functional response model is needed to
predict the number of attacks on lake trout.

A functional response model predicts the number of attacks per predator as a
function of prey densities (Holling 1966). Recent models used to predict the number
of sea lamprey attacks on lake trout on the upper Great Lakes have assumed implicitly
or explicitly a Type I functional response (Sitar 1996; Sitar et al. 1999; Lupi et al.
2003; Stewart et al. 2003). A Type I functional response assumes that the number of
attacks per sea lamprey increases in direct proportion to lake trout density. Under this
type of functional response, the number of sea lamprey attacks per lake trout, and
therefore the per capita sea lamprey-induced mortality rate, remains constant

regardless of lake trout density. A more realistic model of attack rates is a Type II

16

functional response (Holling 1966) in which attacks are assumed to take a known
amount of time, therefore limiting the number of attacks that can occur during a
feeding season. As the density of lake trout increases, sea lamprey become saturated
with possible hosts, and the number of attacks per lamprey reaches an asymptote, the
maximum number of attacks per feeding season. The per capita sea lamprey-induced
mortality rate will decrease as lake trout density increases due to the limited number
of attacks per predator being distributed among a greater number of hosts.

Sea lamprey-induced mortality rates will decrease as sea lamprey densities
are reduced, regardless of which functional response model is used to predict attack
rates. To include the effects of increased lake trout densities and saturation on sea
lamprey-induced mortality rate projections, a Type II functional is needed. This type
of functional response is being used for this purpose on Lake Ontario, and although
such a model had previously been used on Lake Superior, no such models are
currently in use or up to date on the upper Great Lakes (Bence et al. 2003).

In this paper, we develop and parameterize an age-structured population
model that utilizes available fishery, survey, and sea lamprey-related data to estimate
lake trout population and mortality levels for the main basin of Lake Huron. This
approach is an extension of previous statistical catch-at-age approaches to lake trout
in southern Lake Huron (Sitar et al. 1999) and includes parameterization of a sea
lamprey/lake trout Type II functional response model (Holling 1966), similar to one
developed as part of the IMSL process (Greig et al. 1992; Koonce et al. 1993; Bence
et al. 2003). By accurately describing the predator-prey relationship of sea lamprey
and lake trout, we can better model lake trout population dynamics and evaluate the
current and future success of lake trout restoration plans. Our work is unique in the

integration of the parameterization of the lake trout assessment model and the sea

17

lamprey functional response in a unified procedure.

18

Methods

Our statistical catch-at—age model (Megrey 1989) includes a lake trout
population submodel and an observation submodel. The population submodel
describes lake trout dynamics over the observation period (1984-1998), whereas the
observation submodel predicts observed data given those dynamics. We developed an
age-structured population submodel for lake trout and estimated model parameters
that maximized the agreement between observed and predicted values for survey
indices of abundance, fishery harvest, fishery effort, and sea lamprey wounding. In
this section we describe the data and annual inputs used in fitting the model, the
population submodel, and the observation submodel.

The main basin of Lake Huron, excluding Saginaw Bay, was broken into three
regions based on statistical districts (Figure 1). Separate lake trout populations were
assumed for Northern Lake Huron (MH-l and portions of OH-l), Central Lake Huron
(MH-2, portions of OH-l, and OH-2) and Southern Lake Huron (MH-3, MH-4, MH-
5, MH-6, OH-3, OH—4, and OH-S). The estimation/assessment model presented here
includes a model of lake trout dynamics and fishing similar to that used by Sitar et al.
(1999) and described in Bence and Ebener (2002). Our model expands on that work
by simultaneously considering the entire main basin of Lake Huron while including
parameters used to estimate sea lamprey-induced mortality.

We fit the statistical catch-at-age model using a likelihood-based objective
function (Foumier and Archibald 1982; Methot 1990; Methot 2000). Formally, our
point estimates maximized the Bayesian posterior density because we made use of
prior information on key model parameters (see below). Our catch—at-age model
covered 1984 through 1998 and recognized lake trout ages 1-14 and an age-15 and

older “plus” group.

19

The population submodel was developed so as to take advantage of the
available data sets while providing insight to key ecological quantities such as sea
lamprey-induced mortality rates. Based on the spatial sampling design and reporting
of the data used, we modeled the main basin of Lake Huron as consisting of three
areas: north, central, and south (see above and Figure 1). Most parameters were
estimated separately for each region (exceptions are noted) and, subsequent to a
redistribution of stocked fish (see below), there was no migration of lake trout
between the areas. The areas were linked by several shared parameters and the
division of a common total population of sea lamprey among the areas.

Observed data and annual inputs

As noted above, the model was fit by comparing observed and predicted
indices of abundance from surveys, fishery harvest and effort values, and sea lamprey
wounding on lake trout and alternative prey. Survey data were collected by the
Michigan Department of Natural Resources (MIDNR) in the spring from 1984-1998.
The surveys were conducted at fixed stations from April through June using graded-
mesh gill nets. Each sampling unit consisted of nine panels that were 30.5 m long,
ranging in stretched-mesh sizes from 51 mm to 152 mm in 13 mm increments (Merna
et al. 1981; Johnson and VanAmberg 1995). All fish sampled in the survey were aged
and measured in length. Since virtually all lake trout in Lake Huron are stocked, there
is very little aging error due to the use of fin clips. Estimates of the mean log-scale
survey catch-per-unit effort (CPUE) and associated standard error were generated for
each year and area using a mixed-effect linear model (see Sitar et. al. 1999 and Bence
and Ebener 2002 for details). Age compositions (proportions at age) for each year of
the survey data was compiled for each of the three regions in the model, including the

number of aged fish used to determine the age composition.

20

Fishery related data used in the model included (separately for the recreational
and commercial fisheries) total harvest, fishery effort, and fishery age-composition.
The MIDNR, Chippewa Ottawa Resource Authority (CORA), and the Ontario
Ministry of Natural Resources (OMN R) all supplied data on commercial and
recreational harvest and effort for the main basin of Lake Huron. Commercial
fisheries included tribal large-mesh and small-mesh gill nets on northern Lake Huron,
and a gill-net fishery in Ontario. Reported harvest from these sources was combined
to obtain the recreational and commercial totals. The tribal fisheries primarily target
lake Whitefish, salmon, and bloater chubs (Coregonus hoyi), but significant numbers
of lake trout are caught also. In Canadian waters, there is a substantial large-mesh
gill-net harvest of lake trout. Harvest information collected from the northern
Canadian statistical district, OH-l, was divided, based on location of harvest, among
northern and central Lake Huron for modeling purposes. Effort data for the
commercial fishery was determined in feet of large-mesh gill-net set per year (the
primary source of commercial fishing mortality). Recreational fisheries in the main
basin consisted of both charter and non-charter fishermen. The total number of fish
caught, as well as angling hours of effort, were determined from mandatory charter
boat reports and creel surveys. For details on which years of data were collected for
the commercial and recreational fisheries in each region, see Table l.

The model was also fit to data on sea lamprey wounding of lake trout and
alternative prey: lake whitefish and Chinook salmon (Oncorhynchus tshanytscha).
Wounding data for lake trout was based on data collected as part of the spring surveys
by the MIDNR and tribal harvest data (see above). Rutter and Bence (2003)
summarized the number of Al-A3 sea lamprey wounds observed per fish in the spring

using a logistic function of lake trout length, and estimated asymptotic wounding rates

21

(observed wounds per fish on the largest lake trout) as well as the logistic curve's
shape parameters. These logistic model parameter estimates and the associated
variance-covariance matrix were used as data here for fitting the catch-at-age model.
Observed sea lamprey wounds-per-fish for all lengths of Chinook salmon and lake
Whitefish in northern Lake Huron were obtained from Bence et al. (2003) and from
unpublished data provided by Mark Ebener (CORA), and the average number of
observed wounds per Chinook salmon and lake Whitefish were determined (averaged
over all years).

In addition to data directly compared with model predictions, our modeling
required other annual inputs. These were the number of lake trout stocked each year,
the abundance of sea lamprey each year, and the abundance of alternative prey. These
annual inputs were assumed known without error. The number of lake trout stocked
provided an indication of the abundance of each year class of lake trout, whereas the
abundance of sea lamprey and alternative prey for sea lampreys was needed for the
sea lamprey functional response, a key feature of our model.

Our use of stocking as an indicator of recruitment is justified because almost
all of the lake trout harvested in the main basin are stocked fish. From data for both
the number of fingerlings and yearlings stocked in each region in each year, the total
number of “yearling equivalents” stocked was obtained by adding to the number of
yearlings stocked to 0.4 times the number of fingerlings stocked for the same year
class (i.e., in the previous year). A “migration matrix” was used to reassign the
yearling equivalents stocked for each region and year to a recruitment location based
on tagging information on where stocked lake trout were recovered (see Bence and
Ebener 2002 for more details). We obtained the needed stocking information from

the United States Fish and Wildlife Service (USFWS), coordinators of data collection

22

for lake trout stocking in both the United States and Canada.

We assumed that the abundance of parasitic phase sea lamprey was equal to
values estimated by Mullet et al. (2003) for the main basin of Lake Huron from trap
catches of adult sea lamprey during their spawning migration. Population level
estimates of Chinook salmon (total for ages 3 and older) in the main basin of Lake
Huron for each year were taken from Dobiesz and Bence (in review); population
abundances of lake Whitefish stocks (total for ages 4 and older) for each year were
based on statistical catch-at-age assessments conducted for 1836 treaty waters (Ebener
et al. in press).

Population model

Detailed equations for our population model are provided in Table 2, and

individual equations therein are referenced as equation T2.Y, and symbols are defined

in Table 4. Numbers at age (a) and year ()2) for the population model are given by:
Na+l,y+l,r =Na,y,rSa,y,r (l)

where S is the survival for age a fish over year y in region r. This survival is

a, y,r
modeled as a function of natural mortality, fishing mortality and sea lamprey induced
mortality. For each region, the lake trout population was estimated for ages 1 to 14,
and a plus group including ages 15 and above, from 1984 to 1998. Age 1 lake trout
for each year and region were determined by taking the results of the stocking
analysis (see above) and multiplying it by a year and region specific post-stocking

survivability (7y,r ). The year was modeled as consisting of two periods during which

fishing and natural (not including sea lamprey mortality) occurred, separated by a fall
pulse of sea lamprey mortality which was assumed to occur after the first nine months
of fishing and natural mortality. Equations T2. 1, T22, and T23 provide the modeled

number of lake trout of each age surviving immediately before the pulse of sea

23

lamprey mortality, immediately after the pulse of sea lamprey mortality, and to the
start of the next year.

Natural mortality was assumed constant over time, with one rate for age-l fish
and another that applied to all other ages of lake trout. Prior information was
provided on these natural mortality rates and they were estimated during model fitting
(see Fitting the catch-at-age model below).

We included distinct commercial and recreational fisheries in each region of
Lake Huron. The instantaneous fishing mortality rates for each region and fishery
were modeled as the product of an age-specific selectivity and a year specific fishery
intensity (eqs. T24 and T25). In their basic forms (which we modified below to
allow for time-varying selectivity), commercial fishing was modeled as a four-
parameter double logistic firnction of age (eq. T2.6, Figure 2) and recreational
selectivity was modeled as a two parameter logistic function of age (eq. T2.7). The
double logistic functional form of the selectivity curve was chosen because gill net
selectivity is known to generally decrease for the oldest and largest fish. We used a
logistic function for the recreational fishery because preliminary attempts using the
double logistic function produced selectivity patterns that reached an asymptote with
age over the age range we were modeling. The fishing intensities for each of the two
fisheries for each region of the lake and year modeled were parameters estimated as
part of the procedure of fitting the statistical catch-at-age model to the data.

A variety of factors, such as changes in length at age, can act to cause
selectivity to change over time. We therefore modified the double logistic and
logistic functions to allow such variation without unduly increasing the number of
parameters that were estimated. Our approach here was to model the inflection point

parameters of the selectivity functions for each fishery and region which were

24

modeled as random walks to allow selectivity to vary over time (eq. T2.8). This
approach allows selectivity to change over time but penalizes solutions where
selectivity changes sharply from year to year. For the commercial selectivity
function, there were two inflection points for each region (one for the increasing
portion of the function and one for the decreasing portion) and thus two random walks
for each region, whereas for the recreational fishery there was just one random walk
per region. For each random walk a vector of n-l parameters were estimated during
the fitting of the statistical catch-at-age model where n=15 is the number of years
used in the model.

Sea lamprey-induced mortality on a given age of fish was modeled as
proportional to the expected number of attacks by sea lamprey per lake trout over the
feeding season with the proportionality constant being the complement of the
probability of a lake trout surviving a given attack (eq. T2.9, Eshenroder and Koonce
1984, Bence et al. 2003). We assumed the probability of surviving a sea lamprey
attack was a logistic function of lake trout weight (one function for the entire lake)
based on laboratory work by Swink (2003) (eq. T2. 10). For a given age lake trout,
weight at age was determined from region-specific weight at age keys generated by
applying a von Bertalanffy grth model to data collected during spring lake trout
surveys (Bence and Ebener 2002).

The number of sea lamprey attacks (A a, y. r ) per lake trout for a given age,

year, and region was predicted from the density of sea lamprey and the density of
each type of sea lamprey prey (each age of lake trout, Chinook salmon and Whitefish)
in that region using a multi-species type II functional response model (Holling 1959,
eq. T2.] 1). This is a saturating function that reaches an asymptote as prey become

dense. The handling time (or times if they vary among prey types) determines the

25

maximum possible attack rates, whereas the effective search rate (or rates if they vary
among prey types) determines how fast attack rates increase with prey densities. See
Bence et al. (2003) for extensive discussion of application of this functional response
to sea lamprey. Current modeled values of lake trout population abundance at age in
each region, just prior to the pulse of sea lamprey mortality, were divided by lake
trout habitat area for that region to determine the lake trout densities used in these
calculations. The abundance of alternative host species, specifically Chinook salmon
and lake whitefish, were assumed known based on species specific population models
(see Observed data and annual inputs) above. Densities of Chinook sahnon were
determined by dividing the lake wide estimate by the amount of lake wide lake trout
habitat, while Whitefish densities were determined on a region-by-region basis. Lake
trout habitat (in square kilometers) that is shared by sea lamprey and lake trout is
defined as depths less than 40 fathoms, and were taken from Ebener (1998) (see Table
5).

The annual sea lamprey population abundances derived by Mullet et al. (2003)
were treated as known and provided spawning run estimates associated with northern
and central streams (sea lamprey production in the south is negligible). We assumed
that some of the sea lamprey that contributed to northern or central spawning runs fed
further south. During fitting of the statistical catch-at-age model we estimated two
parameters that allocated the observed spawning runs of sea lamprey to obtain
numbers feeding in each of our modeled regions. First we estimated a proportion of
sea lamprey observed in northern spawning runs that fed in either the central or

southern region (ML NtoC ), which was assumed to apply to all years. The total

number of sea lamprey feeding in the central and southern main basin was taken as

the sum of these sea lamprey allocated from the northern streams and the spawning

26

run for the central streams. The second parameter (also assumed to apply to all years)

allocated a proportion of these sea lamprey to southern Lake Huron ( ML S ). The sea

lamprey density was determined after this allocation by taking the estimated sea
lamprey population for each region and dividing it by the area of lake trout habitat in
each region (Table 5).

We attempted to estimate several parameters of the functional response during
the model fitting process, whereas other parameters were assumed known. The length
of the feeding season (5'), was assumed to be 150 days, or 0.41 years (Koonce and
Loci—Hemandez 1989). We attempted to estimate handling time by taking advantage
of prior information (see Fitting the catch-at-age model below). In order to obtain
handling times for alternative host species, the lake trout handling time was multiplied

by an adjustment factor for lake Whitefish (H W ) that was estimated during model

fitting, while Chinook salmon were assumed to have the same handling time as lake
trout (H). We assumed that similar sized lake trout and Chinook salmon would have
nearly identical handling times, while smaller lake Whitefish would have shorter
handling times.

Parameters determining effective search rate were also estimated during model
fitting. We assumed that the effective search rate would be a logistic function of host
length, similar to patterns in observed wounding rates (Rutter and Bence 2003, eq.
T2.12). Based on results reported by Rutter and Bence (2003), we allowed the
inflection point of this function to vary among regions but assumed other parameters
applied to the entire lake. A logistic form of the effective search rate was chosen to
best match results from summarizing observed wounding rates and to fit predicted
wounding rates to observed wounding rates (see Fitting the catch-at-age model

below).

27

Submodels for predicting observed quantities

Baranov’s catch equation was used to predict harvest-at-age for the
commercial and recreational fisheries (Ricker 1975). Due to the fact sea lamprey-
induced mortality is applied as a pulse after nine months, the catch equations need to

be applied separately to the two periods to determine annual catch of each fishery

(C E y. r and C5”, , eqs. T213 and T2.14). Total harvest and proportions at age for

each fishery, region and year were calculated based on the predicted catch-at-age.
For each source of harvest information, commercial and recreational,

predicted fishery effort in a given year and region was calculated. As described in

Population model above, for each fishery in each region and year, fishing intensity

was estimated during model fitting. Predicted fishery effort was determined by

dividing the appropriate fishing intensity by catchability, qf" for commercial and qf

recreational, which were also estimated during model fitting for each fishery in each
region. This approach effectively assumes that fishing mortality will be proportional
(up to a multiplicative error, see Fitting the catch-at-age model) to fishing effort.

In order to match CPUE (K) data collected during spring surveys, the
predicted CPUE was calculated for each age, year and region assuming CPUE would
be proportional to actual abundance (equation T2. 15). Our approach here allowed
survey catchability for “fully selected fish” to vary among regions. In addition, we
modeled relative catchability of different ages by a double logistic function (as for the
commercial fishery selectivity above but with parameters that remained constant over
years), with parameters specific to each region. As with harvest, spring survey
proportions at age was calculated in a similar way for comparison with the observed
values.

Parameter estimates describing how wounding rates varied as a function of

28

lake trout size, and how this relationship varied by region and year (from Rutter and
Bence 2003) were treated as input data. In their analysis, Rutter and Bence (2003)
modeled wounding as a logistic function of host length. For the period we are

modeling, they estimated 45 year and region specific asymptotic wounding rates

(6y, ), three area-specific inflection points (,6,) and one overall logistic slope

parameter (a ), and we generated predictions corresponding to these “observations”.
We used predicted wounding rates on arbitrarily large (1200 mm) lake trout (eq.

T2. 16), as our predictions for the asymptotic rates. Calculations of the predicted
shape parameters was more involved.

Although our predicted wounding rates do not exactly follow a logistic
function of length, because the predictions result from the product of two logistic
functions (one for effective search rate and the other for survival), the resulting
function is similar in form to a logistic function. However, the resulting “logistic -
like” function does not have an inflection point that can be found algebraically.
Assuming the probability of survival is nearly linear with length in the region of the
inflection point, we approximated the inflection point for each region by taking a
Taylor series expansion around the inflection point of the effective search rate and
solving for the lake trout length for which the second derivative is equal to zero for
each region (eq. T2. 17). The predicted value of the slope of the “logistic-like”

function of predicted observable wounding rates, é , was determined by averaging the
slope of the predicted observable wounding rate function at ,3, in each of the three

regions.
For alternative prey, the number of predicted observable wounds on lake
Whitefish and Chinook salmon is calculated from the same functional response model

used for lake trout (eq. T2.] 1). Since length based wounding information is not

29

available for alternative prey, we assumed that the average length of Whitefish is 500
mm and 600 mm for Chinook salmon. In addition, we assumed that the effective

search rate for these species would be equal to the product of what the effective

search rate would be for lake trout of this length in the region (2500,, or 1.600, r) and a

species-specific adjustment factor (rs , estimated during model fitting) that is applied

to all regions (eq. T2.18). The predicted number of observable wounds on alternative
prey is the product of the predicted attack rate and the probability of surviving an
attack (eq. T2.10) where mass was assumed to be 1.5 kg for lake Whitefish and 4.0 kg
for a Chinook salmon.
Fitting the catch-at-age model

Detailed equations for fitting the catch-at-age model are provided in Table 3,
and individual equations therein are referenced as equation T3.Y, and symbol
definitions are continued in Table 4. We followed the likelihood based Bayesian
approach of McAllister and Ianelli (1997) and Sitar et al. (1999). Following this
approach, the point estimates of parameters maximize the likelihood given by the
Bayesian posterior distribution of the parameters given the observed data. Such
estimates were termed joint posterior modal (JPM) estimates (e. g., Tempelman 1998)
and are the mode of the posterior distribution of the parameters. The general form of

the likelihood, up to a proportional constant, is

f(6IX)ocf(XI6)p(6) (2)
where f (6 | X) is the posterior likelihood for parameters given the data, f (xlt9)is the
probability density of the data given the parameters, and p(0) is the prior distribution

of the parameters. In this application, there are a number of independent data sources
used to estimate the model: commercial fishery harvest and effort, recreational fishery

harvest and effort, survey CPUE, age compositions from harvests and surveys, and

30

sea lamprey-induced marking data. Given their independence, it is possible to write
the likelihood of the data given the parameters ( f (X | 6)) as the product of the

densities for each component. A similar approach can be used when the prior

distributions of model parameters for natural mortality are assumed to be independent,
allowing for p09) to be a product of the prior distribution of each parameter. By
taking the natural log of the likelihood, we obtain the log-likelihood (L), up to an
ignored constant, for the model as:

L=L1+L2 +L3 +L4 +L5 +L6 +L7 +L3+L9 +L10 +L11+L12 +L13 (3)
where L1 to L13 were the log-likelihood components associated with the commercial

fishery harvest, the recreational fishery harvest, the index of commercial fishing
effort, the index of recreational fishing effort, the index of survey CPUE, the
commercial fishery age composition, the recreational fishery age composition, the
survey age composition, observed sea lamprey-induced wounding rates on lake trout,
observed sea lamprey-induced wounding rates on alternative prey, post-stocking
survivability of yearlings, commercial selectivity random walk, and Bayesian prior
distributions, respectively.

The first five likelihood components (ignoring some constants) are based on
the log-normal distribution (eq. T3. 1) where observed and predicted values for
components 1 through 5 are the annual commercial fishery harvests, annual
recreational fishery harvests, index of commercial fishing effort, index of recreational

fishing effort, and survey CPUE. For each information source, the number of years of

data (Ti, , Table 1) and assumed variability (log-scale standard deviations, 0t, y’, ,

Tables 6 and 7) varies by source and region, and by year for survey CPUE. The
inverse of the log-scale standard deviation acts as a weight controlling how much

each type of data contributes to the overall likelihood.

31

Age compositions were determined for the commercial fishery harvest (L6 ),
recreation fishery harvest (L7), and survey catches (L8 ), and were assumed to follow

a multinomial likelihood distribution. Likelihoods assume that observed proportions
behave as though they were calculated from multinomial samples with specified
effective samples sizes (eq. T3.2). Similar to the inverse of log-scale standard
deviations for the log normal distributions, the effective sample size acts as a
weighting factor for each component of the log-likelihood. In order to prevent certain

years from dominating the likelihood function (Foumier and Archibald 1982), the

maximum effective sample (J i, y, ,. ) was set to 200. In cases where the sample size

was less than 200, the actual sample size was used.

The likelihood component used to compare observed to predicted wounding

rate parameters (the By, , the ,3, and a ) was based on the multivariate normal

distribution (eq. T3.3), with the variance-covariance matrix treated as known based on
the asymptotic estimate obtained by Rutter and Bence (2003).

An additional component of the log-likelihood (L10) for observed wounding

rates includes wounds observed on alternative prey in northern Lake Huron. Based on
observed data in northern Lake Huron, the average wounds per alternative prey
species was determined for the time period 1984-1998 (0.102 wounds per fish for
Chinook, 0.169 for lake Whitefish), and compared to the same calculation of predicted
wounds per alternative prey over the same time period. Wounds on alternative prey
were considered to be normally distributed with a coefficient of variation of 35% (eq.
T3.4).

While fitting the statistical catch-at-age model, we estimated post-stocking

survivability parameters to allow variation in the survival among year classes of

32

stocked lake trout. We penalized the model for deviations in post-stocking
survivability from 100% by treating the deviations as log-normally distributed with a
median of 1.0 and an estimated standard deviation for each region of Lake Huron (eq.
T3.5). The model was also penalized for large variations in the random walks in the
inflection points for commercial and recreational selectivity functions. We treated the
deviations for each random walk coming from a normal distribution with mean zero
and a standard deviation estimated during the model fitting process (eq. T3.6).
Informative prior distributions for natural mortality of age 1 fish for the entire
lake, natural mortality of age 2+ lake trout for each region of Lake Huron, and (for
our initial variant of the model) the handling time were included in likelihood
component L13 to account for available information on these difficult to estimate
quantities For these priors, we used independent log-normal distributions (eq. T3.7).
Final parameter estimates represent a compromise between these priors and fit to the

data since the posterior likelihood includes a “penalty” when parameters deviate from
the prior medians. For age 1 natural mortality (M 1 ), the mean of the prior was

chosen such that the median of the log-normal distribution was 0.8, a value based on

estimates in Grand Traverse Bay, Lake Michigan by Rybicki (1990). Under a log-

normal distribution, choosing 0M1 to be 0.175 ensures that M will fall between 0.53

and 1.20 99% of the time, and creating a large likelihood penalty if the value of M1

exceeds these bounds. For age 2+ natural mortality in each region, the median of the
log-normal prior was 0.162 with a standard deviation of 0.21. These values are based
on previous estimates of natural mortality for older lake trout in the Great Lakes
(Sakagawa and Pycha 1971; Pycha 1980). Handling time was assumed to have a
median of 13.4 days with a standard deviation of 0.21, and in this case the handling

time range of 7.8 to 23.0 days encompasses 99% percent of the prior probability. We

33

did not specify informative priors for other parameters, although parameters were
bounded during the model fitting process. This implies that all plausible values (e.g.
positive abundance estimates, yet not biologically unreasonable) of these other
parameters on the scale they were estimated were considered equally likely a priori.
In this chapter we express uncertainty in the form of marginal distributions of
the parameters of interest taken from the joint posterior distribution of all parameters
calculated using Markov Chain Monte Carlo (MCMC) analysis (Brooks 1998). The
MCMC chain was created using AdModel builder. This software starts chains at
parameter values that produce the maximum posterior density, and takes multivariate
normal steps based on the asymptotic variance-covariance matrix (Otter Research
2000). After an initial bum-in was conducted, every subsequent 2,000th realization
was saved until a chain of length 100,000 was created. The MCMC analysis creates a
posterior distribution of all the parameters estimated during the model fitting process
in the form of a long, multivariate Markov Chain (100,000 saved vectors of
parameters for this study). In addition to providing a check on inferences based on
asymptotic standard errors, these posterior distributions can also provide the basis for
random draws of parameter values for use in stochastic forecasting simulations of lake
trout dynamics. While we do not present the MCMC results for all estimated
parameters in this chapter, these are central to our approach to stochastic forecasting,

and will be discussed further in chapter three.

34

Results

In a preliminary attempt to fit the estimation model to the data, the model did a
good job of matching the observed data, but failed to produce results that would allow
estimation of the asymptotic standard errors to begin the MCMC process (the Hessian
was not invertible). After examining parameter estimates and residuals, we
determined that in some instances, there was not enough information available to
estimate certain parameters. Given the lack of information on the number of young
(ages 1-3) fish in the lake, the model was unable to estimate both age 1 natural

mortality (M 1 ), and post-stocking survivability. By assuming age 1 natural mortality
to be known and equal to the median of the prior distribution on M1 , the estimation

process behaved better. The model also showed no evidence for a random walk in the
commercial fishery inflection points for central and southern Lake Huron. The
random walks were omitted in these two areas, mainly due to the lack of age
composition data for the first five years in the central region and ten years in the
southern region.

We were not able to estimate plausible values for handling time (H), and
ultimately fixed the value of H at an assumed known 11 days (see below for
explanation), and the objective function was modified by removing the component for
the prior distribution for handling time. Our difficulties with estimating handling time
were associated with difficulties in matching observed wounding on alternate prey in
northern Lake Huron. When attempting to estimate handling time, predicted
wounding on the alternate prey did not approach the observed levels until handling
time approached the smallest allowable value. The resulting estimate of handling
time was determined by the bound of what was meant to be an uninforrnative prior.

When handling time is as low as five days, the number of attacks per lamprey over the

35

feeding season was very high. These results greatly contradict lab results (Swink
2003) from both a handling time and growth perspective, as lamprey with such a high
number of attacks would be larger than observed in the lake.

To explore how handling times influenced estimates of other quantities, we
fixed handling time at three different assumed known values (6.8 days, 11 days, and
20 days) and fit the statistical catch-at-age model for each assumed value. We
summarized the results of these analyses by total number of attacks per sea lamprey
on each prey species, and by age-specific sea lamprey-induced mortality on lake trout.
While the total number of attacks on lake trout varies with the assumed handling time,
the greatest impact of the assumed handling time was on attack rates for alternative
prey (Table 8). In particular, attack rates were substantially lower for lake Whitefish
in the north and Chinook salmon in the south when handling time was assumed to be
longer. Apparently, when estimating handling time, the model is able to match
predicted wounding rates on lake whitefish in northern Lake Huron and produces a
“better” model fit in terms of maximizing the likelihood (eq. 3). It should be noted
that in contrast with the results for lake Whitefish, wounding rates on Chinook salmon
are not matched as well with shorter rather than longer handling times. As handling
time increases, sea lamprey-induced mortality will decrease as the total number of sea
lamprey attacks decreases. Assumed handling time only had modest effects on
estimates of sea lamprey-induced mortality of lake trout (Table 9). The effect of
handling time on estimated sea lamprey-induced mortality on lake trout was the most
pronounced in the northern and central regions. In northern Lake Huron, there was a
modest decrease in sea lamprey induced mortality on age 4 and 9 lake trout as
handling time was increased from 11 to 20 days, and a similar effect is observed for

age 9 lake trout in the central region. Combining the results from this sensitivity

36

analysis with current beliefs that indicate handling times are long but can vary
(Koonce and Locci-Hemandez 1989, Bence et al. 2003), led to our decision to assume
a known handling time of 11 days in our final model. This choice represents a
compromise between the handling time estimated by the model (5 days) and the
maximum assumed value of 20 days (Koonce and Locci-Hemandez 1989).

The quality of the model's predictions of the observed harvest and survey data
was acceptable. Predicted total commercial harvest matched observed levels in
northern and central Lake Huron well (Figure 3), while results in the south were
slightly worse. Total predicted recreational harvests were consistent across the entire
lake (Figure 4), with central Lake Huron producing the best results. Given the fact
that four sources of mortality were estimated by the model, the temporal trends in
harvest were matched fairly well. The model had a more difficult time matching
effort data for the commercial harvest (Figure 5) while the observed recreational
harvest (Figure 6) was matched fairly well by model predictions.

In order to summarize the ability of the model to match observed age
composition data in the fisheries and the survey, average age at harvest was
determined for both the observed and predicted values. Model predictions of average
age at harvest for both the commercial (Figure 7) and recreational (Figure 8) fisheries
matched observed quantities quite well, save for the commercial harvest in central
Lake Huron. For some years, observed age composition was not available, but the
predicted values are shown for reference. The model was unable to match the average
age of surveyed fish (Figure 9), especially in central and southern Lake Huron.

Comparing observed sea lamprey-induced wounding rates to predicted
wounding rates is difficult, given the changing wounding patterns and age

composition of lake trout in Lake Huron. One method to quantify the results of the

37

model fitting process is to compare observed and predicted values of the parameters
of the logistic wounding model as described by Rutter and Bence (2003). Initial
examination of observed and predicted asymptotic wounding rates indicated some
substantial lack of fit (Figure 10). The model was unable to fit the temporal patterns
in wounding, but the overall level of wounding for each region was matched with
some success. Goodness of fit can also be assessed by comparing observed and
predicted wounding rates in a given year by length. Example results for 1987 show
that while the asymptotic wounding rate has not been matched, the pattern of
wounding as a function of length has been estimated with some success (Figure l 1).

We estimated that lake trout abundance was highest in southern Lake Huron,
averaging 3.6 million lake trout (Figure 12). However, the southern region also has
the largest area and the density of lake trout was actually lowest in this region (242
per square kilometer), with northern Lake Huron containing the highest density, with
an average of 270 lake trout per square kilometer. Average density in central Lake
Huron was lower than in the other regions (181 lake trout/square kilometer). While
northern and southern Lake Huron have similar densities of lake trout, the age
composition varies among regions, as seen by the levels of spawning stock biomass
(here calculated as age 7+ fish assuming a 1:1 male to female ratio) (Figure 13).
Northern Lake Huron average spawning stock biomass density (kg/sq km) is 2.30,
with an average density of 7. 19 and 29.54 in central and southern Lake Huron,
respectively. Spawning stock biomass is increasing in all regions of Lake Huron
during the final years of the model, 1994-1998.

Average recreational fishing mortality rates for age 5-10 lake trout are low in
all regions of Lake Huron (Figure 14), as are commercial fishing mortality rates in

central and southern Lake Huron. In northern Lake Huron, tribal gill-net fisheries and

38

a commercial gill-net fishery in Ontario account for the high average commercial
fishing mortality rates. Commercial fishing mortality rates are higher in the north
then average sea-lamprey induced mortality rates, except in 1991 and 1992. The
model estimated a spike in sea lamprey lamprey-induced mortality in 1992 in all
regions of Lake Huron (Figure 15). This increase is associated with a large estimate
of spawning run sea lamprey (Mullett et al. 2003).

The overall yearly trends in sea lamprey-induced mortality, including the
spike in 1992, can be seen in the average total annual mortality of age 5-10 lake trout
for central and southern Lake Huron (Figure 16). In northern Lake Huron, where
average total annual mortality is the highest, the high levels of commercial fishing
mask year-to-year trends in sea lamprey-induced mortality. Total annual mortality
includes the model estimate of background natural mortality for age 2+ lake trout
(Table 10). These model estimates of natural mortality are higher than the median of
the log normal prior 0.162, which was based on previous estimates of natural
mortality on older lake trout (Sakagawa and Pycha 1971; Pycha 1980).

The estimated effective search rate (eq. T2. 12 and Figure 17) differs greatly
from the effective search rate proposed in IMSL (Greig et al. 1992). While the IMSL
model was based on a mechanistic model, the functional form of our effective search
rate was chosen to match the observed patterns in the wounding data for lake trout
(Rutter and Bence 2003). The inflection point of the effective search was allowed to
vary for each region, with the smallest inflection point estimated for northern Lake
Huron (Table 11). The inflection point increased in central and southern Lake Huron,
although these results indicate little difference between the two regions. These results
match patterns observed in the wounding data for lake trout in which the inflection

point for the firnction describing the average number of observed wounds per lake

39

trout as a function of length was smallest in the north and increased in central and

southern Lake Huron.

40

Discussion

In this study, we incorporated a sea lamprey/lake trout functional response
model into a lake trout population model for the main basin of Lake Huron. The
population model was parameterized using statistical catch-at-age techniques that
were extended to include a functional response model. We were able to model
commercial fishing mortality, recreational fishing mortality, sea lamprey-induced
mortality and natural mortality. Expanding on previous efforts to model Lake Huron
lake trout (Sitar et al. 1999), the population model utilized data from the entire main
basin of Lake Huron and estimated lake trout populations in northern, central, and
southern Lake Huron.

Improvements in the sumrnarization of sea lamprey-induced wounding rates
on lake trout (Rutter and Bence 2003) allowed us to parameterize a multi-species
functional response model describing attack rates on lake trout by sea lamprey. This
functional response model can be used to predict sea lamprey-induced mortality as a
function of sea lamprey and host prey densities. This contrasts with previous lake
trout assessments that calculated sea lamprey mortality from observed wounding rates
and did not allow for such predictions without additional assumptions about the
functional response. This functional response model can be used by managers to
predict effects of fishery management and lamprey control programs on sea lamprey-
induced mortality and future lake trout population levels (Ebener 1998).

Fishery stock assessment models have become increasingly more sophisticated
in using observed fishery data by utilizing such techniques as virtual population
analysis and stock-synthesis approaches (Quinn and Deriso 1999). In order to
separate sources of natural (non-fishing related) mortality, there is active research in

incorporating functional response models into population models in an effort to

41

estimate predator-induced mortality. Initial attempts at incorporating these types of
functional response models centered on virtual population analysis (VPA) based
fishery models (Mohn and Bowen 1996; Livingston and Jurado—Molina 2000; Tsou
and Collie 2001) utilizing consumption rate information based on stomach content
analysis. Functional response models have been parameterized using a stock
synthesis, statistical catch-at—age approach (Hollowed et al. 2000; Szalai et al. in
review) by comparing predicted consumption to observed consumption of prey, again
based on stomach content analysis. In our approach, we took advantage of the
parasitic nature of sea lamprey coupled with the quantifiable evidence of sea lamprey
attacks via observed wounding rates to parameterize the functional response model.

Patterns in the observed wounding rates (Rutter and Bence 2003) suggested a
shift in the predatory response of sea lamprey when there is a lack of large trout,
especially in the north. In the estimation model we were able to emulate this, while
retaining a type two functional response, by allowing the inflection point of the
function relating effective search rate to prey size to vary among regions. In northern
Lake Huron, the inflection point was at the lowest prey size, in association with the
greatest scarcity of larger prey. We strongly suspect the difference in inflection points
between the northern and other regions results from sea lamprey responding to a
scarcity of large prey by increasing their search for smaller prey.

An increase in search activity due to an increase in prey density can lead to
Type III functional response (Holling 1966) and is sometimes seen for a particular
prey type when predators switch between types that reside in different areas or require
different search strategies (Akre and Johnson 1979; Dale et al. 1994). Essentially,
time spent searching for one type reduces time spent searching for the other. This

may be occurring for sea lamprey feeding on larger lake trout, with effective search

42

rates on large lake trout increasing with the density of large lake trout over some
range of densities. For smaller lake trout the number of attacks will actually decrease
as the density of large lake trout increases.

The parameterization of the functional response model, combined with the
estimated population levels of lake trout and alternative prey in each region in 1998,
indicates that each region of Lake Huron will respond differently to an increase in
lake trout populations. In northern Lake Huron, sea lampreys are saturated with
available prey, mainly in the form of lake Whitefish. If the lake trout population were
to increase (assuming constant sea lamprey densities), the total attacks per sea
lamprey will decrease as the sea lamprey switch from lake whitefish to their preferred
host, lake trout. The handling time on lake trout is longer than the handling time for
lake Whitefish, and therefore fewer lake trout can be attacked by a single lamprey
during the feeding season compared to lake Whitefish. Under this scenario, sea
lamprey-induced mortality in the north will decrease slowly as the lake trout
population increases (Figure 18). In central and southern Lake Huron, the average
number of attacks per lamprey in 1998 was 9.8 and 10.1, respectively. Assuming a
handling time of 11 days, the maximum number of attacks per lamprey is 11.4.
Therefore, sea lamprey in central and southern Lake Huron are not yet saturated with
available hosts, and the total number of attacks per lamprey will increase as lake trout
densities increase. However, due to the Type II functional response, the increase in
attacks per lamprey is not linear, and sea lamprey-induced mortality rates on lake
trout will decrease as lake trout abundance increases in central and southern Lake
Huron.

As is often the case when fitting fishery models to data, lack of critical data

imposed limitations on what could be estimated. This was particularly true here for

43

the functional response parameters. As shown above, handling time plays an
important role in determining the number of attacks per lamprey. We were unable to
estimate a plausible handling time based solely on data used to fit the statistical catch-
at-age model, and a handling time of 11 days was assumed instead. As described
above, an important aspect of determining handling time is how attacks are allocated
to alternative prey, Chinook salmon and lake Whitefish. The limited data for
wounding rates on alternative prey is a concern, as the number of sea lamprey attacks
absorbed by Chinook salmon and lake Whitefish are quite high. In order to effectively
understand the role of alternative prey, yearly surveys designed to collect wounding
data on alternative prey are needed, as well as more laboratory research that emulates
the current work being done on lake trout with regards to handling time and
probability of survival (Swink 2003). This lack of data makes it difficult to accurately
model the distribution of sea lamprey attacks across all prey types, including lake
trout. The lack of fit associated with some functional response related quantities may
also be attributed to the assumption that sea lamprey and alternative prey populations
were assumed known. By incorporating into the model fitting process estimates of
uncertainty about these population estimates, it may be possible to better match
observed wounding data.

The estimation process did an adequate job of matching observed quantities to
values predicted by the population model. Predicted quantities that are based on
parameters that are estimated separately for each region of Lake Huron, such as
annual commercial and recreational harvest, matched observed data reasonably well.
The model had a more difficult time matching predicted quantities based on
parameters linked across the entire lake, such as those based on observed sea lamprey

wounding rates. Many of the parameters associated with the functional response

model are estimated lake wide, including the asymptote of the effective search rate
and adjustments for alternative prey. The model was forced to match wounding rates
under a variety of predator and prey densities, and did not match observed data as
well as those predicted quantities that were based on region specific parameters.
However, we feel that we have parameterized a functional response model that is
applicable under a wide range of sea lamprey and lake trout densities.

We believe that our study has provided a valuable tool for assessing the future
of lake trout rehabilitation in Lake Huron. The parameterized functional response
model could allow managers to determine sea lamprey-induced mortality rates under
a variety of management scenarios. The main factors in these scenarios are future lake
trout and sea lamprey populations, and, to a lesser extent, the populations of
alternative prey. For example, it would be possible to explore how changes in lake
trout stocking levels influence the mortality they suffer from sea lamprey, and their
future spawning stock size, given status quo control levels for sea lamprey or

alternative levels of control.

45

 

 

 

 

 

 

 

 

Table 1 : Years in which data sources are available for the commercial and
recreational harvest in Lake Huron for each region.

Data Region Years

Commercial yield North 1984-1998
Central 1 984-1998
South 1984-1998

Commercial effort North 1984-1998
Central 1984-1998
South 1984-1998

Commercial age composition North 1984-1998
Central 1990-1998
South 1994-1998

Recreation yield North 1985-1998
Central 1985-1988,1991-1998
South 1 986-1 998

Recreation effort North 1985-1998
Central 1985-1988,1991-1998
South 1 986-1998

Recreational age composition North 1985-1989,1991-1992,1994-1998
Central 1985-1988,1991-1998
South 1985-1997

 

 

 

46

 

Table 2: Model equations describing the catch-at-age model.

C
9 '(Ma y,r+Fa,,ayr+F a,,yrllg2
Na'y,=Na,y’,e

L
9* 9 -Ma, ,r
Nanyera.ey,r y

C R 3

N — N9*e
a+l,y+l,r - a ”We

C _ C C
Fa,y,r — Sa,rfy,r

R _ R R
Fa,y,r — Sa,rfy,r

 

 

 

SC = 1 + 1— 1

ar l+e"wlr(a+w2r) 1+ ‘w3r(a+w4r)
1

SR _

a,r 1+e’aRr(a"flR,r)

fijar = [Bji—lJ + 537m
Méyfi A,,1-ayr( -P(a))

l
P =
s (a) 1 + el.462—.0004lw(a)

 

A : Slhost , rL
has! ,W all hosts

1+ ZHiAlJ‘NflyJ‘

i

y,r

 

 

 

6
,1 = i
1”. 1+8 ‘alil —fl}.,r)
9
C
F ’ZaH—yr
C9 a, ,r
Cauyr= 'y l—e 12 Nauyr
a,y,r

47

(T21)

(T22)

(T23)

(T24)

(T25)

(T26)

(T27)

(T28)

(T29)

(T210)

(T211)

(T212)

(T213)

Table 2 (cont’d)

 

 

 

' 3
C __ _
F Za,y,r
C _ C9 a-Y-r 9*
Ctr,y,r'—Ca,y,r+ 1 1‘3 12 Na,y,r
a,y,r
__ S S
Ka.y.r " qr Sa,y,rNa,y,r
W = 511200,r1«y,r
[200’ y ,r allhosts
1+ ZHl’liJNLyJ'
i
2 2 2
c _ _ '4bps ”apsa). film -bpsal film
r 2
a}. (aps +bp316/1J)
’13,): = 7311;

48

(T214)

(T215)

(T216)

(T217)

(T2.l8)

Table 3: Likelihood equations used to fit the catch-at-age model.

K — -2\

ln[X{’y‘r—]
X- ,
2:2 Tuln __1__ 25 _2;_

r 27:03,)“, y 0.13%!“

 

 

 

 

 

Li = Z ZJi,y,rZPi.,a,y,r ln(Pi,a,y,r)]

f y a

1 _ _ T -1 _ a
“—(X-fll ’3 (JV—ll)
L9 =ln (2n)_m/22_1/2e 2

 

 

 

 

 

1 2010.
L — ln ———-e ”
10 z ,_—2 2
l ”(710,;
f - —Z)
In _l’y,r
L l 1 5 1
: n _ _.
11 2 2,2 5,,
yr \lmllJ ”’
1 - l l
l i 2
— (3
2 (w)
—l/2 25
L12 = 2111 (27128122,) e 12,r
y,r,i

 

 

 

 

49

(T31)

(T32)

(T33)

(T34)

(T35)

(T3 .6)

(T3.7)

Table 4: List of variables and parameters used in the model

Aa,y,r
Ps(a)

Year (1984-1998)

Region of Lake Huron (north, central or south)
Age offish (1-15)

Number of fish of age a in year y and region r

Survival rate for a fish of age a in year y and region r

Post stocking survivability for year y and region r

Number of age a fish after nine months in year y and region r

Natural mortality rate for a fish of age a in year y and region r
Commercial fishing mortality for a fish of age a in year y and region r
Recreational fishing mortality for a fish of age a in year y and region r

Number of age a fish afier nine months and a pulse of sea lamprey-
induced mortality in year y and region r

Sea lamprey-induced mortality rate for a fish of age a in year y and
region r

Commercial fishing selectivity for a fish of age a in region r

Commercial fishing intensity for a fish in year y and region r

Recreational fishing selectivity for a fish of age a in region r
Recreational fishing intensity for a fish in year y and region r

One of four (n=l,2,3,4) shape parameters for model of commercial
fishing selectivity in region r
Shape parameters for model of recreational fishing selectivity in region r

Inflection point of selectivity for fishery i (commercial or recreational) in
year y and region r

Random walk component for inflection points in selectivity models for
fishery i (commercial or recreational) in year y and region r

Number of sea lamprey attacks on a fish of age a in year y and region r
Probability of a fish of age a surviving a sea lamprey attack

Weight of a fish of age a in grams

Length of the feeding season in years
Effective search rate of sea lamprey for the host species i of interest in

region r

Sea lamprey density for a given year y and region r

Handling time in years for lake trout and Chinook salmon

Percent of sea lamprey that migrate from Northern to Central/Southem

Lake Huron
Percentage of sea lamprey in Central/Southem Lake Huron located in the

South

50

Table 4 (cont’d)

Hw
l

9,1
all
18).,r

C9

a,y,r

pa

~fi '3‘: “"1

h ° h

~13 v
‘

'9
{<
‘1

3S.
‘3:
‘I

>.<

:U‘ ~$~
i<
N

in
‘2:
‘1

:0
3:
‘<
\

H1 1:1

Handling time in years for lake Whitefish

Length of fish in mm
Asymptotic effective search rate

Shape parameter for logistic function of effective search rate
Inflection point for logistic function of effective search rate in region r

Commercial catch in the first nine months of the year for age a fish in
year y and region r

Summation of natural, commercial and fishing mortality for age a fish in
year y and region r

Total commercial catch for age a fish in year y and region r

Catchability for commercial and recreational fisheries, respectively, in
region r

Catch per unit effort for spring surveys for age a fish in year y and region
r

Catchability for spring surveys in region r

Selectivity for spring surveys for age a fish in year y and region r

Observed asymptotic wounding rates in year y and region r

Observed inflection point for wounding rates in region r

Observed slope parameter for wounding rates
Predicted number of wounds on a 1200mm lake trout

Predicted inflection point of the wounding rate in region r
Intercept of the linear approximation of the probability of survival

function
Slope of the linear approximation of the probability of survival function

Effective search rate for alternative prey species 3 in region r
Adjustment to the effective search rate for alternative prey species 3
Likelihood component i

Number of years of data for data source i in year y

Variability of data source i in year y and region r

Observed value of interest for data source i in year y and region r
Predicted value of interest for data source i in year y and region r
Sample size of data source i in year y and region r

Observed proportion at age a for data source i in year y and region r
Predicted proportion at age a for data source i in year y and region r

Vector of observed wounding rate parameters
Vector of predicted wounding rate parameters

51

Table 4 (cont’d)

2 Covariance matrix for observed wounding rate parameters

w; Predicted number of sea lamprey wounds on alternative prey i
w,- Observed number of sea lamprey wounds on alternative prey i
§,-,, Predicted standard deviation for data source i in region r

6} Prior mean on parameter i

6,. Estimated value of parameter i

09,- Prior standard deviation on parameter i

52

Area of lake trout habitat (<40 fathoms) in km2 for each region of Lake

 

 

 

 

 

 

Variability (oin’, ) of data for log-normal likelihood components (assumed

 

 

 

 

 

Table 5:
Huron (Ebener 1998).

Mon km2

North 3,432

Central 4,575

South 1 5,210

Table 6:

constant across all year) estimated from the data.

Data Region Standard Deviation

Commercial Harvest North 0.15
Central 0.15
South 0.15

Recreational Harvest North 0.3
Central 0.14916
South 0.1 5

Commercial Harvest North 0.15

Effort Central 0.15

South 0.15

Recreational Harvest North 0.13

Effort Central 0.06991

South 0.15

 

 

 

53

Table 7: Yearly variability (0”,, ) of the log-normal likelihood component of

survey CPUE based on mixed-effect linear model analysis. An entry of
zero indicates no data for that year.

 

Region

Year North Central South
1984 0.84 0.96 0.40
1985 0.84 0.96 0.43
1986 1.37 0.96 0.42
1987 1.37 0.85 0.42
1988 1.87 1.01 0.40
1989 1.37 1.04 0.38
1990 0.00 1.01 0.42
1991 1.37 0.90 0.43
1992 0.60 0.91 0.45
1993 0.82 1.03 0.45
1994 0.91 0.92 0.45
1995 0.81 0.83 0.31
1996 1.08 0.80 0.
1997 0.91 0.83 0.43
1998 0.82 0.83 0.37

 

 

 

 

 

Table 8: Predicted average number of attacks per lamprey (1997-1998) for assumed
handling times of 6.8, 11, and 20 days.

 

 

 

 

 

 

 

Handling Time
Region 6.8 Days 11 [Lays 20 Days
Nonh
Lake Trout 1.11 1.19 1.89
Whitefish 26.60 14.46 1 .66
Chinook 3.29 3.14 3.63
Total 31.00 18.80 7.19
Central
Lake Trout 5.46 4.41 3.04
Whitefish 0.09 0.04 0.00
Chinook 7.00 5.45 3.36
Total 12.55 9.90 6.40
South
Lake Trout 7.53 5.33 3.41
Whitefish 0.05 0.02 0.00
Chinook 6.21 4.96 3.10
Total 13.79 10.31 6.51

 

54

Table 9: Predicted average sea lamprey-induced mortality rates (1997-1998) for age
4 and age 10 lake trout for assumed handling times of 6.8, 11, and 20 days.

 

 

 

 

 

 

 

Handling Time

Egion 6.8 Days 11 Days 20 Days
Nonh

Zl-4 0.135 0.126 0.085
ZI-9 0.428 0.420 0.348
Central

Zl-4 0.000 0.000 0.000
ZI-9 0.317 0.275 0.234
South

214 0.027 0.024 0.021
Zl-9 0.236 0.218 0.203

 

Table 10: Estimated background natural mortality rates for age 2+ lake trout for each
region of Lake Huron.

 

 

mien Natural Mortality]
North 0.2
Central 0%
South 0.2

 

 

 

Table 11: Estimated parameters for the effective search rate for sea lamprey (eq. 12).
95% Bayesian prediction interval based on MCMC analysis.

 

 

 

 

 

 

Parameter Estimate 95% Prediction Interval

91 0.70 (043,166)
at 0.0983 (00810.0.1151)
18,1,North 498.67 (492.32.503.39)
filfientral 562.32 (556.15,566.26)
@130th 548.62 (537.34.555.87)

 

55

FIGURES

Figure 1: Map of Lake Huron with model regions based on the statistical districts listed
(Smith et al. 1961). Northern Lake Huron consists of MH-l and portions of

OH-l. Central Lake Huron consists of MH-2, portions of OH-l, and OH-2.
Southern Lake Huron consists of MH-3, MH-4, MH-5, MH-6, OH-3, OH-4,
and OH-S. Dotted line indicates boundary between Canada and United States.

*‘M
. z . fl' .1 "
orthern. }

‘ N
'0
MH-1 '~. 011-1
0..
Central
'- 011-2
Ml-2

Michigan

MI-3 Southern
M14 ‘. 011-3

{—1 Ontario

 

56

Figure 2: Example of a double logistic selectivity curve estimated by the model for
commercial fishery selectivity and spring survey selectivity.

 

0.8 ~

0.6 -

Selectivity

0.4 ~

0.2 *

 

 

Age of fish

57

Figure 3: Total commercial harvest (in lOOOS of lake trout) for Lake Huron comparing
observed harvest to harvest predicted by the model for each region of Lake
Huron for 1984-1998. Note the difference in scale for each region.

Northern Lake Huron

140
130
120
110a

  

 

100d

90 _ [2] Observed

9 Predicted

 

 

 

80—
705
60~ ‘
50‘ DU
40 r T I r . I . . . r , , r 1

84 85 86 87 88 89 90 9192 93 94 95 96 97 98
Year

  

Number of lake trout (10005)

 

 

 

Central Lake Huron

 

 

 

 

 

 

 

To‘
0
O
S
‘5
E [II Observed
E 9 Predicted
“6
8
E
3
z

0 l l l l l l H l l V I l l l l

84 85 8 87 88 89 90 9192 93 94 95 96 97 98

Year

58

Figure 3 (cont’d)

Southern Lake Huron

 

«h 43 01 01 O)
0 0| O 01 O
1 1 1 1 1

 

 

[2] Observed

35 .
0 Predicted

 

 

 

30
25‘ ‘
20—
15-

10 V l l l l I f l l T l l l l

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

  
 

Number of lake trout (10005)

 

 

59

Figure 4: Total recreational harvest (in 10005 of lake trout) for Lake Huron comparing
observed harvest to harvest predicted by the model for each region of Lake
Huron for 1985-1998. Note the difference in scale for each region.

Northern Lake Huron

 

 

 

 

 

Number of lake trout (10005)

 

 

 

 

 

 

 

 

 

[:1 Observed
0 Predicted
0 I l T T I I l T l l l l l l
85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year
Central Lake Huron
16" D
’13
O
O
S
‘5
.8
a, D Observed
E 0 Predicted
“5
3
E
:3
z
0 l I l l l l l l l T l l I l
85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

60

Figure 4 (cont’d)

Southern Lake Huron

01 01 CD
O 01 O
1 1 1
El
Cl

 

A
U!

 

00 -h
01 O
1 1

El Observed
0 Predicted

 

307

 

 

 

Number of lake trout (10003)

 

 

AANN
01001001
1111

 

l l l I l l l l l l l l *1

85 86 87 88 89 90 9192 93 9 95 96 97 98
Year

0

61

Figure 5: Commercial effort for Lake Huron in 10008 of meters of large-mesh gill-net set

10005 of meters

1000s of meters

 

per year comparing observed effort to effort predicted by the model for each
region of Lake Huron for 1984-1998. Note the difference in scale for each

region.

Northern Lake Huron

 

[:1 Observed
. 0 Predicted

 

 

 

 

 

2500 I T l l l l l l T l 7 l l l 7

84 85 86 87 88 89 90 9192 93 94 95 96 97 9

Year

Central Lake Huron

 

2750 _

2500 -

2250 a

2000 7 El Observed
1750 a 0 Predicted

 

 

 

1 500

 

 

 

1250-
1000 T l i l l l l l I l l T T l
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

62

Figure 5 (cont’d)

10003 of meters

Southern Lake Huron

  

 

 

Cl Observed
9 Predicted

 

 

 

 

0.9 - . 1:1
0.8 . 1:1 D
_ 1 ' :1
0.7 :1
0.6 T l l T l l i l l l l l l l
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

Year

63

 

Figure 6: Recreational effort for Lake Huron in 1000s of angling hours comparing

10006 of hours

1000s of hours

observed effort to effort predicted by the model for each region of Lake Huron
for 1985-1998. Note the difference in scale for each region.

Northern Lake Huron

170~
160—
150n
1405
130-
120‘
110*
100
90
80
70
60

 

 

 

El Observed
0 Predicted

 

 

 

501 l l l l T l l l l

85 86 87 88 89 90 91 92 93 94 95 96 97 98

Year

Central Lake Huron
275—
2504
225-
200a
175-
150-
125-
100

 

 

 

El Observed
9 Predicted

 

 

 

75 l l T T l T l T j

85 86 87 88 89 90 91 92 93 94 95 96 97 98

Year

64

 

 

Figure 6 (cont’d)

10005 of hours

1300
1200a
1100‘
1000—
900~
8005
700—
6005
500~

 

Southern Lake Huron

 

 

400

T

l

T

T

T

l

85 86 87 88 89 90 91

Year

I T l T T T l

92 93 94 95 96 97 98

65

 

 

[3 Observed
0 Predicted

 

 

Figure 7: Average age at harvest in the commercial fishery for Lake Huron comparing

observed average age to average age predicted by the model for each region of
Lake Huron for 1984—1998.

Northern Lake Huron

 

El Observed
0 Predicted

 

 

 

Age in years

 

 

 

3 T T T T T T T T T T T T T l

84 85 86 87 88 89 90 9192 93 94 95 96 97 9
Year

Central Lake Huron

 

[J Observed
9 Predicted

Age in years

 

 

 

 

[:1

 

 

3 T T T T T T T T T T T T l

T
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

66

Figure 7 (cont’d)

Southern Lake Huron

 

[:1 Observed
0 Predicted

 

 

 

Age in years

 

 

 

3 I T T T T T T T T T T T T T

84 85 86 87 88 89 90 9192 93 94 95 96 97 9
Year

67

Figure 8: Average age at harvest in the recreational fishery for Lake Huron comparing
observed average age to average age predicted by the model for each region of
Lake Huron for 1985-1998.

Northern Lake Huron

 

El Observed
6 Predicted

 

 

 

 

Cl

 

 

T T T T T T T T T T T T

T
85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

Central Lake Huron

6.5 -

 

. D,» 1:] Observed
' 6 Predicted

 

 

 

 

Age in years
01

at 01 a:
1 l

  

 

4.5
4
3.5
3 l l T 1 T l T T T T T T T l
858 878 89909192939495969798
Year

68

Figure 8 (cont’d)

Age in years

Southern Lake Huron

 

 

3 V T T T T T T T T l T j T
85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

69

 

 

El Observed
9 Predicted

 

 

 

Figure 9: Average age in the spring graded-mesh gill-net survey for Lake Huron
comparing observed average age to average age predicted by the model for
each region of Lake Huron for 1984-1998.

Northern Lake Huron

 

[:1 Observed
0 Predicted

 

 

 

 

     

   

*E]
3 l T T T T T T T T T T T T T l
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

   
   

     

 

Central Lake Huron

 

El Observed
0 Predicted

 

 

 

 

Age in years

 

 

3 7 I T T T T T T T T T T T T l

84 8 86 87 88 89 90 9192 93 94 95 96 97 98
’ Year

70

Figure 9 (cont’d)

Southern Lake Huron

 

 

 

 

 

 

 

1’.’
8
2‘ El Observed
"' 0 Predicted
85
<

4 _.

3.5 a
3 T T T T j V T T T T T T T T
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

Year

71

Figure 10: Observed asymptotic wounding rates from Rutter and Bence (2003)

Average wounds per lake trout

Average wounds per lake trout

0.8 J

 

  

compared to the asymptotic wounding rates predicted by the model for each
region of Lake Huron for 1984-1998. '

Northern Lake Huron

 

 

El Observed

 

 

 

 

  
 

 

 

 

 

 

0.6 E] 0 Predicted
0.4 ~ El
0.2~ E] El
0 I I I I I I I I I I r j V I
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year
Central Lake Huron
2.25
2
1.75 D
1.5-
1.25 ~ [I Observed
11 O’Fhedkfied
0.75- ‘y  
II]
0.54 C] E]
0.254
O I I I I I T I I I I I I T l

 

84 85 86 87 88 89 90 91

92 93 94 95 96 97 98
Year

72

Figure 10 (cont’d)

Southern Lake Huron

1.75-
1.58

1.25-

 

 

III Observed
0 Predicted

 

0.75 *

 

 

  

0.5 —

Average wounds per lake trout

0.25 ~

 

 

0 I I I I I I I I I I I T j

I
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

73

Figure 11: Observed wounding rates from Rutter and Bence (2003) for 1987 in each
region of Lake Huron compared to wounding rates predicted by the model for
1987. Note the difference in scale for each region.

Observed wounds per fish

Observed wounds per fish

0.40 —
0.35 ~
0.30 ~
0.25 e
0.20 ~
0.15 —
0.10 e
0.05 -*

Northern Lake Huron

 

I I Predicted
m Observed

 

 

 

 

 

0.00 d
400

0.70 -
0.60 ~
0.50 —
0.40 ~
0.30 I
0.20 -
0.10 I

 

0.00 __....__. "

400

600 700 800
Length

Central Lake Huron

 

' I Predicted
-—-- Observed

 

 

 

 

 

74

Figure l l (cont’d)

Southern Lake Huron

 

 

 

 

 

 

5 0.50 1
h:
3 0.40 — ”_va
'O ,1" l .l-.
§ 0.30 ~ .1 . ' - Predicted
3 0.20 - I"?! ' -—~—- Observed
8 /
E 010 ~ --
g I ”Ii/f"
8 0.00 luv—1r .4 . . .

400 500 600 700 800

Length

75

Figure 12: Total number of lake trout (in 10008) age 1 to age 15 predicted by the model
for the three regions of Lake Huron from 1984-1998.

Total Number of Lake Trout in Lake Huron

5000 n
4500 4 ~ \ .. .. g ,

4000 a \ / \

3500~ 3‘ ../\\ .. "~\
3000+ V -. x/ a \ North
2500 ~ ‘ \ Central
2000 ~ 's South

1500-
1000-
5004

0 I I I I I I I I I I I I I I

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

 

 

 

 

10005 of lake trout

 

 

 

76

Figure 13: Predicted spawning stock biomass (biomass of age 7+ female lake trout

10003 kg of lake trout

700-
650~

600 a -

550-
500-
450-
400-
350-
300*
250-
200-
150—
100~
50+
0

 

84 85 86 87 88 89 90 91

assuming 50% gender ratio) in 10003 of kg of lake trout in three regions of
Lake Huron from 1984-1998.

Spawning Stock Biomass

 

\ North
.. \ Central
\ 0° °.‘ SOUth

 

 

 

QWN’

'T'fIIIIITri

92 93 94 95 96 97 98

1 "l r I

Year

77

Figure 14:

Instantaneous mortality rate

Instantaneous mortality rate

 

0.1 r
0

 

84858

1.1“
1..

0.9 r.

0.8 -
0.7 —
0.6 ~
0.5 ~
0.4 -
0.3 r
0.2 r
0.1 a

 

iB‘D-uucvl:--uoo

Predicted average instantaneous mortality rates due to commercial fishing,
recreational fishing, and sea lamprey on age 5-10 lake trout in each region of
Lake Huron from 1984-1998. Note the difference in scale for each region.

Northern Lake Huron

 

\ Corn
\ Rec
.\ Lamp

 

 

 

 

......qq-.. "-"- ' ..
“...—Ifi‘ --...wW-gw-i—u-I—d --.-..._ .. ._ __ _ .‘ ...- o o ‘
“ m "a- “...-... .4."—

...L.-.._....-........_-.- T I I I I I I I

I I I I I I
87 88 89 90 91 92 93 94 95 96 97 98

Year

Central Lake Huron

 

\ Com
\ Rec
.‘s Lamp

 

 

 

W m‘_“~

 

0

84 85 86 87 88 89 90 91

.... ._ ,, " ___-
j I I I I I I I I I I

92 93 94 95 96 97 98

j I I

Year

78

Figure 14 (cont’d)

Instantaneous mortality rate

0.9 4
0.8 a
0.7 4
0.6 -

0.5

0.4 e
0.3 a
0.2 r
0.1 3

 

 

 

 

0

I

I

I

I

Southern Lake Huron

84 85 86 87 88 89 90 91

Year

79

92 93 94 95 96 97 98

 

 

\ Com
\ Rec
”s Lamp

 

 

Figure 15: Predicted average sea lamprey-induced mortality rates on age 5-10 lake trout

Instantaneous mortality rate

Instantaneous mortality rate

for each region of Lake Huron from 1984-1998. The 95% Bayesian
prediction interval of the marginal distribution is shown, with the median of
the marginal indicated by the square.

Northern Lake Huron

1.4 -
1.2 1

0.8—I
3:3: liliTIT

 

 

 

 

0.2I
o . 4 . . . . 1 . . . . . T . 1
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year
Central Lake Huron
1.4-
1.2—
1i I T
0.84
0.6“ i E 11— E ‘1'
0.4~ i E i i i i E i
0.2-
0 . . r . T i T I . . . . . . 4
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Year

80

Figure 15 (cont’d)

Southern Lake Huron

 

$21.4-
3.1%

E 10 I

O

808*

305. E

§04 i i I i I i I i

m o

(O

E 0 I I I I I I I I I I I I I I

 

848586878889909192939495969798

Year

81

Figure 16: Predicted average total (natural, sea lamprey, commercial and recreational

90% n

60% -

50%

40% a
30% r
20% r
10% ~

 

800/0 “W
70% ‘\ /\
. ’.
"a f l 0

fishing) annual mortality on age 5-10 lake trout for each region of Lake Huron
from 1984-1998.

Total annual mortality

 

I. "3’ 1‘
\Auwti‘s . .-'“' I” ‘9 "III-T ‘7‘”: I/Z . I \\\\ 4‘/“ \ Noah
\\-,’ ’ \-:"\.xr’/.\“~--. \ Central
. \ SOUth

 

 

 

 

0%

84 85 86 87 88 89 90 91

I I I I I I I I I I I I I I

92 93 94 95 96 97 9
Year

82

Figure 17: Estimated effective search rate (eq. 12) for central Lake Huron as a function
of lake trout length with the IMSL effective search (Greig et al. 1992) given

Effective search rate

for reference.

Efective Search Rate
0.8 -
0.7 e
0.6 4
0.5 e
0.4 4
0.3 —
0.2 a
0.1 4 ..--r-

0 I I
400 450 500

 

 

I I I

550 600 650 700
Lake trout length (mm)

83

750

 

\ Estimated
\ IMSL

 

 

 

I

I
800 850

I

Figure 18:

Sea lamprey-induced mortality

Sea lamprey-induced mortality

9
a:
1

.0
01
1

.o
1:.
l

_O
03
1

P
N
1

.0
u—L

.0
O)
1

.0
01
1

p
h
1

.9
OJ
1

.0
N
1

9
-L

 

Sea lamprey induced-mortality rates as a function of the percent increase in
lake trout population based on 1998 levels of sea lamprey, lake trout and
alternative prey.
Northern Lake Huron
Age 7 lake trout

 

0%

I j

I I I I I I I T
10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Percent increase in lake trout

Central Lake Huron
Age 9 lake trout

 

 

 

0%

j

I I I I I I I I I
10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Percent increase in lake trout

84

Figure 18 (cont’d)

Southern Lake Huron
Age 9 lake trout

.0
CD
1

.0
01
l

.0
.p.
1

.0
00
l

 

Sea lamprey-induced mortality
9
N
l

 

 

0.1 T I I I I I I I I I
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Percent increase in lake trout

85

CHAPTER 3

PROJECTING THE OUTCOMES OF LAKE TROUT RESTORATION
EFFORTS IN LAKE HURON: INCORPORATING A SEA LAMPREY

FUNCTIONAL RESPONSE AND ACCOUNTING FOR UNCERTAINTY

Introduction

Lake trout (Salvelinus namaycush) historically were an important top level
predator of the Lake Huron fish community, and supported the largest commercial
fishery in the Great Lakes. Following their collapse in the 19505, lake trout were
considered commercially extinct in Lake Huron (Eshenroder et al. 1995) and lake trout
restoration became a goal in Lake Huron and throughout the Great Lakes. Possible
explanations for the rapid decline of lake trout include overfishing (Eshenroder et a1.
1995), predation from the nonindigenous sea lamprey (Petromyzon marinas) (Coble et al.
1990), or a combination of both factors (Hansen 1999). Sea lamprey colonized the upper
Great Lakes by 1947 after migrating through the St. Lawrence River and the Welland
Canal. Following the collapse of lake trout populations, a large scale program of
stocking hatchery-reared lake trout was undertaken, combined with programs to control
sea lamprey, in an effort to restore self-sustaining populations.

Although the sea lamprey control program began in 195 8, control efforts are still
ongoing and sea lamprey predation continues to be a hindrance to successful lake
rehabilitation in Lake Huron (Morse et al. 2003, Bence et al. 2003). Lake trout

populations in the main basin of Lake Huron are subject to predation by “parasitic”

86

juvenile stage sea lamprey that spend up to 18 months feeding before maturing and
returning to Great Lakes’ tributaries to spawn. Juvenile sea lamprey prey on larger fishes
by attaching to and gnawing through their skin to feed on body fluids. Such attacks
frequently result in the death of the prey. Efforts to control sea lamprey by the U.S. Fish
and Wildlife Service and the Department of Fisheries and Oceans, Canada are
coordinated by the Great Lakes Fishery Commission. Control options have concentrated
on preventing larval sea lamprey, called ammocetes, from transforming to their parasitic
phase. In some tributaries, chemicals are used to destroy the ammocetes, and in the St.
Marys River sterile males are released to reduce the percentage of eggs that are fertilized
(see Morse et al. 2003 for details). Other methods of control include placing barriers and
traps on tributaries in order to prevent adult sea lamprey from spawning in the tributaries.
These control programs have been successful in some of the Great Lakes. Lake Superior
currently has self-sustaining lake trout populations, and reduced sea lamprey abundance
has increased the number of larger and older stocked lake trout in Lake Huron.
Cost-effective implementation of sea lamprey control requires an ability to predict
future levels of sea lamprey-induced mortality resulting from different sea lamprey and
lake trout management approaches. In order to forecast sea lamprey-induced mortality
rates, a sea lamprey functional response model is needed to predict the number of attacks
on lake trout (and other prey fish species). An important aspect of sea lamprey attack
behavior that is likely to be important to predictions fi'om a functional response is the
time it takes for the attack to occur. Since the duration of a sea lamprey attack is on the
order of days, and not instantaneous, there is a maximum number of attacks that occur

during a feeding season. If the duration of an attack is ignored the resulting Type I

87

functional response predicts that the number of attacks per sea lamprey will be directly
proportional to host density (Holling 1966). As a consequence, attack rates per host and
resulting per capita sea lamprey-induced mortality will be independent of host density.
To incorporate the duration of an attack, a Type II functional response is needed (Holling
1966; chapter two). With this type of response the number of attacks per host and the
associated per capita sea lamprey-induced mortality will decrease as sea lamprey become
saturated with possible hosts and the maximum number of attacks is approached. While
this type of functional response model was parameterized for the Integrated Management
of Sea Lamprey (IMSL) (Greig et al. 1992) program, it was based on assumed
mechanistic relationships and limited empirical information. Recent work (see chapter
two) has resulted in a multi-species functional response model parameterized within the
framework of a statistical catch-at-age model. This work provides information on the
probable values of parameters for a lake trout population dynamics model including those
controlling the sea lamprey functional response.

It is now clear that management strategies based on models using point estimates
of parameters, acting as though these are known, can often lead to poor results (Smith et
al. 1993, Megrey et al. 1994). One method for addressing this concern is by evaluating
management responses using models based upon Bayesian stock assessments. In this
context, model parameters are reported as posterior distributions (Gelman et al. 2004),
and quantities of interest to managers that are functions of the model parameters can be
expressed as marginal probability distributions. These probability distributions allow

managers to examine the wide range of possible outcomes under different management

88

scenarios and determine the probability of meeting management goals (McAllister and
Kirkwood 1998a and 1998b).

Here we evaluate alternative management scenarios with regard to lake trout
stocking and sea lamprey control taking into account both parameter uncertainty and
stochastic temporal variation in system dynamics. The projection model uses the results
from chapter two in a Bayesian stock assessment context. The stock assessment model
fit in chapter two did not make explicit assumptions regarding how sea lamprey size-
selection would vary in response to lake trout density. However, the differences among
lake regions suggested an intriguing relationship, whereby sea lamprey prey selection
shifted toward larger lake trout when large lake trout were more abundant. Likewise,
although the assessment model did not make explicit assumptions about stochastic
variation in fishing mortality or effective numbers of fish stocked (recruitment), the
results can be used to determine likely levels of future variability. The full model
presented here for evaluating management scenarios incorporated assumptions about
dynamic shifts in prey size selection and stochastic variations in fishing mortality and
stocking. We contrast the predictions of this full model to models without dynamic prey
size selection and stochastic temporal variability to evaluate the importance of these
features. The management scenarios simulated here (see Methods below) were chosen to
represent possible levels of sea lamprey control in Lake Huron as well as changes in lake
stocking regimes in response to successful sea lamprey control and increased lake trout
survival rates.

While self-sustaining populations of lake trout have not been observed in Lake

Huron outside of an isolated embayment of Georgian Bay (Reid et al. 2001), such

89

lakewide stocks are the goal of the rehabilitation program. A variety of milestones have
been developed for lake trout rehabilitation efforts, a number of which are centered on
measures thought to be related to the abundance of mature fish, or factors that would
promote increased spawning stocks (Ebener 1998). Of particular interest here is the
recognition that sea lamprey-induced mortality rates must be reduced in order to
rehabilitate self-sustaining lake trout populations (Ebener 1998). Tied to this is the
objective to maintain total annual mortality for lake trout in rehabilitation zones in Lake
Huron below 45% (DesJardine et al. 1995). We evaluate the consequences of different
scenarios both in terms of lake trout abundance and in terms of mortality caused by sea

lamprey.

90

Methods
Overview

Our simulations were based on the population model and the parameters
described in chapter two. For a given set of parameters a population model was used to
forecast population dynamics over a fifty-year time-horizon. As described below,
parameter values were drawn from a multivariate posterior probability distribution
derived using Markov Chain Monte Carlo methods. By running simulations over the
distribution of parameters we obtained a distribution of outcomes that reflects the
likelihood of different parameter values. .

Forecasts from three different models are presented. Models I and II are
preliminary versions, and their results are contrasted with each other and our third “final”
model as a means for evaluating the influence of the differences among the models.
Models I and II are deterministic, in the sense that once a set of parameter values is
selected the dynamics are determined. Model I is based most directly on the results of
the population model created in chapter two. Of particular importance is that the
relationship between sea lamprey attack rate and host size is assumed to be fixed for each
area based on the relationships estimated for each area in chapter two. Numbers of age-l
recruits, sea lamprey abundance, and abundance of alternative (other than lake trout) prey
are assumed constant over time, and are based on average values observed for the 1984-
1998 period used to parameterize the model in chapter two. Model 11 differed from
model I only in that the assumed relationship between sea lamprey attack rate and prey
size was modeled as a dynamic function of large lake trout density. Model III retains the

dynamic prey size selection of model 11, but allows fishing mortality rates, sea lamprey

91

abundance, and alternative prey abundance to vary stochastically over time, based on
observed variation during the 1984-1998 period.
Population model

The population model used in the simulations describes numbers of age a lake

trout at the start of year y and region r by
Na+l,y+l,r = Na,y,rSa,y,r

where S is the survival for age a lake trout over year y in region r. This survival is

a,y,r
modeled (following chapter two) as a function of natural mortality, fishing mortality and
sea lamprey induced mortality. For each region, the model recognizes lake trout ages 1-
14 and a plus group including ages 15 and above. In order to approximate mortality
events in the lake, the year is separated into two periods in which natural (not including
sea lamprey-induced mortality) and fishing morality occur, separated by a fall pulse of
sea lamprey-induced mortality.

The lake trout population abundance nine months from the start of the year is

modeled as:

C R 9
9 ’(Ma,y,r+Fa,y,r+Fa,y,r)E
Na,y,r = Na,y,re

where M is the natural mortality rate for age a lake trout in year y and region r,

a,y,r

F03 , is commercial fishing mortality rate, and F53, is the recreational fishing

mortality rate. Sea lamprey-induced mortality is then applied and the post sea lamprey

mortality population is given by:

L
9* _ 9 ‘Ma, ,r
Na.y.r-Na.y.re y '

92

The number of lake trout at the end of year y (which equals the number at the start of the
next year incremented by one year in age) is then determined by applying the last three

months of natural and fishing mortality:

C R 3
_(Ma,y’r+Fa,y,r+Fa,y,r)'l—i

9*
N = Na,y.re

a+l,y+l,r
Natural mortality for age 2+ lake trout is based upon a single parameter estimated
during the model fitting process for each region of the lake, while mortality on age 1 lake
trout was assumed to be 0.81 (see chapter two). Both commercial and recreational
fishing mortality are determined as the product of an age-specific selectivity component

and a year specific fishery intensity component:

55,, = Sir/y"...
where S f” is the selectivity for either the commercial (i=C) or recreational fishery (i=R)
on age a in region r and f y’ r is the commercial or recreational fishing intensity in year y

in region r. Commercial selectivity is determined by a four-parameter double logistic

model

 

 

where (02,, and (04’, are the inflection points of the double logistic for each region r,

and £01,, and £03,, and the slopes at the respective inflection points. Selectivity for the

recreational fishery was assumed to follow the logistic curve:

 

R 1
Sam _

_ l+e_aR"(a_'BR7T

93

where fix, is the inflection point of the logistic curve and oak, describes the slope at

the inflection point. The parameterization of the curve starts with a selectivity of zero for
age 1 fish and asymptotes to one for older fish. In the simulations fishing intensity was
either set to a region specific constant for both fisheries or allowed to vary over time (see
below).

Sea lamprey-induced mortality is determined by
Méym = Aa,y,r(1— PS ((1))
where Amy, was the number of attacks by sea lamprey on age a lake trout over the

feeding season in year y and region r, and PS (a) is the probability of an age a lake trout

surviving a sea lamprey attack (Eshenroder and Koonce 1984, Bence et al. 2003). The
probability of surviving a sea lamprey attack was a function of lake trout weight based on
laboratory work by Swink (2003), and this same function applied to all three lake
regions:

1
P a =
s( ) l+el.462—.00041w(a)

 

where w(a) is the lake trout weight in grams at age a. Weight at age differed among the

three regions and thus the probability of surviving a sea lamprey attack varied at the same
age among the lake regions. For each region, the weight at age was based on data
collected during spring lake trout surveys and calculated following procedures described

by Bence and Ebener (2002).

The number of sea lamprey attacks (A0,” ) per lake trout for a given age, year,

and region is determined by a multi-species type II functional response model (Holling

94

1959). As reviewed in Bence et al. (2003), the number of attacks per host (Ahos, ) is

 

predicted by
_ S’Thost L y, r
’103’ — allhosts
1+2th
1

where S is the length of the feeding season (in years), 412051 is the effective search rate of

sea lamprey for the host species of interest, L is the sea lamprey density for a given

yJ'

year and region, H is the handling time (in years), and Di.y,r is the host density in the

year and region of interest. Host density is determined by dividing the host abundance

(N host, y, r ) by the area of lake trout habitat in each region (Table 12). The catch-at-age

model treats each age of lake trout as a separate host type, and in order to incorporate the
affects of alternative host species, estimates of Chinook salmon (Oncorhynchus
tshawytscha) and lake Whitefish (Coregonus clupeaformis) density were included along
with each age of lake trout in the denominator of the functional response. Lake trout
population estimates at nine months, prior to the pulse of sea lamprey-induced mortality,
were used in these calculations. Sea lamprey, Chinook salmon, and lake Whitefish
populations used in the deterministic population model are averages (over years)
calculated from data used to estimate model parameters (see chapter two).
Simulations

Simulations begin with initial numbers at age for the stocks representing each
region of the main basin of Lake Huron and run for 50 years. These initial abundances
were taken from the results of chapter two, using the values estimated for 1998. These

initial age compositions had little effect on the age composition at the end of the 50 year

95

simulations. Projections after the first year require a value for the number of lake trout
that recruited at age-l. For the two preliminary models, recruitment at age one was based
on the 1984-1998 average of “yearling equivalents” stocked multiplied by a post-stocking
survivability of 0.968, the mean post-stocking survivability estimated in chapter two.

The stochastic model allowed for varying recruitment by sampling from normal
distributions centered around the 1984-1998 averages (see Table 13). These are
essentially empirical estimates of “effective stocking” of yearling lake trout, taking into
account the numbers of yearling equivalents stocked, an assumed probability fish stocked
in one region will move to another immediately after stocking and the level of mortality
that occurred immediately after stocking (see chapter two).

The parameters used in each simulation represent one vector of parameter values
from the Markov Chain Monte Carlo (MCMC) (Brooks 1998). The MCMC analysis
results in a joint posterior distribution for all the estimated parameters in the form of a
long, multivariate Markov Chain (100,000 saved vectors of parameters for this study),
with each member of the chain representing a realization from the joint posterior. By
running the lake trout population simulation using parameter estimates from each
realization of the posterior, the distribution of simulation results reflects the variation in
the probability distribution for the parameters.

The MCMC chain was created using AdModel builder. This software starts
chains at parameter values that produce the maximum posterior density, and takes
multivariate normal steps based on the asymptotic variance-covariance matrix (Otter
Research 2000). After an initial burn-in was conducted, every subsequent 2,000th

realization was saved until a chain of length 100,000 was created. Posterior distributions

96

of all estimated parameters were created, as well as marginal distributions of functions of
estimated parameters that represent quantities of interest to managers. Key model
quantities of interest calculated from the MCMC analysis include the number of age 2+
lake trout, the number of age 7+ trout, and the sea lamprey-induced mortality rates on age
5, 8, and 9 lake trout.

To assure that the posterior distribution was a proper one, all parameters were
restricted between upper and lower bounds during estimation. Unless otherwise noted in
chapter two, these bounds effectively served as weakly informative flat prior distributions
between the bounds. All bounds were set to well above and below what was viewed as
possible values for the parameters. Convergence of the MCMC to the posterior
distributions was determined by visual inspection of trace plots and by calculating the
effective sample size. Effective sample size was determined by the methods outlined in
Thiebaux and Zwiers (1984), estimating the autocorrelation function for lags up to 150
steps to insure the autocorrelation function converged to zero.

Comparisons among models

To compare and contrast forecasts from the three models we considered three
scenarios. The first assumed sea lamprey abundance equal to that of the average of the
1984-1998 period (scenario A). The other two scenarios assumed that average sea
lamprey abundance was lowered from this baseline by either 50% (scenario B) or 90%
(scenario C). For comparative purposes we also used model I to make projections based
on point estimates for the model’s parameters. These point estimates were the joint

posterior modal (JPM) estimates (Tempelman 1998) from chapter two.

97

For models I and II, stocking rates were kept constant for all 50 years (see Table
13), as were sea lamprey, lake Whitefish, and Chinook salmon densities (see Table 14).
In chapter two, we assumed that attack rates by sea lamprey would increase with host size
following a logistic function, and based on our analysis of wounding rates (chapter one),
we allowed the inflection point of this function to vary among regions of Lake Huron.
The estimated inflection points were lower in northern Lake Huron than in southern Lake
Huron. A possible explanation for this phenomenon is that there arevery few large lake
trout in northern Lake Huron, and sea lamprey responded by increasing their attack rates
on smaller fish. This explanation argues that the observed patterns in wounding reflects
behavioral responses by sea lamprey to lake trout population characteristics, and that the
absence of similar changes in wounding patterns over time merely reflects a lack of
contrast over the period used to parameterize the model. Therefore, in model II we
allowed the inflection point to vary in response to changes in the age-composition of the
lake trout population in a region. This was done because large changes in lake trout
populations did occur in some of our simulations. Our approach was to make the
inflection point depend upon the density of lake trout longer than 700 mm in the previous

year. The form of this function is

any.” = 0.0052(dr,y,.- - dr,i)+ a,,.-
where fir,y+1,i is the inflection point that applied to region r during year y+1 for
simulation i, d,,y,,- is the density of lake trout greater than 700 mm in length in year y and

region r for simulation i, d r, ,- is the density based on the initial age composition at the

start of the simulation for region r, and ,Br,,- is the initial value of the inflection point at

98

the start of the simulation for region r. Both the ,6,“- and d,’,- are determined directly by

the parameters drawn from the MCMC chain for a particular simulation. The maximum

value of the ,Br,y+1,,- was restricted to 700 mm, to ensure that sea lamprey continue to

attack lake trout greater than that length, regardless of lake trout densities. This
formulation ensures that the initial inflection point is consistent with the both the initial
age composition of large lake trout and the dynamic equation for updating this parameter.
The proportionality constant between the inflection point and density (0.052) was
determined by regressing the maximum likelihood estimate for the inflection point for
each region versus the maximum likelihood estimate of density from 1996-1998 (i.e.,
nine data points used in this regression).

Model III allowed for stochastic temporal variation in fishing mortality, sea
lamprey abundance, lake trout stocking, and in the densities of alternative prey species. In
this model, quantities that had been assumed constant were drawn anew each year from
normal distributions, with means and variances based on inputs and estimates for the
1984-1998 period (see chapter two). The equivalent number of yearlings stocked for
each region, sea lamprey populations for northern and central/southem Lake Huron, lake
wide Chinook salmon populations, and lake Whitefish population for each region were
randomly generated for each year of the simulation (see Table 15). Additionally,
commercial and recreational fishing intensities were randomly generated for each year in
the simulation (see Table 15). The mean and standard deviations were determined from

the estimated values for 1984-1998 in each region of Lake Huron.

99

Scenarios varying lake trout stocking

As described above, we considered scenarios with different levels of sea lamprey
abundance for all three models. In addition to these scenarios, we used our final model
III to explore how changes in lake trout recruitment (stocking) or densities of alternative
prey influenced forecasts. To examine the influence of lake trout stocking we considered
a 50% decrease, a 100% increase and a 200% increase in average stocking under
conditions of a 90% reduction in sea lamprey abundance and a 50% reduction in sea
lamprey abundance (see Table 16). All eight scenarios were repeated with commercial
fishing effort in northern Lake Huron reduced by 56%, corresponding to levels
recommended as necessary for reaching target mortality rates in that region (Bence and
Ebener 2002).
Sea lamprey saturation in northern Lake Huron

Our use of a multi-species type II functional response model implies that the
number of attacks per sea lamprey will be a decelerating function of prey density, with
saturation effects eventually limiting attacks to an upper asymptote at high prey densities.
At relatively low prey densities, saturation effects are minimal and attacks will increase
nearly in proportion with prey density. When saturation effects are evident, sea lamprey-
induced mortality rates will decrease with increases in lake trout density, even without a
decrease in sea lamprey abundance. Lake trout densities, particularly for older fish, had
been extremely low in northern Lake Huron and other scenarios did not suggest
substantial effects of lake trout density on sea lamprey-induced mortality for this region.
We explored the potential for saturation further in northern Lake Huron using model III.

We did this by varying lake trout stocking from baseline levels through a 700% increase,

100

with commercial fishing effort set to zero. This set of scenarios was done using the
baseline sea lamprey abundances (corresponding to Scenario A) and a 90% reduction in
sea lamprey abundance (corresponding to Scenario C) Results were summarized by the
marginal distribution for sea lamprey induced-mortality on age 9 lake trout.
Lake trout damage goals

One possible way to assess the success of the eight simulated management
scenarios is by examining the probability that the scenario reduces sea lamprey-induced
mortality on large lake trout below a target level. The Great Lakes Fishery Commission,
based on existing Fish Community Objectives for the Great Lakes, has proposed a goal of
5 marks per 100 fish as a target for sea lamprey abundance (Mark Ebener, personal
communication). This level of marking corresponds to a mortality rate of approximately
0.05 on large lake trout longer than 700 mm, referred hereafter as large lake trout. Using
model 111 for each management scenario, the marginal distribution was used to determine
the probability that sea lamprey-induced mortality rates would be less than 0.10 and 0.05
on large lake trout.
Marginal distributions for parameters of interest

For simulations using models I and II, the marginal distribution of the number of
age 2+ and 7+ lake trout corresponds to the stable age structure of the population after 50
years. Sea lamprey-induced mortality marginal distributions were determined for age 5
lake trout and for the first age class of lake trout larger than 700 mm. For northern and
central Lake Huron, age 9 lake trout were considered large, while age 8 lake trout were
reported for southern Lake Huron. For model III, the same marginal distributions are

based on the average of the quantities of interest over the last five years of the simulation.

101

For each marginal distribution presented, the 95% Bayesian prediction interval is given,

as well as the median.

102

Results

Comparisons among models

The greatest number of age 2+ (Figure 19) and age 7+ (Figure 21) lake trout
under model I are in southern Lake Huron. Lake trout densities however, were highest in
central Lake Huron (Figures 20 and 22). Commercial fishing mortality rates are much
higher in northern Lake Huron (see chapter two), contributing to lower densities of older
lake trout in the north. Densities of age 2+ lake trout in southern Lake Huron were lower
than in the central region because lower stocking rates were assumed for the simulations.
The number of lake trout yearlings recruiting per square km of lake trout habitat in
southern Lake Huron is approximately half of the levels recruiting in the rest of Lake
Huron (Table 13). Lake trout population levels increased when sea lamprey population
levels were reduced (scenarios B and C). Under scenario C, the number of age 7+ lake
trout in southern Lake Huron increased 117% (Figure 21). Similar results can be seen in
northern (571% increase) and central (711% increase) Lake Huron. Increases in the
densities of age 2+ lake trout (Figure 20) were not as pronounced, as sea lamprey rarely
feed on the smaller lake trout that represent the majority of age 2+ lake trout. Sea
lamprey-induced morality rates on age 5 lake trout (Figure 23) and large lake trout
(Figure 24) decreased sharply in all regions with lake wide decreases in sea lamprey.
Averaged across the lake, a 90% reduction in sea lamprey reduces sea lamprey-induced
mortality 91% on large lake trout under model 1.

Trends in the lake trout populations and sea lamprey-induced mortality rates for

simulations based on the JPM estimates are similar to the results based on simulations

103

using the MCMC results. The width of the 95% Bayesian prediction interval is an
indication of the level of uncertainty that is unaccounted when only the JPM estimates are
used to simulate lake trout populations. The point estimates based on simulations using
the JPM estimates typically fell near the center of the 95% Bayesian prediction interval
for model I (Figures 19-24).

For Model 11, which allows the relationship between attack rate and prey size to
respond to the density of large lake trout, the effects of reducing sea lamprey are similar
to those under Model I, for both numbers (Figures 25 and 27) and density (Figures 26 and
28). There were important differences between model II and model I predictions of sea
lamprey-induced mortality of age 5 lake trout (Figure 29). In central Lake Huron, for
baseline (scenario A) sea lamprey abundance, the median sea lamprey-induced mortality
was 64% higher under model 11 compared to model I. In this region, the density of fish
larger than 700 mm (after 50 years) was lower than that estimated by the catch-at-age
model for 1996-1998. This causes the inflection point of the effective search rate to shift
towards smaller fish in model 11, increasing the mortality rate on younger lake trout. The
opposite effect can be seen in southern Lake Huron under scenario C. The increase in the
density of large lake trout causes the inflection point of the effective search rate to
increase for model II. The shift in the inflection point results in lower sea lamprey-
induced mortality rates on age 5 lake trout (Figure 29) and slightly higher mortality rates
on large lake trout (Figure 30). The same effect is observed in central Lake Huron under
scenarios B and C. In contrast, in northern Lake Huron the density of large lake trout

does not increase enough to lower mortality rates on age 5 lake trout noticeably.

104

The effects of the stochastic components of Model 111, especially the variation in
year to year stocking, are apparent in Figures 7-12. For all three regions, the number of
age 2+ lake trout was greater under model 111 (Figures 25 and 26) than under model 11,
and the width of the 95% Bayesian prediction intervals was greater for model 111. In
northern Lake Huron, the increase in age 2+ lake trout abundance does not lead to a
equivalent increase in age 7+ lake trout abundance (Figures 27 and 28), as high
commercial fishing and sea lamprey-induced mortality rates reduce older lake trout
populations to model 11 levels. For central and southern Lake Huron, which have limited
commercial fishing and lower sea lamprey densities, the increase in age 2+ lake trout is
accompanied by an increase in the predicted abundance of age 7+ lake trout. In southern
Lake Huron, the increase in the density of larger lake trout is sufficient to cause sea
lamprey-induced mortality on age 5 lake trout to approach zero under management
scenarios assuming a 50% or greater decrease in sea lamprey density (scenarios B and C),
with the median number of age 7+ lake trout being 230% greater for model 111 than
model 11 under a 50% reduction in sea lamprey density.

Scenarios varying lake trout stocking

When various management scenarios combining different levels of sea lamprey
control and lake trout stocking strategies are examined, the effects of reducing lamprey
densities are apparent. The number and density of age 2+ lake trout (Figures 31 and 32)
depends largely on the stocking scenario. Scenarios with the largest increase in stocking
(F and H) produce the largest number of age 2+ lake trout, regardless of the level of sea
lamprey population control. This result is not surprising since younger fish make up the

bulk of total abundance and they suffer very low sea lamprey mortality under all

105

conditions. Changes in age 7+ lake trout numbers are influenced by both the level of sea
lamprey control and stocking regime (Figures 33 and 34). In northern Lake Huron, a
90% reduction in sea lamprey combined with a 200% increase in stocking (scenario F)
results in a 1940% increase in the median number of age 7+ lake trout over scenario A
(Figure 33). Similar results are demonstrated in central (2622% increase) and southern
(3442% increase) Lake Huron. Densities of age 7+ lake trout are still highest in the
central region, however, and the absolute gains in the north seem negligible by
comparison (Figure 34). Scenario D demonstrates that a 50% decrease in stocking offsets
the gains of reducing sea lamprey from 50% to 90% of current values in all regions of
Lake Huron. In southern Lake Huron, the difference between a 50% reduction in sea
lamprey (scenarios G and H) and a 90% reduction (scenarios E and F) are minimal in
terms of the number of age 7+ lake trout. This reflects low sea lamprey induced-
mortality rates in southern Lake Huron (Figures 35 and 36).

Sea lamprey-induced mortality on age 5 lake trout approaches zero in southern
Lake Huron for all scenarios involving a 50% or greater reduction in sea lamprey density
(scenarios B-H) (Figure 35). In these cases, the density of older lake trout became large
enough to shift the inflection point of the effective search rate to its maximum value of
700 mm. In northern Lake Huron, sea lamprey-induced mortality rates for age 5 lake
trout did decline with decreases in density of sea lamprey. In this case, however, the
change was approximately proportional to the density of sea lamprey, reflecting a direct
predator density effect rather than a change in size-specific attack rates. In central and
southern Lake Huron, when lake trout stocking is increased from current levels to 200%

of current levels, median sea lamprey-induced mortality on older lake trout decreases

106

(Figure 36), when comparing the same scenarios with the same sea lamprey densities
(Scenarios B,G, and H for a 50% reduction in sea lamprey, Scenarios C,E, and F for a
90% reduction). This indicates a saturation effect for these regions. In the northern
region, sea lamprey are not saturated under these scenarios, and median sea lamprey-
induced mortality is not appreciably altered over the range of lake trout stocking rates
considered here.

When commercial fishing effort in the north is reduced 56%, the median number
of age 2+ lake trout increases, on average, 15% across all eight management scenarios
(Figures 37 and 38), while age 7+ lake trout median abundance increases 335% on
average (Figures 39 and 40). Somewhat surprisingly, under these commercial fishing
conditions, median sea lamprey-induced mortality rates on both age 5 and on older (age
9) lake trout for each scenario are higher than those forecast under historical commercial
fishing effort (compare Figures 41 and 42 with Figures 35 and 36). This increase in
mortality rates as the number of lake trout increases reflects a shift in the inflection point
for the effective search rate function. As densities of large lake trout increased from very
low abundances, the effective search rate on small lake trout (ages 4 and younger)
decreased and these attacks were redirected toward age-5 and older lake trout. Given the
low densities of hosts, saturation effects did not counterbalance the change in effective
search rates. No reduction in sea lamprey-induced mortality was evident with increases
in lake trout stocking for the scenarios with sea lamprey density reduced 90% (scenarios
C,E, and F) and commercial fishing effort is reduced. Similar to the response to reduced
commercial fishing, median sea-lamprey induced mortality rates on age 9 lake trout

actually increase slightly as stocking rates increased for these scenarios. Again this

107

reflects a shift in the inflection point of the effective search rate, with attacks being
further concentrated on older lake trout.
Sea lamprey saturation in northern Lake Huron

When commercial fishing is eliminated in northern Lake Huron, and sea lamprey
mortality is reduced by 90%, model 111 predicted a decrease in median sea lamprey-
induced mortality with increases in lake trout stocking (Figure 43). In contrast, without
the reduction in sea lamprey abundance, lake trout stocking had little apparent effect on
median sea lamprey-induced mortality (Figure 44).
Lake trout damage goals

We do not believe that changes in sea lamprey-induced mortality rates on older
lake trout are meaningful measures of success when commercial fishing mortality is
sustained at historical levels seen in northern Lake Huron. Under these conditions we
sometimes saw increases in sea lamprey-induced mortality as lake trout density
increased. However, these shifts in mortality are occurring when the density of older lake
trout are undesirably low. However, we do note that under these conditions, decreases in
sea lamprey density of 90% (scenarios C-F) suggest a probability ~100% of sea lamprey-
induced mortality on age 9 lake trout being below 0.10 (Table 17). Similar results are
shown when commercial fishing effort is reduced 56%, as the probability of reducing sea
lamprey-induced mortality on age 9 lake trout below 0. 10 ranges from 85% to ~100% for
the same scenarios (C-F). When the probability of reducing sea lamprey-induced
mortality to below 0.05 is examined (Table 18), simulations conducted with a 56%
reduction in commercial effort have a much lower probability of reaching 0.05 or below

than those conducted with historical levels of commercial fishing. The counterintuitive

108

result of apparently higher success when commercial fishing is high in northern Lake
Huron is based on a sea lamprey-induced mortality rate for the very few fish that survive
to these older ages. Since few old fish are present, this mortality rate has little import to
the population. Under these conditions, even a decline in sea lamprey density of 90%
still allows few lake trout to survive to large size. The reason sea lamprey-induced
mortality rates on old lake trout are lower in the face of higher commercial fishing
mortality is because attacks are shifted toward smaller lake trout.

The results for northern Lake Huron with a 5 6% reduction in fishing effort are
more indicative of the effects of sea lamprey control under proposed reductions of
commercial fishing effort (Bence and Ebener 2002), as densities of large lake trout reach
levels observed in other parts of the lake (see Figure 40). These results indicate that
management scenarios (C-F) with a 90% reduction in sea lamprey have a high probability
of lowering sea lamprey-induced mortality rates on large lake trout to below 0.10. Under
these conditions, the probability of reducing sea lamprey-induced mortality in northern
Lake Huron to below 0.05 varies between 5% and 12% (Table 18). When stocking is
reduced 50% (scenario D), the highest probability of achieving mortality rates below 0.05
is indicated. However, the density of older lake trout under this scenario is similar to
scenarios without a 90% reductions in sea lamprey (Figure 40) and is undesirable in
terms of potential spawning stock biomass. Scenarios with lower levels of sea lamprey
reduction have probabilities approaching 0% for reducing sea lamprey-induced mortality
rates to 0.10 or below.

When sea lamprey densities are reduced by 90% in central Lake Huron the

probability of sea lamprey mortality rates being below 0.10 is ~O% with baseline and

109

~100% with substantial increases in stocking. Under these conditions the probability of
mortality rates being below the 0.10 threshold is substantially decreased when stocking
was decreased to 50% of baseline levels. With only a 50% reduction in sea lamprey
density, it appears improbable that the 0.10 level can be reached even with a 200%
increase in lake trout stocking. In central Lake Huron, even a 200% increase in stocking
combined with a 90% reduction is sea lamprey density (scenario F) may be insufficient
for reaching the more stringent sea lamprey mortality target, as the probability of
reaching the 0.05 target was 12% under these conditions.

In southern Lake Huron, where sea lamprey densities are very low, a 90%
reduction in sea lamprey density combined with a 200% increase in lake trout stocking
produces a probability of 61% for meeting a sea lamprey-induced mortality target rate of
0.05 (Table 18). A 50% reduction in sea lamprey, under a 200% increase stocking
scenario in the south, leads to a 87% probability of sea lamprey-induced mortality rates
being below 0.10 for large lake trout. Thus, the southern region has the highest
probability of reaching acceptable sea lamprey mortality through a combination of sea

lamprey control and increased lake trout stocking.

110

Discussion

Effective management of lake trout in the Great Lakes requires projections of how
lake trout dynamics change in response to sea lamprey control and lake trout stocking. In
Lake Huron, previous models used for making these projections implicitly assumed a
type 1 functional response, with sea lamprey-induced mortality a linear function of sea
lamprey abundance, regardless of lake trout density (Sitar 1996; Sitar et al. 1999; Lupi et
al. 2003; Stewart et al. 2003). Given the time sea lamprey spend feeding on a single prey
item, it seems likely that they would become partially saturated at high prey densities,
with sea lamprey-induced mortality rates declining as prey densities increase (see Bence
et al. 2003 for a review). The extent to which this actually would occur under feasible
management conditions clearly has important management implications (Walters et al.
1980; Stewart et al. 2003). The results presented in this chapter suggest that saturation
would likely become a factor in Lake Huron if sea lamprey densities decline by 90%
from abundances seen from the mid-19803 through the late 19905. This is especially true
in central and southern Lake Huron, as sea lamprey-induced mortality rates on lake trout
decrease as lake trout stocking rates increase and sea lamprey populations are reduced
90% from 1984-1998 levels. In northern Lake Huron clear saturation effects were not
evident unless lake trout stocking is substantially increased and fishing mortality was
substantially decreased from the levels of the 1980s and 1990s. Overall, our results
confirm the need to incorporate a sea lamprey saturation component into lake trout
projection models for Lake Huron.

Previous explicit models of the sea lamprey functional response (reviewed in

Bence et al. 2003) have been Type II (sensu Holling 1966). Such models assume

lll

constant effective attack rates while sea lamprey search for prey, with saturation
occurring because less time is spent searching for prey (and more time handling prey) as
prey density increases. To a large extent, previous applications of these models in the
Great Lakes (Greig et al. 1992) set effective attack rates based on assumed mechanisms
of feeding (e. g., based on values for swimming speed and reactive distances). The
resulting effective attack rates were not only constant with respect to prey density, but
increased at an accelerating rate with increases in prey size. Our investigation of the data
suggested that attack rates would be sigmoid functions of prey length, and that the
inflection point of these sigmoid functions respond to changes in the abundance of large
prey (chapter two). This result is similar to a Type III functional response (Holling
1966), which is characterized by increases in prey density resulting in an increase in
search activity, and is sometimes an indication that predators switch between prey types
that require different search strategies (Akre and Johnson 1979; Dale et al. 1994). Here
we show that tying changes in the inflection point of the sigmoid function to the density
of large lake trout has important consequences (by contrasting model I and model 11),
with projected sea lamprey-induced mortality and lake trout densities being altered.
These results need to be interpreted in light of the fact that our understanding of sea
lamprey feeding behavior is very uncertain, however they clearly emphasize the need for
an improved understanding of how sea lamprey feeding may be responding to prey
densities.

As noted above, our functional response model, with the dynamic sigmoid attack
rate, are no longer pure type II functional responses. In many cases, sea lamprey-induced

mortality rates on smaller prey decreased with increases in prey densities while mortality

112

rates on larger prey remained relatively constant, increased, or decreased. For smaller
prey this represented a combination of saturation and a shifting of attacks toward larger
prey (i.e., the change in the inflection point of the sigmoid function, Figure 45). For
larger prey, these patterns reflected the complex relationship between shifts in the
sigmoid attack function, increasing large prey density (Figure 45), and saturation (Figure
46). While the saturation effect built into the type II functional response generally
produces depensatory mortality (Figure 46), our modification allowed for compensatory
mortality on larger prey (Figure 45). The full dynamic consequences of our modeled
functional response on self-sustaining lake trout populations is an area warranting future
study.

We incorporated uncertainty of two forms into our final model. First, rather than
using point estimates, we used estimated probability distributions for the model
parameters. Each simulation represented a draw from the underlying distribution of
model parameters. Second we allowed for stochastic temporal variability in quantities
that were not dynamically modeled, such as recruitment of young lake trout, abundance
of alternative prey species, and sea lamprey abundances. Our inclusion of probability
distributions for the parameters allowed us to consider probabilities of different
outcomes, rather than being forced to rely on point estimates. Our experiences reinforce
current fisheries management ideas that managers need to know the probability that a
management plan will yield the desired effect (Gavaris and Sinclair 1998). However, we
found that probability distributions for forecast quantities were substantially altered when
we acknowledged that some of the unmodeled “constants” actually varied over time.

Ignoring this variation would produce unrealistically tight distributions of our forecasts.

113

We emphasize that our incorporation of both types of uncertainty did more than place a
distribution around the quantity forecast by a point estimate and a deterministic model.
The median or other central measures were also altered. One example of this was the
increase in median sea lamprey-induced mortality rates in northern Lake Huron when
temporal stochasticity was incorporated into the final model. This results from a well
known phenomenon (Jensen’s inequality), whereby a nonlinear function applied to an
expected value produces a different value than the expected value obtained by applying
the same function to each individual input. The potential for this phenomenon is noted in
decision theory literature (Boyce 1992). Here we found that the effects were large
enough to alter point estimates of management quantities in significant ways,
underestimating the density of age 7+ lake trout for example (Figure 8).

Our results indicate that meeting a sea lamprey-induced mortality target of 0.05 in
southern Lake Huron may be possible. However, in northern and central Lake Huron,
reaching this target is a greater challenge. While reductions in commercial fishing
(northern region) combined with increases in lake trout stocking can increase the
probability, the likelihood of reaching the target sea lamprey mortality still remain low.
However, a shift in the overall probability distribution toward lower sea lamprey-induced
mortality rates can be achieved. In order to reduce sea lamprey-induced mortality rates,
our results show that reducing sea lamprey populations has the greatest effect on the
probability of reaching sea lamprey—induced mortality rate goals. Once sea lamprey
populations are lowered, increasing lake trout populations, by either stocking more lake
trout or reducing fishing effort, can lead to additional reductions in sea lamprey-induced

mortality, if a type II functional response, like the one proposed here, is assumed. Since

114

planned sea lamprey control and lake trout management efforts appear unlikely to meet
the existing target, we believe that a cost-benefit analysis is called for (see also Stewart et
al. 2003) to evaluate the ongoing management efforts.

More generally, our modeling and stochastic forecasts provide valuable insight on
the probability of successfully rehabilitating lake trout in Lake Huron. We predicted
probability distributions for sea lamprey induced mortality rates and lake trout stock sizes
based on a population model that incorporated a parameterized functional response
model. Managers could use this same model to evaluate additional management
scenarios and measures of success. Following this approach it is possible to estimate the
probability of meeting quantifiable management goals, and this should help guide lake
trout managers toward actions more likely to facilitate lake trout rehabilitation in Lake

Huron.

115

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 12: Area of lake trout habitat (<40 fathoms) in square km for each region of Lake
Huron (Ebener 1998).
fiegion km2
North 3.432
Central 4,575
South 1 5,21 0
Table 13: The average number of yearling equivalents recruiting in each region of Lake
Huron for models I and II. This is based on the average number of yearling
equivalents stocked from 1984-1998, and takes into account an assumed
probability fish stocked in one region will move to another immediately after
stocking (see chapter two).
Yearling
_Rggion Equivalents
North 357,164
Central 321 ,635
South 594,639
Table 14: Assumed abundances (numbers of fish) assumed constant in the simulation
models based on the average values estimated from 1984-1998 (see chapter
two). For sea lamprey these represent the baseline values, which were
adjusted by stated percentages for different scenarios.
Quantity Region Value
Chinook salmon All 674048
Whitefish North 6219133
Whitefish Central 1034461
Whitefish South 1284019
Sea Lamprey North 212152
Sea Lamprey Central/South 15320

 

 

 

 

116

 

Table 15: Means of the normal distribution used to randomly generated simulation
model inputs for each year in model 111. Chinook salmon, Whitefish, sea
lamprey and yearling equivalents are numbers as defined in Table 14. Fishing
intensity represents the instantaneous fishing mortality rate on a fully selected
age . The mean and coefficient of variation (CV) are based on the values
estimated from 1984-1998 (see chapter two). Coefficient of variation is
reported for each quantity because the CV was held constant when sea
lamprey abundance and stocked yearling equivalents were changed from
baseline values.

 

 

 

 

 

 

 

Quantity Region Mean CV
Chinook salmon All 674048 0.32
Whitefish North 6219133 0.28
Whitefish Central 1034461 0.53
Whitefish South 1284019 0.20
Sea Lamprey North 212152 0.32
Sea Lamprey Central/South 15320 0.46
Yearling Equivalents North 357164 0.35
Yearling Equivalents Central 321633 0.29
Yearling Equivalents South 594640 0.42
Commercial Fishing

Intensity North 0.63 0.32
Commercial Fishing

Intensity Central 0.03 0.24
Commercial Fishing

Intensity South 0.04 0.20
Recreational Fishing

Intensity North 0.02 0.29
Recreational Fishing

Intensity Central 0.02 0.32
Recreational Fishing

Intensity South 0.04 0.28

 

117

Table 16: Descriptions of management scenarios simulated in terms of changes in sea
lamprey abundance and lake trout stocking.

 

 

Percent Change Percent Change in
Scenario in Lamprey Lake Trout StockinL
A 0% 0%
B -50°/o 0%
C -90% 0%
D -90% -50%
E -90% 100%
F -90% 200%
G ‘-50 % 100%
H -50% 200%

 

 

 

 

 

118

Table 17: Probability of reducing sea lamprey-induced mortality on age 9 lake trout in
northern Lake Huron below 0.10 based on marginal distribution determined
under model 111 for each management scenario described in Table 16.

 

 

 

 

 

 

 

 

Region
North North Central South
56%
Historical reduction in
commercial commercial
Scenario effort effort
A 0% 0% 0% 0%
B 0% 0% 0% 33%
C 99% 97% 76% 56%
D 99% 98% 94% 32%
E 100% 92% 93% 84%
F 99% 85% 99% 97%
G 0% 0% 0% 62%
H 0% 0% 0% 87%

 

 

Table 18: Probability of reducing sea lamprey-induced mortality on age 9 lake trout in
northern Lake Huron below 0.05 based on marginal distribution determined
under model 111 for each management scenario described in Table 16.

 

 

 

 

 

Region
North North Central South
56%
Historical reduction in
commercial commercial
Scenario effort effort
A 0% 0% 0% 0%
B 0% 0% 0% 6%
C 26% 5% 0% 16%
D 95% 12% 0% 6%
E 32 % 6% 0% 35%
F 40% 10% 12% 61%
G 0% 0% 0% 16%
H 0% 0% 0% 33%

 

 

 

 

119

 

Figure 19: Number of age 2+ lake trout in northern, central, and southern Lake Huron for
model I. The 95% Bayesian prediction interval of the marginal distribution of
age 2+ lake trout is shown, with the median of the marginal indicated by the
square. The point estimate of age 2+ lake trout based on the joint posterior
modal estimates from chapter two is indicated by the gray circle. For
information on scenarios, see Table 16. Note the difference in scale between

 

 

 

 

regions.
Northern Lake Huron
g 400 —
g A 380 I
_e 8 360 — T.
e e a40~ T
S V _
g 320
z 300 r i I
A B C
Scenano
Central Lake Huron
j; 650 .
E g 600 ~ T
~05 8 550 t
g c 500 - I
g 450 ‘ E":
z 400 r ' ‘
A B C
Scenano

Southern Lake Huron

 

 

'3 1300 .

g A 1100 - f

59 8

“(3 § 900 ~ it

E“ 700 ~ i

g 500 . . .
A B c

Scenano

120

Figure 20: Density of age 2+ lake trout in northern, central, and southern Lake Huron for
model I. The 95% Bayesian prediction interval of the marginal distribution
for age 2+ lake trout density is shown, with the median of the marginal
indicated by the square. The point estimate of the age 2+ lake trout density
based on the joint posterior modal estimates from chapter two is indicated by
the gray circle. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

 

 

 

 

E 150 —

8

5 100 ~ 1 i i

0.

fig 50 ~

.12

3 O I I I
A B C

Scenano
Central Lake Huron

E 150 -

O' E

"’ i..:.

0') 100 — I.»

Q

5 50 —

£2

3 O I I I
A B C

Scenano
Southern Lake Huron

E 150 -

8 100

8. -

s . '

g 50 t I“?

3‘2

3 O I I I
A B C

Scenafio

121

Figure 21: Number of age 7+ lake trout in northern, central, and southern Lake Huron for
model I. The 95% Bayesian prediction interval of the marginal distribution of
age 7+ lake trout is shown, with the median of the marginal indicated by the
square. The point estimate of age 7+ lake trout based on the joint posterior
modal estimates from chapter two is indicated by the gray circle. For
information on scenarios, see Table 16. Note the difference in scale between
regions.

Northern Lake Huron

 

 

 

 

E3 12 —
2.. 13 *
fl 8 "
“5 8 :5 “
S I
Z O T T l
A B C
Scenario
Central Lake Huron
a: .. .,
a g. 150 T
“5 8 100 .
.é " 50 1 EA
lie
2 O I I I
A B C
Scenario

Southern Lake Huron

400 - -..

300 - T.

200 —

100 — is.
0

Number of lake trout
(1000s)

 

4.

 

I I

A B C
Scenano

122

Figure 22: Density of age 7+ lake trout in northern, central, and southern Lake Huron for
model I. The 95% Bayesian prediction interval of the marginal distribution
for age 7+ lake trout density is shown, with the median of the marginal
indicated by the square. The point estimate of the age 7+ lake trout density
based on the joint posterior modal estimates from chapter two is indicated by
the gray circle. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

 

 

 

 

 

5 50 —
8 40 ~
3 30 —
‘53 20 -
E 10 ~
g 0 +21- ? P'
A B C
Scenano
Central Lake Huron
E 50 -
9; 40 -
8' 30 _‘ E12:
‘5 -
g 20 I1.-
3 1° ‘ .
3 0 I I I
A B C
Scenario
Southern Lake Huron
E 50 - .. ..
8 40 ~
3 30 -
g 20 ~ E
3 1O 7 1.. [5:
3 0 We 1 r
A B C
Scenano

123

Figure 23: Sea lamprey-induced mortality on age 5 lake trout in northern, central, and
southern Lake Huron for model I. The 95% Bayesian prediction interval of
the marginal distribution of mortality is shown, with the median of the
marginal indicated by the square. The point estimate of mortality based on the
joint posterior modal estimates from chapter two is indicated by the gray
circle. For information on scenarios, see Table 16. Note the difference in

scale between regions.

Northern Lake Huron

 

 

 

 

 

 

1.0 a
Q)
E 0.8 -
E 0.6 — £5
E 0.4 — h
2 0.2 _ -
0.0 l T $~Li
Scenano
Central Lake Huron
£3
‘3 0.010 0
g
g 0.005 0 l
2 =3; ;
0.000 I l :—
Scenaflo
Southern Lake Huron
1.0 V. _ ,,,,,
9 J
‘3 0.8 E
g. 0.6 ~
5 0.4 ~
2 0.2 0
0.0 I I :35;
A B C

Scenano

124

Figure 24: Sea lamprey-induced mortality on lake trout for the youngest age lake trout
with mean length exceeding 700 mm in northern (age 9), central (age 9), and
southern (age 8) Lake Huron for model I. The 95% Bayesian prediction
interval of the marginal distribution of mortality is shown, with the median of
the marginal indicated by the square. The point estimate of mortality based on
the joint posterior modal estimates from chapter two is indicated by the gray
circle. For information on scenarios, see Table 16. Note the difference in

scale between regions.

Northern Lake Huron

 

 

 

 

 

 

0.8 — »- , .
Q)
E 0.6 ~ E
136‘ 0.4 -
1:
C q 22:
2 0.2 E
0.0 1 I 1"”
A B C
Scenario
Central Lake Huron
m .
E 0.6 ~ E
3% 0.4 - i
t -_i:.
o .
2 0.2 II
0.0 , I f
A B C
Scenano
Southern Lake Huron
0.8 1"
a)
E 0.6 — I
g 0.4 a if E
o _
2 0.2 I
0.0 , I f“
A B C

Scenano

125

Figure 25: Number of age 2+ lake trout in northern, central, and southern Lake Huron for
three different models. The 95% Bayesian prediction interval of the marginal
distribution of age 2+ lake trout is shown, with the median of the marginal
indicated. The models with an adjusting [3 (model 11), a fixed [3 (model I), and
the stochastic simulations with an adjusting [3 (model III) are shown. For
information on scenarios, see Table 16. Note the difference in scale between

regions.

600
550

450
400
350
300

Number of Lake Trout
(10003)

900
800
700
600
500
400

Number of Lake Trout
(10003)

500 .,

Northern Lake Huron

 

 

 

300 ,

 

I Adj. Beta
. »- Fixed Beta
E - - Stochastic
i i” '
A B C
Scenario
Central Lake Huron
I Adj. Beta
leed Beta

 

 

- Stochastic

 

 

Scenario

126

Figure 25 (cont’d)

Number of Lake Trout

(1000s)

1800

1 600
1400
1200
1000
800
600

Southern Lake Huron

 

I Adj. Beta
= leed Beta
- Stochastic

 

 

 

Scenario

127

 

Figure 26: Density of age 2+ lake trout in northem, central, and southern Lake Huron for
three different models. The 95% Bayesian prediction interval of the marginal
distribution for age 2+ lake trout density is shown, with the median of the
marginal indicated. The models with an adjusting [3 (model 11), a fixed 0
(model I), and the stochastic simulations with an adjusting [3 (model III) are
shown. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

 

 

 

 

 

 

 

 

 

 

’- i
3 150 4 S .
53 { j 3 I Ad]. Beta
-‘:'i 100 . 1:: i. g . leed Beta
g i - Stochastic
g 50 ~
(0
_‘I O l l T

A B C

Scenario
Central Lake Huron

E 200 ~
x .
3 150 ~ j .
a“: i i I Adj. Beta
‘3 100 1 I i” i . leed Beta
g - Stochastic
m 50 .—
x
(O
_J O I l l

A B C

Scenano

128

Figure 26 (cont’d)

Southern Lake Huron

 

 

 

 

 

 

E zoow-u ~
x
5” 150 — .
g 100 — i j . Fixed Beta
g E n I" -Stochastic
a) 50 — rs ' 5
x
(U
.1 0 l T l

A B C

Scenario

129

Figure 27: Number of age 7+ lake trout in northern, central, and southern Lake Huron for
three different models. The 95% Bayesian prediction interval of the marginal
distribution of age 7+ lake trout is shown, with the median of the marginal
indicated. The models with an adjusting [3 (model 11), a fixed [3 (model I), and
the stochastic simulations with an adjusting [3 (model III) are shown. For
information on scenarios, see Table 16. Note the difference in scale between

 

 

 

 

 

 

 

 

 

 

 

 

 

regions.

Northern Lake Huron
S 12 _ .......
o
I: 10 ‘ F s
g 2,? 8 . ‘ . Adj. Beta
.11 8 6 _ I .- leed Beta
0 ‘— . - 1 .
h": V 4 - {I i - Stochastic
_o 1
E 2 1 ._ “ ' ,
2 o I}; . . i

A B C
Scenario

Central Lake Huron
2
i; 200 - g
.. 8 » . leed Beta
0 \— 100 ~ .
L V g - Stochastic
2 50 I L i
E .3; l
3 in i
Z 0 I V I I

A B C

Scenario

130

Figure 27 (cont’d)

Southern Lake Huron

 

 

 

 

2

I- 400 -

3 A I Ad' Beta
3 g 300 ~ E. "

“5 8 200 . . leed Beta
23 : : - Stochastic
E 100 7 i f’ I

E o 5?“;- . . i

 

 

Scenano

131

Figure 28: Density of age 7+ lake trout in northern, central, and southern Lake Huron for
three different models. The 95% Bayesian prediction interval of the marginal
distribution for age 7+ lake trout density is shown, with the median of the
marginal indicated. The models with an adjusting [3 (model II), a fixed [3
(model I), and the stochastic simulations with an adjusting B (model III) are
shown. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

 

 

 

 

 

 

 

 

 

 

E 60 a t
'2‘: 50 1 .
l”
33 40 a - Adj. Beta
33, 30 — . Fixed Beta
g 20 — - Stochastic
3‘2 10 - ‘

A B C

Scenario
Central Lake Huron

E 60 a
i? 50— .
(I) i
5 40 a E - Adj. Beta
.3 30 . E g . leed Beta
g 20 « E i - Stochastic
OJ -' i
x 10 ~ i
3 o ' I; . .

A B C

Scenario

132

Figure 28 (cont’d)

Southern Lake Huron

 

I Adj. Beta
. leed Beta

 

 

 

E I E . - Stochastic

 

 

Lake trout per sq km
—I N (to h 01 O)

O O O O O O O

l l l l l l l

Scenario

133

Figure 29: Sea lamprey-induced mortality on age 5 lake trout in northern, central, and
southern Lake Huron for three different models. The 95% Bayesian
prediction interval of the marginal distribution of mortality is shown, with the
median of the marginal indicated. The models with an adjusting [5 (model 11),
a fixed B (model I), and the stochastic simulations with an adjusting [3 (model
111) are shown. For information on scenarios, see Table 16. Note the
difference in scale between regions.

Northern Lake Huron
1.2 -
1.0 l
0.8 E - Adj. Beta

 

. Fixed Beta
- Stochastic

 

 

0.4
0.2
0.0 i . '1

 

Mortality rate
0
03

PH
l—f-—i

Scenario

Central Lake Huron

0.030 - ~
0.025 -

0.020 g W
0015 s » Fixed Beta
0.010 - Stochastic
0.005 .

0.000 I r -.

 

 

Mortality rate

Scenario

134

Figure 29 (cont’d)

Southern Lake Huron

1.2 ~
1.0 ‘

 

0.8 - E . Adj. Beta
0.6 . . leed Beta
0.4 - i l - Stochastic

 

Mortality rate

 

 

0.2 -

0.0 I I: Ji— %

 

 

Scenano

135

Figure 30: Sea lamprey-induced mortality for the youngest age lake trout with mean
length exceeding 700 mm in northern (age 9), central (age 9), and southem
(age 8) Lake Huron for three different models. The 95% Bayesian prediction
interval of the marginal distribution of mortality is shown, with the median of
the marginal indicated. The models with an adjusting [3 (model 11), a fixed [3
(model I), and the stochastic simulations with an adjusting [3 (model III) are
shown. For information on scenarios, see Table 16. Note the difference in
scale between regions.

Northern Lake Huron

 

 

 

 

 

 

 

 

 

 

 

 

1.0 -
g 0.8 —
9 : I Ad'. Beta
3 0'6 _ 3 Fl): d B t
a 04 — :‘. ' e e a
E ' ‘ (I: y — Stochastic
2 0.2 — I" a
0.0 I , "T;
A B C
Scenano
Central Lake Huron
1.2 -
a, 1.0 ~ ;
E 0.8 ~ ? - Adj. Beta
:3 0.6 — if . leed Beta
5 0.4 ~ i I - Stochastic
5 0.2 ~ I :
0.0 . . "i '
A B C

Scenaflo

136

Figure 30 (cont’d)

Southern Lake Huron

1.2 ,
1.0 r

 

0.8 - I - Adj. Beta
0.6 ~ II . Fixed Beta
0.4 _ 5 - Stochastic

 

 

 

Mortality rate

_ i
0.2 I . .. I

0.0 I 7 ‘

 

Scenano

137

Figure 31: Number of age 2+ lake trout in northern, central, and southern Lake Huron for
model 111 under a series of proposed lake trout management scenarios. The
95% Bayesian prediction interval of the marginal distribution of age 2+ lake
trout is shown, with the median of the marginal indicated by the square. For
information on scenarios, see Table 16. Note the difference in scale between
regions.

Northern Lake Huron

2000 — .
1500 — I I
1000 ~ I

500‘.i i i

Number of lake trout
(10005)

 

 

Scenafio

Central Lake Huron

 

 

g 2000- I
gA1500— l
«28 :
~§§1oooa I E I
mv

E i i i - :
g 0 T l l l l T l l

A B C D E F G H
Scenano

Southern Lake Huron

 

 

23'; 6000 _..,. _ 7 7 , 7 7 7 7. , ... ,
7: 5000-«
E g 4000 r i i
*5 g 3000 — I I g
E I: 2000 « i i

E 1000 ~ i I

2 o . . f . , . . .

A B C D E F G H
Scenano

138

Figure 32: Density of age 2+ lake trout in northern, central, and southern Lake Huron for
model 111 under a series of proposed lake trout management scenarios. The
95% Bayesian prediction interval of the marginal distribution for age 2+ lake
trout density is shown, with the median of the marginal indicated by the
square. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

g 600

g 500 ~

‘2’ 400 - I I

«o-a 300 "

§ 200 d i I E

g 100 - i i I I

3 O I f l 1 l l l l
A B C D E F G H

Scenario

Central Lake Huron

 

 

 

 

E 600 — .. w .
g 500 — I I
‘- 400 “ I
g 300 - E i I
g 200 - I I
g 100 ~ 1 E . I
3 0 r l I T , , I I
A B C D E F G H
Scenano
Southern Lake Huron
5 600
g 500 a
‘ 400 ~
g 300 - I E
_9‘) 100 4 I I i I
N O r r l y r r r I
_J
A B C D E F G H
Scenario

139

Figure 33: Number of age 7+ lake trout in northern, central, and southern Lake Huron for
model III under a series of proposed lake trout management scenarios. The
95% Bayesian prediction interval of the marginal distribution of age 7+ lake
trout is shown, with the median of the marginal indicated by the square. For
information on scenarios, see Table 16. Note the difference in scale between

 

 

regions.
Northern Lake Huron
g 40 I ..
a) _
a 2‘ 3° I
*5 8 20 ‘ I
8 " 10 ~ I
E i i E i E
Z 0 I I I I I I r 1
A B C D E F G H
Scenano

Central Lake Huron

 

 

g 1000» , ~ ******* l
E 800 ~
4‘ 1? ;
I? 8 600 r I I
E $2 400 ~ I I l

V l
g 200 4 i i
3 . I I ;
Z 0 f I l I I I I 1

A B C D E F G H
Scenario

Southern Lake Huron

1500 —‘

IIII

 

 

Number of lake trout
(10003)

500 ~ i i
0 J t T I i I I I T l
A B C D E F G H
Scenano

140

Figure 34: Density of age 7+ lake trout in northern, central, and southern Lake Huron for
model 111 under a series of proposed lake trout management scenarios The
95% Bayesian prediction interval of the marginal distribution for age 7+ lake
trout density is shown, with the median of the marginal indicated by the
square. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

 

 

 

 

E 200 —- -
O'
2 150 -
8
.. 100 —
3
2
*5 50 -
E 0 ‘ T I r I I ' I I I I I t I ._1
A B C D E F G H
Scenano
Central Lake Huron
g 200 I
U’
3 150 « ¥
8
3
g 50 +
% 0 I i i I i
_J I I I I T r I 1
A B C D E F (3 H
Scenaflo
Southern Lake Huron
g 200 'I”""'
U'
2 150 ~
8' 100
o
:3 50 ~ i E i
0 i
g l I I
—’ O . I I I "— I I I I I
A B C D E F G H
Scenafio

141

Figure 35: Sea lamprey-induced mortality on age 5 lake trout in northern, central, and

1.2
1.0
0.8
0.6
0.4
0.2
0.0

Mortality rate

Mortality rate
0
O)

oo—x—s
mono»

Mortality rate
0
'4:

.09
on

southern Lake Huron for model III under a series of proposed lake trout
management scenarios. The 95% Bayesian prediction interval of the marginal
distribution of mortality is shown, with the median of the marginal indicated
by the square. For information on scenarios, see Table 16. Note the difference
in scale between regions.

Northern Lake Huron

 

I l l I I I I l I I I l
A B C D E F G H
‘ Scenario

Central Lake Huron

! I
- -
1 I I I I - I - I I l

A B C D E F G H

Scenario

Southern Lake Huron

WI—HWII—IW—H—IWIW
A B C D E F G H
Scenario

142

Figure 36: Sea lamprey-induced mortality for the youngest age lake trout with mean
length exceeding 700 mm in northern (age 9), central (age 9), and southern
(age 8) Lake Huron for model III under a series of proposed lake trout
management scenarios. The 95% Bayesian prediction interval of the marginal
distribution of mortality is shown, with the median of the marginal indicated
by the square. For information on scenarios, see Table 16. Note the difference

in scale between regions.

Northern Lake Huron

 

 

 

 

1.0 ~* A
a: _
E 0.8
g. 0.6 -
7“ 0.4 - i
1:
§ 0.2 - i i 1
0.0 l 7 ' I . I . I ! I I I
A B C D E F G H
Scenano
Central Lake Huron
g 1.0 « I:
9 0.8 + { }
g 0.6 - l
t 0.4 r E i E I
0 l
2 0'2 - I I I a I
0.0 I r I T I l I i
A B C D E F G H
Scenano

143

Figure 36 (cont’d)

Southern Lake Huron

1.0 - l
0.5 -

0.0

Mortality rate

 

C D E
Scenano

144

Figure 37: Number of age 2+ lake trout in northern Lake Huron for model III under a
series of proposed lake trout management scenarios. Commercial fishing
rates are reduced 56%. The 95% Bayesian prediction interval of the marginal
distribution of age 2+ lake trout is shown, with the median of the marginal
indicated by the square. For information on scenarios, see Table 16.

Northern Lake Huron

 

 

a 2000 L”...

g a 1500 — i i

~5§ 1000 ~ I u

.8 v 500 r I I .

E I

2 O I j I T I I I I
A B C D E F G H

Scenario

Figure 38: Density of age 2+ lake trout in northern Lake Huron for model 111 under a
series of proposed lake trout management scenarios. Commercial fishing
rates are reduced 56%. The 95% Bayesian prediction interval of the marginal
distribution for age 2+ lake trout density is shown, with the median of the
marginal indicated by the square. For information on scenarios, see Table 16.

Northern Lake Huron

500 -

i ,
400 — i
300 4 ' I
200 ~ I
100 - ' . ;

 

Lake trout per sq

 

O
1
4

Scenario

145

Figure 39: Number of age 7+ lake trout in northern Lake Huron for model 111 under a
series of proposed lake trout management scenarios. Commercial fishing
rates are reduced 56%. The 95% Bayesian prediction interval of the marginal
distribution of age 7+ lake trout is shown, with the median of the marginal
indicated by the square. For information on scenarios, see Table 16.

Number of lake trout

 

Northern Lake Huron

A B C D E F G H

Scenano

Figure 40: Density of age 7+ lake trout in northern Lake Huron for model 111 under a
series of proposed lake trout management scenarios. Commercial fishing
rates are reduced 56%. The 95% Bayesian prediction interval of the marginal
distribution for age 7+ lake trout density is shown, with the median of the
marginal indicated by the square. For information on scenarios, see Table 16.

m

.2

Lake trout per sq

 

Northern Lake Huron

 

150 — I
100 — ‘
50 — g
0 j I I I ! I I I I I i I Q I I fl
A B C D E F G H
Scenano

146

Figure 41: Sea lamprey-induced mortality on age 5 lake trout in northern Lake Huron for

Mortality rate

1.2 1 I. ,, ,, .7 , .....
1.0 4

0.8
0.6
0.4
0.2

model 111 under a series of proposed lake trout management scenarios.
Commercial fishing rates are reduced 56%. The 95% Bayesian prediction
interval of the marginal distribution of mortality is shown, with the median of
the marginal indicated by the square. For information on scenarios, see Table

16.

Northern Lake Huron

-1

A I IIl

i I I l I

 

Scenano

Figure 42: Sea lamprey-induced mortality on age-9 lake trout in northern Lake Huron for

Mortality rate

1.0
0.8
0.6
0.4
0.2
0.0

model III under a series of proposed lake trout management scenarios.
Commercial fishing rates are reduced 56%. The 95% Bayesian prediction
interval of the marginal distribution of mortality is shown, with the median of
the marginal indicated by the square. For information on scenarios, see Table

16.

Northern Lake Huron

_.

3i; II

—1

 

 

I I i i
A B C D E F G H
Scenano

147

Figure 43: Sea lamprey-induced mortality on age 9 lake trout in northern Lake Huron for
model III as the number of lake trout stocked is increased. Sea lamprey
populations are based on a 90% reduction in the average value observed in
Lake Huron from 1984-1998.

Mortality rate

Sea Lamprey-Induced Mortality

0.20 ~
0.15 e
0.10 e
0.05 a

I I

 

 

0.00

1 00 300 500 700
Percent Increase in lake trout stocking

Figure 44: Sea lamprey-induced mortality on age 9 lake trout in northern Lake Huron for
model III as the number of lake trout stocked is increased. Sea lamprey
populations are based on the average value observed in Lake Huron from

Mortality rate

1984-1998.

Sea Lamprey-Induced Mortality

0.8 ~
0.6 ~
0.4 .
0.2 -

 

 

0.0

100 300 500 700
Percent Increase in lake trout stocking

148

Figure 45: Age specific sea lamprey-induced mortality rates for lake trout in northern
Lake Huron under scenario C (90% reduction in sea lamprey) and scenario F
(90% reduction in sea lamprey and a 200% increase in stocking) using model
III. The median of the marginal distribution of mortality is shown.

Northern Lake Huron

9 0.08 —
g 0.07 ~
= 0.06 ~
€323 “
3 0.03 4
dc) .

g 0.02 ~
*5; 0.01 —

c .f'u
— 0.00 "M I F I I I I I I I I T I

12 3 4 5 6 7 8 9101112131415
Age

 

war-- Scenario C
-0- Scenario F

 

 

 

 

 

 

Figure 46: Age specific sea lamprey-induced mortality rates for lake trout in southern
Lake Huron under scenario C (90% reduction in sea lamprey) and scenario F
(90% reduction in sea lamprey and a 200% increase in stocking) using model
111. The median of the marginal distribution of mortality is shown.

Southern Lake Huron
0.12 -

0.10 -
0.08 4
m 0.06 -
c 0.04 a
m 0.02 —
" 0.00 ~

Ity rate

mortal

 

mar- Scenario C
-0- Scenario F

 

GOU

 

nta

nst

 

 

 

12 3 4 5 6 7 8 9101112131415
Age

149

 

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