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This is to certify that the
dissertation entitled

Uncertainty in the population dynamics of alewife (Alosa
psuedoharengus) and bloater (Coregonus hoyr) and its effects
on salmonine stocking strategies in Lake Michigan

presented by
Emily 8. Szalai

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Fisheries and Wildlife

UMRW

0 Major Professor’s Signature
jw\\1 ’3‘ L 100 3
U

Date

MSU is an Affirmative Action/Equal Opportunity Institution

‘.

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

. ,. QATEPUE DATE DUE DATE DUE

.
a V T; 'r
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6/01 cJClRC/DatoDuepes-p. 15

UNCERTAINTY IN THE POPULATION DYNAMICS OF ALEWIFE (ALOSA
PSUEDOHARENG US) AND BLOATER (COREGONUS HOYI) AND ITS EFFECTS
ON SALMONINE STOCKING STRATEGIES IN LAKE MICHIGAN.

By

Emily B. Szalai

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

2003

 

mt

Pie

ABSTRACT
UNCERTAINTY IN THE POPULATION DYNAMICS OF ALEWIFE (ALOSA
PSUEDOHARENGUS) AND BLOATER (COREGONUS HOYI) AND ITS EFFECTS
ON SALMONINE STOCKING STRATEGIES IN LAKE MICHIGAN.
By

Emily B. Szalai

The dynamics Of alewife (Alosa psuedoharengus) and bloater (Coregonus hayi) in
Lake Michigan were investigated and the implications of uncertainty in these dynamics
on the outcomes of stocking strategies for salmonines were Simulated using a stochastic
model. I also analyzed the long-term trends in bloater size at age with a dynamic growth
model for evidence of density-dependent growth regulation.

I fit a dynamic von Bettalanffy model and length-weight relationship with time-
varying parameters to mean length and weight at ages of bloater from annual surveys.
My results support a positive relationship between asymptotic length, L... , and the Brody
growth coefficient, k, indicating that under conditions supporting larger L... , individuals
approach L... more rapidly. I explored the relationship between year-Specific growth
parameters and bloater abundance indices and found evidence of density-dependent
growth. However, in the most recent years, L... and yearling length have remained low
despite low bloater abundances, suggesting a potential shift in the food web.

I reconstructed the population dynamics of alewife and bloater by fitting dynamic
models to historic prey fish survey data (bottom trawl and hydroacoustic indices). These
models allowed recruitment variation and accounted for mortality due to salmonine

predators. Chinook salmon predation followed a Type II functional response while other

predators were assumed to be consuming at a constant rate. Estimates of consumption
based on existing assessments of predators were also used in model fitting. The joint
posterior distribution Of the stock-recruitment parameters for alewife and bloater and a
key parameter of chinook salmon’s functional response was approximated using Markov
Chain-Monte Carlo methods. While the amount Of uncertainty in the parameters Of the
stock-recruitment relationship for alewife and bloater was large, the uncertainty in the
parameter Of the functional response was moderate.

To assess the impacts of these uncertainties on the outcomes of salmonine
stocking strategies in Lake Michigan, I constructed a stochastic simulation model to
forecast the outcomes of stocking strategies on both alewife and chinook salmon
population dynamics. Uncertainty in the stock-recruitment relationship for alewife, the
functional response of chinook salmon and the response of chinook salmon mortality
rates to decreases in growth rate were included in the model. I investigated both fixed and
dynamic Stocking Strategies; for the latter, stocking rates responded to changes in the state
of the system. The outcomes of all stocking strategies I considered were highly variable,
with a large proportion of undesirable outcomes. The abundance of other salmonine
predators had large effects on the dynamics of the system. This suggests that the
salmonines require integrated management and our ability to maintain a desirable state
long-term through stocking strategies alone may be limited. Removing uncertainty in
stock-recruitment parameters for alewife caused the probability of undesirable outcomes
to decrease suggesting that ignoring uncertainty in these parameters will cause overly

Optimistic predictions Of future outcomes.

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ACKNOWLEDGMENTS

I would like to thank my adviser, Jim Bence and my committee members, Mike
Jones, Gary Mittelbach, and Rob Tempelman for their assistance and guidance in
completing my dissertation. I would also like to thank the vessel crew and science staff
Of the USGS-Great Lakes Science Center, particularly Guy Fleischer and Chuck
Madenjian, for collecting the long-term forage fish survey data and for providing
invaluable advice regarding this data set. John Netto and Wenjing Dai provided
programing support for the simulation model and have my Sincere thanks. Finally, I
would like to thank my friends and family for their support throughout my graduate
career.

Financial support for this project was provided by the Michigan Sea Grant
College Program, project number R/GLF-46, under grant number N A76RG0133 from
the Office Of Sea Grant, National Oceanic& Atmospheric Administration (NOAA), US.
Department Of Commerce and Michigan State University through the University

Distinguished Fellowship program.

 

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TABLE OF CONTENTS

LIST OF TABLES

 

LIST OF FIGURES

 

CHAPTER ONE
MODELING TIME-VARYING GROWTH USING A GENERALIZED VON
BERTALANFFY MODEL WITH APPLICATION TO

BLOATER(C OREGON US HOYI) GROWTH DYNAMICS

IN LAKE MICHIGAN

 

CHAPTER TWO
QUANTIFYING UNCERTAINTY IN LAKE MICHIGAN ALEWIFE
AND BLOATER POPULATION DYNAMICS

 

Introduction

 

Methods

 

Prey fish surveys

 

Predator abundance and consumption

 

Alternative prey

 

Estimation model

 

Results

 

Discussion

 

CHAPTER THREE

EVALUATION OF THE EFFECTS OF UNCERTAINTY IN
ALEWIFE AND CHINOOK SALMON POPULATION DYNAMICS
ON THE OUTCOMES OF STOCKING STRATEGIES IN

LAKE MICHIGAN

 

Introduction

 

Methods

 

 

Predator population models
Prey population models

 

Model scenarios and stocking Strategies

 

Results

 

Harvest

 

Alewife biomass

 

 

Chinook mean Size at age and mortality events
Discussion

 

APPENDIX A

 

vii

14
14
20
20
23
24
25
34
4O

7O
70
75
75
81
83
87
87
89
9O
92

123

APPENDIX B 143

 

APPENDIX C 165

 

LITERATURE CITED 189

 

vi

LIST OF TABLES

Table 1. List of variables and parameters used in the
estimation model (a: age, y: year)

 

Table 2. Model equations describing the alewife and
bloater population dynamics

 

 

Table 3. Model equations used in the Observation sub-model

Table 4. Values for parameters assumed known during
model fitting (LT: lake trout, CHS: chinook salmon,
CO: coho salmon, ST: steelhead, and BT: brown trout)

 

Table 5. Negative log likelihood components utilized
during model fitting. CL indicates the likelihood component
was incorporated using the concentrated likelihood form

 

Table 6. Mean, variance, 95% credibility intervals (CI) and
effective sample size ( N efl ) for the posterior distributions
Of all estimated parameters

 

Table 7. Correlations between pairs of parameters in
MCMC samples drawn from the posterior distributions

 

Table 8. List of variables and parameters used in the
simulation model (a: age, y: year)

 

Table 9. Model equations describing the population dynamics
of chinook salmon and lake trout in the simulation model

 

Table 10. Model equations governing prey species dynamics
in the simulation model (3: species, a: age, )2: year)

 

Table 11. Numbers at age (in thousands) and length at age
(mm) of coho salmon, brown trout and steelhead in the
simulation model

 

Table 12. Model scenarios

 

Table 13. Stocking (millions) policies for lake trout

vii

47

48

49

50

51

52

53

98

100

101

102

103

 

Tal
the

T111
XIII

Tal
Chi

Tat
slat
(mt

 

and chinook salmon

Table 14. Average cumulative harvest (numbers in thousands)
of chinook salmon in 30 years under eight different stocking
policies for different model scenarios. Stocking policies with
the highest average cumulative harvest are in bold

 

Table 15. Average cumulative harvest (numbers in thousands)
of chinook salmon in 30 years under eight different stocking
policies when uncertainty is ignored in key parameters.
Stocking policies with the highest average cumulative harvest
are in bold

 

Table 16. Average cumulative harvest (numbers in thousands)
of chinook salmon in 30 years under five different Stocking
policies when uncertainty is ignored in key parameters

for the Simplified chinook-alewife model. Stocking policies

 

with the highest average cumulative harvest are in bold

Table 17. Percentage of Simulations years that fall alewife
biomass (mt) was in each category for different model
scenarios and stocking strategies

 

Table 18. Mean, standard deviation, 0.1,and 0.90 quantiles for
the duration (years) of chinook salmon mortality episodes

 

under different model scenarios and stocking strategies

Table 19. Equations defining the Lake

 

Michigan stock assessment model

Table 20. Definition of symbols used in equations for

 

chinook salmon stock assessment model

Table 21. Parameter estimates and their asymptotic
standard errors for the models described by equations 1
(measurement error model) and 2 (process error model)

 

viii

104

105

106

107

108

109

178

180

182

 

 

 

 

 

 

Spr

Fig

pr
lar

LIST OF FIGURES

Figure 1. Observed (symbols) and predicted (lines) fall
bottom trawl survey indices for age 0 (squares and solid
line) and age 3+ (circles and dashed line) alewife in Lake
Michigan, 1962-1999

 

Figure 2. Observed (symbols) and predicted (lines) trawl
survey indices for (a) age 0 (squares, solid ), age 1 (circles,
dashed), (b) age 2 (squares, solid), age 3 (circles, dashed),
age 4 (triangles, solid), and (c) age 5 (squares, solid), age
6 (circles, dashed), and age 7 (triangles, solid) bloater in
Lake Michigan, 1962-1999

 

Figure 3. Observed (symbols) and predicted (lines) fall
hydroacoustic biomass estimates of (3) age 0 and (b)
age 1+ alewife in Lake Michigan, 1993-1996

 

Figure 4. Observed (squares) and predicted (line)
hydroacoustic biomass estimates for bloater in

 

Lake Michigan, 1993-1996

Figure 5. Observed (squares) and predicted (line)
consumption of all prey types by all five salmonine
species in Lake Michigan, 1965—1999

 

Figure 6. Observed (symbols) and predicted (lines)
proportion of small alewife (squares, solid line) and

large alewife (circles, dashed line) in the total consumption
by all five salmonine species in Lake Michigan, 1965-1999

 

Figure 7. (a) Observed (symbols) and predicted (lines)
consumption per predator for age 1 (squares, solid line),

age 2 (circles, dashed line), and age 3 (triangles, solid line),
(b) predicted proportion of maximum consumption achieved
(solid line) and 95% credibility intervals for age 3 chinook
salmon in Lake Michigan, 1968-1999

 

Figure 8. Predicted instantaneous predation rates (P) on
(a) age 0 (squares, dashed line), age 1 (circles, solid line),
and age 2 (triangles, dashed line) and (b) age 3 (squares,

ix

54

55

56

57

58

59

60

solid line), age 4 (circles, dashed line), age 5 (triangles,
solid line), and age 6+ (diamonds, dashed line) alewife
in Lake Michigan, 1965-1999

 

Figure 9. Predicted instantaneous predation rates (P) for (a)
age 0 (squares, solid line), age 1 (circles, dashed line), age 2
(triangles, solid line), and age 3 (diamonds, dashed line),

and (b) age 4 (squares, solid line), age 5 (circles, dashed line),
age 6 (triangles, solid line), and age 7+ (diamonds, dashed line)
bloater in Lake Michigan, 1965-1999

 

Figure 10. Posterior density functions of the catchability
coefficients Of (a) age 0 and (b) age 1+ alewife hydroacoustic
survey in Lake Michigan, 1993-1996, and (c) age 0 alewife fall
trawl survey in Lake Michigan, 1991-1999

 

Figure 11. Posterior density function of the catchability
coefficients of the bloater hydroacoustic survey in Lake
Michigan, 1993-1996

 

Figure 12. Posterior density function of the length-based
scalar of the effective searching efficiency on an optimal sized
prey for salmonine predators in Lake Michigan

 

Figure 13. Posteriogdensity function of the (a) 1n(aaw) ,
(b) flaw , and (c) Jaw, r parameter of the Ricker
stock-recruitment function for alewife in Lake Michigan

 

Figure 14. Posterio density functions of the (a) 1n(abl) ,
(b) flbj , and (c) dbl, r parameter of the Ricker

 

stock-recruitment function for bloater in Lake Michigan

Figure 15. Maximum posterior estimates Of the stock-recruitment
relationships for (a) alewife (Stock size is the number of age 2+
fish divided by 1x10”). Recruitment is the number of age 0

fish divided by 1x109), and (b) bloater (Stock size is the number
Of eggs divided by 1x10”. Recruitment is the number of age

0 fish divided by 1x109)

 

Figure 16. Posterior density function of the instantaneous
mortality rate (S67 ) on age 1+ alewife during the dieoff
in 1967 in Lake Michigan

 

61

62

63

64

65

66

67

68

69

 

 

 

 

 

 

Fi
wi
SCI

Fig

Figure 17. The probability Of an alewife dieoff as a

function of stock size (numbers times 10”) for model
scenarios: baseline (solid triangle), DA (Open circles),
DB (x’s), DC (solid diamonds), and DD (solid squares)

 

Figure 18. Distribution of the numbers of chinook salmon

harvested in 1000 simulations for the baseline scenario with
the (a) Status quo stocking policy and (b) feedback stocking
policy with one year lag

 

Figure 19. Distribution of cumulative harvest for 1000 simulations
with a feedback stocking policy with a one year lag for model
scenarios (a) FR and (b) SR

 

Figure 20. Distribution of cumulative harvest for 1000 simulations
with a feedback stocking policy with a one year lag for model

 

scenarios (a) M1, (b) M2, and (c) M3

Figure 21. Distribution of cumulative harvest for 1000 simulations
of the simplified decision model with status quo stocking for model
scenarios (a) baseline, (b) FR, and (0) SR

 

Figure 22. Distribution of cumulative harvest for 1000 simulations
of the simplified decision model with status quo stocking for model
scenarios (a) M1, (b) M2, and (c) M3

 

Figure 23. Chinook salmon average Spawning weight at
age 3 with status quo stocking for model scenarios
(a) baseline, (b) FL, and (c) FH

 

Figure 24. Chinook salmon average Spawning weight at
age 3 with status quo stocking for model scenarios
(a) DA, and (b) DC

 

Figure 25. Chinook salmon average spawning weight at
age 3 with status quo stocking for model scenarios

 

(a) DB, and (b) DD

Figure 26. Chinook salmon average spawning weight at
age 3 with a feedback stocking policy with one year lag
for model scenarios (a) baseline, (b)FR, and (c) SR

 

Figure 27. Chinook salmon average spawning weight at

xi

110

111

112

113

114

115

116

117

118

119

 

 

CJ
("‘1

f0

for

age 3 with a feedback stocking policy with one year lag
for model scenarios (a) M1, (b)M2, and (c) M3

 

Figure 28. Distribution of the number of mortality events
in each 30 year simulation time period for the baseline
scenario with (a) status quo stocking, and (b) feedback

 

stocking with a one year lag

Figure 29. Distribution of the number of mortality events
in each 30 year simulation time period with feedback
stocking with a one year lag for model scenarios

(a) M1, (b) M2, and (c) M3

 

Figure 30. Temporal patterns in weight at annulus
formation for age 3 (W, diamonds) and model estimated
natural mortality rate at age-2 (M, squares)

 

Figure 31. Relationship between changes in mortality at
age-2 from year y-l to year y and weight at annulus
formation in year y+l

 

Figure 32. Estimated posterior density for ,0 ,

which is the effect Of last year’s natural mortality
year effect on this year’s year effect (eq. 2)

 

Figure 33. Estimated posterior density for ,8 , the effect

Of age-3 weight at time of annulus formation in year y+1
on the year effect for natural mortality in year y (eq. 2)

 

Figure 34. The estimated posterior density for 0'2 ,

 

the variance for the process errors (8) in equation 2.

Figure 35. Probability of transition from high to low mortality
regime for age-2 predicted given weight at age-3 (annulus
formation) by equation 6, with b , = 3.0 and b2 = 4.0

 

xii

120

121

122

183

184

185

186

187

188

CHAPTER ONE

Szalai, E. B., Fleischer, G. W., and Bence, J .R, 2003. Modeling time-varying growth
using a generalized von Bertalanffy model with application to bloater (Coregonus
hoyi) growth dynamics in Lake Michigan. Can. J. Fish. Aquat. Sci. 60: 55-66.

 

.m rmmwmm >11 1.1.. a.

 

55

Modellng tlme-varylng growth uslng a gonerallzod
von Bertalanffy model wlth appllcatlon to bloater
(Coregonus hay!) growth dynamlca In Lake

Mlchlgan

mummu.oww.mammn.m

 

WAWWhWWmmhmmdM(Conmmh
WWMwWWWMWWoWMWMWW
.mcmammmmmwmmmmmmmmmmm
weight-”ages (ages 1-7) from annual survey: (1965—1999). Wemodeledyearlinglmgflr. aaymptnticaiu( L.),andtho
parametenofapowetrelafionahipbetweenmeanweightandmeanlength(aandB)ndrangingalowlyovertimo
uaingarandomwalkmodel.TheBmdyyuwdroocficienuhwumodeledualimfmaimofLwflhyw-
apecificrandomdeviafionaOmreafltsmppatapoaitivcrdaflomhipbetween Lmhindicafingdlatnndercondi-
dmwgmwmmmmwmuymmwy.mapmmm
afipbuwwnywapxificmwammfimdmummamfomdwidmmd
dendrydependentgrowhfimfinthcmostmmtym Landyearlinglengdrhavemainedlawinhb
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W:Chubdmdefump(Cangmhoyi)dnlnhfiehigamhodnddemod’dea
l’abondancedanamutblacetd'unedimimtiondelatailleaunlgedmnélaiueuoimfil’med'mrégulaflm
fihmmfihmfiNmmmmmeajmmm
WfldehmflmW—mmmmwmmumm
donnéudelongmmoyenncetdemaunlgedonneflgu l-Dpovenmd’inventairuannnela(1965—l999).
Nouaavonamodéliaélalongmdajeuneadelmhlongueual’uympm(LJetleapuambuud’unerd-ionda
puissanoecntnlamauemyenneetlalongmmoya-a(aetB)arleafaiaantvariulanmdana1etunpal
l’aided’unmodeledemarchealéamire.ucoefidentdeaoimdofimdy(k)awmodfliaeomuncfom
lineairudeLnavecdeadeviationaaléamimaspécifiqneaal’mnée.Nmr6mltauindiannemhdonpoaifiwm

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lumwmmmmmml'ermfidmaqumMIme

danaleréaemalimtain.
madnitparlakédaction]
Introduction

Dynarnicgruwdrdfishuhnmanyinflicafiommdappfi-
mindnmagenmtoffiahaimandmyecolog'ml
arllpaaicalimightscanbegaincdbyexminingpmin
fiahgmwth(e.g.,Fa1uiand'Ihylcr 1996;quandenlnnd
mwmmmzoooymmam
munlhmaongrowthlnvebemexamineddmughbodru-

puimentalmanipuhtiouuxiobaervafionalmluflmgh
hmmtalappoaehaflomhstmogutinfamitia
ofimnotfusibleforpopulatiomorspeciesofmln
mycaseswhenanobaavmimalappoadrismy,a
fina-as-ofobsavatianofsin-at-ageandvimnmalal
Wmmm(e.g.,MalletetaL 1999;Millaretal.
1999;13argoandKr-onlnnd NW). Becmeofmehighvariabfl-
ityin-a-sizb-at-ageeatimatuaboutmepopullion’suue

 

ReceivedzmymWENOWMMWmuNRCWMWOmeMkajfumum

13 Fallacy 2003.
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East Lamina. MI 48824, U.S.A.

G.W. M. NCAA Fiaheriea NWFSC - FRAM Division, 2725 Mondale Boulevard B. Seattle. WA 98112, USA.

‘Correaponding antha' (o-rnail: Wfimedn).

Can. J. Fish. Aquat. Sci. ‘0: 55-66 (2N3) doi: 10.1139/903-003

OWNRCCanada

a may mmwanaa mammwmmmmamma mmnmmumnenpw Emmaammwummmm «mm was

 

56

memdu-‘mitisafuudvmmidufifymfita
gmwdrmdelmdredatampavidesmoethedestimd
grawthavertime.

Such a model-based analysis afdatafmman observa-
tionalstudy generallyelmeMs atraditionalgrawthmodel,
umllydrevonBertalanffymodeLmincludetime-varying
parameeersfittothetimeseriesafobservations(e.g.,Mallet
etal. 1999;Millaretal.1999;Fargoandenlnnd20(X)).
'I'hetimeseriesofobsavatiansusedtofitdiemodelisgen-
erallyoneoftwotypee.size—at-agedataorgrowdrinaement
datacalarlaeedfi'omsize-at-agedata.
modelsintwageneralways,dumghexplicitfunctiansafan
environmental variable or by estimating, independently of
euvironmentaldata,year-specificparametetsfurdiegrowth
madeLApplicationsafdiefirst (MillarandMyers
1990;Malletetal. 1999;Mi11aretal.1999)usuallymodel
eitherthe harkpuameMOfavonBertalanfl’ygmwdr
curveasfunctionofanenvironmentalfactor,usuallytemper-
factaranddregmwthparameten.1ftheconnectionbetwem
growthanddreenviranmentismisspecified, itispassible
thatspmiauspattemsinyear-specificgrawdwouldbeiden—
tified.

Thesecondapproachistafitagmwthmodelmdataon
size-at-ageindependentlyforeachyearfarwhichdataare
available (e.g., Zhaaetal. 1997; FargoandKronlund 2W).
'I'hetemporalpattemsintheresulringparametaestimam
canmenbeexaminedinrelationtatemporalpattansinfac-
marathatmightcausethemtavary.Byindependentlyfitfing
agrawthmodeltaeachyearafsize—at—agcdatghowevu,
thisapproachignorestheinterdependenceofgrawthbetwaea
years Immemodelrsdescnbmgthepauemmsrze-at
agenrithinayeasnatmeangrawdrafagivenagebetween
yearyandyeary+1.'1hetwoareonlyequiva1entwhen
grawdrratesarenatchangingovertim Hence,thisap-
proachcausesdificultiesininterpretatian.

Hetewepresentanaleemativemethodforapplyingme
aecendqipruachWeuseavanButalanfiygrwdrcurve
withtime- ' parametenmpredictgruwdrincremenn
flunaneyeutotheanekeepu-ackofthedynamicafly
changingpedictiansafsize—at-ageandcomparemesepe-
dictionawidratimeseriesafsize—at-ageabservatianstoesti-
mmepuametasofammadeLThuathetime-varying
parameeetsareestimatedindependenflyofanypoposedm-
vhonmennlmechanisminconn'astwiththefirstmethad,
andthepanmeeersdescribegrowdrratherthanyear-spedfic
pademsindze—d-agauinmanyapplicatiansofthesecand
methodOurtechniqueusesatimesetiesapproachmmodel
theclnngeeinyowdrparamemnavatimeandestimams
theeepuameuaunng amaxinnrmlikelihoodframework.
Weapplythisapp-aachmdsegrawthdynamicsoftheblom
(Conganushayi)populatianinukeMichigan.Wedrenex-
plosethepoeentialmechanismsfarchangesinblaaeu'gmwth

Blaaeu'isanimpartantmemberafdrelakeMichigaueca-
symhImNfichigamdriscoreganineisdresalemain-
ingmemherafanatiginalendemiccamplexafdeepm
fauna ('lbdd et a1. 1981). Hisdnically, blow populations

Can. J. Fish. Nun. Sd. Vol. 60. 2003

supportedvalnablecammucialfishedesmruwnetallm;
Fleischul992)andwaefongefathenativelakeuaut
(Salveh'm nanaycush) and bulbot (Iota Iota) populflims.
Gmenfly,theyareforagefardrearrayofsmcked
salmonids, whenblaawrareyoungaandsmalla
(HalcyetaL1998; Madenjianeta12002). Changesinblaamr
populations within Lake Michigan have been dramatic.
Blamwundanceina'easedfmmnearminIWOmdom-
inatedreplankfivorebiamassafthelakebythelatelm
(Madenjianeta12m2)(Pig.la).Blaamabundancere-
mainedhighdmingmastafdrel9903andsubsequenflyda-
clinedmlawlevelsagainCI‘ekaeletaLM).
Concunentudthdaeselngechmguinabundmblaam
1mgdr-andwdghtat-agealsowentdu'oughlargechangea
(Figs.1b—1d).’1hesechangesinsize—at—ageappearedeabe

might '

growthCI‘eW'mkeletaLZOOZ).Becansebloamlength-and

' -at-age havebeenmanitored confinnouslysince 1965
topsesenflexcept l966and 1998),thesedatapavidedus
meappamnitymexplarethechangesthathaveoccunedin
thegrowthofbloaeeravertimeanddiepatenfialferdensity-
dependmtgrowdrregulation intheukeMichiganbloater
population.

mum

ThebkeMichiganblaaterpopulafionhasheenasaeaaed
annuallyaincel962bytheU.S.GealagicalSurvey(USGS)-
GreatlakeScienceCentexinafallbottamtnwlsurvcy.
Mmyiscanduceedatfixedlacatiansthraughauthke
Michiganandpravidesinfonnationanthesiuaengthand
weight), age composition, and abundance (dnough catch-
pa-unit-efiart (CPUE)) of the bloater population along with
theexotic alewife (Alosa puudahamgus) population. We
useddresesurveydataeaconsu'uctatime-varyinggrawdr
modelfortheblaawtpopulatianandexplareddierehioa-
ship betweenthe estimatedgmwdrparameten and abun-
danceafbothblaatermdalewife.

Fallu'awlsuneyddgn

’Ibcalculatemeanlength- andweight-at-ageandabun—
danceindimweanalyzedcamhdatafi'amannnalbomum
u'awlsurveyscondnceedbytheUSGS—GreatlakesSciencc
Ceneaeachfall(e.g.,1-1aechetal.l981).'1'hesurveywas
inifiaeedinl962,buthlaaterwerefirstagedinl965and
hareewerestrictedauranalysismthel96S-l999period.
‘IhwlswesegenetallyIOmininlengthandusedaifiYan-
keeSnndardNa.35hommtrawl(12mheadmpe.15.5m
faouepeandIS-mmmeshinthecodend)draggedoneon-
tour-duringthedayasdeaaibedbyHaechetaL(l981).Sam-
plingwudaneafishmeoffixedshmelocafimsfign.
Generallytawswetemadeateachlacatimatdeepdlin-
tervalsanddresampleddepdrsrangedfiam6uo128mNat
anlocatiansweresampledineachyeannsnallyrelaeedm
weatherconditionsmurwerealldepdnsampledateachla-
ctionbecauaeofin'egularbotmmfeatures. Althoughthe
lacatianswerefixed.thenurnberandidmtityafsamp ledla-
cadonsdidchangeavetdminitially,fmml962ml966.
tnwlsmmadeanlyafiSaugahrck.Mich.(Pig.2),andin

OWNRCCIIada

Weld.

57

Pk.L(a)Relativeahundaneeofyearlingblosmr(Carugmhayi)(solidcincles),adn1t(age2+)blostu(solidtl'iflslfl).andadnlt
alewife(AlasapsaIdolIarmgns)(opensquues)inukemdriganfmml965m1999.Obsaved(symbols)andpedictnd(lines)mean
length-at-agefor(b)age1(solidcircles, lawcline)andag92(solidtriangles,upperline),(c)ageS(solidcircles.law¢line)andage
4(solidtriangles,upper1ine),and(d)age5(soflddschs,bwafine),age6(soliduimgles.nuddbfine),andage7(nfidsquuu,

upperline)avutimeinLakeMichigan.
10

 

Relative abundance
O

.5.1

 

 

'10 I’ I I
1965 1970 1975 1980 1985 1990 1995 2000

275-
255-

4
235-

1
215-

Lengm (mm)

195d

175
1965 1970 1975 1990 1995 1990 1995 2000

 

 

 

1967,0ueeadditionallocationswereincluded8tartingin
1973,anaddidonaldneelocadonsweseaddedforatotalaf
sevenfixedshore locations.
Bloaterscaleswerecollectedannnallyforagedetamina-
tianssince 1965 (exceptfor19669nd1998).Bloatexscale
samplingwaslimitedtofishcollectedatfomdesignatedlo—
cations (Frankfort, Mich, Saugatuck, Mich, Waukegan. 111.,
andManistique,Mich.)r-atherthanatalllocations (exceptin
1999whenscaleswaealsosampledfiomhldinztomhfich.,
andPortWaahingtan, Wis). From l965to 1982, scaleswene
sampledfromfishselectedatrandomfi'amtheentirebloam
sizedisuibutionateachofdredesignatedlocationsJIaw-
ever, since 1983, scale samples have been collected follow-
ingastntifiedsamplingdesign. Scaleswenecollectedfram
armximnmfixednumba'offishforeachlo-mmlength

agesandanageclassdesignationwasbasedonenumeration
ofohservedannuli. BloaterrangedfiomOtolZyearsof
age.AgeingerrarwasnotincorpoI-atedintheanalysisaf
size-at-agedstaScor-edageswereassumedtobetrueages

   

 

 

205-4
g 1.5-
g 165-1
_I
145-
125 I I I I I
1965 1970 1975 19180 1985 1990 1995 2000
340-
320-l
300-
2801
260
240-1
220‘
20° 1 I I f I I

 

 

 

I
1965 1970 1975 1980 1985 1990 1995 2000

Year

fasllfish'lheamountofageingeuorinscaleageingaf
bloaterpopulationsindieGreatlakeshasmtbeeninvesd-
gatedandsmdiesafageingeuorinothueoregonidspecies
havesuggesteddratdleamauntofageingeuordependson
Mumwithhighammwsinslower-gmwingpopu-
latians(Raitaniemieta1.1998).Fewindividualsover-age7
wee sampled, andscales werenoteonsistently collected
franindividualslessthanlmmminlengm'l‘lmgour
growdrmadelinguseddataforageslto7on1y.

Meanlargth-andwdght-at-age
Becausescalesampleswerenotcollectedatalllocations
fordreeatiretimeperiod(l965—l999),itwasnotpossihleto
calculate location-specific mean size-at-age. Previous analysis
afthefallbottomtnwldatahassuggesteddratthemeansize-
at-ageofbloamdifi'essacrossdielakemrausel999).For
thismasombloafiagedatawerepooledintoanorthernre-
gion (Port Washington. Wis, Sturgeon Bay, Wis, Maniatique.
Mch,Prankfort.Mich..andlndinsmn.Mich.)andsouthan
region(Saugatuck,Mich.,andWaukegan,Ill.)tocalculatere-
gional mean lengths- and weights-at-age (Fig. 2). Blasters
weresampledinbothregionsforsllyearsexceptl965when
onlythesoudremregionwaswveyed. From 1965m1982,
whenagestrucnneswaesampledmndomly,drems-lengd|-
andweight—at-ageineflyearandregionwerecalculatedby
takingdiemeanoftheleagthandweightrespectively,ofsll

GNNRCCanads

(1)

7.1 am

 

gal
wwwwmwm

m

58

mimedahorelocstioasinlakeMidn'ganforU.S.Gealogi-

calSnrvey-OreatlahesScieneeCmfallhottamtrawlsuvey.

(Imet)MapoftheGreatIabswithdreinternationalbormdary

betweerICanadaandtheUnitedSmsafAmericaindicsbd.
s_7:4s' arse- saw

‘4— :—

 

      
   
   
   

9's»:-

. 9a
3, Manistique

Ludlngton

 

  
   

Port
Washington

 

 

 

*42'15‘

 

 

 

sr'ss' ssTso' ss‘rs
agedfiahineachregion.Proml983tol_999whenstratified

sampling was used, mean length-at-age “9.114) in each year
andregionwasestirnatedby

 

2 phi-‘prJJ id.’,1,]
(1) za.," = I
ZPCJJJPLU
1
"mp, iltheproportionofdrebloaterinlengthbinj
fromthe mkforyearyandregioni,

puuistheproportionafageainlengthbinjfromthescale
sampleforyearyandregianiJndlamfljisthemeanlenng-
at-ageoin binjforyearyinregioni(Gavarisand
Gavaris1983).Forlargecatches(e..,g greaterthanmkg),
thetotalsizeofthecatchanddrenumberofblaawrineach
lengdrbinwereestimsredfromsubsamplesafdretotalcatch
(seeKrause(1999)fordetails). Mean weight-at-age (IV—..,“)
ineachregionandyearwasestimatedsimilarlywithi,,;j
WWW...“ (unweight-at-aseainlensmbinjin

yearyandregionr. Becauaetheblaaterpowlatiouisgena-
allytreatedasonestockratherthanastwodistinctstocks,
dieregionalestimatesofsize-at-agewereaveragedtaobtain

Cm.J.FIsh.Aqual.Sd.Vol.m.2003

alakewideestimateofmeansiae-at-age(£., andW,,,re-
spectively)fordiebloaterpopulation.

Abundaneelndieesofhbaterandalewlfe
CPUEdatafromtheannualUSGSfalltrawlsur-veywere
analyzedtoderiveyear-specificindicesofrelativeabun-
dmcefmaduh(age2+)bloateranddewifedmgwidr
yeariingbloater.’l‘henaturallogoftheCPUEdatawasfitto
agenerallinearmodelinearporatingefiectsforyear,loca-
dommddepthandaflauringcouelafionsamongobsava—
tionsfi'omthesamelocmianwithinayear(seeKranse(1999)).
'I'heestimatesofthefixedefl’ectsforyearwereusedasan
indexofrelativeabtmdance.'1hesere1ativeindicesareex-
pressedonthenatmallogarithrnicscaleanddifierfromdre
natmnllogofCPUBbyanadditiveconstanLWeusedthese
indiwstoexploredrerelatianshipbetweendiechangesin
bloater overtimeandabtmdanceofyesrlingand
adultbloater,alongwithadultalewife.

Gnawthmodel
Weappliedagrowdrmadelthatallawedmeanlength-n-
agetochangebetweenyears.Wealsomodeledthe1ength—
weight relationship in a way that allowed weight-at-length to
varyavertime.1ncombination,thisallawedustoconsider
how both weight- and length-at-age changed over time. Dy-
namicsinmeanlength-at—ageweremodeledbygeneralizing
theincrementalvonBertalanfl'ygrawthmadeltoincorporate
time-varying growth parameters. Thus, predicted mean
length-at-ageinagivenyeardependedonmeanlengdiaf
dresamecohortinthepreviousyearandthegrawthparame-
lengths-at-age were compared with the observed lengths-£-
ageova'allyearsandadjusunentsinthegrowdiparametea
weremade,usinganumericalsearch,taprovidethebest
agreement between the observed and predicted
leagths-at-ageWewerealsointerestedinhadeerelation-
ship between mean weight-at-age and mean length-at-age
changedovertimeanddmsrnadeledmeanweightasa
pawafimctionafmeanlengthwithparmdratwereal-
lawedtovaryovertime.The determiningthis
time-varyingrelationshipwereestimatedbyusingthemto
predict mean weight-at-age using observed mean length-at-
ageandcomparingthesepredictionstodieabservedmean
weight-at-age.Theparametersadeefinalmodelarede-
saibedin'lhblel.’1hedynamicequationsusedtopredict
meanlength-andweight-at-ageandchangesinthegrowdr
overtimearedescribedin'lhblel'l'heequatiam
presentedinthesetablesarereferencedbyTaxwherexis
dretablenumbermdyistheequationnumberwitldn'lhble
x.

nodal

In1966.-sanlenng-st-ageforallageswaspledictedby
diestandardformofthevonBertalanfl’ygrowthmodel
(eq.'1‘2.l).Forallotheryears,meanlength—at-agefarages2
andalderwaspredictedbydieincrementalformafthevan
Bertalanfiy model (eq. T22). Ifthe incrementalformpre-
dictedanegativegrowthinaemeatthenthegrawthincre-
mentwassettozero.’1‘hissitrmionneveroccunedoncedie
parametervalueshadeonvergedtodiebestestimBe-
cansechangesinyearlinglengdiovertimeconldnotbede-

OMBNRCCanada

&Mad

scribedwellbytheabovemodel,wenmdeledyeading1ength
separatelyasarandomwalkonthelogscale(eq.'l‘2.3).'1‘he
randomwalkwasstartedinl965attheyearlinglengthpe-
dimdbyeq. 12.1.1‘hisapproachallowsyearlinglengdrm
changeovertimebutassumesdratyeu'linglengthinagivar
yearwouldtendtobesimilsrtothevahrefromdreprevious
year.
Similarly,wemodeledL_asarandomwalkoatlnlog
scdewhaegisanndomchmgeesdmatedbydlemadel
inevuyyearanddrerandomwalkwasstartedatL.inl965
(eq.T2.5).'Iheparameters Luandkwerenatallowedto
varyfreelyfromoneanodrer.ratherkwasassumedtobea
linear function of L... with a year-specific random (white
noise) deviation (eq. '1‘2.6).Thisrelatianshipwouldallow L...
andktohaveeitherapositiveornegativerelationahip.The
parametusofthisrelationshipandeachyear-specificrm—

Wands!
Meanweight-at-ageinevgyyearwaspredictedfi'omob-
servedmeanlength-at-age(l.,,,)usingalmgth—weightrela-
tianship with time-varying parameters (eq. 12.4). Both 01,
andB, wereassumedtofollowarandomwalkonthelog
scalewithdiestartingvaluesinl965estimatedasparame-
ters during model fitting (eqs. T27 and 12.8, respectively).

Moddapdllizadou
ThemodelwasfittedusingADModel Buildersoftware
(0tterResearch2000).ADModeanilderisasupersetof
CH that uses automatic difi'erentiation in the application of
aquasi-Newtonnrethodtofitfingnonlinearmodekbasedon
a user-specified likelihood equation. In the fitting ofthis
model, the negative log-likelihood function was minimized
toobtainparameterestimates.’l‘he1og-likelihoodconsisted
ofsevencamponents:
(D €=Q+Q+Q+Q+Q+Q+Q

whichcouespondtodiasefa'theobservedmeanlength-
andweight-at-age, therandomwalks, andthewhite-noise
deviations (Table 3). Mean length- and weight-at-age were
both assumed to follow a lognarmal distribution (eqs. T3.1
and'1‘3.7).'1herandomdeviationsfmmtherandomwalks
andfromtherelationshipbetweenL..andkwereassumedto
follow normal distributions (eqs. 13.2-13.6). Because the
variancemrmsforel-87wereeidierestimateddirectlydruing
modelfittingorprovidedbeforefitting.thelikelihoodcom-
ponentsareself-weightingandnoadditianalweigluingfac-
torswereused.

'I'hcvarianceterm.o§,,for€1 (eq. '1‘3.1)wasage-and
year-specificandcornposedoftwovariancecomponents,

surementerrorparametercapmresthechangesrnthereli-
abilityofthemeanleagths-at-ageassamplesizechanges.
'I'heprocesserrorparametercapuuesanyerrorsdratarein-
dependent of sample size (e.g., the spatial distribution of
catainsizesofbloaterindielakedifl'eringbetweenyears).
'l'hemeasurernenterror(onthelagscale)wasnotfitteddur-
ingtheminimiudonprocesaratheranestimatewasob-
tainedbasedanthevariationinobsavedlengths-at-agebia
0.0041)andtreatedasaconstant.

59

For€‘(eq.'1‘3.4),thevariancetmn,a§,cmldnmbeesti-
matedduringmodelfittingnthesametimeasoi.Bothaf
dresevariancesrefertoprocesserrorsforunobservedstate
variables,andhencethereisasingularityinthelikelihood
mfacewhentheirratioapproacheszemorinfinity($chnute
andRichardsl995).Asanalwrnative.wemodifiedastrat-
egyusedbySchnuteandRichards(l995).Pirstweassumed
dratoiwasproportionaltooiwidrmepropa'tionalitycon-
stant,p,thatwasfixedbeforemodelfitting.DIuingmodel
fitfing,o§,wasestimatedasaparamemrandtheq?ropriate
vahreofofiwascalculatedusingdieestimateofowandthe
fixedvalueofp.’1bchooseavalueforp,weusedaniteutive
approachFustwefitthemodelusingawidemngeofval-
uesforptaobtainestimatesofoiandoi.Wealsocalcu-
latedthe“observed”varimsbasedonthevariationinom
estimatesafgande,foreachvalueofp.Bycomparingthe
observedvariancesintherandomdeviationstotheestimnes
ofthevafiancecompmenmwefoundthesewasonlyone
valueofpforwhichtheaequantitieswereapproximately
equalanduseddrisvalueofpinourfinalnrodeLAtthis
valueofp.thelikelihoodprofilewithrespecttopisnearly
flat‘l‘heresultingvariancesappearedreasonabletousand
theresultingmodelpredictionsofmeanlength-at-agewere
notva'ysensidvetospecificchoieesforpAlthoughourap-
proachisarepeatablemediodforobtainingtheratioofthese
variances,itisanadhocapproach.Methodsforweighting
likelihoodcomponentswhenprocessmareinvolvedis
clearlyanareaforfurtherresearch.

Thevariancecomponents, a}, sndo’fort’sandfg
(eqs.'1‘3.5and'1‘3.6,respectively),alsocoufdnothothbees-
timateddmingthefittingprooess.lnstead,asdescribedfor
variancecomponentsof€4,weallowedo§tobeestimd
duringdremodelfittingprowssandthenassumedthatog
wasproportionaltoofiwidrapropationalityconstantofcp
thdwasfixedduringmodelfitting.’1‘hevalueofrpwsscha-
musingthesameprocedmedescribedforfi.

The L.,a,andBinthefirstyear,theintercept
ofthefinearrdadonshipbetweenLuandhandthevariance
componentsmustbepositiveandwereresuictedtotheposi-
tiverealnumbersduringmodelfittingbyestimatingthemon
drelogscale.Allotherparameter-swereestimatedandre
arithmeticscale.

Othermodelvarlantsesplared

Thepocess ofchoosingthemodel structurepresented
abovewasbasedonexploringmanyvariantsofastandard
vonBatalanfl'ygrawthmodeltafindamadelstrucunethat
adequately described the patterns found in the observed
lenng-andweight-at-agedauWeatternptedtofinddIe
mostparsimoniousmodelthatprovidedgoodpredictionsaf
meansize-at-ageovertime.Weassessedmodelfitbyvisu-
allyinspectingresirhralplotsandbyusingAkaike’sinfonna—
tionaiterion(A1C)tocoranuedifl'erentmodels.IndIis
sectionwewilldescribeseveralofthedifl’erentmodelstruc-
anesthatweinvestigatedtohelpguidemodelchoicc.

Inaninitialattempttomodelthechangesinlength-at-age.
weallowedonlyL..tobediflerentineachyear,keepingk
conshntoverdreendredmepaiadwhichproducedsuoag
pawnsintheresidualsthatwedeemedunacceptable.8ub-
sequently,weallowedbothkandL.tovarywithtimeinde-
pendentlyofoneanother,butthismadelfai1edtoconverge

02003NRCCuada

Cm.J.Fleh.Aquat.Sd.Vol.60,2003

mammamwtmmmmmmmmmm

 

 

'1 l I l‘ 1'.
Parameter Description
I...” Lformeanlengtb—at-ageinl965

m
V

Ageatlength01n1965

~93”?

GQQJ“
guuuau*

Q
-N

D

Randomwalkdeviationsfa’l...

Intereeptoflineufimctionbetween kandL.
SlapeoflinearfimctionhetweenksndL
Year-specificdeviationsforlinecfunctionbetween kandL...
Randomwalkdeviationsforlq,
“Measuremenf'en'orinlengdi-at-agedata‘
“Proms”errurinlength-at-agedata
VariancecomponerrtforLrandornwalk
VariancecomponentforLlJrandornwalk
Proportionslityconstantforvarianceconmanentfa e.‘

(1196, afiomtheweight-lengthrelationshipinl965

13m, swam—mammal”

u, Randomwalkdeviationsfora

5, Wwalkkviaflonsftrfl

oi Varianceoornponentforweight-at-agedata

ofi Vuiancecomponentfathearandomwalk

«p Proportionalitycomtantforthevariancecompenentofdrefirandomwalk‘
n Numberofyears

n” Samplesizeoffiahagedineachageandyeu

 

Tumfixadduingmodel-fitdngproeesaSeetenforfmtherexphnadon.

'lhhleLDynamieequationsusedtopedictmean
length-andweight-at-ageandchangesingrawdrpa—
rametersova'time.

 

Mean length- and weight-stage

n1 Lu,“ .., Luau _ ‘4rm““o))

T22 Lao-1,»! 3‘ L1,, 1' (L, " ..,)O " 94’)
12.3 Wimp“): 10804,) + i,

17.4 W., =01 ,1...)3

Growth prams

T25 |Os(L.,.,)=los(L..,)+6,

12.6 k, a b + 1111..., + 9,

117 ”((1,4) g 10801,) + U,
'rzs logtB,.r)- 10303,) + 5,

Natal.” isthepedictedmeanlengdr—at-ueainyeuy;
Z” istbobsavedmeanlength-at-ageainyeary;wu is

drepredictedmeanweightatobsuvedmenlengtb-at-agea
bye-Ink, hummhmnkhb
Brodymwdreoeffidflinyeuxanda, andB, arethe
mdthewa’ght—lengthrelatiomhipinyeary.

onasolufionlnanatmInpttaallawbothLmdktovar'y
avertimebutinacouelatedmsnner,we Unincor-
pontedrefunctionalrelationshipsbetweenL.andk(odee
generalformk=qLZ£)proposedbyJensen (1996), butnone
ofdresemoddswouldconva'gemasoludmapparently
becausethe sssnmednegativerelationahip does notagree
withdieobser'veddataWedIenfittedamodelwherekwas

alinearfunctianL,whichwasabletoconvergeonasolu-
tionbrrtstillhadsuongpatternsindieresidualaparticulariy
foryearlings(age l).Torernovetheaepatternsintheresidu-
alaweurivedatthemodelpesentedaboveWealsoat-
Wedmcapmredrechangeinmeanweight-at-ageby
allowingonlyaorBtovarywidrdmebutwefoundthatthe
AICforthemodelinwhichbothaandBwereallowedto
varywithtimewaslowerthanehhaofdiesesimplermod-
els.

Finally, wealsofitacolnrt-basedmodel(i.e.,growthpa—
rametusdifieredbetweencohatsbttremainedconstantaaoss
agivencohort’slife)becansethegrowthpstternsameared
tobecorrelatedacrossyears. Howevu',wefonnddratthis
modelstructureproducedverystrangpattemsindreresidu-
alsandwasunabletocaptrn'ediechangesingrawththatoc-
curredinthelmel990s.Fathisreason,wedeemeddris
typeafmodelunaeceptable.

Moddllt
Aldroughthemeaslnementeuurvariance(o.)was6.8l
timeslargerdrantheprocessermr(o,;‘lhhle4),theconui-
butionofmeasurementerrortototalvarianoeinpredicted
usalength-at-agedeclinesrapidlywithincreasingsample
size.Forexample,forasamplesiaeoflO, themeasurunent
errorcontribules +10% tothe totalvarianee inpredicted
meankngth-at-age,butitscontributiondeclinesto~21%fu
asamplesiaeaf25.Acrossallyearsandages,28%odee
ouervedsamplesizes(650f234age-yeu'samples)were

OMNRCCnada

Szalaletal.

61

mam-ermmnabmmmr

 

T3.1 6,: mean length-at-age

13.2 (‘2: random walk in L.

T33 (3: random walk in L1.)

13.4 6‘ year-specific deviations in k
T35 6,: random walk in a

113.6 (‘6: random walk in B

13.7 (7: mean weidrt at mean length

 

Z Z _0 (M02 ..,) + (108(La.y) - lo8(I:a,y))2]

a",
_ 2 2
a” — o, + 0...,InM

"171°“ ”“2" 3??"
17-10803?)- g—fiéfi
’
-§|os(01)—g—f-Ze§
7
-10.5
‘12—108038)‘ 1?”,
n_-110.5
2 «ms .63.;

 

2
2!.

”my

:2-.. rainwawuv
'5,

’

 

MWmdefinedindietextandTablel.

less ofsamplesiaes
less 251ntheyearling(33.,3%)andolderages(age5
2.4%;age6, 45%; age7, 72%)thanintheintermediateages
(ageZ,15%; age3, 6%; andage4, 3%). Themodelfitindi-
catesapositivecorrelationbetweean andkCI‘able3),im—

and k (Pauly 1980; Jensen 1996).

Theresidualsforbothmeanlmgth-u-ageandmeanwdght-

at-agefromthismodelwere moderateinsimwith
mostwid:in:t2standarddeviatians(SD),andonlyintlnee
caseswasdievalueoutsidedierangeflSD.Moreover,the
residualsgmerallyexhibitednopatternwithyearateach
ageWeexaminedthosespecificcasesinwhichthereap-
pearedtobeapotentialternparalpatternintheresiduals.
Thefirstcasewaameanlength-at-age4,wherediereap-
peuedtobeaslighttrendofnegativeresidualsintheearlim
yeuachangingtowardspositiveresidualsinmelateryears.
Becausedieseresidualswererelativelysrnallinmagnimde,
dieymaysimplyrepreaentaminordepartnrefmmtheaa-
sumedvonBertalanflygrowthmodeLWedidnotdeemthis
minordepartm'eafsuficientmagnimdemexploreamore
flexible growth model such as the model proposed by
Schnute (1981). The second andthirdcases, mean weight-
at-ageslmd7,shawedoppositetrends.1npredictingna-
wa'ght-at-age 1, the model slightly overestimated men
weightintheearlieryearsandunderestimateditforlater
yearsintheseries.Formeanweight—at-age7,themodel
wndedtounderestimatemeanweightindieearlieryearsand
overestimateitinlateryeara.WeinMpret®aeresultatoin-
dicatedntourmodelmaynothavecapturedallofthecom

Mll’arameterestimatesand

standard «rats (in pumdreaes) (p = 18.16
and (p = 2.75 x 10").

 

 

 

Prametm a}, :3 0.1!)41

to -3.641 (0.477)

b 2.164 x NT” (7.65 x 10'”)

In 5.98 x 10" (7.68 x 10")

of, 7.40 x 10-4 (1.90 x 104)

a}. 4.51 x 10" (1.21 x 10”)

of 2.56 x 10“ (1.03 x 10")

of, 0.199 (2.66 x 10")

a: 8.76 x 104 (3.19 x 104)
Note: This table contains all of the paramenrs

that do not vary with time.

plexities In changes In the bloat

relmonship
overtimeandthatthefactorsdminflnencedtheweightd
Isaalengthofyoungerandalderbloatersmayhedifl’erent.

Bloatergrowthdynanda
'l'hetrendsinestimatesofbadrgandyearlinglength
showasubstantialdeclineoverdrelatterportionofthestndy
period (Figs. lb, 3a).Estimatesof L..peakedinl976—l977
atavalueof354mm,thendeclinedmadilytlnoughthe
1980sandl990sto302mmbyl999(Frg.3a).Yeading
lengthdeclinedfi'omamaximumof183mminl965to
l35mmin1987butsubsequentlyhasincreasedslightly
(Fig.1b).'lheestimatesofkshowaaimilardeclineindre
1980s(‘Pig. 3b). Predictedandobau'vedrneanlenng-at-ages
2—7ahawedaninaeasing-a-lmgthdnoughthel970s

OMSNRCCuIada

 

62

(c)

 

 

62

Cm.J.Hsh.Aquat.Sd.Vol.m.2003

ulfistimamsof(a)l.., (b)k,(d)a. (¢)B,and(0pedicmdmeanwdglnofammmbloata(Congmhay0mdme.
(c)1heesdma0edrdadomhipbuweenkndLinmelabmduganbloamrpopdafim

360-
350.
340..
330-
320-
310-
300-4

290

T I I l l I 1
1965 1970 1975 1980 1985 1990 1995 2000
Year

(a)

L,ID (mm)

(

 

 

0.35-
o.30-
0.25-
0.20.. . .. .°
0.15- " ‘

0.10.. .

(e)

.1)

I; (year

0. 05-1

000 T I
290 300 310 320 330 340 350 360
L00 (mm)

 

 

3.46-

3.45-

(Q. 3.444

3.43‘

 

 

3.42 , , i

l r I 1
19651970 1975 1980 1985 1990 1995 2000
Year

andthendeclinedsharplythrouflnthel980sCPigs. lb-ld).
Inrecentyemmeanlcngmmrallagesappearsmbein-
creasingahghtly.
Esdmatesofaandflfromthelength—weiglnrelationahip
showsimilarpattermofchsngeovertimefigs. 3d. 3:).
Bedlinaeasedchn-ingthelancl960ssndremainedrclatively
highduoughoutthe19‘700. Duringdie1980stheydeclined
initiallyandthenincreased thelatel980sbefose
decliningagaininthel9900.’1‘hepredictedweightofm
internmdiate—sizedbloamr(200mm)ovadmeincreased
thronghtheearly l980sandmbsequa1tlydeclinedafiera
slightrecovesyintheeu'ly l990s(F1g. 3f).
'lbassesstherolethatdensin-dependcntmechanismsnny
playinregtdningbloatergmwth,wegrsphicallyaplored
therelationshipsbetweendseesdmatesofgrwthparameten
andbloater abnndmce.Estima0eaofL..andthe
relativeabnndanecofadult(age2+)bloaterappearsmbein-

0.35..
0.30-
0.25..
0.20-
0.15-
0.10-
0. 05-

0.00

1965 1970 1975 1980 1985 1990 1995 2000
Year

(b)

1)

7: (year

A

 

 

0.81 -
(d)

0.79 -

0.77-1

& (us-mm‘)

0.75 ..

 

0.73 r , r

1965 1970 1975 1980 1985 1990 1995 2000
Year

 

75' m

70-

65..

Weight (g)

60..

 

 

55 I l I I I I

l
1965 1970 1975 1980 1985 1990 1995 2000
Year

vuselyrelateddmingmostofthestndypedodfig.4a).
However,asadultblo¢erabundancepeakedanddleude-
clinedduringthel990s,estimatesofl._didnotrespondinan
andcipateddensity-depcndentfashionlncontastahowsa
(Fig.4b)’l‘hereappearstobeaslightchangeindlisuendat
highrelaliveadultabundanclewefacmrouttheefl’ectsof
chmguianyonlylookingattheestimatedrandomdeV/ia-
tions(e,,thisfeauueismoreapparent(Pig.-1c).
Thetemporalpauerninthebloateslengm-weightrelation-
shipduoughmemid-l980salsosuggestsdensity-depmdmt
eflects. Oln’estimatesofthemeanweightatZNmmfrom
1965—1984apparmfollowasimi1arpauemwith
weightdmingpesiodsoflowerbloaterabundancefig. 4d).
However,dmingmelamel980sandearlyl990s, maaaweight
inaeaseddespiuethehighuadultbloamrabundances. This

uanaientmmporalresponseofmemweingZNmmwas
ammo-ne-

Szalaiatd.

m4.Relationahipbetweenestimamsof(a)l..,(b)k,and(c)deviations(¢,)fromme1inearrelationahipbetween Landkand
(Qpedictedmeanweigluofammmbloau(Corsgmshayi)andmerelniveabondanceofarh1t(sge2+)bloaulinLake

 

 

 

 

 

 

 

 

0.35-(b)
0.30-
025 "h.
A ' _. .73 O
L 7:275.';§{:1 s10
8 0.20'l fit..sn °”7°.0‘9§
s 1
3" 0.15- “"3 i”Ila:
.g
0.10“ s“
0.05-
0-00 r r 1 1 1 f r 1 r 1
-5 4 -3 2 -1 0 1 2 3 4 5
75-(d)
O
A70_£2gsn. .
E 1-
2 "
965- u. 'N 0 1
i) 6” as?
. 03
60.. "“13
0“ 0.9:.”
55 r r 1 I 1 r r r r 1
-5 4 -3 -2 -‘| 0 1 2 3 4 5

Adult abundance

Wehiml965—1999.
380~(a)
m-mn. n
O
340- auB‘IW-os
A Q”
E 330- "1
E “‘91.
9 320~ :3
sud 'W
310‘ 0.3.9‘
300.. '99 “5:“
290 T l I I 1 If 1 I V 1
5 4 -3 2 -1 0 1 2 3 4 5
0.10- (c)
"I
ns
0.05-
U) s73. . .
.5 “,5. 11. u-
a 74 ”sue .WOOO
g 0'00“ 70' '"oss' .
' 097”” in01
8 s s” a“
0.05.
'94
.0'10 I I I T T I I I T
5 .4 -3 2 -1 0 2 3 4
not seen in the von Bertalanfl‘y . Subsequently,

parameters

meanweightatZOOmmretrunedtolowlevelsandlikeL“
didnotincresseinthelsmel990sdespitelowerblomrabun-
dance.

Density-(bpmdentgmwthalsoappearsmoccurforyeading
blodus. The relatiomhip between estimatedyeadiug length
amiyeadingrelativeabundsnceshowsdecreasedyesrlingsize
withincreasedabrmdaneeofyearlingsfig. 50). However,in
memostrecentyears(l994—l999),aswasseenfordieadult
growdiparamemmisrelationshipappearstohavebroken
downluvenilebloatergrowthandadultalewifeabundance
areposirivelyconelatedfioml965ml999,withlarger
yeadinglengthaehievedinumesofhighalewifeabundsnce
(Fig.5b).

Discussion

Wewereabletocapmrethedramaticchangesinbloder
m—mwmodefimchmsuhkhmflmdyadins
hogmmmodeLingenaaLpredictedmeanlengm-and
weight-stagematsgreedva'ywenwithobsa'vedMSome
MWeredetectedindieresiduals,butthesepaMmsap-
pectobeonlymincrdeviafiomfromomMmodeLAl-
moughlimitswithregardtoestimabilitywereencountesed
(e.g.,wecouldnotestimatekindependentlyforeachyear),
wewesestillabletocapmrethedynamicsandinferme
mechanismsofthechangesinblostu'grawthomtime.

Ourmodelsharesfeauueswithsomeprevimsmuuptsmfit
dynandcgrowdrmodelsto-s-lmgdi-at-agedatamar
andMyersl990;Millaretal.1999).Uniquefesunesofour

approachinclndedusingadmeseriesapproaehmdescribe
changesingrowthparametersovertime(i.e.,randomwalks)

tamerthanassumingrelaticnshipswith environmentalfac-
mrsandcombininghngth-andweight-at-ageinfonnadonin
ingrowthovertime. Byusinga

eliminating
observed when calculated directly from observed size-st-
age.Furthermore,theparametersofdiegrowthmodelpo-
videausefirlandparsimoniouswayofsummarizinghow
gmwthischangingovertimeandallowresesrcherstoinves-
tigatemechanismsforchangesingrowthovertimeusingin-
formationfromallagesradierthansselectedfew.
Theinaememalgrowthmodelpresennedhereallowedall
fishofdnmsinmgardlessofdaeirprevicnsgmwthhis-
tory,toschievethesamegrowth.'l‘hisfeauueallowedfcr
fishtorecoupsize—at—ageevenafierpreviouspoorgrowdrif
cmdifiomimprovedAsmchmismodelwouldnotbeable
to emulate stunting (the phenomenon of previous growth
historyinfluencingcuuentgmwthpotenfial;Ylihrjulaetsl.
1999). Almoughwedidnotseeevidenceforsmntinginme
lakehfichiganbloawrpopuladmourabilitymdetectdlis
conditionmaybeobscuredbecauscoftherelativelyslow
changeinconditionsovestimeandtheobservationthatmost
individualsexperieneingpoorgrthhconditionsinearlylife
dsoexpedencepoorgrowthcondifimsusnadnklffmdier
evidencesuggestsdntsmntingmay beimportsntinblodu
growth dnnthedynamicgrowthmodelwould
needtobesdjustedmincorporatethisprocess.Millarand
OMSNRCCIIada

 

(a) r

('8':
1965

Length (mm)

   

Length (mm)

 

64

Fig.5.Relationshipbetweenestimatedyearlinglengthand
(a)relativeablmdanceofyearlingbloates(Coregonushayi)in
lakeMichigan.1965—l999,and(b)relativesblmdanceofa¢hrlt
(1.9? 533mm (Alma pseudahaungus) in Lake Michigan.

 

 

190- (a)
670 68' 56650
E _ 77.
g 78;”
g, 80-
g 081
_1 150— 987. '82
96° . 09333.3
9504-
4111117
130 I l 1 I 1 l
-5.0 -2.5 0.0 2.5 5.0 7.5 10.0
Yearling abundance
190— (b)
.8268 was
9
E 170_ 77.74%‘75'.73M
E
go so-
: 81-
3 150- 132-99,
. 931'
110-1"
0786'
130 I I l l I l I l

 

 

-2 -1 0 1 2 3 4 6

Alewife abundance

5

Myers (1990) have explored one model that incorporams
surntingandcouldbeusedinananalysissimilartoours.
Ourmalysis indicates apositive relationship between L.
andkintheLake bloamrpoptfladonPrevious
studiesoffishgrowthhaveshownthatme be-
tweenL. andkis generally negative (Pauly 1980; Jensen
1996). However, Pauly (1980) looked at this relationship
scrosspopuldmns, whereas we lookedwithinoneindividual
fiahpopulationWehaveuiedseveralfunetimalformsto
fmceanegafiverdadonshipbetweenhandhhoweva,
noneofthesemodelswasabletoconvergeonasolution.
Thepodtiverdadonshipthatweobsavedalthonghinitially
conntaintrfitivesimplyimpliesdiatwhengrowdlconditicns
uefavmbbfmblmtaetheyrespondwidibothfastgrowth
ratesandlargermaximumsius.
Ouranalysiscorroborawsaprevioualyreporteddenaity-

depemgruwthresponseofbloamrinlskemchigan
('IbWinkeletaLZOM). Allofmegrowthparamemrsesti-

ll

Can.J.Flsh.Aquat.Sd.Vol.60,2003

matedseanedtocbclineinconcutwidiincreasingbloater
abundancellowevensinccthebeginningofthe 1990s.s
changeindiisrelationshipappearstohaveoccurredDuring
ddsdmbloatcr'gr'owthinlengthandmeanweightat
Mmmremainedlowdespiterelativelylowabundancesof
both adultmdjuvenile bloam. We notethatthese recent
changesinbloatergrowthwerecoineidentwiththeinvasion
andexpansion of zebramussels(Dreissena polymorpha)and
medisappearanceofDiporeiaspp.inLakeMichigan(Naleps
etaL2000;Fleischeretal.2001).Diporeiarepresentsams—
jorlinkbetweenpelaficproductionanduppertrophiclevels
inlakehfichiganandisalsoanimpcrtantcomponentof
adultbloawrdietinlakeMichigamexhibitinghigherlipid
conmntthanotherbenthicmaaoinvertebrates(GudneretaL
1985;Randetal. 1995;Davisetal. 1997).Zebramussels
mayalsobehavingprofoundefl’ectsontheprimaryp'oduc-
tivityofLakeMichiganbecauseofenergeticdenmds
(Madenjian 1995; Stoekmann and Garton 1997) thatmay
manifest in lower productivity in lower trophic level fishes.
huesfinegJhechangesinbloamgrowthinthelm
areevidentindieestimatesofldandyeadinglengdibutnot
intheestimsMofk.‘lhissuggeststhatalthoughalltluee
growthparameteramaybeafiectedbyinuaspecificdensity,
changes in the food web can influence them difierently.
Suchadifierencecouldariseifdiechangeinthefoodweb
causedapproximatelythesarneefl’ectonirmemerualgmwth
(inmosth).inespectiveoffishsize(seeWslMsandPost
1993). Only future monitoring of bloater length-at-age will
tellifthesenendscontinueindiefuuueandifdiesechanges
inbloatergowthareahsrbingeroffuunechangestothe
LakeMichiganecosystem.
Ominsbilitytomodelchangeaingrowthcfbodradult
andjuvenilebloaterwithacommonsetofparameterssug—
geststhattheselifestagesofbloatermayberespondingto
difl’esentenvironmentalconditicns.’l‘hisresultisnotunex-
pectedbecauseofdiedifi'erentbaflrymetriehabims occu-
piedbythetwolifestages.luvenilebloateroccupymae
nearshore,shallowerdepthsandfeedpdmarilyonzooplank—
mmwhu'easadultblosterarelimitedtoahypolimneticdis-
uibutionandfeedprimarilyonhypolimnetiepeymaviset
aL199‘7).Ccnsequa1tly,inuaspecificdemity-depmdentgrowth
inblostersappearstobelifestsgespecific(i.e.,juvenih
growthsppears torespondsu-ongly tojuvenileabundance
butnottoadultabundance,andach11tgrawdrseemstore-
spondprimarilytoadultabundance).
Previoussmdieshsveconcludeddiattheinvasionoftheex-
odeskwifeflbsapseudohmmgufihsshadprofoundefiects
on the bloater population in LakeMichigan (e.g., Crowrhr
and Crawford 1984; Eek and Wells 1987). For example,
(310de and Crawford (1984) suggested that competition
betwssathepelagicjuvenilebloaterandadultalewifefor
pelagicresoureeshascauaedbloatertoswitchtobenthicre-
sourcesatanearlierage(fromage3toage2).lfcompeti-
tionbetweenjuvenilebloamrandsdultalewifewassuong,
wewouldexpectthatintimesofhighalewifeabundmce,
juvemlebloatergrowdiwouldberedlicdlnfactyoung
bloater growth rate, as indicated by predicted yearling
lengmmaeasedincoucutwidiadultalewifeabundsnce.
'l‘hissuggeststhatanynegstiveintersctionbetweenjuvenile
bloaterandsdultalewifedoesmtappeartoafl’ectjuvenile

OWNRCCnada

Eel

11H: h

 

Szalaletal.

blosterlength,atleastwidiinthedensitiesofalcwifeand
bloataobserveddufingthesmdyperioda965-l999).
The implications of density-dependent growth in bloater
formelskeMichiganecosystemmaybeprofoundsmce
memid-l980s,concernabomdiebalancebetweentheped-
atorydemandsofstockedsalmonidsandtheproductivityof
diefomgebssehssrisenSeveralobservationsindicatemat
dwforageavailabletothesalmonidaparticularlythechi-
nooksalmon(0ncorhynchus tshawytxha), may notbesufi-
cienttomaintainthe populationsofthel9808(Stewart
and lbarra 1991; Holey et a1. 1995). Bloom; My
smallerindividualuepreseutapotentiallyimportantalterns-
tive forage item for the salmonids (Madenjian et a1. 1998).
Thereforeindiefiiuueanundmtandingoftheforeesgov-
erningbloaterdynamicswillbeimportsntinbalancingpred—
atorydemandandforageavailability.Chsngesinsize-at-age
mayhevemanyefl’ectsondwirpopulationdynamicsdn—
cludingmcrtalityrawsandfecundity.0urabilitytosccu-
ratelysssesstheavailabilityoffongeforsalmonidsdepends
inpartbnourunderstsndingofblowpopulationdynamics.

Acknowledgment

ThisstudywasunderwrMnbytheU.S.Fisheriesand
WildlifeServiceaJSFWS) AidinSportfish Restoratioan-
ject F-80-R-3 (Michigan) and the Michigan Department of
Natural Resources (DNR) Fisheries Division, by Michigan
Sea Grant College Program Project No. R/GLF-46, under
Grant No. NA76R60133 from the Ofice of SeaGrant, Na-
tional Oceanic and Atmospheric Administration (NCAA),
US.
the State of Michigan. We also acknowledge die contribution
ofbaselinedalafromlskeMichiganasdwresultofthe
continueddedicationofvesselaewandsciencestsfl’ofdie
US. Geological Survey (USGS)—GreatLakesScienceCen-
teroverthepast40years'l'heGovernmentoftheUnited
Statesisauthorizedtoproduceanddisuibutereprintsfor
governmentalpmposesnotwithstandinganycopyrightnota—
tionappesringhereon'lhisarticleisConuibutionNo. 1230
oftheUSGS-GrestlabsScienceCenter.

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OMNRCCmada

||l

 

Chapter 2

QUANTIFYING UNCERTAINTY IN LAKE MICHIGAN ALEWIFE AND

BLOATER POPULATION DYNAMICS, 1962-1999

Introduction

With the increasing acknowledgment of the importance of ecological interactions
in the management of fisheries resources, focus has shifted from single-species stock
assessments to integrated assessments of the effects of management actions on fish
communities and ecosystems. Such assessments require not only the ability to assess the
current status of a key species of interest but also the ability to assess other species
involved in interactions with this species and the form of these interactions. Recognition
that predation mortality can play an important role in the dynamics of a fish population by
altering natural mortality rates over time has led to the development of several approaches
to incorporating this source of mortality in fisheries stock assessments. These approaches
have ranged from extending traditional stock assessment methodologies, such as virtual
population analysis (Tsou and Collie 2001) or statistical catch-at-age analysis (Livingston
and Methot 1998), to incorporate the effects of predation to the development of new
trophic mass-balance models, such as Ecopath (Cox et a]. 2002) for fish communities.

One of the first approaches expanded virtual population analysis (VPA) into
multi-species models that can incorporate species interactions including predation
commonly known as multispecies virtual population analysis (MSVPA ; Pope 1991; Tsou

and Collie 2001). This method uses catch at age and stomach content data to reconstruct

l4

the population abundances of each species over time. Predation mortality for a given prey
type is calculated from the abundance of each predator type, the consumption rate of each
predator type, a prey-specific suitability index, and the abundance of all prey types.
Consumption rates (per predator) of each predator species are assumed known and
estimated externally to the MSVPA. In practice, these consumption rates are generally
assumed constant over time. While MSVPA provides a reconstruction of the population
abundance of the species of interest that accounts for predation, it suffers from inability to
capture how predator consumption rates change with changes in prey abundance (e. g. a
functional response), assumes that consumption rates are known without error, and lacks
a strong statistical framework to derive uncertainty estimates about the population
reconstruction.

A second approach to incorporating predation mortality into stock assessments is
to generalize the statistical catch-at-age framework and treat predators as another type of
fishery operating on the species of interest (Livingston and Methot 1998; Hollowed et al.
2000). Predator abundances indices play an equivalent role to fishery effort data while
data on the age composition of the stomach contents of the predator plays an equivalent
role to catch age composition data. Predator consumption rates were initially assumed by
Livingston and Methot (1998) to increase linearly with increases in prey abundance as in
a Type I functional response (Holling 1959) but Hollowed et al. (2000) has extended the
approach to incorporate asymptotic consumption rates. Additionally, because of the
likelihood-based approach used in model fitting, it is possible, although not necessarily
simple, to obtain uncertainty estimates for model predictions. While the applications of

Livingston and Methot (1998) and Hollowed at al. (2000) incorporated multispecies

15

 

5P

at-

me

prc

0ft

 

 

interactions through the effect of several predators on a single prey population, they did
not incorporate how changes in the alternative prey availability affect these interactions.

Here we present an extension of the Livingston and Methot (1998) approach to
incorporate the dynamic links between two species sharing a common suite of predators.
We utilize a multispecies Type H functional response to model changes in predator
consumption rates with changes in the abundance of both species and incorporate
estimates of the biomass consumed by predators from bioenergetic models as an
additional source of data. Incorporation of this novel source of data allows us to estimate
key parameters of the functional response. Additionally, our formulation lacks a fishery
operating on either species of interest and predation mortality serves as the sole time-
varying mortality source. Consumption estimates along with fishery independent survey
data provide enough information to reconstruct the historical abundances of the two
species of interest. This suggests that the basic approach underlying the statistical catch-
at-age methodology may have applications to non-fishery based populations where a
measure of the absolute abundance of a source of time-varying mortality exists (e. g.
predator abundance).

In particular, we applied our methodology to reconstruct the historical abundances
of the bloater (Coregonus hoyi) and alewife (Alosa pseudoharengus) in Lake Michigan.
Alewife and bloater serve as important prey items for the five stocked salmonine species
in Lake Michigan (Madenjian et al. 2002). Because the abundance of all five salmonine
species is maintained primarily through stocking, the natural feedback between the
abundance of prey and the abundance of predators does not exist. Therefore, concern

that excessive stocking could lead to a collapse of the prey base arose (Stewart et al.

16

dt
5;:
st:
thc
c0

1(

| |
I
r . . l.

1981; Stewart and Ibarra 1991). Additionally, a potential imbalance between predatory
demand and prey production was suggested by the collapse of the Lake Michigan chinook
salmon (Oncorynchus tshawytscha) fishery in the late 19808. This collapse coincided
with an outbreak of bacterial kidney disease, believed to be aggravated by nutritional
stress (Holey et al. 1998; Benjamin and Bence in press a and b). The ability to manage
the Lake Michigan ecosystem to support a successful salmonine fishery without
compromising the prey fish community depends critically on understanding the dynamics
of the prey fish community and the dynamic link between the prey fish and their
salmonine predators.

Alewife are an exotic invader of Lake Michigan and are believed to have had
profound effects on the Lake Michigan ecosystem. Alewife originally invaded Lake
Michigan from their native marine environment in the late 19408 through the Welland
Canal from Lake Ontario (Smith 1970). By the late 19608, the alewife population had
exploded and become the dominant species in the prey fish community. Concurrent with
the high abundance of alewife, declines in several native prey species (e.g. bloater,
emerald Shiner (Norropis atherinoides) and yellow perch (Perca flavescens» were
observed, presumably due to egg and larval predation by the exotic alewife (Brown et al.
1987). Alewife dieoffs occurred during the late 19608, with dead alewife littering
beaches and interfering with water intake systems (Brown 1972). Fishery managers
responded to the problem of abundant alewife with a plan to stock Pacific salmonines
into Lake Michigan with the goal, in part, of controlling alewife abundance (T ody and
Tanner 1966). Since the late 19608, alewife abundance has declined dramatically and has

remained at relatively low levels since the late 19808 (Madenjian et al. 2002).

17

Nevertheless, alewife remain the most important prey item in the diet of the five stocked
salmonines (Madenjian et al. 2002).

Although bloater does not comprise a large fraction of salmonine diets, juvenile
bloater are thought to be an important alternative prey source for the stocked salmonines,
particularly during times of high bloater recruitment (Elliot 1993). Bloater populations in
Lake Michigan have gone through several cycles of increases and declines since the late
19608 (Brown et al. 1987; Madenjian et al. 2002). Concurrent with these large shifts in
abundance have been large changes in the growth rates of bloater suggesting density-
dependent regulation (Szalai et al. 2003).

Early attempts to assess the effects of the salmonine community on the prey fish
in Lake Michigan relied on the comparison of estimates of predatory consumption from
bioenergetics modeling to estimates of the lakewide biomass of prey fish from fall trawl
surveys (Stewart et al. 1981; Stewart and Ibarra 1991). However, this approach was
limited because it could not dynamically predict how prey populations would respond to
changes in salmonine abundance in that there was no underlying model of prey fish
dynamics or a link between predator consumption and prey abundance. Additionally,
since consumption by the salmonine predators was expressed on an annual basis and the
abundance of prey was expressed as biomass rather than production, it was difficult to
determine if there was sufficient prey production to sustain the estimated level of
consumption.

Jones et al. (1993) recognized the need for a dynamic model of salmonine and
prey fish populations to assess the effects that changes in stocking levels would have on

the dynamics of the prey fish community. Koonce and Jones (1994) constructed

18

 

 

al
ke

int

pa
Stat

int

 

 

 

multispecies dynamic models, called the SIMPLE models, of Lakes Michigan and
Ontario that incorporated dynamic links between predator and prey population through a
functional response model. Through the construction of these models, they recognized
the importance of understanding the dynamics of the prey fish population and emphasized
the need to investigate this area further. Additionally, they emphasized the effect that
incorporating uncertainty in these dynamic models may have on the outcomes of different
stocking scenarios.

To address these two needs, we have attempted to quantitatively reconstruct
alewife and bloater populations in Lake Michigan from 1962 to 1999 while estimating
key parameters governing their dynamics using available survey data. We have
incorporated a functional response model into our estimation model to capture the
dynamic link between alewife and bloater and their salmonine predators. Additionally,
by using information from predator assessment models, we were able to estimate key
parameters of this dynamic predator-prey relationship. We have also used Bayesian
statistical techniques to quantify the uncertainty in all of our estimated parameters for use

in future investigations of stocking policies in Lake Michigan.

19

Methods

We reconstructed alewife and bloater population dynamics in Lake Michigan,
accounting for the effects of predation by the stocked salmonine populations. We did this
by modifying the statistical catch-at-age (SCAA) approach used in fisheries stock
assessment to incorporate predation mortality as the primary time-varying mortality
source. Our estimation model contains two sub-models, a dynamic population model for
alewife and bloater and an observation sub-model. The dynamic population sub-models
track abundance at age of both alewife and bloater using predictions of recruitment,
natural mortality rates and predation mortality rates over time. Predation mortality was
modeled using a Type II functional response, which allowed mortality rates to respond to
changes in both prey and predator population abundances. The observation sub-model
predicts values for the survey and assessment data used in model fitting based on the
current abundances at age for alewife and bloater. We then compare our predictions to
observed survey indices for both prey populations and assessment estimates of salmonine
consumption to estimate key parameters governing alewife and bloater dynamics,
including those describing the stock—recruitment relationships and the functional
response. Uncertainty in these parameters was assessed using Bayesian statistical
techniques to describe the joint posterior probability distributions of all estimated
parameters.
Prey fish surveys

The United States Geological Survey-Great Lake Science Center (USGS-GLSC)
has been monitoring the prey fish community in Lake Michigan since 1962. A lakewide

fall bottom trawl survey has collected information on age, length, weight and abundance

20

of alewife and bloater as part of the ongoing prey fish assessment. In addition, from 1992
to 1996, the prey fish community was also sampled using a fall hydroacoustic survey.
Both of these surveys provide information on the changes in abundance, age composition
and length distribution over time.
Fall bottom trawl survey

Fall bottom trawl surveys have been conducted annually since 1962 at fixed
locations throughout Lake Michigan by the USGS-GLSC. The surveys provide
information on size (length and weight), age composition, and abundance (through catch
per unit effort-CPUE) of the alewife and bloater populations along with several other prey
fish species (deepwater sculpin (Myoxocephalus thompsom'), rainbow smelt (Osmerus
mordax), and slimy sculpin (Cottus cognatus». Trawls were generally 10 minutes in

length and used a % Yankee Standard No. 35 bottom trawl (12-m headrope, 15.5-m

footrope, and 13-mm mesh in the cod end) dragged on contour during the day as
described by Hatch et. al (1981). Sampling was done offshore at up to seven fixed shore
locations distributed geographically around the lake. Generally tows were made at each
location at 9 m depth intervals and the sampled depths ranged from 6 m to 128 m. Not all
locations were sampled in each year, usually related to weather conditions, nor were all
depths sampled at each location due to irregular bottom features. All trawl catches were
processed to estimate the total weight of the catch by species (Krause 1999).

Both alewife and bloater populations were sampled for age determination using
the fall bottom trawl survey. Scales were used to determine age from 1965 to 1982.
Since 1982, otoliths rather than scales have been used for alewife age determination while

scales continued to be used for bloater. All fish below a length cutoff which varied over

21

 

 

f1

01

0W

 

 

time (100-120 mm for alewife, 100-140 mm for bloater) were assumed to be young of the
year (Krause 1999). Ages ranged from zero to nine for alewife and zero to twelve for
bloater. However, very few fish over age 6 were captured for alewife and few fish over
age 7 were captured for bloater.

The age samples were used to construct age-length keys for alewife and bloater
for each year (and where appropriate lake region) and these keys were used to convert
catch per length bin to catch per age class (Krause 1999). Catch per unit effort (CPUE)
by age class for alewife and bloater were analyzed to derive year-specific relative indices
of abundance. The natural log of the CPUE by age class data was fit to a general linear
model incorporating effects for year, location and depth allowing for correlated errors
among samples from the same location within a year (Krause 1999). The estimates of the
fixed effects for year were used as an index of relative abundance at age for both alewife
and bloater from 1962 to 1999. These relative indices are expressed on the natural
logarithmic scale and differ from the natural log of CPUE by an additive constant.

Because very few bloater over age 7 were captured in the trawl survey, all bloater
over age 7 were combined to produce a relative index of abundance for age 7+ bloater.
Concerns over the large aging errors in adult alewife when scale structures were aged, as
observed by O’Gorman et al. (1987), caused us to analyze the adult alewife CPUE data as
a composite age 3+ age class rather than individual age classes. Additionally, age 1 and
age 2 alewife are incompletely sampled by the bottom trawl survey, so the CPUE data for
these age class may not reflect trends in true abundance (C. Madenjian, USGS-GLSC,
Ann Arbor, Michigan, personal communication). Therefore, we only utilized age 0 and

age 3+ relative indices of abundance for alewife in our model.

22

 

 

 

 

 

“’11!

I011

 

Both alewife and bloater length and weight at age were calculated from the fall
bottom trawl data. Since bloater weight and length at age has varied substantially over
time, we used the predicted mean weight and length at age from Szalai et al. (2003)’8
time-varying growth model. Alewife mean length and weight at age was calculated from
the fall bottom trawl surveys by averaging across all years.

Hydroacoustic survey

From 1992 to 1996, the Lake Michigan prey fish community was assessed by the
USGS-GLSC using a fall hydroacoustic survey. Acoustic measurement were made at
night along a selected transect with a second vessel following to perform a midwater
trawl to determine species composition (Argyle et al. 1998). Survey transects were
located throughout Lake Michigan (excluding Green Bay and Grand Traverse Bay) and
were selected to provide good geographic coverage of the lake basin. Alewife abundance
estimates were divided into two life stages (young of the year and age 1+) while bloater
abundance estimates were combined across all age classes (Argyle et al. 1998). Variance
estimates were then calculated for the lakewide estimates of abundance (Argyle et al.
1998). Due to inclement weather, the number of transects completed in 1992 was
insufficient to provide lakewide spatial coverage and subsequently we chose not to utilize
these estimates during model fitting (Argyle et al. 1998).

Predator abundance and consumption

Estimates of age-specific abundance of lake trout (Salvelinus namaycush, ages 1-
10+), coho salmon (0. kisutch, ages 1-2), chinook salmon (ages 0-5), brown trout (Salmo
trutta, ages l-5+) and steelhead (0. mykiss, ages 1-5+) at the beginning of the year (prior

to any mortality occurring) were obtained from the most recent predator assessments for

23

 

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SIT,
(1)1

bo

ab

Sm
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are

Lake Michigan (Appendix A). In our calculations of the mortality rates on alewife and
bloater due to predators, we used geometric mean predator abundances (over the year)
derived from these assessments (Appendix A). Chinook salmon weight and length at age
was varied over time, whereas the other predators were assumed to have constant size at
age in all years, because available size at age data did not indicate trends over time
(Appendix A). Predator length and weight at age were obtained from the same stock
assessments (Appendix A).

Estimates of total fish consumption and total consumption by prey type, small
(<120 mm) alewife, large alewife, and other fish, in metric tonnes by all five salmonid
predators were obtained from Madenjian et al. (2002), derived using a production-
efficiency method as described by Ney (1990).
Alternative prey

Four other types of alternative prey besides alewife and bloater were included in
the predation model. These prey types included small (<lOO mm) and large rainbow
smelt, slimy sculpin and deepwater sculpin. These species were not modeled
dynamically, instead their abundance, as estimated by swept-area methods using the fall
bottom trawl survey data from 1972-1999, were treated as known inputs (Madenjian et al.
2002). The average of their abundance from 1972-1977 was used as an estimate of their
abundance from 1965-1971. The average length and weight of small and large rainbow
smelt, slimy sculpin and deepwater sculpin was estimated as the average across all years
(1972—1999) of length and weight data from the fall bottom trawl survey. Abundance in
numbers for each prey type was calculated by dividing the biomass estimates by the

average weight of each prey type.

24

 

 

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10

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Pit
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1:1

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Estimation model

Symbols used in model development are defined in Table 1. Equations governing
alewife and bloater population dynamics and for the prediction of survey indices and
assessment estimates of consumption are in Tables 2 and 3. These equations are
referenced in the text by Tx. y, where x is the table number and y is the equation number
within Table x. Appendix B contains the data used in model fitting and the values of
assumed known constants.

Numbers at age for alewife and bloater were modeled from 1962-1999. For
alewife, the ages modeled ranged from age 0 to age 6+, where the final age group
accumulated all older surviving alewife. For bloater, the ages modeled ranged from age 0
to age 7+, where the final age group also accumulated all older surviving bloater.
Recruitment to age 0 for both species was estimated as a free parameter for each year.
Numbers at age in the first year for both species were treated differently. For bloater, the
number at age in the first year for ages 1 to 7+ were estimated independently as model
parameters. For alewife, the abundance of age 1 was estimated as a model parameter, and
the abundances at ages 2 to 6+ was assumed to follow a stable age distribution predicted
by background natural mortality rates and recruitment at age 1 equal to the estimated age
1 abundance for the first year.

Numbers at age of alewife and bloater for later years and ages greater then zero
were calculated by eq. T2.1. Older alewife and bloater were accumulated in the last age
group (I) by eq. T2.2.

Mortality rates

Instantaneous total mortality rates for alewife and bloater consisted of two

25

 

 

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ra
b(

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de

lar

se .
Pre
Cor
WC)
Dre.
199
bee

funK

additive sources, background natural mortality rates and predation mortality rates (eq.
T23). Background natural mortality rates were not estimated during model fitting but
supplied as known quantities. We assumed natural mortality took one value for age 0 fish
and another for all older fish. Initial estimates for age 1+ background natural mortality
rates were derived from Pauly’s equation and for age 0 the background natural mortality
rates were double that of age 1+ (Pauly 1980). Several scenarios were investigated where
both the natural mortality rates of age 0 and age 1+ for alewife and bloater were increased
and decreased by 25% and 50%. Final estimates of alewife and bloater natural mortality
rates were chosen by the scenario where the objective function was lowest and are
reported in Table 4. We also explored several models where natural mortality rates
declined gradually with age but these models performed poorly. Natural mortality rates
were assumed constant over time for both alewife and bloater with one exception. The
large dieoff in 1967 for alewife was simulated by estimating an additional instantaneous
mortality rate to decrease the survival of age 1+ alewife in 1967 by eq. T2.4

Predation mortality rates for each alewife and bloater age class were calculated
separately for each age class of the five salmonid species. Consumption rates per
predator (kg) of lake trout, brown trout, coho salmon and steelhead were assumed to be
constant from 1965 through 1999 because no consistent trend has been detected in their
weight at age from 1965 to present (Appendix A). Therefore consumption rates per
predator by predator type j were set at the average consumption per predator for 1965-
1999 from Madenjian et al. (2002). We refer to the rate for predator type j as Cmax, j
because we assume that these predators are consuming at or near the asymptotes of their

functional responses. These consumption rates were then distributed among the various

26

prey types and converted to numbers based on the predator’s attack rate for each prey type
and the abundance of each prey type by eq. T2.6a.

Consumption rates per predator for chinook salmon were assumed to follow a
Type H functional response as these predators showed a declining size at age with
declining alewife abundance. The instantaneous consumption rate ( Aid-J) for an age j
chinook on prey type i in year y is predicted by eq. T2.6b. The maximum consumption
rates per predator of chinook salmon were estimated from the maximum observed
consumption per predator by Madenjian et al. (2002).

Attack rates for each predator-prey combination were the product of three
components as in Jones et al. (1993): the effective searching efficiency of each predator
type on an optimal sized prey, the size preference of a predator for a prey type, and the
habitat overlap between the prey and predator type (eq. T2.7). The effective searching
efficiency of each predator was calculated as a length-based scalar, y , which was
estimated as a model parameter, times the length of the predator. As in Jones et al.
(1993), the size preference function is based on the length ratio of the prey type to the
predator type, 3 1.; by eq. T2.8. It is a bell-shaped function that peaks at a preference of l

at the optimal ratio, 6 of 0.25 and the width of the bell is controlled by the parameter

opt ’
07 , which was fixed at 0.01. To account for predator growth throughout the year, we
used the geometric average length at age (of beginning of the year length and end of the
year length) of a predator to approximate its mid-year length.

Habitat overlaps were assumed to be constant across predator ages and only varied
with predator species and prey type and age categories. Habitat overlaps for all predator

species with all ages of alewife were assumed to be one based on the observation that

27

 

al
by
m

3P1

CO.

 

Cor

ale'

 

alewife remained an important component of the predator’s diet even when abundance
was low. Similarly, habitat overlaps of all predators with both small and large rainbow
smelt were assumed to be one. Habitat overlaps between all predators with the exception
of lake trout and brown trout, and adult bloater (age 2+), slimy, and deepwater sculpin
were assumed to be zero based on the different bathymetric distributions of these species.
Both brown trout and lake trout had nonzero habitat overlaps (0.1 and 1, respectively)
with adult bloater due to the observation that these two species have the potential to
occupy similar bathymetric distributions as adult bloater. Only lake trout had a nonzero
habitat overlap with slimy and deepwater sculpin (0.1 for both prey types) as they are the
only predators that feed at the depths occupied by these species. Since juvenile bloater
(ages 0,1) are pelagic, they are potentially available as prey for all predators. The habitat
overlap between lake trout and juvenile bloater was assumed to be one while the habitat
overlap between the other predators and juvenile bloater was assumed to be 0.1.

Instantaneous predation rates on alewife and bloater by age were calculated by eq.
T2.5 using the “average” abundance of each prey type. The consumption of age 0 to 6+
alewife and age 0 to 7+ bloater was calculated using Baranov’s catch equation (eq. T3.1)
by applying these rates to beginning of the year abundance. We devised an iterative
method to find an “average” abundance of alewife and bloater that produced the
appropriate amount of consumption. The following steps were repeated until total
consumption in biomass of alewife and bloater changed by less than 1% of the total
consumption calculated in the previous iteration :

l. Instantaneous predation rates were predicted using “average” abundance for

alewife and bloater (beginning of the year abundance in the first iteration)

28

2. The instantaneous predation rates were used to calculate end—of-year
abundances of alewife and bloater along with total consumption of alewife and bloater.

3. The geometric mean abundance of alewife was calculated and used as the new
“average” abundance in step 1.

The total consumption of alewife and bloater ( Cawbl,j ) by predator type j is
predicted by summing across the consumption of each species and age predicted by eq
T3.1. However, since rainbow smelt, slimy sculpin, and deepwater sculpin are not
modeled dynamically, the consumption of each predator type on these species cannot be
predicted using eq. T3.l. From eqs. T2.6 a and b, we can calculate the relative proportion
of each prey type in the diet. These relative proportions can then be used to approximate
total consumption of all prey species by eq. T3.2. Total consumption by all predator
types (6y ) is then predicted by eq. T3.3 for comparison with observed quantities.
Additionally, we also predicted the consumption (in weight) per predator (C211; ) for ages
1-3 chinook salmon for comparison with observed quantities (eq. T3.4). We also
predicted the proportion of total consumption (by weight) on small (ages 0 and 1) and
large (ages 2+) alewife, and other fish (6Ay ) for comparison with observed quantities.
Predicting survey abundance indices

The observed indices of abundance for the fall bottom trawl survey were on the
log-scale and were calculated relative to the abundance in 1999, and the trawl survey
usually occurred in October of each year (but see below). Therefore, we predicted indices
as the log of the ratio of abundances at age in each year to the abundance at age in 1999,
after accounting for mortality that occurred over 10/12ths of the year. Indices for ages 0

to 7 bloater and for ages 3+ alewife for 1962—1997 were predicted by eq. T3.5a.

29

The timing of the fall bottom trawl survey shifted from an October mid-date in the
19608 through 19808 to a mid-date of mid-September in the 19908. This shift appears to
have influenced the catches of alewife (particularly age 0 alewife) caught in the trawl
survey which seems plausible if the vertical distribution of young alewife changes during
this period (Figure 1). For this reason, we attempted to account for the apparent change
in the catchability of age 0 alewife by estimating a relative catchability coefficient for age
0 alewife after 1990 and modifying the predicted trawl indices from 1962-1990 as in eq.
T3.5b.

Fall hydroacoustic survey indices were predicted by the biomass of bloater (all
ages), age 0 alewife, and age 1+ alewife present at the time of the survey (again after

10/12ths of the years mortality). The hydroacoustic indices were assumed to be only

relative indices of abundance so a catchability coefficient (‘11) was estimated for young

of the year alewife, age 1+ alewife and bloater.
Estimating model parameters

The parameters of the model were estimated using AD Model Builder software
(Otter Research 2000). Using automatic differentiation, AD Model Builder fits statistical
nonlinear models with user specified likelihood equations and performs Markov Chain
Monte Carlo (MCMC) sampling using a Metropolis-Hastings algorithm to sample from
the joint posterior distribution of the specified parameters. All parameters were estimated
on the log scale and diffuse, flat, bounded priors (on the log scale) were utilized so the
resulting posterior distributions would be driven by the data used in model fitting. The
log likelihood for our model consisted of eight likelihood components: one for each of the

seven data sources and one penalty function to prevent the model from converging on

30

 

 

([5

cf
pr
EX
f Ur

lnj

impossible solutions (Table 5).

N 6
62-51LOg(Z€,)+€7+€8 (1)

i=1
The first six components (6, — Z 6) were all assumed to follow normal or log-
norrnal distributions and were incorporated into the log likelihood by a concentrated
likelihood. This allowed us to easily specify weighting factors ( ,1’ ) for each component,
while the overall scale of the variances was set to be consistent with model fit. The
bottom trawl survey indices and the hydroacoustic indices were weighted by the inverse
of the observed variances of each index (If; , If; , 0:1 , 1);"; ; Table 5). Since no
uncertainty estimates were available for the consumption data provided by predator
assessments, the weighting factor for these components (AC and A” ) were fixed at one,
which gives these data sources approximately equal weight as both the survey data
sources. The final data source, the proportion of total consumption from small and large

alewife and other fish was assumed to follow a Dirchlet distribution (Table 5; Williams

and Quinn 1998). The effective sample size of a Dirchlet distribution is equal to y - l

and we modified McAllister and Ianelli’s (1997) iterative method to find the appropriate
effective sample size (38.54). The final likelihood component was a penalty function to
prevent the estimated recruitments of alewife and bloater in 1998 and 1999 to be
extremely different from the average recruitment between 1994-1997. This penalty
function was necessary early in model fitting to prevent the model from becoming trapped
in implausible areas of the parameter space and was not included during the final phase of

model fitting, so it had no impact on final parameter estimates.

31

A stock-recruitment function for both alewife and bloater was fit externally to the
estimation model using the estimated series of abundances at age. The stock-recruitment
function for both species was fit to the linearized version of a Ricker function and the

parameters of as , ’65, and 0'2

S r were estimated for both species (Quinn and Deriso

1999). For alewife, stock size was indexed by the abundance of age 2+ alewife and for
bloater, because size-at-age changed over time, stock size was calculated using the
equation reported by TeWinkel et al. (2002) which relates the weight of mature females
to the number of eggs produced.

Posterior distributions of the estimated parameters, those from the population
model and the stock-recruitment functions, were generated using a two-step MCMC
procedure as in Haeseker et al. (in press). First, the Metropolis-Hastings algorithm within
AD Model Builder was used to sample the parameters estimated directly in the population
model along with the stock size of alewife and bloater in each year. The MCMC chain
was run for 3 million samples saving every 300‘h sample with the step sized scaled to
produce moderate acceptance rates (0.25-0.5) for a total saved sample size of 10,000.

The second step was used to add the stock-recruitment parameters to this MCMC
sample. We appended a single sample from an approximate posterior distribution for the
stock-recruitment parameters to each of the 10,000 saved MCMC samples, given the time
series of stock sizes and recruitments associated with each sample. First, maximum
likelihood estimates of as , fls and 0'52, r were obtained from each MCMC sample by
linear regression as described above. Then, we drew samples of as and fig for each
species by drawing one sample (for each species) from a multivariate normal distribution

with a mean and covariance matrix set at the maximum likelihood estimates for a given

32

 

 

dc“

 

stock-recruitment time series (i.e., one of the 10,000 samples). We also generated a
single sample for each of the 10,000 saved MCMC samples from the posterior of
air for each species by drawing a sample from an inverted scaled I 2 distribution with
degrees of freedom of 34 (number of observations minus 2) and the scale parameter equal
to the maximum likelihood estimate of 0'3, (see Gelman et al. 1995, pg. 480 for details).
Samples taken during the bum-in period (1000 of the saved steps in the MCMC
chain) were discarded. MCMC chain convergence was checked with three methods.
First, trace plots of each estimated parameter were constructed to ensure the chain was
well-mixed and not exhibiting any substantial “stickiness”over long portions of the chain.
Second, the chain was divided into thirds and the mean of each third was calculated and
the distributions plotted. Substantial differences in the means or distributions would
indicate that the chain has not converged upon the posterior distribution. Finally, the
effective number of samples for each parameter was calculated following the procedure

described by Theibaux and Zwiers (1984).

33

Results

To assess overall model fit to the observed data, we compared the predicted
values based on parameters that maximize the posterior likelihood. Predictions of fall
bottom trawl survey abundance indices generally matched the observed values well
(Figures 1, 2). For adult alewife (age 3+), the predicted trawl indices did not decline as
rapidly as the observed trawl indices during the 19708 and then increased more rapidly
during the 19908 (Figure 1). The predicted relative size of the 1967 year class was much
larger than observed values and the predicted relative year class sizes were generally
lower than those observed for late 19708 (Figure 1). For age 0 bloater, the predicted trawl
indices were higher than the observed trawl indices during the early 19608 and the
predicted indices for age 1 bloater did not reach as high a peak as the observed indices in
the late 19808 (Figure 2a). The predicted trawl indices for age 2+ bloater generally
matched the trends in the observed indices well (Figures 2b, c). However, the predicted
indices for age 7+ bloater did not decline as rapidly as the observed indices in the 19708
(Figure 2c).

Fit to the trends in the hydroacoustic survey indices was in general not as good as
to the trawl indices (Figures 3, 4). While the trends in the age 0 alewife hydroacoustic
index were reflected in the predicted values, the predicted values of the age 1+
hydroacoustic index did not reflect the increased abundance observed in the
hydroacoustic survey in 1995 and 1996 (Figure 3). For bloater, the predicted
hydroacoustic survey indices suggested a more gradual decline in the abundance of
bloater than that observed (Figure 4).

Predictions of total consumption by all salmonine predators matched well with the

34

time series of observed values (Figure 5). However, the model slightly overestimates
consumption during the final three years (1997-1999, Figure 5). The model had more
difficulty matching the proportion of total consumption by prey type (Figure 6). The
predicted proportion of large alewife was larger than the observed proportion in both the
early 19808 and the 19908 but the proportion decreased more dramatically and to a lower
level than observed during the mid 19808 (Figure 6). The predicted proportion of small
alewife generally remained more constant than the observed proportion of small alewife
(Figure 6). The model was able to capture the dynamic changes in chinook salmon
consumption per predator at age over time for ages 1 through 3 (Figure 7a). There were
some difficulties in predicting consumption per predator for age 2 chinook, with the
model consistently overestimating consumption for the entire time series (Figure 7a).
Additionally, the predicted consumption per predator for age 1 chinook salmon was
consistently higher than the observed values from mid 19808 on. The predicted
consumption per predator for age 3 chinook salmon was consistently lower than the
observed values from the 19608 through 1980 and the predicted consumption per predator
for age 3 did not reach the high level observed in some years in the mid 19908 (Figure
7a).

Estimates of the instantaneous predation rates by age on alewife and bloater, at the
parameter values that maximize the posterior likelihood, allow us to summarize the
effects predation has had on the dynamics of these two populations. In general, alewife
have sustained much larger predations rates than bloater throughout the time series
(Figures 8, 9). Predation rates on adults (age 1+) of both species peaked in the mid 19808

concurrent with the peak in predator abundance in Lake Michigan (Figures 8, 9).

35

Additionally, predation rates on adults (age 1+) in the late 19908 are similar to or larger
than those observed during the peak predator abundance in the mid 19808 (Figures 8, 9).
Peak predation rates on age 0 alewife and bloater occurred during the late 19708. For
alewife, instantaneous predation rates have been greater then two times background
natural mortality rates since the 19808 (Figure 8). Instantaneous predation rates on
bloater, even in the mid 19808, have never been larger then one half the background
natural mortality rates (Figure 9).

Uncertainty in the key model parameters was quantified using Bayesian
techniques to estimate the posterior distributions for each parameter and we summarized
these distributions using 95% credibility intervals (Table 6). In general, the effective
sample sizes from the MCMC chain were several thousands with smaller effective sample
sizes from the posterior distributions of the stock-recruitment parameters for alewife
(Table 6). The trace plots for each parameter generally showed no long range (> 120,000
chain steps) autocorrelation, suggesting that the chain was sampling the entire range of
the posterior, and the beginning, middle and end thirds of the chains had similar means
and distributions of the sampled parameters, suggesting that the chain had converged
upon the posterior distribution. There was covariance among parameters in the posterior
distribution, and this is summarized by the correlation matrix among the different
parameters in the MCMC sample (Table 7). In general, the correlations were very low,
with the exception of high correlation observed between the parameters of the stock-
recruitment function for each species and a high correlation between the survival of age
1+ alewife during the 1967 dieoff and the stock-recruitment parameters for alewife.

The estimated posterior distributions for the catchability of bloater and alewife in

36

the hydroacoustic survey suggest that the survey measures a much higher proportion of
the true abundance of age 0 alewife and bloater than it does for age 1+ alewife (Figures
10, 11). The uncertainty in the catchability of age 1+ alewife, with a coefficient of
variation (CV) of 39.1%, is higher than that for the catchability of age 0 alewife, with a
CV of 18.3%, and bloater, with a CV of 26.0%. The effect of the shift to an early start for
the fall bottom trawl survey appears to be large, with the posterior of the catchability for
trawl survey being skewed strongly towards zero and very little density above a value of
0.1 (Figure 100). However, the uncertainty in this parameter is large with a CV of
106.5%.

The estimated posterior distribution of the length-based scalar ( 7 ) for the
effective searching efficiency on an optimal sized prey of chinook salmon suggests the
parameter is fairly well-determined with a CV of 17.2%. The posterior distribution for
this parameter is relatively symmetric suggesting there is an approximately equal chance
of the value of the parameter being either above or below the maximum posterior
estimate (Figure 12). To assess the degree of food limitation in the chinook salmon
population over time, we calculated the proportion of Cmax,chs consumed each year by
an age 3 chinook salmon based on the parameters that maximized the posterior likelihood
along with 95% credibility intervals (Figure 7b). During the early 19708 age 3 chinook
were consuming at annual rates close to their maximum. However, as the abundance of
alewife declined in the late 19708 and early 19808, the proportion of Cmax,chs consumed
declined quickly to a low of 0.31 in 1986. After a slight recovery in the late 19808, the

proportion of Cmax, ch S consumed has remained relatively constant at approximately 0.5

Of Cmax,chs -

37

 

re.

 

The estimated posterior distributions for the stock-recruitment parameters for
alewife in Lake Michigan suggests there is considerable uncertainty remaining in these
parameters, particularly in the degree of compensation (Figure 13). For alewife, the CV
of mm”) is relatively low at 23.9% while the CV’s of 03W (41.9%) and flaw
(65.4%) are larger. There are also differences in the shape of the posterior distributions
for these parameters. While the posterior distribution for ln(aaw) is relatively
symmetric about the maximum posterior estimate, the posterior distribution of flaw is
skewed highly towards low values, indicating a significant probability of relatively weak
compensation at high stock sizes (Figures 13a, b). The posterior distribution of the
parameters describing variability in recruitment about the stock recruitment
relationship( Jim) is also skewed, with a long tail extending towards high levels of
recruitment variability (Figure 130). The maximum posterior estimates of the stock-
recruitment relationship for alewife reveals a moderate amount of recruitment variability
unexplained by stock size (Figure 15a).

The estimated posterior distributions for the parameters of the bloater stock-
recruitment parameters are all relatively symmetric with only the posterior of
031$ having an extended tail towards large values (Figure 14). The level of uncertainty
in the parameters of the bloater stock-recruitment is generally higher than that for the
parameters of the alewife stock-recruitment. In particular, both the parameters describing
the productivity at low stock size (ln(a'b1) ) and the degree of compensation (flbl ) have
CV8 larger than 100% (158.7% and 107.5%, respectively). The estimates of the

parameter describing amount of variability about the stock-recruitment relationship

(073-1 r ) have much lower uncertainty with a CV of 27.8%. The maximum posterior

38

estimates of the stock-recruitment relationship for bloater shows a high level of
recruitment variability and a low degree of compensation at high stock sizes (Figure 15b).
The estimated posterior of the survival of age 1+ alewife from the 1967 dieoff
confirms that the alewife population most likely suffered a large dieoff (Figure 16).
However, the degree of uncertainty in this parameter is high with a CV of 84.5%.
Therefore, the estimated posterior distribution suggests that there is also a positive

probability that the alewife population might have only suffered a mild dieoff in 1967

(Figure 16).

39

Discussion

We were able to achieve our goal of reconstructing alewife and bloater dynamics
in Lake Michigan from current prey fish survey data and predator assessment models by
modeling the predation process using a dynamic multispecies functional response that
allowed the instantaneous predation mortality rates to respond to changes in both predator
and prey abundance. Through this modeling process we discovered that there are many
gaps in our knowledge regarding how the predation process occurs and that the
assumptions made in the face of this lack of knowledge can have important consequences
on the reconstruction process. Clearly, reconstructing the population dynamics of a fish
species that is strongly influenced by interactions with other species requires a suite of
information not commonly available for most species (Kitchell et al. 1999; Hollowed et
al. 2000; Cox et al. 2002; Link 2002). The availability of a long-term monitoring
program for the prey fish in Lake Michigan and up-to-date assessments of the main
predator species were invaluable in this process. As fisheries management continues to
focus on ecosystem and food web management, the need for these types of assessments
will increase (Link 2002).

The ability to predict how changes in abundances of predator and prey populations
will effect the consumption rates of the prey population remains an area of active
investigation in fisheries research (Eby et al. 1995; Hollowed et al. 2000; Cox et al. 2002;
Essington et al. 2002). Maintaining a balance between predatory demand and prey
production to support satisfactory growth rates of predators and preserve diverse prey
populations relies on our ability to make these predictions (Jones et al. 1993; Spencer and

Collie 1997; Heikinheimo 2001; Cox et al. 2002; Essington et al. 2002). However,

40

attempts at estimating parameters governing the functional response of a fish predator
from large-scale observational data rather than small-scale experimentation have been
limited and the uncertainty associated with the estimated parameters has not been
quantified (Eby et al. 1995; Cox et al. 2002). In our approach, the utilization of both
current predator stock assessments and existing prey fish assessments allows us to
estimate some of the parameters governing the predation process and quantify our
uncertainty in these parameters.

The Lake Michigan ecosystem provides an interesting opportunity to investigate
how predation structures a pelagic prey fish species. Prior to 1965, the invasive alewife
existed in a system essentially lacking large piscivores. With the introduction of
substantial numbers of five salmonine species through stocking, predation pressure rose
rapidly, causing declines in the overall abundance of alewife in the system (Figures 1, 8).
These declines in alewife abundance led to consequent declines in the consumption rates
of chinook salmon (Figure 7b). This apparent food limitation of chinook salmon
coincided with collapse of the chinook salmon population in the late 19808 (Holey et al.
1998). Our current estimates of predation pressure show that levels in the late 19908 are
rapidly approaching the levels seen during the period preceding the chinook salmon
collapse (Figure 8). If, as suggested by Holey et al. (1998), the collapse of the chinook
salmon population was, in part, caused by food limitation, then the system may again be
approaching conditions where such a collapse is a serious risk.

Both adult alewife and bloater sustained peak levels of predation during the late
19808, while age 0 alewife and bloater predation rates peaked in the 19708. The

difference in the timing of the peak predation pressure between age 0 alewife and older

41

alewife results primarily from two different sources. First, the predation rates on age 0
alewife rise more rapidly in the late 19608 and early 19708 because the predator
population is dominated by younger fish, which preferentially prey upon the age 0 alewife
because of their smaller size. Secondly, while age 1+ alewife abundance declined
steadily from the late 19708 through the mid 19908, the abundance of age 0 alewife
increased from the 19708 through the 19908 causing a decreasing per capita predation rate
despite the growing abundance of salmonine predators. Similarly, the abundance of age 0
bloater declined throughout the 19708 causing increasing per capita predation rates during
this time. However, in the mid 19808 during the period of increased salmonine
abundance, the recruitment of bloater to age 0 increased substantially to cause a
decreasing per capita predation rate.

The full implications of our uncertainty about prey fish dynamics in Lake
Michigan on the consequences of different stocking policies for salmonid predators
remains to be seen. Although the expected recruitment of alewife at low stock sizes is
well estimated, large variations in alewife recruitment that are not explained by the stock-
recruitment relationship suggest that this process variation may play an important but
unpredictable role in the future dynamics of alewife population (Figure 13). Thus, it may
not be possible to maintain the alewife population at a relatively constant level by
selecting an “optimal” stocking level for predators. Effective stocking policies may need
to be responsive to changes in alewife abundance. A formal evaluation of the
consequences of our uncertainty regarding prey fish dynamics in Lake Michigan on
stocking policy decisions, using techniques such as decision analysis (Raiffa 1968), will

be necessary to answer these questions.

42

The use of functional response in ecological modeling has recently drawn
criticism from Walters (2000) because of the lack of fish captured with full stomachs and
low proportions of maximum consumption estimated by bioenergetics modeling . This
suggests that the phenomenon of satiation and the tradeoff between time spent handling
prey and time spent searching for prey may not be applicable to some aquatic ecosystems.
Rather, Walters (2000) argues that fish consumption is driven by a predator balancing the
need to search for food versus the risk of being consumed during foraging activities,
producing rates of consumption that are driven not only by the abundance of food but also
by the energy state of the predator and the level of risk. While Walters’ (2000) arguments
may apply to many aquatic ecosystems, this lack of evidence for satiation does not appear
to apply in the Lake Michigan ecosystem. The lack of a predator on large salmonines in
Lake Michigan suggests that the influence of predation risk on foraging of Lake Michigan
salmonines should be minimal and would only remain through previous evolutionary
pressure on salmonine species to avoid high predation risk. Diet studies of Lake
Michigan salmonine predators have found predators with full stomachs, indicating that
satiation can occur (Elliot 1993). Further, most salmonine diet assessments in Lake
Michigan target actively searching salmonines, which are less likely to be satiated, so
indices of stomach fullness in Lake Michigan are most likely biased towards unsatiated
fish (R. Elliott, USFWS, Green Bay, Wisconsin, personal communication). Bioenergetics
models also suggest that some Lake Michigan salmonines (e. g. coho salmon) also
consume at rates of 70-80% of the possible maximum consumption (personal
observation). Additionally, the consumption rates of chinook salmon are highly correlated

with the abundance of their primary prey, alewife. These observations suggests that the

43

use of a saturating functional response model is appropriate for modeling Lake Michigan
salmonine predation.

There is, however, significant uncertainty in the form of the functional response
that describes process of predation by salmonines in Lake Michigan. Our modeling has
assumed that chinook salmon consumption rates follow a Type 1] functional response
while all other salmonine predators consume at a constant rate. Both of these assumption
may be inappropriate. Clearly, for lake trout, brown trout, steelhead, and coho salmon,
the use of a completely flat functional response is invalid across all potential prey
abundances. However, across the wide range of prey abundances observed in Lake
Michigan since the 19608, lower growth rates for these predators have not been linked to
lower prey abundances (Appendix A). Thus, the prey abundance at which the
consumption rates of these predators declines remains unknown.

Some lack of fit observed when fitting our model could be explained by a
departure from a Type II functional response for chinook salmon. Our predation model
underestimates the contribution of alewife to total salmonine consumption during the
collapse of the alewife population in the mid to late 19808 and overestimates this
contribution during the recovery of the population in the 19908 (Figure 6). This suggests
the possibility of an increase of preference (relative search rate) for alewife when they
become scarce. This could, for example, result from concentrated feeding in areas where
alewife density remains high. The assumptions made in this study were chosen to
represent our current understanding of the mechanisms governing predator searching
behavior in the system. However, these assumptions have an uncertain basis and the data

we used in model fitting were uninforrnative on this topic. More detailed diet

information combined with a quantitative analysis of the abundances of prey types in the
lake could provide more information on the foraging behavior of salmonine predators.

Additionally, there is an apparent conflict between the observed alewife trawl
survey abundance and the abundance of alewife necessary to produce the patterns in
salmonine consumption used in our model. Our model estimates indicate that alewife
abundance declined less rapidly and recovered more quickly than the observed trawl
survey data suggests (Figure 1). One potential explanation is that predators, particularly
chinook salmon, had lower energy density when alewife abundance was declining, rather
then the constant energy density assumed in the bioenergetics models used as a basis for
estimating consumption in the predator assessments. If energy density was declining, this
assumption would cause us to overestimate consumption that was occurring during this
period and consequently overestimate the abundance of alewife in the system needed to
support our estimates of consumption. Additionally, if the energy density of chinook
remained low despite increasing growth rates during the 19908, this could also account
for our overestimate of the abundance of alewife in the late 19908. Lipid levels and
energetic status of fish have been demonstrated to vary spatially and temporally and these
variations are linked to overall fish health (Adams 1999; Madenjian et al. 2002). In Lake
Michigan, recent evidence suggests that chinook salmon energy density has changed from
year to year and is now low enough to be a potential fish health concern (A. Peters,
Michigan State University, East Lansing, Michigan,unpublished data). Thus, historical
changes in the energy density of chinook salmon are plausible.

The role that predators play in structuring fish communities has been shown to be

important to understanding the ecosystem consequences of fisheries management (Cox et

45

al. 2002, Essington et al. 2002, Link 2002 , Link and Garrison 2002). The modeling
approach presented here provides an extension of statistical catch-at~age methodology for
exploring predator and prey assessment data simultaneously and provides a methodology
for quantifying uncertainty in the resulting parameter estimates. It does however, require
large amounts of data from both predator and prey fish assessments. The availability of
long term monitoring of both predator and prey populations may be a significant
limitation to the application of this approach to other systems (Link 2002). Additionally,
even in systems such as Lake Michigan where these types of data are available, several
key uncertainties remain, particularly regarding the dynamic link between predator and
prey populations. Clearly, the uncertainties surrounding this dynamic link are not unique
to the Lake Michigan ecosystem and our analysis has highlighted several areas of future
research to further understanding of predator-prey interactions in pelagic fish
communities. Additionally, the consequences of these uncertainties on the management
practices to balance predatory demands and prey production remain unknown. However,
the Bayesian statistical framework utilized in this modeling effort allows the qualification
of some of these uncertainties for future formal analysis of their effects on management

decisions.

46

Table 1. List of variables and parameters used in the estimation model (a: age, y: year).

NS,0.y
28909))

3,0

Beginning of the year numbers at age of prey species 3
Total instantaneous mortality rate for prey species 8 (y")
Background instantaneous natural mortality rate for prey species s (y“)
Instantaneous total predation mortality rate for prey species 3 (y")
Instantaneous mortality rate associated with the 1967 dieoff (y")
Maximum annual consumption rate (kg y'l ) per predator by predator type j

Instantaneous consumption rate (in numbers per year) per predator of predator
type j on prey type i

Instantaneous consumption rate (in numbers per year) per predator of predator
type j on prey species 3

Mid-year weight (kg) of prey type i
Proportion in weight of alternative prey in diet of predator type j
Instantaneous attack rate (y") of predator type j on prey type i
Approximate mid-year abundance of prey or predator type i
Length-based scalar for a predator’s effective search area (cm2 y")
Length ratio between prey type i and predator type j
Mid-year length of predator type j (cm)
Size preference of predator type j for prey type i
Habitat overlap of predator type j and prey type i
Optimal predator-prey length ratio
Parameter controlling the width of the size preference function
Predicted consumption (kg) of species s by predator type j
Predicted total consumption (kg) of all prey types by all predator types
Total consumption (kg) of all prey types by predator type j
Predicted proportion of prey category 1' in C? y
Predicted consumption (kg) per chinook salmon predator
Predicted trawl survey index for species 3 and age category k

Predicted hydroacoustic survey index for species s and age category k

BS
k ’)’ Bigmass 9f species 3 and age categon k at time 8f hydroacoustis survey

47

Table 2. Model equations describing the alewife and bloater population dynamics.

Population dynamics model

 

 

 

—Z (T2.l)
Ns,a+1,y+l : Ns,a,ye 3,61,),
—Z J- 1, -Z ,1, (T22)
Ns,l,y+l : Ns,l- Lye S y + Ns,l,ye s y
25,), = Mm + P,“ y at 1967 r) s at aw (T23)
(T24)
Zaw,a,l967 = Mama + Paw,a,1967 + 567
~ (T25)
ASJIJJ *NJ'J
Ps,a,y = Z N
j s,a,y
Predator other than chinook salmon:
~ (T2.6a)
A. . : Cmax,j ai,jNi,y
My , ~
Why 2 ai,jNi,y
l
a“. fiw (T2.6b)
Chinook salmon: Aid-J) = ~
' 0’1 'Ni “’1‘
1+ 2[ 9., 9y % J
i Cmax,i
aid-z 7*lj*Fi,j*H0i,j (T27)
2 T2.8
F- --ex (gm—[0171) ( )
I)! " p spw

 

48

Table 3. Model equations used in the observation sub-model.

 

 

 

 

 

 

 

Asa '
9 9],)’ .2
CM, m = TNWJU— e Siaiy)*w,-,y (T3.1)
say
2 Z Cs,a.j.y (T32)
C . = S a
101,} Poth,j,y
” _ T3.3
Cy ' Z Ctot,j ( )
J
"chs _ ~ (T3.4)
Ca,y _ Ctot,chs/Nchs,a,y
/ 10
Z Ns,a,ye- /122s,a,y
7:3 : L0 a (T353)
k’y g - 1%2Zs,a,1999
Z Ns,a,l999e
\ a J
For age 0 alewife, 1962-1990:
/ _10 z (T3.5b)
Z Ns.a.ye /12 s,a,y l
fks ___ [40g (1
9y _10 Z
9 91
Z qtrN ”1.19999 /12 s“ 999
K a /
(T3.6)

S _ S
Hk,y ‘ quk.y
—

49

Table 4. Values for parameters assumed known during model fitting (LT: lake trout,

CHS: chinook salmon, CO: coho salmon, ST: steelhead, and BT: brown trout).

Species

Alewife

Bloater

LT
CHS
CO
ST
BT

Natural mortality rates per year

age 0 age 1+
0.44626 0.22313
0.47237 0.47237

Maximum annual consumption rates (kg) by age
0 1 2 3 4 5 6 7 8
n/a 0.495 1.98 3.59 4.93 5.37 6.17 6.69 6.99
2.15 9.30 26.7 55.1 108.7 n/a n/a n/a n/a
n/a 2.46 5.14 n/a n/a n/a n/a n/a n/a
n/a 1.54 6.39 4.58 7.8 n/a n/a n/a n/a
n/a 1.44 5 .09 4.98 0.02 n/a n/a n/a n/a

50

9
12.78
n/a
n/a
n/a

n/a

Table 5. Negative log likelihood components utilized during model fitting. CL indicates

the likelihood component was incorporated using the concentrated likelihood form.

Component

Alewife trawl
survey

Bloater trawl
survey

Alewife
hydroacoustic
survey

Bloater
hydroacoustic
survey

Total
Consumption

Chinook
consumption
per predator

Consumption

prey type
composition

y

Recruitment
penalty function

Equation

 

k=1
1
aw _
4%,), _ aw
Tk,y
_ bl "bl bl 2
£2 - z ’ia,y(Ta y - 721 y)
a,y
[lb] _ L
0.)! _ b1
Ta’y

2
£22 gamma )_1n(ng;»2
y

 

k=1
2*” — 1
,y‘ aw
Uk,y

422 “(111017) - 1n(Hjj’ ))32

lb!“ 1
y —Ubl
y

= Z ,1C(1n(éy)-1n(cy))2

y

£6 = Z 1"”(é‘zf‘5- 65W

a,y

)7. -2 1nr(y)- §[1nr(yjy)+(7j,-y 9'" 913w]

j= l

A

(Jay = yam

1 " '— . " — 0
£8 = Z E((R;"” - R“”)2 + (Rj’ - Rb’)“)

Distribution

Normal
(CL)

Normal
(CL)

Lognormal
(CL)

Lognormal
(CL)

Lognormal
(CL)

Normal
(CL)

Dirichlet

Normal

 

51

Table 6. Mean, variance, 95% credibility intervals (CI) and effective sample size ( N eff )

for the posterior distributions of all estimated parameters.

Parameter Mean Variance 95% CI Nefl
2.456 0.34 (1.44, 3.38) 878
1n(a/aw)
0.306 0.04 (0.66, 0.03) 620
flaw
2 4.07 2.91 (1.99, 7.33) 1201
0aw,r
1mg“) 0.315 0.25 (-0.51, 1.12) 8602
0.186 0.04 (-0.14, 0.54) 5087
.301
2 5.39 2.24 (3.42, 8.08) 6560
0b1,r
y 1.54E-06 7.0E-14 (1.29E—06, 1.8E-06) 2178
(lad 0.140 0.003 (0.07, 0.24) 3005
q 0.771 0.02 (0.51, 0.97) 6111
yoy
4b! 0.665 0.03 (0.39, 0.95) 1980
CI" 0.023 0.0006 (0.007, 0.062) 1668
-1.65 1.35 (-3.91, —0.15) 1723

Sb

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53

Relative abundance
w
I

 

 

'2 I 1 l I I I 1 1

1960 1965 1970 1975 1980 1985 1990 1995 2000
Year

Figure 1. Observed (symbols) and predicted (lines) fall bottom trawl survey indices for
age 0 (squares and solid line) and age 3+ (circles and dashed line) alewife in Lake

Michigan, 1962-1999.

54

Relative abundance

 

 

 

1

1

Relative abundance
on N A O A N w J: 01 O)

 

 

 

'5 T l T 1
1960 1970 1980 1990 2000

Relative abundance

 

 

 

1960 1970 1980 1990 2000
Year

Figure 2. Observed (symbols) and predicted (lines) trawl survey indices for (a) age 0
(squares, solid), age 1 (circles, dashed), (b) age 2 (squares, solid), age 3 (circles, dashed),
age 4 (triangles, solid), and (0) age 5 (squares, solid), age 6 (circles, dashed), and age 7

(triangles, solid) bloater in Lake Michigan, 1962-1999.

55

160000 — (8)
140000 «
120000 .
100000 ~

80000 4

Biomass (mt)

60000 ~

40000 ‘

20000 ~

 

 

0 I I 1
1993 1994 1995 1996
Year

100000 ~ (0)

90000 ~ '
80000 ~
70000 A
60000 -
50000 ~
40000 ~

Blomass (mt)

 

30000 —
20000 ~ '
10000 .

0 I l 1
1993 1994 1995 1996
Year

 

 

Figure 3. Observed (symbols) and predicted (lines) fall hydroacoustic biomass estimates

of (a) age 0 and (b) age 1+ alewife in Lake Michigan, 1993-1996.

56

600000 1

500000 -

400000 -

300000 -

Blomass (mt)

200000 ~

100000 -

 

 

1993 1994 1995 1996
Year

Figure 4. Observed (squares) and predicted (line) hydroacoustic biomass estimates for

bloater in Lake Michigan, 1993-1996.

57

160000 -

140000 —

 
  
 
 
 
 
   
 
 
 

120000 0

100000 -

80000 -

60000 -

Consumption (mt)

40000 -

20000 -

 

0 l l l l I l l
1965 1 970 1975 1980 1985 1990 1 995 2000

Year

 

Figure 5. Observed (squares) and predicted (line) consumption of all prey types by all

five salmonine species in Lake Michigan, 1965-1999.

58

1 -
0.9 ~
0.8 -
0.7 -
0.6 d . 3-.

0.5 #0.

Proportion
\
O
O

0.4 d

c
0.3

 

0.2

0.1 *

 

 

0 l l l l I l l l

1983 1985 1987 1989 1991 1993 1995 1997 1999

Year

Figure 6. Observed (symbols) and predicted (lines) proportion of small alewife (squares,
solid line) and large alewife (circles, dashed line) in the total consumption by all five

salmonine species in Lake Michigan, 1965-1999.

59

(a)

AAA‘AAAAA

0')
O
1

0"
O
1

.5
O
1

   

000000000

Consumption (kg)
0.)
O

 

 

20 - ° ' ‘ - , ..............
10 1M
0 l I l
1968 1978 1988 1998
Year

.0 .o
(D (O —‘
1 1 1

0.7 4

.0 .0
U1 0)
1 1

0.4 4

 

9.0.0
AND»)
111

 

Proportion of maximum consumption

 

O l l 1 i I l
1970 1975 1980 1985 1990 1995 2000

Year

Figure 7. (a) Observed (symbols) and predicted (lines) consumption per predator for age 1
(squares, solid line), age 2 (circles, dashed line), and age 3 (triangles, solid line), (b)
predicted proportion of maximum consumption achieved (solid line) and 95% credibility

intervals for age 3 chinook salmon in Lake Michigan, 1968-1999.

60

25 ' (a)

 

 

 

1965 1970 1975 1980 1985 1990 1995 2000
Year

0.80 ~ (b)
0.70 4
0.60 —
050 ~
0.0.40 .
0.30 —
0.20 ~
0.10 ~

 

 

 

0.00

1965 1970 1975 1980 1985 1990 1995 2000
Year

Figure 8. Predicted instantaneous predation rates (P) on (a) age 0 (squares, dashed line),
age 1 (circles, solid line), and age 2 (triangles, dashed line) and (b) age 3 (squares, solid
line), age 4 (circles, dashed line), age 5 (triangles, solid line), and age 6+ (diamonds,

dashed line) alewife in Lake Michigan, 1965—1999.

61

013T
(125 ~
032—
n-OJES-
04 a

(105 —

 

O

1965

(1045 -
(1040 a
(1035 d
(1030 a
(1025 a
(1020 a
(1015 1
(1010 -
(1005 _
(1000

a.

 

1965

(a)

O
9
’0‘
9 e

5 9 o

. a 9'
9 06‘ 9
v o
o 3
3- p°°w 609
30 && w» 1'
l r 3 C 3 ‘ x 6-0 9 " v s ‘I

1970 1975 1980 1985 1990 1995 2000

Year

I

(b)

. 5
0‘0. Q
0

09990
$ ‘- C

. v
v

Imp-11C f

I

.v
- ‘

1970

 

I

1990

1995 2000

I I I

1980 1985
Year

1975

Figure 9. Predicted instantaneous predation rates (P) for (a) age 0 (squares, solid line),

age 1 (circles, dashed line), age 2 (triangles, solid line), and age 3 (diamonds, dashed

line), and (b) age 4 (squares, solid line), age 5 (circles, dashed line), age 6 (triangles, solid

line), and age 7+ (diamonds, dashed line) bloater in Lake Michigan, 1965-1999.

62

2.5

1.5

0.5 1

O-‘NQAU’ICDVCDCOO

80
70
60
50
4O
30
20
1 O

 

 

 

 

 

_ (a)
///
0 02 0.4 0.6 08 1
gaw,y0y
1 (b) -.
.0 /‘“‘
4 / \
i / \H
—1 //
2 “\\v\\
0 0.1 0.2 0.3 0.4 0.5
qaw,ad
" (c)
. \-\

 

0 0.025 0.05 0.075 0.1 0.125 0.15
qtr

Figure 10. Posterior density functions of the catchability coefficients of (a) age 0 and (b)

age 1+ alewife hydroacoustic survey in Lake Michigan, 1993-1996, and (0) age 0 alewife

fall trawl survey in Lake Michigan, 1991-1999.

63

2.5 -

 

 

2 -
1.5 .
1 -1
0.5 .
0 T ‘ 1 I a
o 0.2 0.4 0.6 0.8 1

CIbl

Figure l 1. Posterior density function of the catchability coefficients of the bloater

hydroacoustic survey in Lake Michigan, 1993-1996.

64

3.0E+06 -

2.5E+06 -

2.0E+06 -

1.5E+06 -

1.0E+06 1

5.0E+05 -

 

 

0.0E+00 1 1 1 1 fl
0.0E+00 5.0E—07 1.0E-06 1.5E—06 2.0E-06 2.5E-06

7

Figure 12. Posterior density function of the length-based scalar of the effective searching

efficiency on an optimal sized prey for salmonine predators in Lake Michigan.

65

(a)
0.7 1

0.6 1
0.5 1
0.4 1
0.3 1

0.2 — 1
0.1 - //

 

 

0 *f/ T I I \fI\—‘ 1
0 1 2 3 4 5
ln(aaw)
2.5 1
(b)
2 /\
\m\

1.5 ‘/ \/'\\

0.5 1 \

 

 

 

 

0 1 1 1 ”M 1

o 0.25 0.5 0.75 1 1.25
aw

0.35 — (c)

0.3 — ,

0.25 _ /

0.2 1 1

0.15 « / \

- \

0.05 « \‘k
0 1 1 1 .1»- T .

o 2.5 5 7.5 10 12.5 15
UEWJ'

Figure 13. Posterior density function of the (a) 1n(aaw) , (b) flaw , and (c) 0-2

parameter of the Ricker stock-recruitment function for alewife in Lake Michigan.

66

(a)

\
/ \
0.54 /
0.25- /
/ \
-2 -15 -1 -0.5 0 0.5 1 1.5 2

ln(ab,)

0.75 - jfl/

 

 

2.5 1 (b)

2-.

1.5 1

1 _1

0.5 1 /
fl/

0 {-o/ u
1 I l T r r I 1

-O.75 -0.5 -0.25 O 0.25 0.5 0.75 1 1.25

IBbl

 

 

0.35 — (C)
0.3 1 1y
0.25 «
0.2 «
0.15 ~

0.1 ~ \
0.05 ~ /
,/ ‘

O 2 4 6 8 10 12 14

2
0'bl ,r

Figure 14. Posterior density functions of the (a) 1n(abl) , (b) :Bbl , and (c)

 

0731 r parameter of the Ricker stock-recruitment function for bloater in Lake Michigan.

67

Recruitment

Recruitment

180 — (a)
160 - .
140 -
120 J
100 -
801

 

 

O‘Ra'l R:~fl

201

151 ‘

101

20 25 30

 

 

 

Stock size

Figure 15. Maximum posterior estimates of the stock-recruitment relationships for (a)

alewife (Stock size is the number of age 2+ fish divided by 1x10“). Recruitment is the

number of age 0 fish divided by 1x109), and (b) bloater (Stock size is the number of eggs

divided by 1x10”. Recruitment is the number of age 0 fish divided by 1x109).

68

0.4 1

0.35 1

0.3 1

0.25 1

0.2 1

0.15 1

0.1 1

0.05 1

 

 

Figure 16. Posterior density function of the instantaneous mortality rate (S67 ) on age 1+

alewife during the dieoff in 1967 in Lake Michigan.

69

CHAPTER 3

EVALUATION OF THE EFFECTS OF UNCERTAINTY IN ALEWIFE AND
CHINOOK SALMON POPULATION DYNAMICS ON THE OUTCOMES OF

STOCKING STRATEGIES IN LAKE MICHIGAN.

Introduction

Acknowledging uncertainty in the outcome of fisheries management requires
managers and decision makers to formally and quantitatively incorporate uncertainty in
the decision making process (Punt and Hilbom 1997; Varis and Kuikka 1999; Harwood
2000; Peterson and Evans 2003). In natural systems where our level of uncertainty is
often quite high, the influence of this uncertainty on the outcomes of our management
decisions may be substantial. Ludwig (1996a) and Ludwig (1996b) found that failure to
incorporate uncertainty in the prediction of outcomes, by only utilizing point estimates in
simulation models, leads to an overestimate of the resiliency of a stock to exploitation.
Therefore, in an era where risk assessment and a precautionary approach to fisheries
management are being advocated (Francis and Shotton 1997), acknowledging and
incorporating uncertainty in fisheries management decisions has become paramount.

Bayesian decision analysis provides a framework to explicitly incorporate
uncertainty into our predictions of the outcomes of different management actions and to
choose between management actions based on the management objective specified
(Raiffa 1968; Punt and Hilbom 1997). Additionally, it provides the tools to evaluate how

reductions in different sources of uncertainty will affect the outcomes of management

70

actions and provides a quantitative methodology for ranking the importance of reducing
uncertainty from different sources (Raiffa 1968). The first step in decision analysis is the
identification of the management objectives for the fishery and of potential management
actions. Then, areas of key uncertainty are identified and quantified. A simulation model
that incorporates these uncertainties is then built and the performance of each
management action is assessed (Punt and Hilbom 1997). This simulation model can then
also be used to address questions regarding the importance of different sources of
uncertainty on the resulting decisions.

The application of decision analysis to fisheries management decisions has
increased dramatically in recent times (e. g., Schnute et al. 2000; Peters and Marmorek
2001; Peterson and Evan 2003). However, most of these applications have focused on
single-species management decisions and have not incorporated ecological interactions
with other species. As the importance of the consideration of ecological interactions in
fisheries management decisions increases and as we acknowledge our uncertainty about
these interactions (Link 2002), the need for decision analyses for these types of decisions
will grow. Here we present an attempt, using the techniques of decision analysis, to
inform us of the effects of uncertainty on management outcomes for a predator-prey
system, where the abundance of predators is primarily controlled through stocking. Due
to the lack of a natural feedback mechanism between the abundance of predator and their
prey, it is necessary, through management of stocking levels, to provide a balance
between the desire for large predator populations to support a sport fishery and ability of
the prey fish production to support the predatory consumption demand.

The history of the Lake Michigan fish community has mirrored the changes seen

71

in the other Great Lakes during the last century with the virtual extirpation of the native
piscivores, lake trout (Salvelinus namaycush) and burbot (Lota Iota), and chub species
(Coregonus spp.) and the invasion of the exotic planktivores alewife (Alosa
pseudoharengus) and rainbow smelt (Osmerus mordax). Alewife, in particular,
dominated the fish community in the early 196OS and caused severe economic damage, by
littering beaches and clogging city water intake lines with dead fish, and ecological
damage, through predation on native fish eggs and larval stages (Brown 1972; Brown et
al. 1987). The need to reduce alewife stocks, the desire to establish a successful sport
fishery, and the desire to re-establish the native lake trout population led to the stocking
of lake trout along with chinook salmon (Oncorhynchus tshawytscha), coho salmon (0.
kisutch), brown trout (Salmo mum) and steelhead (0. mykiss) in the late 19603 (Tody and
Tanner 1966). The numbers of stocked salmonines quickly increased from 1.3 million
fish in 1965 to over 16.5 million fish in 1985 (Benjamin 1998; Bence and Benjamin in
press b).

The stocking program in Lake Michigan was successful in establishing an
economically important sport fishery and has dramatically reduced alewife abundance
lakewide ( Bence and Smith 1999; Madenjian et al. 2002; Chapter 2). The stocked
salmonines feed primarily on alewife, but also consume the native coregonid bloater
(Coregonus hoyi), the exotic rainbow smelt and lesser numbers of a variety of other
species (Jude et al. 1987). Chinook salmon have dominated consumption in Lake
Michigan due to their fast growth rates and large population sizes (Madenjian et al.
2002). Concern over a potential imbalance between the consumption demand by the

predators and prey fish production arose after declining growth rates of chinook salmon

72

and large mortality events, thought to be caused by bacterial kidney disease, were
observed in the late 19805 (Holey et al. 1998; Stewart and Ibarra 1991).

The need to balance predatory demand with prey production led to a series of
investigations of both the amount consumed by salmonine predators through
bioenergetics modeling (Stewart et al. 1981; Stewart and Ibarra 1991) and the potential
production of the alewife population (Eck and Brown 1985; Eek and Wells 1987). Jones
et al. (1993) and Koonce and Jones (1994) recognized the need to develop dynamic
models where both prey and predator populations responded to changes in stocking
policies and developed the SIMPLE models for Lakes Michigan and Ontario. Jones et al.
(1993) also noted substantial uncertainty about the ecological interactions between the
salmonine predators and their prey fish and called for improved stock assessments to
address these uncertainties. With the recent development of a statistical stock assessment
of alewife and bloater populations in Lake Michigan (Chapter 2) and improved stock
assessments for the salmonine predators, particularly chinook salmon (Appendix A), our
ability to construct a dynamic stimulation model that acknowledges uncertainty in our
understanding of the Lake Michigan salmonine community and their prey has
substantially increased.

Here we present the development of such a model and evaluate its behavior in
response to changes in stocking policy. We have explicitly incorporated both parameter
and model structure uncertainty in three primary areas: the stock-recruitment relationship
of alewife, the functional response of chinook salmon to their prey, and the potential for
episodic mortality events in chinook salmon during periods of poor growth. These

processes have been identified by Lake Michigan managers, decision makers, scientists

73

and stakeholders as the key uncertainties affecting management decisions for chinook
salmon stocking (Jones and Peterman 2000). We also examine the model’s sensitivity to
assumptions that are uncertain, but for which we lack sufficient data for quantitative
analysis. The purpose of this project is not to determine the optimal stocking strategy for
stocking salmonines in Lake Michigan but rather to explore the importance of our
uncertainty on the outcomes of salmonine stocking decisions and explore the performance

of several potential stocking strategies in the face of these uncertainties.

74

Methods

We developed a suite of age-based stochastic population models to describe the
interactions between salmonine predators and their prey in Lake Michigan following the
example of Jones et al. (1993) and Koonce and Jones (1994). Predation between the
salmonine predators and their prey was described using a Type H function response
(Hollings 1959). The effects of different stocking policies were simulated for a 30 year
time period. Several alterative forms of the model, called model scenarios, were also
investigated for a more limited suite of stocking policies. The 30 year time series for
each model scenario and stocking decision was simulated 1000 times (called trials) to
incorporate the effects of stochastic variation and quantified uncertainty in model
parameters. For parameters where posterior distributions describing our uncertainty were
available, a new set of parameters (sample) was selected from the posterior distributions
for each trial but the same set of samples were used for each scenario to eliminate
variation among scenarios due solely to sampling variability. The parameters and
symbols used in the model equation are defined in Table 8. The equations describing the
dynamics of the model are in Tables 9 and 10 and the equations will be referenced in the
text as T x. y where x is the table number and y is the equation number within Table x.
Predator population models

Chinook salmon (ages 0-5), coho salmon (ages 1-2), lake trout (ages 1-9), brown
trout (ages 1-4) and steelhead (ages 1-4), were included in the simulation model. For
coho salmon, brown trout, and steelhead, size at age and mortality rates were assumed to
be constant over time because there is little evidence of decreased growth rates or

increased mortality rates with the declines in prey abundance in Lake Michigan

75

(Appendix A). Mortality rates and length at age were based on the most current stock
assessments for these predators (Appendix A). Because we assumed a constant stocking
rate over time for these species and a constant level of natural recruitment (for coho
salmon and steelhead), this leads to constant numbers at age over the thirty year
simulation period (Table 11).

For both chinook salmon and lake trout, we investigated several different stocking
policies (Table 12). The recruitment to the first age of lake trout (age 1) and chinook (age
0) was calculated by eq. T9.1. Post-stocking mortality, H S , was assumed to be 0.4 for
both species. Natural recruitment was assumed to be constant over the 30 year simulation
period at zero for lake trout and 960,000 for chinook salmon (Appendix A). Numbers at
age for older ages in each year for both lake trout and chinook were calculated by eq.
T92. Maturation mortality was assumed to be zero for lake trout (Appendix A). For
chinook salmon, maturation mortality was assumed to be 1 for all mature fish and the
proportion of mature fish at age was predicted by a logistic function of spawning weight
(eq. T9.3, see Appendices A and C for more details). The proportion mature at age was
capped at a maximum of 15.8% for age 1 chinook, and 42.9% for age 2 chinook,
regardless of size. Total instantaneous mortality rates for lake trout were the sum of
background natural mortality rates and fishing mortality rates (eq. T9.4b). Background
age-specific natural mortality rates from the most recent assessment were used and were
assumed to be constant over the simulation time period (Appendix A). Fishing mortality
was predicted as the product of age-specific vulnerability (Appendix A) and effort (eq.
T9.5). Effort was assumed constant over the simulation time period at 5.1 million angler

hours. Lake trout growth rates were assumed to be constant over time and mid-year

76

length at age was set at the values used in Chapter 2.

Total instantaneous mortality rates for chinook salmon were the sum of three
components, background natural mortality rates, fishing mortality rates, and a time-
varying natural mortality rate (eq. T 9.4a). Background mortality rates from the most
recent assessment were used and were assumed to be constant over the simulation time
period (Appendix A). As for lake trout, fishing mortality was predicted as the product of
age-specific vulnerability (Appendix A) and effort (eq. T9.5). Harvest (in numbers) of
chinook salmon was predicted using the Baranov catch equation (eq. T9.13). The final

component, DC was incorporated to allow for increased mortality on chinook

h,a,y ’
salmon during periods of poor growth as was observed in the late 19808 (Holey et al.
1998; Bence and Benjamin in press a). Because only one incident of this increased
mortality rate was observed in chinook salmon in Lake Michigan, there are several
competing hypotheses of how this mortality rate is related to chinook salmon growth
rates. We chose to represent several of these hypotheses in three different models

describing the dependence of Dc on chinook salmon growth rates. For all models,

h.a.y

Dch,a,y was decomposed into an age and age-year effect (Appendix C):

Dchs,a,y : AaKaJ (1)-

In the first model (M1), the age-year effect (Ka’ y) was assumed equal for all ages
as a year effect ( Ky ) and was modeled on the log-scale using an autoregressive time
series model. The year effect was a function of the year effect in the previous year, the

weight achieved by an age 2 fish at then end of the current year and a year-specific

77

process error term (for more details see Appendix C). A joint posterior distribution for
all parameters was available and 1000 samples (one for each model trial) was drawn from
it (see Appendix C for details). The second and third mortality models (M2 and M3,
respectively) are based on stronger assumptions about the relationship between growth
and mortality events. Both models are based on the assumption that there are two distinct
mortality states, one with high levels of mortality and one with low levels of mortality.
The transition probability between these states depends upon the current mean weight at
age of chinook salmon. These two models differ in their description of how weight affects
the transition between these two states.

For M2, each episode of high mortality was assumed to last at least five years.
After being in a high mortality state for 5 years, the transition probability between high
and low mortality rates was assumed to be logistic function of end of the year weight.
The transition probability from a low mortality state to a high mortality state, was
assumed to be the complement of a logistic function of end of the year weight. In the
final mortality model (M3), the transition from a high mortality state to a low mortality
state is assumed to be deterministic after the end of the year weight at age exceeded a
threshold value. The transition probability from a low mortality state to a high mortality
state, was assumed to be the complement of a logistic function of end of the year weight.
For both M2 and M3, the time-varying natural mortality was calculated using eq.1
with K“ a, y equal to 1.27 for high mortality states and 5E-5 for low mortality states.
During each trial, a mortality model (M1, M2, or M3) was chosen, with equal

probabilities, and the chosen mortality model was used during the trial’s 30 year

simulation time period.

78

For both M2 and M3, the parameter values were chosen to reflect observed
patterns in the mortality event of the late 19805. With only one high mortality event in
Lake Michigan, it was not possible to statistically estimate these parameters and therefore
we have not quantitatively described uncertainty with regard to these parameters (for
more details, see Appendix C).

For all three mortality models, it is still possible for chinook salmon to reach an
unreasonably small size without the occurrence of a mortality event, particularly with M1.
To prevent this situation, we included a logistic model that increases the probability of a
mortality event with decreasing end of the year weight at age if a mortality event has not
occurred. With this model, the probability of invoking a mortality event, if one is not
occurring, for 4.9 kg age 3 chinook is close to zero but approaches 1 as weight of an age 3
chinook declines to 3.4 kg.

We modeled how chinook salmon’s and the other predator’s consumption
responded to changes in the abundances of prey using a multi-species Type [1 functional
response (eq. T9.6; Holling 1959). For all predators, we followed the general approach
used for chinook salmon in Chapter 2. The instantaneous attack rate of all predators on all
prey types was assumed to be the product of four components (eq. T9.7): the predator’s
mid-year length, the size preference function, the habitat overlap between each predator
and prey type and the length-based scalar for a predators effective search rate on an
optimal prey. The habitat overlaps between the salmonine predators and the prey types
were as in Chapter 2. As in Chapter 2, the size preference function (eq.T9.8) is a bell
shaped function with the optimal prey length of 25% of the predator’s length. A posterior

density for the length-based scalar for a chinook salmon’s effective search rate on an

79

optimal prey, ychs , was estimated in Chapter 2 and a value was sampled from this
distribution for each simulation trial to account for the uncertainty in this parameter.

Search rates for the other predators were not statistically estimated because their
growth rates have not varied with changes in prey abundance (Appendix A). Instead, we
assumed that all other predators had the same effective search rate ( y j ) and that this
effective search rate was 1.5 times as large as 7chs- Using the maximum posterior
parameter estimates from Chapter 2, this was the lowest effective search rate that leads to
lake trout consumption at age being at least 75% of Cmame for all ages in 1986, the year
with the lowest alewife abundance. We felt that if the actual value of the effective search
rate was lower than this value, a relationship between lake trout growth rates and prey
abundance would have been evident.

The approximate mid-year abundance at age , 1V,” , for all predators except
chinook salmon was calculated as the geometric mean abundance at age by projecting the
end of year abundance using the instantaneous mortality rates (Appendix A). For chinook
salmon, the total instantaneous mortality rate depends in part on the consumption of
chinook salmon through Dch, a, y (eq. T9.4a). Therefore, the geometric mean abundance
at age could not be calculated prior to calculating chinook salmon consumption. To
approximate mean abundance, we assumed that Dch, a, y would be the same
as Dch,a,y—l and used this approximate total instantaneous mortality rate at age to
calculate to the geometric average abundance at age.

To allow for competition among predators for prey, consumption per predator at
age for chinook salmon was predicted for each prey type using the Baranov catch

equation (eq. T99) and summed across all prey types. The proportion of was

Cmax,chs,a

80

calculated (eq. T910) and a bioenergetics model (Appendix A) using this proportion as
the proportion of maximum ration and the current weight at age at annulus formation (or
WC‘Z:Z_1,y_ 1 ) was used to predict the amount of growth achieved during the year, 031;.
Current end of the year weight, “€3.72, )1 , and spawning weight at age, VVCShp, a, y , were
predicted using eqs. T911 and T9.12. Similar calculations were not done for the other
predators which were assumed to have a constant size at age schedule.
Prey population models

Alewife dynamics are strongly influenced by the predatory demand of the
salmonine predators in Lake Michigan and chinook salmon population dynamics appear
to be strongly influenced by the abundance of alewife (Madenjian et al. 2002; Chapter 2).
However, the influence of the other main prey species, bloater and rainbow smelt, on
chinook salmon appears to be limited (Madenjian et al. 2002). Therefore, we chose only
to model alewife dynamically and bloater and rainbow smelt abundance was assumed to
be independent of predator abundance and constant for the 30 year simulation period.
The abundance of age 1 through 7+ bloater and both small ( < 120 mm) and large
rainbow smelt was set at the average abundance from 1995 to 1997 from Chapter 2, as
was the mid year length at age. The approximate mid-year abundance ( 1V i.y ) of bloater
and rainbow smelt was assumed to be equal to the beginning of the year abundance. To
calculate consumption of these prey species for chinook salmon (eq. T9.9), total
instantaneous mortality rates were assumed to be equal to the instantaneous predation
rates (eq. T10.5) with no background mortality rates.

Alewife (ages 0 to 6+) numbers at age were modeled dynamically to respond to

changes in the demand of salmonine predators. Recruitment of alewife was predicted

81

using a Ricker stock-recruitment function where stock size was defined as the number of
age 2+ alewife (eq. T10.1). The year-specific random errors, 8y , were assumed to be
independent and identically distributed as a normal distribution with a mean of zero and a
variance of 03. A sample of parameters was selected for each simulation trial from the
joint posterior distribution of aaw , flaw , and 0-3 estimated in Chapter 2. The 5y
were then drawn randomly from a normal distribution with mean zero and variance equal
to the value of 0'} selected from the joint posterior distribution. Numbers at age for each
year were then calculated using eq T10.2. Numbers at age 6+ were the sum of numbers
surviving from age 5 and the numbers of age 6+ surviving from the previous year (eq.
T10.3). For alewife, the total instantaneous mortality rates were the sum of background
natural mortality rates and instantaneous predation rates (eqs. T104, T105). To calculate
the approximate mid-year abundance ( [171.0, ) of alewife used in the functional response
(eq. T96), the iterative method proposed in Chapter 2 was used. The following steps
were repeated until total consumption in biomass of alewife changed by less than 1% of
the total consumption calculated in the previous iteration :

1. Instantaneous predation rates were predicted using “average” abundance for

alewife (beginning of the year abundance in the first iteration)

2. The instantaneous predation rates were used to calculate the end of the year

abundance of alewife along with the total consumption of alewife.

3. The geometric mean abundance of alewife was used as the new “average”

abundance in step 1.

A large dieoff in the alewife population occurred in 1967 and this dieoff was

thought to be caused, in part, by the high abundance of alewife in the mid-19603 (Smith

82

1972). We attempted to capture this behavior by incorporating a stochastic dieoff
mortality model for alewife. The probability of an alewife dieoff occurring was assumed
to follow a logistic function of the stock size (eq T10.6). Because only one major dieoff
of alewife has occurred in Lake Michigan, the parameters of this model could not be
estimated statistically. Rather, the parameters were chosen such that the probability of a
dieoff occurring at a stock size of approximately 4 x 10'2 (the estimated stock size in
1966 and 1967; Chapter 2) was 50%, since a dieoff occurred in 1967 but not in 1966.
Additionally, the slope parameter was adjusted so that the probability of a dieoff
occurring at stock sizes less than 2 x 1012 was small (< 0.1) since a dieoff did not occur in
the early 19603 or early 19703 when alewife stock sizes were estimated to be
approximately 2 x 1012 (Chapter 2). The dieoff affected age 2+ alewife only and the
proportion of alewife surviving the dieoff ( S die) was 0.65 , the maximum posterior
estimate of the survival of the 1967 dieoff from Chapter 2, if a dieoff occurred and 1
otherwise.
Model scenarios and stocking strategies

The outcomes of different model scenarios and different stocking policies were
evaluated using system variables that relate to the major objectives of Lake Michigan
salmonine management. In particular, we looked at total cumulative harvest (in numbers)
of chinook salmon over the 30 year simulation time period to assess effects on the
chinook salmon fishery. We also looked at the frequency and duration of chinook salmon
dieoffs (Dch.2,y > 1.25) along with the average spawning weight of an age 3 chinook
salmon during the 30 year simulation time period to determine the health of the chinook

salmon population. Finally, we measured the frequency of alewife biomass in each of

83

three categories, low ( < 100,000 metric tons), moderate (100,000 - 2.5 million metric
tons), and high (over 2.5 million metric tons), over the 30 year time period. These
categories were chosen to approximate the three general states of the alewife population
during the late 19803 (low), the mid 19703 (moderate) and late 19603 (high).

T o assess the sensitivity of the decision model to assumptions made during model
construction, we tested its performance with alternative values for some model
parameters that were not statistically estimated (Table 12). In the baseline scenario, the
values of all of the parameters that were not statistically estimated were set at our best
guesses and all three chinook mortality models were used with equal probabilities. We
assessed the effect of the search rate of predators other than chinook by increasing their
relative search rates, so that they are 3 times more efficient predators than chinook
salmon of the same size at low prey abundance and by decreasing their relative search
rates, so that they search with the same efficiency as chinook salmon of the same size.
Additionally, we tested the model’s sensitivity to the occurrence of alewife dieoffs by
altering the parameters of the logistic model to make dieoffs both more likely and less
likely at lower alewife abundances (Figure 17). We also evaluated the performance of the
decision model when the uncertainty in key parameters representing the chinook
functional response, the alewife stock-recruitment relationship and the chinook mortality
model, was ignored by setting the parameters of interest to the value with maximum
density in the posterior distribution.

A total of eight stocking strategies were evaluated to determine the differences in
the outcomes of the model caused by stocking strategy (Table 13). We evaluated two

main types of stocking strategies, a fixed stocking strategy where the number of chinook

84

salmon and lake trout stocked are constant and does not respond to changes in the state of
the system and a feedback stocking policy, where the number of chinook salmon stocked
respond to a measure of the current state of the system. The four fixed stocking strategies
included the current status quo stocking rates (SQ), a reduction in chinook stocking by
25%, a reduction of chinook stocking by 50%, and status quo stocking of chinook salmon
with a doubling of lake trout stocking rates (Table 13).

Two general classes of feedback policies were used, both with status quo or
increased stocking of lake trout. Both feedback policies responded to changes in chinook
salmon end of the year weight at age 3. When end of the year weight at age 3 fell below a
critical point (7 kg), stocking of chinook salmon was decreased by 50% from the status
quo level. When chinook salmon end of the year weight at age 3 increased above 7 kg,
stocking rate was returned to the status quo level. The difference between the two
strategies was the lag time between the measurement of the state of the system and the
implementation of the stocking increase or decrease. In the first set of feedback stocking
strategies (F 1 and FlLT), the response in stocking rate to the state of the system had no
delay. That is, when end of the year weight of an age 3 chinook salmon reached the
critical level, the appropriate change in stocking was implemented in the following year.
In the second set of feedback stocking strategies (F3, F3LT), we included a delay of three
years to occur between the measurement of the system and the implementation of a new
stocking rate to simulate practical constraints associated with substantial changes in
stocking levels.

We also constructed a simplified decision model where only alewife and chinook

salmon population dynamics were included and alternative prey and the stocking of all

85

other salmonines was excluded to determine if the behavior of our decision model was
sensitive to the inclusion of these other species. We evaluated the performance of this
simplified model for five different stocking strategies (the strategies with changes in lake

trout stocking were excluded).

86

Results
Harvest

The average cumulative harvest in the baseline scenario is highest for the
feedback policy with a one year lag (F 1, Table 14), however, average cumulative harvest
for both status quo stocking and the feedback policy with a three year lag were within 2%
of this level. The feedback policy with a one year lag reduces the occurrence of low
harvests, in comparison with status quo stocking, however the distributions of potential
cumulative harvests for both policies were wide and flat (Figure 18). A 25 % reduction in
chinook stocking (R25) led to a 13% decrease in average cumulative harvest and a 50%
reduction led to a 26 % reduction in average cumulative harvest relative to status quo
stocking. Doubling the number of lake trout stocked without reducing chinook salmon
stocking (DLT) leads to a decline of approximately 14% in average cumulative harvest of
chinook salmon relative to status quo stocking (Table 14). When combined with a
feedback policy, doubling lake trout stocking reduces average cumulative harvest of
chinook salmon by 12 % and 10% for the 1 and 3 year lag.

The effects of alternative assumptions for the search rate of lake trout, coho
salmon, brown trout, and steelhead on average cumulative harvest are substantial. By
assuming that these predators are only as efficient as chinook salmon of the same size at
low prey abundance (FL), the average cumulative harvest increases for all stocking
strategies between 12 and 14 % (Table 14). If the search rate of lake trout, coho salmon,
brown trout, and steelhead is assumed to be three times more efficient than chinook
salmon of the same size, the average cumulative harvest decreases for all stocking

strategies by 12 to 15% (Table 14).

87

The effect of changes in the probability of an alewife dieoff was less pronounced.
If the probability of an alewife dieoff occurring at lower alewife abundances was
increased (DB, DD) relative to the baseline scenario, then the average cumulative harvest
decreased for all stocking strategies and the feedback policy with a one year lag increases
harvest by 3.4 % (DD) or 2.1 % (DB) relative to status quo stocking. When the
probability of an alewife dieoff occurring at lower abundances is decreased, the status quo
stocking policy has a higher average cumulative harvest than the feedback policy with
one year lag (Table 14).

To assess the effects of including uncertainty on the cumulative harvest of
chinook salmon, we ignored the uncertainty in both the effective search rate of chinook
salmon and the stock-recruitment parameters of alewife. If we ignored the uncertainty in
the stock-recruitment parameters of alewife and the maximum posterior estimates were
assumed to be the true values, the status quo stocking policy rather than the feedback
policy with a one year lag provided the highest average cumulative harvest (Table 15).
Using a feedback policy with a lag of one year, the distribution of cumulative harvests
ignoring the uncertainty in the stock-recruitment parameters is more sharply peaked and
the probability of extreme harvests are reduced when compared to the baseline scenario
(Figures 18b, 19b). Not including uncertainty in the effective search rate of chinook
salmon and assuming the maximum posterior estimate is the true value leads to a
significantly lower average cumulative harvest (Table 15). The distribution of harvests
with F1 stocking remains wide with a slightly higher occurrence of low harvests than the
baseline scenario (Figures 18b, 19a).

The different models for chinook salmon mortality also have large effects on the

88

average cumulative harvest predicted by the decision model. The first mortality model
(M1) is characterized by much lower levels of harvest then either of the other two models
(Table 15). The other mortality models (M2 and M3) predict similar levels of cumulative
harvest with M3 providing slightly higher average cumulative harvest (Table 15). The
differences in the three mortality models are also reflected in the distributions of
cumulative harvest, however all three mortality models still result in a wide distribution
of potential harvests (Figure 20). For all three mortality models, stocking policy Fl
provides the highest average cumulative harvest (Table 15).

In the simplified decision model with only chinook salmon and alewife, the
average cumulative harvests for all five stocking policies were larger than in the full
decision model (Tables 15 and 16). The status quo stocking policy produced the largest
average harvest in all model scenarios (Table 16). The distribution of cumulative
harvests for the simplified model were more sharply peaked than their counterparts in the
full decision model (Figures 21 ,22) and the effect of removing uncertainty from the
stock-recruitment parameters on the distribution of harvests was not as large as in the full
model (Figure 21).

Alewife biomass

In the baseline scenario, the percentage of occurrences with alewife in either the
low or high abundance category was approximately 45% for the status quo stocking
policy and slightly lower (41%) for the F1 policy (Table 17). Alternative assumptions
about the alewife dieoff model affected the distribution of alewife biomass as expected
with scenarios with a decreased chance of dieoffs at low abundances having less

occurrences of low biomass and more occurrences of high biomass relative to the baseline

89

scenario (Table 17). The effects of the effective search rate of lake trout, brown trout,
steelhead and coho salmon were large with the higher search efficiencies leading to more
occurrences of low alewife biomass (Table 17). The functional form of the chinook
mortality model used had very little effect on the distribution of alewife abundances.
Ignoring the uncertainty in the chinook salmon effective search rate slightly increased the
occurrence of low alewife biomass. Ignoring the uncertainty in alewife stock-recruitment
parameters had the largest effect on alewife biomass distribution with the occurrence of
low alewife biomass declining by over one half from the baseline scenario with F1
stocking.

Chinook mean size at age and mortality events

The distribution of average spawning weight for an age 3 chinook salmon was
wide for all model scenarios with both status quo stocking and F1 stocking, ranging from
less than 4 kg to over 16 kg (Figures 23-27). The percentage of simulations with average
weight at age 3 less than 4 kg was smaller with F1 stocking than status quo stocking for
the baseline scenario (Figures 23a, 26a). Not surprisingly, as the searching efficiency of
other salmonines, relative to chinook salmon, is increased (Figure 23), or as the
probability of an alewife dieoff at lower abundances increased (Figures 24, 25), the
occurrence of small chinook salmon also increased.

Ignoring the uncertainty in chinook salmon’s effective search rate slightly
increases the occurrence of small chinook salmon with F1 stocking while ignoring the
uncertainty in alewife’s stock-recruitment parameters had a large effect, reducing the
occurrence of small (< 4 kg) chinook from 20% to 7% (Figure 26). The effect of the form

of the chinook mortality model on the distribution of the average spawning weight of an

90

age 3 chinook salmon was slight (Figure 27).

The frequency of chinook salmon mortality events in the baseline scenario was
similar for both status quo stocking and F l stocking with over 90% of all simulation trials
having at least one mortality event during the 30 year simulation period and over half
having one or two events (Figure 28). The functional form of the mortality model had a
substantial effect on the distribution of the number of mortality events with model
scenario Ml producing a higher occurrence of multiple (3 or more) mortality events in a
30 year simulation period (Figure 29).

The average duration of the mortality events ranged from 5.63 to 13.33 years but
in all scenarios, the durations were highly variable (Table 18). A feedback stocking
policy (Fl) appeared to reduce the average duration of mortality events in the baseline
scenario slightly but both stocking strategies had outcomes with prolonged (> 20 years)
mortality events. Any model scenario that increased the occurrence of low alewife
biomass (DB, DD, FH) also increased the average duration of mortality events (Table 18).
Ignoring the uncertainty in alewife’s stock-recruitment parameters had the largest effect
reducing the average duration of an event to 5.6 years and reducing the variability in
duration time (Table 18). Model scenario M2 produced slightly longer duration times

due to its assumption of a minimum duration of 5 years.

91

Discussion

The management of predatory demand in Lake Michigan through the stocking of
five species of salmonines poses an interesting challenge to decision makers. The
original goals of the salmonine stocking program to support a successful salmonine sport
fishery and to control the exotic alewife abundance have been achieved (Madenjian et al.
2002) and managers are faced with a objective of maintaining the valuable sport fishery
while controlling the predatory demand to maintain moderate levels of alewife abundance
(Eshenroder et al. 1995). This objective requires the understanding of the dynamics of
the alewife population along with an understanding of the dynamic link between
salmonine predators and their prey (Jones et al. 1993). Recent advances in the
quantitative stock assessments of both the salmonine predators and alewife has led to an
increased understanding of the systems dynamics along with a quantitative evaluation of
our uncertainty in key parameters, such as the alewife stock-recruitment parameters, and
the effective search rate of chinook salmon. By incorporating these uncertainties in a
stochastic decision model, we were able to evaluate the effects of these uncertainties on
system dynamics and investigate which stocking strategies may hold the most potential
for achieving the management objective in the future.

Perhaps the most important outcome of our analysis is that all model scenarios
and stocking strategies investigated in the full decision model result in a wide range of
potential system states in the future. All model scenarios and stocking strategies we
investigated had a non-trivial chance of producing a future state of the system where
alewife abundance and chinook harvest were low, chinook mortality events were either

frequent or prolonged and the average spawning weight at age of an age 3 chinook was

92

unacceptably low (< 4kg). Additionally, a non-trivial chance of a future state of the
system where alewife abundance is unacceptably high for these same scenarios and
stocking strategies was predicted. This suggests, given our current understanding of the
system, that management of the system through the stocking of salmonines may not be
able to meet all of our management objectives without accepting a level of risk for
undesirable future outcomes. Effective communication of this outcome to stakeholder
groups and decision makers so that the economic and social costs of these risks can be
incorporated in the decision making process may be vital (Hilbom and Walters 1997;
Krueger and Decker 1999).

Our analysis also reveals the importance of lake trout, brown trout, steelhead, and
coho salmon in predicting the future dynamics of the system. The response of these
predators to low alewife abundance remains a significant uncertainty regarding the
dynamics of predator-prey interactions in Lake Michigan and the assumptions made about
the relative search efficiency of these predators has a large impact in the potential
outcomes of management actions. While chinook salmon is the dominant consumer in
Lake Michigan (Madenjian et al. 2002), the effect of these other predators on alewife
dynamics, should alewife abundance become low, appears to be important, with increased
searching efficiency of these predators increasing the chance of undesirable outcomes.
Additionally, the results of our simplified chinook-alewife model show that the chance of
low cumulative harvests is substantially underestimated if the effects of the other
salmonine predators are ignored. This implies that the suite of salmonine predators must
be managed in concert and changes in the management of one species, e. g. lake trout,

may have important effects on the chinook salmon fishery.

93

Our examination of the importance of different sources of uncertainty on the
outcomes of management actions revealed that ignoring the uncertainty in alewife stock-
recruitment parameters leads to an overly optimistic prediction of the outcomes of future
stocking decisions. Ignoring our uncertainty in the stock-recruitment parameters of
alewife also suggested that the status quo stocking policy would provide higher average
cumulative harvests of chinook salmon than a feedback policy. This results suggests that
the failure to include our uncertainty, as in Ludwig (1996a) and Ludwig (1996b),
camouflages the true level of risk in management decisions.

Additionally, the chance of an alewife dieoff occurring in the future similar to the
dieoff that occurred in 1967 also has noticeable effects on the outcomes of management
actions. Any scenario where there is an increased chance of dieoffs occurring at lower
population abundances leads to an increased frequency of undesirable outcomes. These
results strongly suggest that understanding the dynamics of the alewife population is key
to managing the Lake Michigan salmonine community. The need for future research that
further investigates both the stock-recruitment dynamics and potentially the mechanisms
triggering large alewife dieoffs is paramount.

However, the dominance of the importance of the dynamics of alewife in the
performance of management objectives suggests that our ability to control the system
through the stocking of salmonine predators may be limited by stochastic variation in
alewife recruitment that is not predictable. An additional concern for future management
is the profound changes in the lower trophic levels of Lake Michigan with the invasion of
zebra mussels (Dreissena polymorpha) and the disappearance of Diporeia (Madenjian et

al. 2002). Recently, declines in the growth and condition of alewife in Lake Michigan

94

have been noted (Madenjian et al. in press) and the impact of these declines on alewife
population dynamics remains unknown.

Given the large sources of uncontrollable variability, focusing on stocking
strategies that optimize cumulative chinook salmon harvests over time using a risk
neutral utility function may not address the needs and concerns of all stakeholder groups.
Rather, policies that provide lower chances of certain unfavorable outcomes (e. g. low
chinook salmon size at age) or that minimize the interannual variability in the state of
certain system variables may be preferred (Quinn and Deriso 1999). Choosing an
objective function that minimizes variability in annual chinook salmon harvests (e.g,
Quinn et al. 1990) or incorporates a risk-averse utility function (e.g., Mendelssohn 1982)
could provide a quantitative technique for incorporating these objectives.

In this analysis, we chose to investigate the performance of two classes of
stocking strategies, a static stocking strategy and a responsive stocking strategy. We
initially believed that a stocking strategy that responds to the state of the system would
provide some feedback to stabilize the predator-prey interaction as in a natural system.
We chose chinook weight at age as a trigger for changes in the stocking policy because
we felt weight at age integrated the overall state of the system and would be easily
measured and interpreted by managers and decision makers. The results of our decision
model suggest, that while a feedback policy does impart some benefit in the baseline
scenario, the gains are often slight and are not robust to changes in model assumptions.
There are several potential explanations for the unexpected poor performance of our
feedback stocking policies. One is that weight at age 3 does not respond quickly enough

and that previous years of good growth can mask changes in growth rates making the

95

feedback trigger too slow to stabilize the system. Using a more responsive trigger such as
the proportion of maximum ration consumed by chinook salmon may allow managers to
respond more quickly to changes in the system. Additionally, triggers related to the
alewife population rather than the chinook salmon populations may allow managers to
anticipate future changes in the chinook salmon population and be pro-active in
management actions. However, the use of more responsive triggers may be hindered by
the difficulty in obtaining measurements of these quantities that are precise and accurate.
Clearly, a more thorough investigation of potential triggers for a responsive stocking
policy is warranted.

While our investigation has incorporated some of the key uncertainties in the
salmonine predator-prey system in Lake Michigan, there are many uncertainties that were
not be included in our analysis. As discussed in Chapter 2, the form of the functional
response of chinook salmon remains a key uncertainty. In historical reconstructions of
the Lake Michigan salmonine predator-prey system, the Type H functional response
tended to underestimate the contribution of alewife to salmonine consumption at low
alewife abundances. Given the importance of the response of chinook salmon to low
alewife abundances in predicting future outcomes, these deviations from the Type H
functional response could have profound effects on future outcomes.

Secondly, the link between consumption, energy density and chinook salmon
growth rates remains an area of uncertainty. Current investigations into the energy
density of chinook salmon in Lake Michigan suggest that energy density can vary from
year to year and has been low enough in the recent past to be a potential concern for fish

health (A. Peters, Michigan State University, personal communication). Given the

96

importance of energy status on fish health and population dynamics (Adams 1999), these
changes in energy density may have implications for the response of the chinook salmon
population to changes in stocking rates and alewife abundance. A greater understanding
of the link between consumption, energy density and chinook salmon population
dynamics may be necessary to improve our ability to forecast the response of the system
to changes in stocking rates.

While significant uncertainties about the Lake Michigan salmonine community
remain, the decision model presented here provides managers and decision makers with
the opportunity to explore potential stocking strategies and when used in concert with
other analyses may provide a useful vehicle for exploring the future management of Lake
Michigan salmonines. Acknowledgment of uncertainty in some key parameters has
revealed areas for future investigation and led to recognition of the difficulty in

prescribing one management policy that meets all management objectives.

97

Table 8. List of variables and parameters used in the simulation model (0: age, y: year).
—

N s’ao, Beginning of the year numbers at age of species 3
R s, y Natural recruitment of predator species 3
S s, y Annual stocking (numbers) for predator species 3

M st,s Post-stocking mortality for predator species 3

Z s, a, y Total instantaneous mortality rate for species 3 (y")

M S, a Background instantaneous natural mortality rate for species s (y")
Dch,a, y Time-varying instantaneous mortality rate for chinook salmon (y")
F S, a, y Instantaneous fishing mortality rate for predator species 3 (y")
P5733, Proportion of predator species 3 dying due to maturation

Ps’a,y Instantaneous total predation mortality rate for prey species 5 (y")
S die Proportion of alewife age 2+ surviving a dieoff

C max, j Maximum annual consumption rate (kg y'l ) per predator by predator type j

Instantaneous consumption rate (in numbers) per predator of predator type j

on prey type i
ACh.S Instantaneous consumption rate (in numbers) per chinook salmon predator
any on prey type i
A S, a, 13)) Instantaneous consumption rate (in numbers) per predator of predator type j
on prey species 3
w

s,a, y Mid-year weight (kg) of species 3
Mid-year weight (kg) of prey type i

“281:2, y End of the year weight (kg) for chinook salmon
Wsp Spawning weight (kg) of chinook salmon

ai‘j Instantaneous attack rate of predator type j on prey type i

NW Approximate mid-year abundance of prey or predator type i

71' Length-based scalar for a predator j’s effective search area (cm")
6 i, j Length ratio between prey type i and predator type j

[j Mid-year length of predator type j (cm)

9" 1 Size preference of predator tme " for prey tme i

98

Table 8, cont.

6

.

opt

Habitat overlap of predator type j and prey type i
Optimal predator-prey length ratio
Parameter controlling the width of the size preference function

Parameter for alewife Ricker stock-recruitment function

Compensation parameter for alewife Ricker stock-recruitment function
Alewife recruitment variability unexplained by stock size

Process error for alewife Ricker stock-recruitment function

Vulnerability of predator species 3 to fishing

Fishing effort

Consumption per predator (kg) of chinook salmon

Proportion of maximum consumption ration consumed by chinook salmon
Annual change in weight (kg) by chinook salmon

Annual harvest (numbers) of chinook salmon

Inflection point for the chinook salmon maturation model
Slope for the chinook salmon maturation model
Probability of an alewife dieoff occurring

Slope of the alewife logistic dieoff model

Inflection point of the alewife logistic dieoff model

Pro ortionalit constant for for lake trout, brown trout, coho salmon
P y

and steelhead (i.e. 71- = pijhs)

99

Table 9. Model equations describing the population dynamics of chinook salmon and

lake trout in the simulation model.
—

 

 

 

 

NW3), = R5,, + swe' MW 1': 0 or 1 (T91)
‘2 , , m t
Ns,a+l,y+l : Ns,a,ye 5a y(1_ Ps,aa,y) (T92)
mat 1 T9 3
Pch,a,y = Sp ( - )
1+ CXP(—(77a + wa *Wch,a,y))
Zchs,a, y = M chs,a + F chs,a, y + Dchs,a, y (T943)
th,a,y = Mlt,a + Flt,a,y (T9413)
— >l<
Fs,a.y — Us", E y (T95)
A - ai’jfiiy
My ‘ ‘1'
1+ 2 (ai‘jNi’yWiy J (T96)
i Cmaxj
ai,j=}’j*lj*9i,j*H0i,j (T9-7)
2
(li,j ' lopt)
’ (U
Cgf’; = Z fina- e "y )*w,-
i try
chs (T9.10)
a,y
91 2113 :
y Cmax,chs,a
end _ end chs (T9.l l)
vvchs,a,y — chs,a—l,y—l + Ga,y
SP _ end chs chs (T9.12)
“10723210; " Wchs,a,y _ (1" PG,fall) *Ga,y
(T9.13)

F
chs chs,a,y ‘ Zchs,a,
Hy =Zzh Nchs,a,y*(1-e y)
H c s,a,y

100

Table 10. Model equations governing prey species dynamics in the simulation model (3:

species, 0: age, y: year).
—

 

Naw,0,y+l =aawexp[' flaw *Saw,y 1' 8y] (T101)
Z (T102)
_ " aw,a,
Naw,a+ 1,y+1 ‘ Naw,a,ye y *Sdie
Z Z (T103)
‘ ,l- l, " .1.
Naw.l.y+l : (Nan- Lye aw y 1' Naw.l.ye aw y ) *Sdie
_ T10.4
Zaw,a,y — Maw,a + Paw,a,y ( )
1 (T105)
Psfld’ : N 2: Asflszst)’
s,a,y j
1 (T106)

 

P- =
dze,y 1+ exp(-d1(Saw,y - (12))

 

101

Table 11. Numbers at age (in thousands) and length at age (mm) of coho salmon, brown

trout and steelhead in the simulation model.

—
Numbers at age

Coho salmon

1 2

818.2 672.7

Steelhead

l 2 3 4

925.5 853.9 588.4 293.7

Brown trout

1 2 3 4

966.7 547.6 200.7 51.6
Length at age

Coho Salmon

1 2

280.2 546.2

Steelhead

l 2 3 4

162.4 506.4 694.1 800.6

Brown trout

l 2 3 4

277.0 445.5 600.0 658.2

102

Table 12. Model scenarios
_

BL baseline scenario

DA (11 increased 25 %

DB ' d1 decreased 25 %

DC (12 increased 50 %

DD d2 decreased 50 %

FL ,0], = 1

FH '0}, = 3

M1 chinook mortality model 1
M2 chinook mortality model 2
M3 chinook mortality model 3
FR no uncertainty in 7chs

SR no uncertainty in am“, ’Biili’ and 0‘2

103

Table 13. Stocking (millions) policies for lake trout and chinook salmon.
_

Policy Chinook Lake trout
Status Quo (SQ) 5.5 2.5
25% Chinook Reduction (R25) 4.125 2.5
50% Chinook Reduction (R50) 2.75 2.5
Doubling Lake Trout (DLT) 5.5 5.0
Feedback policies
Lag Weigh < 7.0 111,131,, > 7.0
F l 1 2.75 5.5 2.5
F3 3 2.75 5.5 2.5
F 1LT 1 2.75 5.5 5.0
F3LT 3 2.75 5.5 5.0

104

Table 14. Average cumulative harvest (numbers in thousands) of chinook salmon in 30

years under eight different stocking policies for different model scenarios. Stocking

policies with the highest average cumulative harvest are in bold.

BL
FL

FH
DA
DB
DC

SQ

5,574.6
6,302.9
4,789.8
5,829.8
4,889.0
5,131.4

R25

4,851.7
5,546.8
4,128.0
5,144.6
4,323.4
4,567.1

R50

4,1 12.4
4,667.1
3,588.0
4,344.3
3,740.2
3,870.3

DLT

4,798.0
5,833.0
3,856.6
5,098.6
4,241.4
4,583.1

F 1

5,604.1
6,310.2
4,610.5
5,778.4
4,991.7
5,073.2

F3

5,517.0
6,122.2
4,618.1
5,641.5
4,922.0
5,035.1

F 1LT

4,958.4
5,912.4
3,774.7
5,107.0
4,304.9
4,555.6

F3LT

4,914.7
5,677.4
3,828.8
5,006.0
4,361.6
4,571.1

DD 3'917.4 3'620.4 3.1250 3‘3004 4I2§1i2 3'9862 3'474.7 3‘3448

105

Table 15. Average cumulative harvest (numbers in thousands) of chinook salmon in 30

years under eight different stocking policies when uncertainty is ignored in key

parameters. Stocking policies with the highest average cumulative harvest are in bold.
—

BL
SR
FR
M1
M2

SQ

5,574.6
6,204.4
4,706.1
4,720.1
5,814.5

R25

4,851.7
5,450.1
4,213.2
4,723.8
5,180.8

R50

4,112.4
4,745.7
3,639.9
3,595.9
4,371.2

LT

4,798.0
5,033.7
4,132.5
4,188.5
5,040.9

F1

5,604.1
5,331.9
4,703.5
4,816.2
5,910.3

F3

5,517.0
5,170.1
4,729.6
4,743.7
5,695.6

F 1LT

4,958.4
4,513.0
4,163.2
4,289.0
5,138.4

F3LT

4,914.7
4,454.8
4,196.7
4,242.2
5,043.0

M3 5'950.2 5'2695 4‘4786 5.1587 $026.1 5.8758 5'316.1 5'2048

106

Table 16. Average cumulative harvest (metric tons) of chinook salmon in 30 years under
five different stocking policies when uncertainty is ignored in key parameters for the
simplified chinook-alewife model. Stocking policies with the highest average cumulative

harvest are in bold.
—

SQ R25 R50 F1 F3
BL 7,460.4 6,417.2 5,351.4 7,381.2 7,237.5
SR 8,188.5 6,924.8 5,596.3 7,921.1 7,855.0
FR 7,553.3 6,453.0 5,311.5 7,314.3 7,299.3
Ml 6,587.6 5,615.1 4,614.6 6,321.2 6,245.2
M2 7,964.1 6,827.6 5,561.7 7,797.3 7,725.9

M3 8,133.4 7.0009 5.7029 8.0093 7‘863.4

107

Table 17. Percentage of simulations years that fall alewife biomass (mt) was in each

category for different model scenarios and stocking strategies.

_
Status Quo Stocking

Model Scenario < 100,000 100,000 - 2,500,000 > 2,500,000
BL 23.18 55.04 21.78
DA 21.38 54.53 24.09
DB 29.83 53.65 16.52
DC 18.98 53.33 27.68
DD 40.62 49.40 9.98
FL 14.51 61.18 24.31
FH 33.55 48.34 18.11

Feedback stocking with 1 year lag
BL 17.95 59.32 22.73
MI 18.89 57.37 23.74
M2 18.07 58.64 23.29
M3 18.97 57.55 23.48
FR 23.10 55.10 21.81
SR 7.30 86.88 5.82

108

Table 18. Mean, standard deviation, 0.10, and 0.90 quantiles for the duration (years) of

chinook salmon mortality episodes under different model scenarios and stocking

strategies.

Status Quo Stocking

Model Scenario Mean Standard Deviation Quantiles
BL 9.62 8.60 (1.5, 27)
DA 9.44 8.68 (1.5, 27)
DB 10.84 9.23 (2, 27)
DC 8.20 8.45 (1.5, 27)
DD 13.33 10.06 (3, 29)
FL 7.87 7.34 (1.5, 21)
FH 12.14 9.95 (2, 28)

Feedback stocking with 1 year lag
BL 8.51 8.32 (1.4, 26)
M1 8.24 8.60 (1.33, 26)
M2 10.03 7.88 (5, 27)
M3 8.25 8.34 (1, 27)
FR 9.18 9.11 (1.5, 27)
SR 5.63 5.81 1151121

109

Probability

 

 

 

l

O 5 1 O 1 5
Stock Size

Figure 17. The probability of an alewife dieoff as a function of stock size (numbers times
10“) for model scenarios: baseline (solid triangle), DA (open circles), DB (x’s), DC
(solid diamonds), and DD (solid squares).

110

Percentage
00
1

 

Q Q 0 0 Q Q Q 0 0 Q
9 (t9 «9 .9 «9°69 «9 69 69.6929 0°

Numbers harvested (x 10M)

16 7 (b)
14 4
12 ~
10 ~

Percentage
00

 

00 N

oooooooooooo
\0‘19‘50996960’\°%°Q°\63\\°\‘19

Numbers harvested (x10M)

Figure 18. Distribution of the numbers of chinook salmon harvested in 1000 simulations
for the baseline scenario with the (a) status quo stocking policy and (b) feedback

stocking policy with one year lag.

lll

16 (a)
.41

Percentage
on

 

 

0 0 0 0 0 0 0 0 0 0
\° 9° (5° 0° <63 123° 4° 95° 9° No°
Numbers harvested (x10’\4)

16 1 (b)
14

1

12~
10*

Percentage
G)

 

l 1

0000000000
\°‘t9'b°bt°<oo 6949 69 69,69

 

Numbers harvested (x10’\4)

Figure 19. Distribution of cumulative harvest for 1000 simulations with a feedback

stocking policy with a one year lag for model scenarios (a) FR and (b) SR.

112

16“ (a)
14 «
12 ~
10 ~

Percentage
co

 

o N & a,
1 1 1 1

°~<Pr1°° «9° 9°69°§49°66369°§¢9°§°

Numbers harvested (x10M)

16 — (b)
14 1

Percentage
a:

 

11.9.9.9 6.99.96.96.99

Numbers harvested (MOM)

Percentage
co

 

04
° 8° 69° 69° 9° «9° 69 «<9 69 §\§\~@\m§
Numbers harvested (x10M)

Figure 20. Distribution of cumulative harvest for 1000 simulations with a feedback

stocking policy with a one year lag for model scenarios (a) M1, (b) M2, and (c) M3.

113

     

20 ‘
§ 15‘
E
(1?: 1O
5 i | 11 1
0 . ..f. n I lllll ll _ Lfifiififj
° 9° 9° 69 9° 9° 69° «9° 69 §K§¢§4§P§Q§
Numbers harvested (x1044)
25 1 (b)
20 ~‘
§ 151
E l
E 101
0..-...1-1111111 In- ,5
c90°§d°o°bo°§9§o°§§t9c9d°c9
Numbers harvested (x10M)
25
1 (C)
20 .
§ 151
8
gm

,M,__....r.111111111.-

M99w9999999s§§

Numbers harvested (MOM)
Figure 21. Distribution of cumulative harvest for 1000 simulations of the simplified

decision model with status quo stocking for model scenarios (a) baseline, (b) FR, and (c)
SR.

114

20 .
o l
g 15‘
E
8
g: 10 ~
9 9 9,,9,9,,9,9°9 9,9 9 9
Numbers harvested (x1044)
25 (b)
201
§ 151
E
8
g 101
5
0
o ’59 moo ‘5“9 99° 659 29° «0° 29° 9&\d§\\&{b§{b&\§9
Numbers harvested (x10M)
25— (c)
201
“g: 15-
is“,
g .0.
5<
0’

 

° 9° 9° 9° 9° 9° 9° 9° 9° 9§\C§P\\§\W§{b§\§

Numbers harvested (x10M)

Figure 22. Distribution of cumulative harvest for 1000 simulations of the simplified

decision model with status quo stocking for model scenarios (a) Ml, (b) M2, and (c) M3.

115

50 1 (a)
45 1

4O

35 '

30 7

Percentage
10
m

20‘
151
101

5.

0. Jul
5 6 7

<4

Percentage
N
0|

151
10‘

o.
<45

JI IIIII%

11 12 13 14 15
Weight(kg)

IIIIIII

11 12 13 14 15 16 17
Weight(kg)

_ I I“ , I , I , I I [LII IJV- .

<45

10 11 12 13 14 15 16 17 18
Weight(kg)

Figure 23. Chinook salmon average spawning weight at age 3 with status quo stocking

for model scenarios (a) baseline, (b) FL, and (c) FH.

116

5° 1 (a)

0.) ii ->
0 U! 0 0'1
“44 A .L 41 1

Percentage
—e N 10 00
0|

1

1.4] LI I IOJ I I LIJ 1:,

<45678 111214151617
Weight(kg)

_5
0010010

Percentage

<45 6 7 8 9101112131415161718
Weight(kg)

—‘ —* N N 00 CD A A (J1
o 01 o 01 0 U1 0 01 0 0'1 0
1 1 1 1 1 1 1 1 1 1 1

 

Figure 24. Chinook salmon average spawning weight at age 3 with status quo stocking

for model scenarios (a) DA, and (b) DC.

117

#01
0'10

Percentage
-* _. 10 M w 00 A
0 U1 0 U! O 01 O

5.

,1 _I,I.I_I,II,IIIIIIk
<456789101112131415161718
Weight(kg)
50~ (b)
459
405
359

Percentage
—* 10 10 ca
01

 

5..
10 11 12 13 14 15 16 17 18
Weight(kg)

Figure 25. Chinook salmon average spawning weight at age 3 with status quo stocking

for model scenarios (a) DB, and (b) DD.

118

25 2 (a)

15*

10‘

Percentage

 

<45 6 7 e 9101112131415161718
Weight(kg)

25 (b)
1
20 -

15*

Percentage

107

 

O1
<456789101112131415161718

Weight (kg)
25 ~

(C)
20 «

151

10*

Percentage

 

<45 6 7 8 9101112131415161718
Weight(kg)

Figure 26. Chinook salmon average spawning weight at age 3 with a feedback stocking

policy with one year lag for model scenarios (a) baseline, (b) FR, and (c) SR.

119

20-

15‘

Percentage

 

<45 6 7 8 9101112131415161718
Weight(kg)

25 7 (b)

207

15‘

10“

Percentage

 

<45 6 7 8 9101112131415161718
Weight(kg)

25 7 (c)

20~

Pecentage

10‘

 

<45 6 7 8 9101112131415161718
Weight(kg)

Figure 27 . Chinook salmon average spawning weight at age 3 with a feedback stocking

policy with one year lag for model scenarios (a) M l, (b) M2, and (c) M3.

120

50 w (a)

30

259

201

2'1
2!.' .j,_,_,,m,.7,1

8 9 10

WPercentage

Number of 5Episodes7
50 1
45
40
35
30

20
151
10%|
5
012.!I ET

Number of 5episodes

Percentage
10
U1

Figure 28. Distribution of the number of mortality events in each 30 year simulation time
period for the baseline scenario with (a) status quo stocking, and (b) feedback stocking

with a one year lag.

121

Percentage
(J A
o 0
V 4L.—l—%L— #241

20
to
0
o 1 2 3 4 5 6 7 8 9 10
Number of Episodes
60 1 (b)

Percentage
(a
O

50

40

20‘

103

o, i . I1_1_,,,_,ir ”W
O 1 2 3 4

5 6 7 8 9 10
Number of episodes

50

40

30

l
1 l
1
20~
o- . a I-.- ,,,,,
o 1 2 3 4 5

6 7 8 9 10
Number oi episodes

Percentage

Figure 29. Distribution of the number of mortality events in each 30 year simulation time
period with feedback stocking with a one year lag for model scenarios (a) Ml, (b) M2,

and (c) M3.

122

Appendix A

Here I provide more detail on methods used to produce estimates of salmonine
dynamics, including biomass, production and consumption, which were reported by
Madenjian et al. (2002), and used to construct the prey fish population estimation model
reported in Chapter 2 and construct the simulation model presented in Chapter 3.
Overview of methods

We estimated biomass, production and consumption of the five major salmonine
species (lake trout, chinook salmon, coho salmon, steelhead, and brown trout). Age-
specific population models were parameterized and track abundance at age over time.
Together with mortality and growth rates, we used these abundance estimates to calculate
gross production over time for each species. Using estimates of gross conversion
efficiencies (GCE) from bioenergetics modeling, total consumption was calculated using
the production-conversion efficiency method (Ney 1990,1993) where the gross
production is divided by the GCE to obtain consumption estimates.
Population models

For all species except chinook salmon, we used structure of the population models
currently implemented in SIMPLE (Koonce and Jones 1994) for Lake Michigan. We
updated the parameter values in these models to incorporate the most recent information
available for each species. These population models operate on annual time steps.
Numbers at age in each year for each species except chinook salmon were calculated

from estimates of recruitment to age 1 in each year using eq. 1:

_Zay
Na+l,y+l:Na,ye (I‘Pm,a) (1)

123

where Zay is the instantaneous mortality rate for a given age and year and Pma is an age
specific pulse of maturation mortality at the end of the year. Maturation mortality was
assumed to be zero for all ages of lake trout. The instantaneous mortality rate was broken

into mortality sources by eq. 2:

Zmy = Ma + LN + FM (2)

where M a is the age-specific instantaneous natural mortality rate, L0,), is the instantaneous
sea lamprey mortality rate in a given age and year and F my is the instantaneous fishing
mortality in a given age and year. Sea lamprey mortality was assumed zero for all ages
and years for all species except lake trout. The details on how mortality over time was
parameterized and estimated are described below.

For chinook salmon, the population was modeled in the stock assessment with
two time periods within the year with a pulse fishery and a pulse of maturation mortality
occurring in month seven (see Benjamin and Bence in press a). Note, this differs from
the models presented in Chapters 2 and 3, where fishing mortality is assumed to occur
throughout the year and maturation mortality is assumed occurs as a pulse of mortality at
the end of the year. Numbers at age at the beginning of the year in the stock assessment

are calculated using eq.3:

—M
Na+ l,y+1 : Na,ye a,y (1’ PF,a,y)(1— Pm,a,y) (3)

where PR0”), and P are the proportions of fish that die due to fishing or spawning.

m, a, y

Natural mortality rates were allowed to vary over time with

124

M = Ma + MTVM’a,y (4)

a,y

where M a is the baseline mortality rates and MTVM’ ad, is age- and year-specific stress

related mortality. The number at age at the end of the first period and at the end of the
second time period, which are needed in calculations of gross production, are given by

eqs. 5 and 6:

-7/12M
N;,y,1 = Na,ye my (5)

N 2 : N:,y,l(1— PFa,y)(1— Emmy) (6)

my,

where N a, y, 2‘ indicates the numbers for period 1' and the plus indicates the numbers are for
the end rather than the beginning of the period. Maturation mortality was modeled to
occur immediately after fishing mortality and before additional natural mortality. Details
on how chinook salmon mortality was parameterized are described below.

The geometric average abundance at age during the year for each species for use
in the estimation model presented in Chapter 2 was calculated as follows from the stock
assessment estimates. For all predators except chinook salmon, the geometric mean

abundance for predator j was calculated as

 

_Z . a
Law = \/Nj.a,y *Nmo’e J y (7)

2

125

where Njflay is the beginning of the year abundance and Zj,a,y is the total
instantaneous mortality rate at age a in year y. For chinook salmon, fishing and
maturation mortality were modeled in the stock assessment to occur as a pulse at the end
of July. Therefore, the average abundance used in the model presented in Chapter 2 was
calculated as a weighted average of geometric mean abundances for the time periods
before and after fishing and maturation mortality occurred.

Mortality Estimates

Lake trout .

Lakewide age-specific natural mortality rates and age-and year-specific sea
lamprey mortality rates were estimated from the catch-at-age models (CAA) constructed
as part of the stock assessments in the 1836 treaty ceded waters, generally following
methods similar to those used by Sitar (1999) and Bence and Ebener (2002). These
models were management unit specific models for the Michigan waters of Lake
Michigan, so the mortality rates in each area had to be combined to produce a lakewide
rate. Additionally, the number of ages used in these models (15) was larger than the
number of ages employed here (10), so a composite mortality rate for age 10+ lake trout
was needed. First, within each area the age 10+ natural and sea lamprey mortality rate
were estimated for each year by weighting the CAA model mortality estimates by the
number of fish in each age class. This produced a year specific age 10+ natural and sea
lamprey mortality rate in each area. Then lakewide age- and year-specific natural and sea
lamprey mortality rates were estimated by taking a weighted average with each mortality
rate weighted by the number of fish in that age class in each area. For sea lamprey

mortality rates, these estimates represented the age- and year-specific mortality rates used

126

in modeling lake trout abundance and consumption. For sea lamprey mortality rates prior
to 1981 (the first year included in the CAA models), an average of the 1981—1983 rate
was used. To estimate a lakewide average age-specific natural mortality rate, each
lakewide age- and year-specific natural mortality rate was averaged by weighting by the
total number of lake trout in each age class for each year.

Using the lakewide natural and sea lamprey mortality rates, a lakewide fishery
vulnerability function was estimated from the parameter estimates of the CAA models.
First, the predicted catch at age for ages 1-10+ in each area in 1998 ( the last year
modeled in the lake assessment models used here) was calculated using the Baranov catch
equation with the area specific natural mortality rates and sea lamprey mortality rates.
These catches were then summed across all areas and the fishing mortality rates (F a, 1998)
needed to produce the predicted total catch at age were estimated using the Baranov catch
equation with the lakewide estimates of natural and sea lamprey mortality rates from
above. The estimated F a. [998 were then divided by the maximum Fa, ,998 to give the
vulnerability pattern.

To estimate year-specific fishing mortality on a fully vulnerable age (fy), an age
structured model was built to tune these fishing mortality rates against observed harvest.
Using the number of yearling equivalents stocked the number at age in a given year could

be calculated by:

Na+l.y+1 = Nay exp(-Za.y) (8)

where Z‘Ly is the instantaneous mortality rate consisting of the lakewide age-specific

natural mortality, lakewide age- and year-specific sea lamprey mortality and fishing

127

mortality (F a ), which is the product of the year-specific fishing mortality (fy) and the

.y
lakewide fishery vulnerability function (S a).

Year-specific fishing mortality rates (13,) could then be estimated by tuning the
predicted total catch to the observed catch. Total lakewide harvest including commercial,
recreational, incidental and assessment catches for 1985-1998 was used as complied by
the Lake Michigan Technical Committee for the Lake Michigan Committee meeting
March, 2000. Since no harvest information was available prior to 1985, some
assumptions were needed to describe how fishing mortality behaved prior to 1985. Sport
fishing effort in Wisconsin’s waters of Lake Michigan increased more or less gradually
from the mid-1960s until the 19803 (Hansen et al. 1990). Hence, we assumed the sport
fishing mortality rates increased linearly from zero from 1965 to 1985. Based on patterns
in total mortality estimates and anecdotal descriptions in areas subject to tribal
commercial fishing (IAAWG 1979; Technical Fisheries Review Committee 1992) we
assumed that commercial fishing mortality was negligible prior to 1977 and increased
linearly from 1978 to 1984. We note in passing that there may have been significant
unreported commercial harvest in the state licensed Whitefish fishery in earlier years.

From the total harvest data, we determined that in 1985 approximately 66% of the
fishing mortality was recreational. Therefore, from 1985 to 1965, we allowed the
recreational portion of the fishing mortality to decline linearly to zero from the 1985
value. The 1985 commercial harvest was higher than in 1986 or 1987 and we felt this
high value reflected a unique event associated with the 1985 consent decree. To adjust

for the above average commercial harvest in 1985, we assumed that 27% rather than 35%

of the mortality in 1985 declined linearly to zero from 1985 to 1977 and then was zero for

128

the remaining years. We based the 27% figure on the ratio of the average commercial
harvest from 1985-1987 to the total harvest in 1985.

Since we required effort and catchability rather than year-specific fishing
mortality rates (fy), catchability was estimated by dividing the fishing mortality rate in
1996 by the targeted effort for salmonines in 1996 from Benjamin and Bence (in press b).
Relative effort was then generated for all other years by dividing the fishing mortality rate
(Q) by the estimated catchability.

Steelhead, Brown trout and Coho salmon

Natural mortality rates and maturation mortality for each species were taken from
the CONNECT model (Rutherford 1997). To estimate vulnerability for each species, the
age-specific instantaneous fishing mortality rate in CONNECT was divided by the
maximum instantaneous rate for that year and the average was taken over all years.
Catchability was estimated for each species by dividing the instantaneous fishing
mortality in 1996 for a fully vulnerable age in CONNECT by the targeted salmonine
effort in 1996 from Benjamin and Bence (in press b). Relative effort for each species
from 1985-1995 was then calculated by dividing the instantaneous fishing mortality of a
fully vulnerable age by the estimated catchability. Effort prior to 1985 was assumed to
decline linearly to zero in 1965. To obtain relative effort for 1997-1998, relative effort
estimates were tuned to produce harvest at the level of the observed lakewide total
harvest, excluding the weir fishery, for 1997 -1998 (as complied by the Lake Michigan
Technical Committee for the Lake Michigan Committee meeting March, 2000). Maturity
schedules and mortality were implemented as in Koonce and Jones (1994) with coho

salmon, brown and steelhead suffering a pulse of spawning mortality at the end of the

129

year after all other sources of mortality had occurred. Sea lamprey mortality was assumed
to be zero.

Chinook salmon

Chinook salmon population dynamics was modeled using a modified version of
the age—structured model described by Benjamin and Bence (in press a) and mortality
rates were either assumed known or were estimated by fitting the model to fishery data.
Mortality components include baseline instantaneous natural morality, time—varying
natural mortality (stress related modeled from 1985-1996 to coincide with observed BKD
mortality), fishing, and maturation (spawning) mortality (eqs. 3 and 4).

Age- and year-specific natural mortality rates were assumed to occur throughout
the year at constant rates and consisted of age—specific constant baseline mortality and
age- and year-specific stress-related mortality (Benjamin and Bence in press 3). Age-
specific baseline mortality rates were model inputs and were set to 0.7, 0.3, and 0.1 for
ages 0, 1, and 2-5, respectively. Stress-related mortality was modeled as the product of an
age and year parameters. Independent parameters for stress-related age-effects were
estimated for ages 0, 1, 2, and 3-5. Ages 3-5 were assumed to experience the same
natural mortality rate. Independent year-effect parameters were estimated for 1985-1996.

Fishing mortality in the stock assessment model was estimated using similar
methods to those described in Benjamin and Bence (in press a), and was modeled as an

instantaneous event occurring at the end of July in each year. Fishing mortality (F) was

defined by the assumption that the proportion surviving the pulse fishery was e]: and that

F was a function of age-specific vulnerability (Sa) and year—specific fishing intensity (fy):
FM = S.,,f,.. (9)

130

The model estimated vulnerability for ages 0-2, while ages 3-5 were assumed to be fully

selected to the fishery. Fishing intensity was calculated by the model:

fy = qu -devy, (10)

where q is a constant catchability coefficient parameter, E is observed fishing effort, and
ln(devy) is a normally-distributed process error parameter such that the sum of the log-
scale devy parameters equals zero. The ln(devy) parameters were estimated for 1985-
1999. For 1967 to 1984, fishing intensity was assumed to increase linearly from zero to
the 1985 level estimated by the model (IAAWG 1979). Fishery harvest is the proportion
of the population abundance remaining at the end of July that dies from fishing, as
described in Benjamin and Bence (in press a). Fishing mortality in the simulation model
presented in Chapter 3 was assumed to occur throughout the year and the instantaneous
fishing rate, F, for each age in each year was calculated by eq. 9.

Mature chinook salmon will enter streams to spawn, and will inevitably die during
or after the spawning run. Observed data from the recreational fishery and the spawning
weirs allowed us to separately estimate maturation (spawning) mortality. Similar to
Benjamin and Bence (in press a), maturation mortality in the stock assessment was
modeled as an instantaneous event occurring immediately after fishing mortality and
before additional natural mortality. Age-specific maturation was estimated independently
for ages 1-4. Age-0 fish were assumed to have no mortality due to spawning, while all
age 5 fish surviving to the time of spawning run were assumed to die at that time. All
chinook salmon were assumed to have reached maturity by age 5 because few age 6 fish
are observed in the fishery and fishery-independent surveys. From 1985 through 1999,

maturity was assumed to be a logistic function of weight at age during the summer- fall:

131

 

1
Pm,a,y : 1+ e'(:&),a+fl1,aWHAR’a’y) (11)

Note that the parameters of this function (130,“ and [31,0 ) were age-specific, so the logistic
function varied among ages. The source of summer-lfall weight at age data (WHAR,a,y) is
described below (“Weight at age and Growth”). For years prior to 1981, a constant
maturity schedule was used based on results reported by Stewart (1980): (PM = 0, 0.12,
0.33, 1.0, 1.0, 1.0 for ages 0 through 5 respectively). From 1981 through 1984 the
maturity at each age was linearly interpolated between the value assumed for 1980 and
the value estimated for 1985. Th same maturation model (eq. 11) was used in the
simulation model presented in Chapter 3 but the timing of the maturation mortality pulse
was moved to the end of the year.

The stock assessment model was fit to estimates from observed fishery data and
included: (1) annual total harvest from 1985-1999, (2) annual fishing effort directed at
chinook salmon from 1985-1999, (3) age-frequency compositions of the annual total
harvest from 1985—1997, (4) age-frequency compositions of the annual total harvest of
mature fish from 1985-1997, and (5) age-frequency compositions of the annual weir
harvest of fish captured during the spawning run from 1985-1996. Parameters were
adjusted while fitting the model in order to best match model estimates to observed
fishery data, using a maximum likelihood approach (see Benjamin and Bence in press a).

Recruitment

132

Recruitment was quantified as the number of individuals (or smolt—equivalents)
entering the lake fishery, and equaled the sum of hatchery and naturally-reproduced
production. Recruitment was defined as age 0 for chinook salmon and as age 1 for other
species. Records of hatchery plants by all agencies were summarized by Holey (1995)
and Benjamin (1998). We used this information and unpublished updated summaries
(USFWS, Green Bay, WI). Natural reproduction of chinook and coho salmon was
estimated from regression analysis (Smith, K.D., Michigan Department of Natural
Resources, unpublished manuscript ), stream surveys (Carl 1980), weir harvest records
(e. g., Pecor 1992; Hay 1992), and by the ratio of wild to hatchery adults sampled in the
lake (Patriarche 1980; Hesse 1994) or in tributary streams (MDNR unpublished data),
assuming equal survival between hatchery and wild fish from smolt-to-adult. Numbers of
steelhead smolt-equivalents were estimated by Rand et al. (1993) for 1975 to 1990, and
then updated. Brown trout (Rutherford 1997) and lake trout (Holey et al. 1995) were
assumed to have no natural reproduction.

To estimate numbers of hatchery smolt-equivalents for all species, actual numbers
of fall fingerlings or yearlings stocked were adjusted for survival from stream to the lake,
or for survival immediately post—stocking. These adjustments and survival rates varied by
species and size of fish. For lake trout, stocked numbers of yearling equivalents were
calculated from stocking information provided by the USFWS Great Lakes by R. Elliott
(GBFRO) by adding the number of yearling stocked in a given year and the adjusted
numbers of fingerlings stocked in the previous year. Fingerlings were adjusted by

multiplying by 0.4. Yearling lake trout were assumed to have 100% survival during

133

stocking. Stocking numbers (in yearling equivalents)for steelhead, coho salmon and
brown trout for 1965-1996 (except brown trout, see below) were taken from CONNECT.
Stocking numbers for 1997 were obtained from Jory Jonas (Michigan Department of
Natural Resources, Charlevoix, Michigan) as compiled for the State of the Lake and for
1998 from the summary of stocking in Lake Michigan distributed at the Lake Michigan
Lake Committee Meeting (March 2000). For steelhead, the numbers stocked were
multiplied by 0.5 and wild recruits were added to correspond with the treatment of
stocking for the 1990's in the CONNECT steelhead model. For coho salmon, the number
stocked were multiplied by 0.5 to estimate yearling equivalents stocked and wild
production, 5% of the yearling equivalents stocked, was added to correspond with the
treatment of stocking in the 1990's in the CONNECT coho model. For brown trout,
stocking numbers of fingerlings and yearlings from Benjamin and Bence (in press b) from
1965-1996 were used to estimate stocked numbers of yearling equivalents. Stocked
numbers of yearling equivalents for a given year was estimated by the sum of the number
of fingerlings stocked in the year before with 25% survival and the number of yearlings
stocked for that year. Stocking in 1997 and 1998 was assumed to have the same
proportions of yearlings and fingerlings as in 1996 (46% fingerlings, 54% yearlings), and
the number of yearling equivalents stocked was estimated as above. Yearling brown trout
were assumed to have 100% survival during stocking. Recruitment of chinook salmon to
age 0 was a model input calculated as the sum of estimated wild smolts and stocked
fingerlings multiplied by an assumed post-stocking survival of 0.75. Stocked fingerling

data were obtained from Benjamin and Bence (in press b) and updated for 1997-1999

134

using the same source as for other species. Estimated wild smolts were obtained from
CONNECT.
Weight at age and growth

For coho salmon, mean weight at age was determined from samples of harvest
and at weirs. There were distinct modes separating fish in their first (age 1) and second
(age 2) lake year. Weight at age for the harvest (used in tuning models against observed
harvest biomass) was estimated as the average weight of a harvested fish (averaged over
reporting agencies and weighted by the number of fish harvested). Differences in weights
of harvested fish in different jurisdictions reflect the seasonal progression of the fishery as
coho salmon migrate around the lake, and weight at age for the start and end of the year
for age 2 fish was extrapolated from these patterns.

For chinook salmon, weight-at—age (ages 1 and above) in the 19608 and most of
the 1970s is based on observed mean weight at age reported for spawning run fish by
Rybicki (1973). Weight at age starting in 1985 is based on mean weight at age from
biological samples of sport harvested fish in Michigan (unpublished Michigan DNR
data). Standard harvest/spawning weights in the modeling calculations were taken as
harvest weights for ages l-2 and spawning run weights for ages 3 and above. Conversion
was based on ratios calculated from unpublished Michigan DNR weights of both types in
the same years. Estimation of weight-at-annulus formation from harvest/spawning
weights was based on the proportion of annual growth through harvest, estimated from
backcalculated weights reported by Wesley (1996). Age 0 fall weight was assumed

constant and was based on the assumption that the same proportion of annual growth

135

occurs by fall for age 0 as was observed for age 1 by Wesley (1996), and average age 1
growth. Weight at age for the period between 1978 and 1985 was interpolated between
values used in 1978 and 1985.

Lake trout weight at age was based on observed weight at age in Michigan DNR
spring surveys. These surveys provided no evidence for large changes in growth over
time, and weight at age was assumed constant over time. Fish in the southern part of the
lake grew somewhat faster and weight at age schedules for the north and south were
combined, weighted by estimates of abundance in different areas of the lake. To obtain
weight at age for the age 10+ age group, all fish older than age 10 were grouped together
and the average weight was calculated using a weighted average with weighting factors
used in the average being an increasing power of 0.5 for increasing ages (i.e., the weight
at age 11, 12, 13 had the corresponding weighting factors of 1, 0.5, 0.25, ). This
procedure assumes that each subsequent age in the 10+ group was half as abundant as the
previous age.

Steelhead length-at-age were obtained from back-calculated growth curves
(Seelbach 1993), and weight at age was obtained from a length-weight relationship. For
steelhead, all fish older than age 5 were grouped together and the average weight for the
5+ group was calculated using the same weighted average procedure used for lake trout.
Data on weight at age for brown trout were available from samples from Wisconsin and
Michigan DNR creel surveys (Michigan and Wisconsin DNR unpublished data). Weight

at age was assumed constant over time.

136

Gross conversion efficiency

For lake trout, chinook salmon and coho salmon, bioenergetics models based on
Stewart and Ibarra (1991) were used to estimate individual gross conversion efficiency at
age for each species. Temperature, energy density, and diet information (1978-1981)
from Stewart and Ibarra (1991) were used to estimate gross production and consumption
by age for each species using weight at age schedules described above. Weight at
annulus formation for age 0 chinook was assumed to be 4.54 g as in Stewart and Ibarra
(1991). Mean weights at harvest were used as ending weights for the last age of chinook
and coho while a weighted average of the mean weight at annulus formation of age 11+
lake trout was used as the end weight for age 10+. Weighting factors for this 11+ average
were assigned as increasing powers of 0.5 as previously described for the 10+ group, such
that age 11 was weighted 1, age 12 weighted 0.5 etc.

The lake trout model was extended to include age 10 lake trout by assuming the
diet of age 10 lake trout was the same as age 9 lake trout. Spawning losses for lake trout
were incorporated as in Stewart et al. (1983). The chinook model was extended in a
similar manner to include age 4 by extending the Stewart and Ibarra (1991) age 3 chinook
model to a full 365 day year by assuming that the diet remained unchanged after day 214
and by assuming age 4 chinook had the same diet as age 3 chinook, and spawned and died
on day 214. To account for the declining growth of chinook salmon during the mid
1980s, two sets of GCEs were estimated using the same diet information as above (1978-
1981) but using two weight schedules, one that applied prior to 1979 and one that was the

average weight at age observed during 1985-1999. Because of the earlier maturity

137

observed in the 1970s GCE was not estimated or used for ages 4 or 5 prior to 1979.
During 1979 through 1984 the same GCE was used for these ages as was estimated for
the 1985-1999 period. For ages 1-3 GCE was interpolated linearly from the value
assumed for 1978 and the one assumed for 1985. Because we assumed that fall weight at
age zero was constant, the same GCE was used throughout the assessment period for that
age.

For steelhead and brown trout, no species specific bioenergetics models for Lake
Michigan were available. Therefore, the average gross conversion efficiencies were
calculated from GCEs of similar sized lake trout and chinook. Using the weight at age at
annulus formation schedule in CONNECT for each species, the GCE for a similar sized
lake trout and chinook salmon were averaged together to produce a GCE by age for each
species. For age 5+ steelhead, a weighted average, as described above for lake trout, was
taken using the weight at annulus formation for all steelhead age 5 and above.

Diet Composition

We estimated consumption of small alewife, large alewife (size categories as
defined by Stewart et al. 1981) and other fish using diet composition information
summarized by Rand et al. (1993), Stewart et al. (1981) and Stewart and Ibarra (1991),
and Elliott (unpublished data; U. S. Fish and Wildlife Service, Green Bay, WI) for
chinook, coho and lake trout and steelhead. We assumed brown trout to have similar
diets as steelhead.

For coho salmon, lake trout, and steelhead we calculated a weighted (by number

of days in each seasonal period) average percent (by weight) contribution to diet based on

138

information reported by Stewart and Ibarra (1991), Stewart et al. (1981), and Rand et al.
(1993). For chinook salmon, we calculated a weighted average percent for each year (or
time period since diet information for 1981-1983 and 1994-1995 was available in pooled
form) because of the more comprehensive sampling for this species and because it is the
dominant consumer of alewife. Before 1981, the first year with diet information, the
1981-1983 diet composition was used. In 1993 and after 1995 the 1994-1995 diet
composition was used. Diet composition from 1989-1992, when no data were available
were interpolated between 1988 and the assumed diet for 1993.
Consumption Estimates
Lake trout, Brown trout, Steelhead, and Coho salmon
Biomass at age was calculated using the weight at age at annulus formation.

Consumption can be calculated from estimates of gross production per year and GCE.
Gross production each year is estimated as the sum of yield, production lost to death, and
change in standing stock biomass. Since constant instantaneous fishing mortality rates
were assumed, production at age lost to death and yield in each year can be calculated
simultaneously by

Day: Za,y*Na'y* Wa*(1/(Ga-Za’y)*[exp(Ga-Za,y)-l] (12)
where Ga is the instantaneous growth rate, estimated by

Ga = ln(Wa+1/Wa) (13)
Instantaneous growth rate can not be estimated for the last age so species specific
assumptions were made for Ga of the last age group. For brown trout, steelhead and lake

trout, Ga was assumed to be zero for the last age group. For coho salmon, which grow

139

 

considerably during their last year, Ga was estimated using the weight at harvest averaged
over all years for age 2 coho as Wa+ 1. For coho salmon, brown and steelhead, the
production lost to spawning mortality was then estimated by

R“ = Na,y*exp(-Za,y)*Wa+1*Pm (14)
Weight at the next age was used to estimate biomass lost because spawning was assumed
to have occurred after all growth during the year was completed. For lake trout, no
biomass was lost due to spawning. For the last age of coho and brown trout, production
lost to spawning was not added because all fish were assumed to die spawning.

Total gross production is then the sum of D R0,), and the change in standing

(1,)"

stock which can be estimated by Ba+1,y+1'Ba,y- For the last age group, Ba + 1'er I was

estimated as
Ba+1,y+l = Ba,y*exp(Ga'Za,y) (15)

This age-specific production was then divided by the age-specific GCE and summed over
all ages to obtain total consumption. Year-specific consumption of each prey type was
estimated by multiplying the year-specific proportion of each prey type in the diet by the
total consumption for each year.

Chinook Salmon

Production was estimated using an approach similar to the methods described
above for the other salmonines, with the added complication that harvest and maturation
were modeled as a pulse 7/12ths of the way through the year. Production was estimated
as a function of fishery yield, production lost to natural in-lake mortality, production lost

to spawning mortality, and change in biomass of the standing stock:

140

 

X0», = Ya», + Dem, + R0,), + Ba+1,y+1- Bad, (16)
Yield was estimated as the product of harvest mean weight at age at harvest
(WHAR,a’),)and harvest-at-age (Cay):

Ylw : WHAR,a,yCa.y' (17)
Because fishery harvest and spawning mortality were modeled as instantaneous events
occurring 7/12ths of the way through the year, production lost to natural in-lake mortality

was first estimated for the first 7 months prior to fishing and spawning, and then for the

last 5 months beginning immediately after fishing and spawning:

00,), = Ma,yNa,yWANN,a,y(l/(Ga,y- Ma’y))(e(ca,y‘Ma.Y)_ J , (18)

where N a, y was used for the first 7 months, and adjusted for the last 5 months to account
for natural mortality, fishing, and spawning mortality from earlier in the year. For the

first 7 month period, the instantaneous growth rate (Gay) was calculated as:

Ga,_v : Ln(WHAR,a.y/WANN,a.y)/(7/12)’ (19)

where WHARfly is the mean weight at age at harvest, and WANN'M is the mean weight at
age at annulus formation, defined as the start of the year for age a fish. For the last 5

month period, GM was calculated as:

Ga,y = Ln(WANN,a+1,y+l/WHAR,a,y)/(1_ 7/12)- (20)

141

The estimate of production lost to spawning mortality (Ra,,y) was modified from equation

14 to account for the difference in the way fishing mortality was estimated, such that:

- M
Ra,y : Na,ye( a,y)(1_ PF,a,y)WHAR,a,me,a,y (21)
As for other salmonines, age-specific production (Xay) was divided by age-specific gross
conversion efficiency (GCE) to obtain estimates of age-specific consumption.
Consumption was summed over ages to obtain annual totals. Year—specific consumption
of each prey type was estimated by multiplying the year-specific proportion of each prey

type in the diet by the total consumption for each year.

142

Appendix B
Below are the five .dat files used in fitting the parameter estimation model
presented in Chapter 2. They contain all the data used in the likelihood equations along

with all the other information assumed known.

Awbl.dat

# DATA FOR LAKE MICHIGAN ALEWIFE AND BLOATER CAA
# BY EMILY B. SZALAI LAST MODIFIED 7/13/01
# First year of trawl survey

1962

# Last year of trawl survey

1999

# First year of predator abundance data
1965

# First year of hydroacoustic survey data
1993

# Last year of Hydroacoustic survey data
1996

# First alewife age

0

# Last alewife age

6

# First bloater age

0

# Last bloater age

7

# first bloater age in trawl survey

0

# last bloater age in trawl survey

7

# number of rainbow smelt ages 0-5

6

# number of slimy sculpin ages 0-5

6

# number of deep water sculpin ages 0—5
6

# first lake trout age

1

# last lake trout age

10

# first coho age

1

143

 

# last coho age

2

# first chinook age

0

# last chinook age

5

# first rainbow trout age
1

# last rainbow trout age
5

# first brown trout age

1

# last brown trout age

5

#

#Habitat volume of Lake Michigan
4800

Altprey.dat

#Data for alewife-bloater CAA Model by Emily B. Szalai

#Altemative prey biomass from swept-area estimates provided by Guy F. 12/99
#Updated 7/16/2001 For 1998 used average of 94-99

#

## Biomass(kg) of small ( < 90 mm)rainbow smelt by year

# For years 1965-1972 used avgerage biomass estimate from 1973-1977
477200 477200 477200 477200 477200 477200 477200

477200 224000 282000 788000 991000 101000 970000

659000 3603000 2100000 1948000 885000 920000 1620000

1281000 1211000 4168000 2518000 180000 1190000 944000

1101000 360000 375000 629000 133000 456600 786000

# Biomass (kg) of large ( > 90 mm ) rainbow smelt by year

# For years 1965-1972 used avgerage biomass estimate from 1973-1977
10666400 10666400 10666400 10666400 10666400 10666400 10666400
10666400 10894000 9560000 13437000 9609000 9742000 12585000
13884000 15717000 24397000 29246000 18970000 9983000 15168000
16731000 14975000 24265000 9323000 11596000 19850000 17741000
16879000 7802000 4058000 6104000 3726000 5205200 4336000

# Biomass(kg) of slimy sculpin by year

# For years 1965-1972 used avgerage biomass estimate from 1973-1977
3073400 3073400 3073400 3073400 3073400 3073400 3073400
3073400 962000 3007000 4537000 5487000 1374000 866000

1568000 1366000 1720000 397000 564000 214000 295000

259000 698000 850000 513000 315000 1314000 1220000 914000
1566000 1537000 2867000 2253000 2559600 4575000

144

# Biomass(kg) of deepwater sculpin by year

# For years 1965-1972 used avgerage biomass estimate from 1973-1974

8473000 8473000 8473000 8473000 8473000 8473000 8473000 8473000

5612000 11334000 30767000 33901000 26970000 30928000

50432000 85352000 66965000 49663000 98550000 72210000 85173000 58132000
91194000 63147000 35108000 72281000 31438000 40638000 34188000
21505000 27686000 43991000 49513000 38011200 47361000

Awblerror.dat

#Data for alewife-bloater CAA Model by Emily B. Szalai

#Estimated errors in trawl and hysroacoustic survey data

#Updated 7/13/01

#Estimated standard error of alewife year effects for trawl data 1962-1999

1.736417 1.195118
1.629004 1.122164
1.535537 1.058777
1.535537 1.058777
1.620708 1.116983
0.999023 0.688154
0.972835 0.670262
0.97768 0.673572
0.967961 0.666915
0.96323 0.663672
0.946262 0.65205

0.824182 0.567752
0.816109 0.562196
0.818989 0.564292
0.840549 0.579152
0.840708 0.57914

0.820051 0.565098
0.821836 0.566327
0.825017 0.568315
0.823426 0.567177
0.865093 0.595715
0.836531 0.575982
0.83822 0.577129
0.834897 0.574867
0.838558 0.57737

0.834724 0.574751
0.834782 0.574812
0.847297 0.583298
0.870009 0.598967
0.86267 0.59398

0.872408 0.600609
0.888822 0.611767

145

0.894472 0.615579
0.923623 0.635743
0.907138 0.624221
0.891516 0.61361

9999 9999
9999 9999
#

#Estimated standard errors of bloater year effects for trawl data 1962-1999
1.278789 1.451733 1.328923 1.382948 1.360281 1.404705 1.463497 1.497112
1.198713 1.358631 1.362316 1.417823 1.394716 1.438675 1.498968 1.533402
1.134043 1.279884 1.257983 1.309913 1.289666 1.336069 1.390992 1.420699
1.134043 1.279884 1.257983 1.309913 1.289666 1.336069 1.390992 1.420699
1.198713 1.358631 1.362316 1.417823 1.394716 1.438675 1.498968 1.533402
0.723792 0.820982 0.782276 0.813903 0.800374 0.825571 0.860283 0.880398
0.71193 0.806663 0.767489 0.798738 0.785763 0.81163 0.845507 0.864717
0.71193 0.806663 0.767489 0.798738 0.785763 0.81163 0.845507 0.864717
0.708183 0.802138 0.761322 0.792382 0.779578 0.805553 0.839108 0.858031
0.704555 0.797762 0.755399 0.78627 0.773627 0.799699 0.832947 0.851598
0.691602 0.782101 0.734453 0.764594 0.752506 0.778979 0.811158 0.828891
0.598259 0.678163 0.594073 0.617971 0.607721 0.627899 0.654114 0.66914
0.593316 0.672268 0.588036 0.611843 0.601843 0.622374 0.64823 0.662835
0.594566 0.673843 0.586751 0.61033 0.60029 0.620582 0.646399 0.661057
0.609389 0.690262 0.605899 0.630327 0.620043 0.641288 0.667912 0.682923
0.600446 0.680573 0.579361 0.602528 0.592524 0.612529 0.638049 0.65264
0.595643 0.674872 0.579114 0.602288 0.592386 0.612649 0.638109 0.65254
0.598099 0.677799 0.580879 0.604065 0.594112 0.614359 0.639909 0.654422
0.600317 0.680448 0.58585 0.609404 0.599385 0.619764 0.64554 0.660168
0.597934 0.677592 0.580958 0.604342 0.594402 0.614686 0.640238 0.654731
0.623957 0.70753 0.601987 0.626274 0.615939 0.636798 0.663299 0.678377
0.606486 0.687786 0.587166 0.610986 0.600919 0.621351 0.647211 0.661916
0.606486 0.687786 0.587166 0.610986 0.600919 0.621351 0.647211 0.661916
0.606486 0.687786 0.587166 0.610986 0.600919 0.621351 0.647211 0.661916
0.606486 0.687786 0.587166 0.610986 0.600919 0.621351 0.647211 0.661916
0.606486 0.687786 0.587166 0.610986 0.600919 0.621351 0.647211 0.661916
0.606367 0.687625 0.58705 0.610842 0.600779 0.621198 0.647049 0.661745
0.616007 0.699201 0.594001 0.617952 0.607523 0.627375 0.653703 0.669033
0.629085 0.713666 0.611825 0.636657 0.626174 0.64737 0.674329 0.689667
0.625535 0.709347 0.605609 0.630181 0.6198 0.640892 0.667565 0.68273
0.627303 0.711469 0.607271 0.631896 0.621466 0.642546 0.669306 0.68455
0.625535 0.709347 0.605609 0.630181 0.6198 0.640892 0.667565 0.68273
0.625535 0.709347 0.605609 0.630181 0.6198 0.640892 0.667565 0.68273
0.672716 0.764304 0.6417 0.667514 0.656165 0.677527 0.705974 0.722608
0.642627 0.729917 0.613349 0.638041 0.627229 0.647761 0.67493 0.690767
0.642627 0.729917 0.613349 0.638041 0.627229 0.647761 0.67493 0.690767
#

#CV for adult alewife hydroacoustic estimates

146

0.40 0.25 0.25 0.32

#CV for young of the year alewife hydroacoustic estimates
0.36 0.45 0.15 0.54

# CV for bloater hydroacoustic estimates

0.16 0.29 0.27 0.20
Awblinit.dat

#Initial parameter values for alewife-bloater CAA model

# Updated 2/23/2003

#log_q for adult alewife hydro survey

0

#log_q for yoy alewife hydro survey

0

#log_q for bloater hydro survey

0

#log_init_pop_aw

#21.47 20.03 21.79 21.33 16.65 16.65

#log_init_pop_bl

11.0411.1617.03 169219.21 19.21 19.21

#log_init_recruit_aw

25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
25 25 25 25 25 25 25 25

#log_init_recruit_bl

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
20 20 20 20 20 20 20 20

F uncresp.dat

# Data for alewife-CAA Model by Emily B. Smith
Parameters for functional response.

All from SINH’LE And SCOL II (GCE)
Updated 12/13/02

search constant per predator body length
98E-6

HABITAT OVERLAPS FROM SIMPLE

ALEWIFE-OVERLAP FOR AGE 6+ SET EQUAL TO AGE 5 FROM SIMPLE
Habitat overlaps for alewife by age with lake trout

1 1 1 1 1 1
Habitat overlaps for alewife by age with coho

l 1 1 1 1 1 1

# Habitat overlaps for alewife by age with chinook

1 l 1 l 1 l 1

#
#
#
#
#
1.
#
#
#
#
#
1

#

147

# Habitat overlaps for alewife by age with rainbow trout

1 1 1 1 1 1 1

# Habitat overlaps for alewife by age with brown trout

0.7 1 0.8 0.8 0.8 0.8 0.8

#

# BLOATER - OVERLAPS FOR AGE 6+ SET EQUAL TO AGE
5

# Habitat overlaps for bloater by age with lake trout

l l 1 1 1 l l 1

# Habitat overlaps for bloater by age with coho
0.10.1000000

# Habitat overlaps for bloater by age with chinook
0.10.1000000

# Habitat overlaps for bloater by age with rainbow trout
0.10.1000000

# Habitat overlaps for bloater by age with brown trout
0.10.10.10.10.10.10.10.1

#

# RAINBOW SMELT

# Habitat overlaps for rainbow smelt by age with lake trout
0 0.5 1 1 1 1

# Habitat overlaps for rainbow smelt by age with coho

1 0.6 0.6 0.6 0.6 0.6

# Habitat overlaps for rainbow smelt by age with chinook

l l 1 l 1 l

# Habitat overlaps for rainbow smelt by age with rainbow trout
1 0.75 0.75 0.75 0.75 0.75

# Habitat overlaps for rainbow smelt by age with brown trout
1 0.75 0.75 0.75 0.75 0.75

Habitat overlaps for slimy sculpin by age with lake trout

1 l l 1 1
# Habitat overlaps for slimy sculpin by age with coho
0.10.10.10.10.10.1
# Habitat overlaps for slimy sculpin by age with chinook
#0.10.10.10.10.10.1
# Habitat overlaps for slimy sculpin by age with rainbow trout
0.10.10.10.10.10.1
# Habitat overlaps for slimy sculpin by age with brown trout
0.2 0.2 0.2 0.2 0.2 0.2
#
# DEEPWATER SCULPIN
# Habitat overlaps for deepwater sculpin by age with lake trout
0.3 0.3 0.3 0.3 0.3 0.3

#
# SLIMY SCULPIN
#
1

148

# Habitat overlaps for deepwater sculpin by age with coho
0 0 0 0 0 0

# Habitat overlaps for deepwater sculpin by age with chinook
0 0 O 0 0 0

# Habitat overlaps for deepwater sculpin by age with rainbow trout
0 0 0 0 0 0

# Habitat overlaps for deepwater sculpin by age with brown trout
0 0 0 0 0 0

#

#

# MAXIMUM GCE's FROM SCOL H

# lake trout

0.205037 0.19238 0.161662 0.143462 0.144346 0.128036 0.115909 0.104652 0.095365
0.084538

# coho

0.302 0.234

# chinook (Max. GCE's from Jim's CAA model)

0.2707 0.2786 0.1877 0.1037 0.0369 0.0369

# rainbow trout

0.237766 0.221432 0.212774 0.154487 0.144215

# brown trout

0.2314372 0.221432 0.212774 0.154487 0.154487

# Gmax(kg) for CHS

0.5815 2.5896 5.0175 5.717 4.010.01

# consumption by predators from SCOL H

599.6469

3332.011

9027.261

15248.91

23718.82

34322.44

39205.89

47507.23

59576.74

68551.65

75099.37

83745.64

89890.04

89565.63

85412.86

91588.05

91794.15

94391.64

90299.26

92598.92

92568.5

149

86332.33
99031.48
82531.94
81188.48
84149.73
91114.74
91238.23
81015.6

79512.46
1013702
1200142
95729.01
1202955
1202955

# Proportion of small alewife, large alewife, and other fish in consumption from SCOL H

0.30609877
0.27163269
0.2844447

0.31737736
0.26314987
0.24291606
0.27500902
0.26159143
0.23648702
0.26193975
0.26961594
0.25632055
0.23758517
0.23963457
0.28125929
0.27335491
0.27440399
0.27320033
0.35443351
0.29684841
0.36227697
0.37521 12

0.43588714
0.42589115
0.39572388
0.36483412
0.32590392
0.28962396
0.26797463
0.26865688

0.039344
0.175473
0.239608
0.251924
0.333768
0.396242
0.367528
0.401525
0.450722
0.42448

0.431732
0.454864
0.48366

0.483218
0.419997
0.437747
0.458109
0.448743
0.326868
0.396998
0.330637
0.352151
0.349209
0.340368
0.367143
0.396304
0.443795
0.477718
0.48078

0.476627

0.654557
0.552894
0.475947
0.430699
0.403082
0.360842
0.357463
0.336883
0.312791
0.313581
0.298652
0.288816
0.278755
0.277148
0.298744
0.288898
0.267487
0.278057
0.318698
0.306154
0.307086
0.272638
0.214904
0.233741
0.237133
0.238862
0.230301
0.232658
0.251246
0.254716

150

0.24985764
0.23041592
0.25087924
0.2171374 0.57993 0.202933
0.2171374 0.57993 0.202933
# Proportion of consumption from chinook and other predators

0.520116
0.556449
0.509482

0.230026
0.213135
0.239639

0.144736842
0.034980535
0.48990813

0.578734499
0.435767733
0.507276683
0.59686009

0.554063588
0.54022633

0.572158747
0.623988955
0.617688883
0.513785974
0.536917821
0.564917283
0.534548034
0.32791402

0.457490147
0.344643068
0.387662449
0.376866305
0.374889423
0.415864292
0.46756353

0.542749741
0.612762391
0.656281678
0.655369883
0.677835275
0.705107526
0.68015993

0.72615712

0.526315789
0.637747446
0.292420216
0.219821306
0.330701 123
0.278331922
0.208172247
0.240784695
0.251899335
0.228239685
0.188499222
0.191410492
0.271930907
0.254231038
0.25470048

0.252260345
0.393398253
0.287065202
0.38991063

0.408983869
0.502845828
0.511555509
0.45072031

0.388710063
0.322447331
0.259282833
0.209206369
0.209782983
0.194374602
0.174175813
0.190343093
0.158494851

0.328947368
0.327272019
0.217671654
0.201444195
0.233531144
0.214391395
0.194967663
0.205151717
0.207874335
0.199601568
0.187511824
0.190900625
0.214283119
0.208851 141
0.180382236
0.213191622
0.278687728
0.255444651
0.265446302
0.203353682
0.120287867
0.113555067
0.133415397
0.143726407
0.134802928
0.127954776
0.134511952
0.134847134
0.127790124
0.120716661
0.129496977
0.11534803

0.667644449 0.143208865 0.189146685

# Lake trout other fish diet proportions 1981-1993

0.40 0.56 0.51 0.51 0.42 0.42 0.42 0.42 0.42 0.42

# consumption per predator from SCOL 11 based on bioenergetics

# lake trout

0.917385252 2.065542461 3.587326355 4.924855488 5.366366166 6.195763697
6.686746743 6.994411536 12.78158861 0

# coho

151

3.2802841 11
# brown trout

5.136837009

2.668738727 6.519267979 6.734972567 0.027245395 0

# steelhead

2.842515897 8.193812042 6.182241894 10.57000433 0

# proportion of diet from fish from SCOl H

# lake trout

0.540.9611111111

#coho

0.75 1
#chinook

l 1 1 l 1 1

# brown trout

0.54 0.78 0.74 0.74 0.74
# rainbow trout

0.54 0.78 0.74 0.74 0.74
# Consumption per predator for chinook ages 0-4, 1967-1999

L978891146
.L978890315
L978892909
L978895634
L978894065
L978890594
L978885376
L97889229
L978887695
.L978895703
L9788909
L978890657
L978892365
L978893536
L978895282
L978891016
L978891721
L978891552
L978891938
L978819294
L978813463
L978768248
L978734062
L97875516
L978697167
L978678934
L978678507
L978775141
L978865009

0 0
8.693059654
8.688400387
8.683723737
8.679020673
8.674353504
8.669697182
8.665000457
8.66036226
8.655693399
8.651079156
8.31551978
7.955871908
7.603370335
7.248262532
6.899129571
6.540079132
6.163903391
5.201034512
4.930392218
5.200035642
4.870060899
5.41 1685914
5.027166207
6.433954302
6.727827425
4.331012451
4.279067342
6.027428883

0 0

0 0
22.26011038
22.24631347
22.2326376
22.21894389
22.2053333
22.19169119
22. 1780674
22. 1645938
22.1511 118
21.06108817
19.40729125
17.82167571
16.28707367
14.74489523
13.19606062
11.61843138
9.744509684
9.163040409
l 1.67446825
10.98378161
11.20630935
12.72350936
13.22799412
14.37061702
16.25097974
13.98907699
15.203 87846

0

0 0
55.12805633
55. 12789775
55. 12795265
55. 12794289
55. 12804715
55. 12785279
55. 12783641
55.12806458
55.12784326
46.911 16294
41.59774426
35.35974072
29.940962]
25. 18750632
21.04892144
16.81998295
14.60133572
22.72630248
21 .4179121
24.959265?
28.77450344
27. 15270433
34.7419529
30.92707125
26.4449114
31.21 191059

152

OOOOOOOOOOOO

4.285760879
9.771665738
14.67298063
19.59144921
17.06347176
51.55550823
36.96693704
53.166273

55.00858998
50.04514546
61.08497558
108.7906829
103. 1448733
54.82412876

1.978888575 4.419542096 15.30541331 40.67952634 5644471421
1.97889256 3.209207024 11.29596384 15.5466301 14.14092567
1.978893137 3.253922745 12.80065959 26.81120879 26.77203207
1.978888865 3.256717708 12.82649997 26.77096576 26.80437077

Predabund.dat

#Data for alewife-bloater CAA Model by Emily B. Szalai

#Predator abundance from 1965—1999 from SCOL H

#Updated 09/ 17/2002

#

#Average Number of lake trout ages 1 to 10

986580.4544 0 0 0 0 0 O 0 0 0

1367845.]33 678954.7688 0 O 0 0 0 0 0 0

1436468531 941171.588] 535384.990] 0 0 0 0 0 0 0

1629225688 9882152193 741346.4387 4168940047 0 0 0 0 0 0

1548715598 1120624759 777554.760] 575097.877] 316680.0278 0 0 0 0 0
151787124 1065060259 880778.5086 600914.4449 433747.54 236115.5303 0 0 0 0
1653797862 1043664672 8361953047 678124.5525 4499954676 320593.8804
174862.4842 0 0 0

2016025829 1136925532 8185054515 641374.346] 504202.7385 329716.4897
235413.7529 129192.562] 0 0

1836334.]38 1385700037 8906758203 625441.273 473486.4463 366228.4278
240061.0947 1725549574 95621.34843 0

1747869541 1261968.]3 1084386046 6780250254 458440.2485 340932.7555
2643855284 174571.0143 126802.7358 0

1988747.721 1200961919 986484.1876 822377.239] 493448.873 327233.8642
244038.7926 190740.438 127367.067] 0

201908941 1366228876 937773.652] 745312.475 594248.0362 3491660319
232248.4862 174670.1554 138169.3195 0

1835309862 1386828823 1065661557 705841.825 534730.9225 416842.4172
2457147065 164917.7776 125623.642 0

1930765461 1260376362 1080552189 799079.0826 502810.7259 3718380033
2908543618 173101.3582 117761.6758 0

1860407255 1325441558 979897.6565 8045095091 5623544938 3448176515
256029.7332 202369.1385 1222457864 0

1990526632 1276654158 1028052521 723621.8249 557479.2711 378429.196]
233083.3384 175100.3139 1407081522 0

1694929843 1365423615 987860.0602 7527742422 493363.2659 368177.7708
2510049054 1567051563 1199048173 0

1752166693 1162211909 1054066797 7176489656 505662.184 319341.9614
2399740408 165820.9364 1055851881 0

1929842097 1201000513 895101.0927 759728.791 474997.2742 3216283286
2041962124 155859.0863 110079.434] 0

876221.6735 1322280902 9228001806 6404305383 495444.313] 296627.219
202023.0258 1303594308 1016404772 0

153

2169118942 599927.6608 1011713728 6528169816 410292.6275 302632.3922
1822548618 1262956883 83523.39997 0

2274344812 1484592209 4578801218 7094478447 411558.753 245631.4896
1819400138 1114302946 79133.5561 0

1782350946 1555394257 1127865.]41 3163384349 4361963397 2398649769
1439106699 1084297375 68163.0089 0

1496064366 1218437489 117842434 7703228519 190047.7757 246381.1513
136351.6065 83430.34101 6468658419 0

1745686602 1022981845 9239783752 8037903057 4601465368 1046802268
1366043683 7709200914 48569.17854 0

2063640917 1194946812 780354.562] 639200.414] 4854921824 259606.099]
5849591206 7754095453 4468457074 0

2151839237 1412739.]58 913108.237] 5475940204 397777.6536 2805235393
1476270432 3435793411 4616891328 0

2137937332 1475195.]61 108781638 653124.0119 3523701677 241667.0337
1683479245 88233.03659 21339.9412 0

2296657976 1465510839 1136407428 784611.3992 429279.0025 2191408923
1502823813 1051548408 5555206403 0

1932577502 1574615789 1130244871 8246265962 523523.7974 2726824317
138141.4073 960259752 6797384458 0

2173798841 1324392.]01 1211193856 810711.7083 5393219428 326332.669]
1679467402 8464852001 599853123 0

1526493449 1491271976 1024577677 885161.7299 543341.286 3412699079
2041380874 1043787992 5243489826 0

1775198038 1047257744 1154917625 7548481894 6056570076 3515815254
215689.2174 1292951315 66221.27067 0

1782426116 1216762504 8066563898 8341320333 5020675634 3830365379
2157857928 129651.2915 7895529229 0

1782426116 1216762504 806656.3898 8341320333 502067.5634 383036.5379
2157857928 129651.2915 7895529229 0

#

# Avg Abundance of coho by age (1-2) and year

0 0

2504186575
6569528067
4565387009
1265325552
137785868

1 1 14898.29]
1069484649
9332362934
1475452418
9855475348
1222020423
1253628589
1093568468

0
1770045856
459363.4697
315794.335]
865 83 1 .8466
9326966919
7465784583
7084664046
61 15623764
9564864346
632027.0915
775248.654
7867484885

154

1662703762 6789182885

122299033 1021153365
1023209035 7430250958
9053260063 614963.4619
9591651203 5382628763

1228181672 5641404644

1078133409 7145966537
9264410084 5731244442
932399.4919 5036057561

1300029983 5200615769
9452228541 7256523382
943554.974] 5686977412

980669.806 5892264106
1087055939 5773398956
6770141562 6388321363

579745.333 3339881667
947909.3758 314443.6153

1224305452 6298267434

1043621031 867681.6928
8181942303 6726723452
8181942303 672672.3452

#

# Avg Abundance of chinook
000000

000000

429751.2784 0 0 0 0
3684084936 2453704902 0 0
3850324576 210151.4279 158441.2588
1026395716 219431.194 1351459868
1269320094 5844058234 140539.310]
1122457203 722055.066 372775.621
1697667809 6379218798 458714.165]
2026007186 9639469309 403629.4763
2445217084 1149323688 6074559218
1982570377 1385862172 721371.1395
1853775769 1122624255 866349.2855
3188190301 1048737985 6989904299
3010689265 1802014712 6503844095
3703347284 1700140805 1113099532
3189112662 20969989 1057151.4
408205906 1810706402 1321474061
431119091 2323916558 1156079879
4965603086 246089195 1502853341
4051969612 2841915173 1611050744
3963833395 2331802189 1497003.]86
4048552403 2276407418 120000333

by

age
0
0 0
0 0
6496292883

55117.10267
57011.18668
1504122876
184096.853]
161120.3693
241179.427

284862.168]
3402654429
273045.885

2526795668
446948.973]
448943.501

591499.6074
543714.3663
740517.9896
860037.936]
5922520005

155

(0-5) and

OOOOOOOOOOOOO
OOOOOOOOOOOO

2255963468
43108.20827
8121730867
9508433988
1547780017
2474148814

year

0

1508621403
4199902272
1024840617
1438365872
5077163587

3869179732
5231981496
588006047
4710639987
4253186233
4134976941
4004861308
4885886584
4653799. 166
4413337. 125
3547260774
3591298. 191
#

# Avg Abundance of

9416691032
1858682759
114746.098

278985.5744
286521.7531
232366.1034
4147245259
508811.6815
8290182778
5541898962
7198188688
951388.0674
6460329257
5241786624
952481.414

893116.514

742419.060]
4912318344
8207136704
1385347136
9081242999
1113754867
1122129.] 13
6736495967
8606710797
8718365085
101222653

1001480075
9936184467
112957827

1019053. 143
987606.575]

2336007435
223830432

3045870626
3402790668
2740] 12.238
249036468

2417279. 143
2343437377
2845656426
2727144554
258400429

2079345439

0 0
81221.81787
159981.1513
9855772663
2391242727
245069.185
198332.] 103
3532394097
4324696849
703156.3484
4690677587
607980.1265
8018867243
5433743455
4399598284
797773.105]
7464834252
6192276646
4088621258
681665.375]
1148226.] 17
752235.669]
9225676808
929504.334]
558010.884]
712928.2716
715714. 1252
8230356953
8123490986
805972.1939
916255.745]
8266034226

1012565746
9386407398
9895069017
111320735
1137334687
9010902665
1185067358
1585523297
15016321
2081548841
2034785613
1931751036

rainbow

0 0

0 0
6258584628
1227823389
7533923738
182061.4268
185843.1013
1498008863
265738.251
324044.1443
5247639536
3486673493
4501203246
591310.8767
3990851543
3218422035
581263.416
541723.1255
4475805085
294348.0813
4887864455
8220999306
538581.2007
660534.524]
6655008473
399521.251
502838.698]
495308.1036
566176.463]
5588249714
5544382212
6303037815

[10111

436943.1432
2984093898
234587.907]
2687054534
214482.4773
1768722278
124243.94

3584560879
7030735479
6494334228
1234036032
1229832353

by

0

0 0
3380633936
7008062852
4865700855
1038803926
111243.5715
9303 1 . 1912
1520541836
1880332794
2964650596
2152736583
2601630283
335234.61
2429663827
194076.1315
318625.1547
3091805545
2607555942
177684.799
265571.293]
4390676247
316313.702]
367489.1416
3738024923
2379842585
269277.1912
264234.425]
297453.719
297134.3839
2954058325

156

1393997018
9640961742
5722833483
3789618332
36881.2459

1866658013
1297747485
9519.093357
44753.17194
4229291725
132458.0066
2954424884

age(1-5) and

COOOOOOOOOOOOOOOOOOOOOOOOOOOO

5097924755
3343523047
9930458786
455.2515258
2370208604
13 1 . 1703553
4.469931963
3393849274
1982004419
1534405784
3409226585
3133717274

year

1028813851
925526.184
925526.184
#

8176334953
8538577758
853857.7758

#Avg Abundance of

year
0 0

18 199.75 833
3354769574
6901033578
1068322788
1207488959
168316.4402
281719.71 17
6527046166
5790340646
4540838052
5637045343
744754.8149
623532.934
6367567089
602151.897
6512559599
6577040353
876689.936
8751422432
7993307871
831697.7368
592463.139]
8057426232
791292.718]
860887.331
817925.5082
8439482445
9035076991
957875.826]
9813029246
1023791.]24
9860794709
9666964304
966696.4304

Survey.dat

0 0

0 0
102027496
1874305071
3842533857
5928325385
66778.8] 162
92770.10176
1547478802
357314.0845
315910.123
246900.1873
305466.024
4022079785
3356006062
3415567296
3219002794
3469708446
3492189134
4639158909
4615277869
4214383017
441193.8779
316162.9615
432687.6514
4286999397
4654025572
443367.105
458086.5145
491282.] 165
5219555927
5348086593
5579362876
5476084695
5476084695

5883531602
5883637438
588363.7438

brown

0

0 0

0 0
3787654729
6906009985
1405200809
21517.23621
24056.19346
3316881997
549136692
1258460697
1104299386
8566007965
105184.981]
137459.661]
1138363837
1149886479
1075591202
1150674893
1149452892
1515536625
150301.3085
138424.095
1469173036
106724.526]
1486400016
1479188599
160782.918]
1538547413
159510.9928
171835.] 193
1829279392
187419.185
2007336613
200733.6613

3070176886
2937185432
293718.5432

trout by

0

0 0
1076980662
1946.510791
3926088603
5959369094
6604391294
9026679053
1481393842
336528477
2927258566
2250841016
2739757796
35491.61803
2913558297
2917357692
2705042207
28686.10169
2840548215
3728837607
37297.74512
3491029964
37641.42857
2789157458
39146.85581
3896153363
42588.56201
40911.90394
4263407424
46056.16894
490258408
51566.80855
51566.80855

# Data for alewife-bloater CAA Model by Emily B. Szalai
# Observed year effects from trawl survey data for alewife and bloater

157

0
0
0

age

COOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

(1-5)

and

# Observed biomass from hydroacoustic surveys of alewife and bloater

# Alewife year effect for year 1962-1997 for age 0 and age 3+ (9999 missing value)

# Updated 2/13/2003

#

#

# TRAWL SURVEY DATA
#

0.734297 2.620397
1.793844 3.33543
3.087469 3.043394
2.017563 4.358424
1.185049 4.78334
3.80697 1.69733
2.836418 2.106245
4.150209 2.279781
4.17984 1.966759
1.893811 2.225125
3.889474 0.616442
2.683019 1.825276
5.315345 0.541599
3.267781 1.41973
4.537853 0.916014
3.24033 0.19451
2.775378 0.2579
3.701179 0.194624
4.886939 0.51194
5.080376 0.094754
2.126993 0.60361
3.242949 .1.50225
3.310497 -1.61399
3.243975 4.55501
2.159304 0.55087
3.534968 0.70734
1.428037 -1.3384
4.254836 -1.56671
4.052675 4.23399
1.099938 -1.1 1332
0.983756 0.68027
1.128417 0.60926
0.303298 0.94897
0.549285 -0.22869
0.92334 0.767569
0.86026 0.975965

# Day Between mid-date for trawls in each year and mid-date in 1999 (1962-1997)

36
36

32
37
43
35
27
28
22
32
30
27
21
14
21
14
17
18
20
20
15
16
16
13
12
15
10
18
15

-6

# Bloater year effects of year 1962- 1997 by age and year ages 07

-2.58311 4.24064 3.469584 3.009534 0.854588 -1.58769 -3.3lO83 -3.81203
-3.03938 3.227216 2.630348 2.347304 0.360088 -1.887 -3.26671 -3.75564
-2.5365 2.019093 1.576195 1.579327 0.403835 -0.98703 —2.43l63 -2.7093
-2.5365 2.755128 2.008125 1.951677 0.878591 -0.63643 -1.55506 -5.43293
-3.03938 1.706991 1.178604 1.313358 0.144438 -0.98054 -1.81262 -2.27832
-2.64066 -3.57842 -0.46735 0.740767 0.302877 -0.09573 -0.93588 -1.82394

159

-2.38749 -0.60047 -0.59139 0.132853 0.010813 -0.43781 -0.77233 -1.7534
-2.01807 0.969002 -1.04662 0244 -0.18132 -0.29976 -0.68829 -1.19583
0.240135 1.294968 -0.21516 -0.13782 -0.2495 -0.17792 -0.59767 -1.2922
-1.03324 0.955049 -0.52378 -0.50136 -1.0398 -0.97053 -1.09923 -1.71363
-1.76614 -0.85467 -1.15602 -1.07557 -1.74695 -2.06985 -2.29477 -3.83795
-0.22343 -1.30849 -2.61213 -1.70017 -2.37509 -2.84302 -4.26828 -5.06295
0.1536 -0.10841 -3.18717 -3.21642 -3.52722 -3.51038 469167 -4.92461
0.7897] 0.143055 -1.86713 -2.72443 -3.57652 -3.67201 -4.58903 -5.46906
-1.45128 -1.28576 -2.11152 -2.38404 -3.65023 -4.14623 -4.84672 -5.47183
-0.29556 -1.92146 -2.81144 -2.76236 -3.42681 -4.40254 -4.96206 -5.16076
1.663658 1.425841 -2.95504 -2.80625 -3.10534 -3.41625 -4.83329 -5.35987
1.606874 2.407765 -0.07598 -2.12433 -3.1413 -3.73225 -4.28322 -5.19096
3.069752 2.52047 0.410453 -0.49271 -2.69961 -3.32243 —4.65733 -5.59793
3.109964 4.714235 1.048391 0.612625 4.23605 —3.3097 —4.82132 -5.5911
4.561486 4.066298 1.205095 0.14762 -1.1136 -2.60003 -4.49858 -5.30809
5.561741 5.919077 1.592817 1.428036 0.307418 -2.26196 -4.21643 -4.32975
3.603663 6.572836 2.81723 1.614216 0.200633 —1.15029 -2.96047 -4.48751
3.98628 6.791133 3.79989 2.110115 0.676546 -0.79984 -2.33447 -3.92441
4.265112 7.422017 4.150544 2.960131 1.531155 0.044426 -2.01652 -4.17933
4.113977 8.81286 4.452841 2.967903 1.419252 -0.05986 -l.28691 —4.70743
2.792367 8.474901 5.265237 3.856757 2.536243 0.812665 -0.3493 -2.71183
3.840411 7.923474 5.123194 4.068321 2.155665 1.04406 -0.73206 -2.29548
2.007523 7.393716 4.118784 3.797856 2.646337 1.320423 0.8314 -0.75446
-0.00445 6.774735 4.352125 4.40746 2.56765 1.711743 0.912068 -0.38826
-0.3453 5.660682 4.637553 3.826487 2.742895 1.923687 1.129808 0.330104
0.17635 4.742612 3.958893 4.28284 3.262435 2.269607 1.790752 1.179496
-l.0692 -0.57718 0.851064 2.963586 2.639851 2.108901 2.215243 1.845146
-1.99965 -1.74369 -0.45629 2.238882 2.159272 1.706011 1.167072 1.335689
-1.06292 -1.3266 1.650619 2.396154 2.377201 2.292263 1.866024 1.663835
-0.31017 0.407103 -0.07066 1.326799 1.25166 1.442647 1.471476 0.920408
#

#

# HYDROACOUSTIC SURVEY DATA- From 1998 An Integrated Acoustic and Trawl
Based #Prey Fish Assessment Strategy for Lake Michigan pg. 77

#

# Adult Alewife biomass (metric tons) by year(1993-1996)

13384 21695 45049 91333

# Young of year Alewife biomass (metric tons) by year 1993-1996

31027 13984 148394 8407

# Bloater biomass(metric ton) by year 1993-1996

476927 251196 239735 275387

Weightdat

# Data for alewife-bloater CAA Model by Emily B. Szalai
# Updated 1/11/2000

160

# WEIGHT AT AGE FOR PREY- based on trawl survey data

#Weight at age for predators based on SCOL H

# alewife

0.0002 0.01 0.02 0.032 0.04 0.047 0.054 0.07

# LENGTH AT AGE 19 FOR BLOATER FROM growth modeling 1965-1999
183.154 207.393 227.792 244.96 259.409 271.569 281.803 290.416 297.664
181.863 209.142 229.276 246.221 260.482 272.484 282.585 291.086 298.24
180.549 208.31 230.95 247.66 261.724 273.56 283.521 291.905 298.96
178.875 209.309 232.016 250.535 264.203 275.707 285.388 293.536 300.393
174.003 207.159 232.245 250.962 266.226 277.493 286.975 294.955 301.671
173.002 206.143 232.935 253.206 268.33 280.664 289.768 297.429 303.878
172.715 207.486 233.912 255.275 271.439 283.499 293.334 300.593 306.703
172.412 209.029 236.471 257.328 274.189 286.946 296.464 304.226 309.956
172.463 211.742 240.101 261.355 277.508 290.566 300.446 307.817 313.829
171.034 208.495 239.716 262.258 279.15] 291.99 302.37 310.223 316.082
171.245 208.422 238.025 262.699 280.512 293.862 304.008 312.211 318.417
171.758 216.258 244.095 266.261 284.735 298.073 308.069 315.666 321.808
167.25 217.362 250.499 271.228 287.734 301.491 311.424 318.868 324.525
162.381 207.06 246.19 272.066 288.252 301.141 311.883 319.639 325.452
159.889 202.348 237.341 267.988 288.254 300.932 311.026 319.44 325.514
157.051 201.522 234.365 261.434 285.14 300.817 310.623 318.432 324.94
153.963 195.734 230.786 256.674 278.01 296.696 309.052 316.781 322.936
149.623 190.085 223.537 251.607 272.339 289.425 304.389 314.284 320.474
143.816 182.755 215.836 243.186 266.136 283.086 297.056 309.29 317.381
139.267 173.675 206.291 234.001 256.909 276.133 290.331 302.032 312.28
136.998 171.833 200.198 227.087 249.93 268.816 284.664 296.368 306.015
134.988 171.934 200.187 223.193 245 263.528 278.845 291.699 301.191
134.357 169.674 199.647 222.568 241.232 258.924 273.955 286.382 296.81
136.181 165.951 195.108 219.855 238.778 254.187 268.794 281.203 291.463
138.103 169.535 193.67 217.309 237.371 252.713 265.206 277.048 287.109
137.331 164.114 190.766 211.231 231.275 248.287 261.295 271.888 281.93
137.605 165.025 187.419 209.703 226.815 243.574 257.798 268.675 277.532
141.012 163.345 186.511 205.431 224.258 238.715 252.875 264.892 274.082
141.637 165.178 184.136 203.802 219.864 235.847 248.119 260.139 270.341
140.379 156.531 177.842 195.005 212.808 227.348 241.817 252.927 263.808
140.595 167.846 181.177 198.767 212.933 227.627 239.628 251.57 260.74
144.081 166.212 188.992 200.136 214.841 226.682 238.965 248.997 258.98
146.142 168.091 186.725 205.906 215.289 227.669 237.639 247.982 256.428
147.157 171.864 190.038 205.467 221.349 229.118 239.369 247.624 256.188
148.192 172.63 193.095 208.149 220.928 234.083 240.518 249.009 255.847
#

# WEIGHT BY YEAR (1965-1996) FOR ALTERNATIVE PREY

# small (<90 mm)rainbow smelt (avgweight of lifestage(ls) 0 rs from trawl data
1966-1997)

0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527
0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527

16]

0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527 0.00527
0.00527 0.00527 0.00527 0.00527 0.00527

# large (>90 mm) rainbow smelt (avgweight of ls7 rs from trawl data 1966-1997)
0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709
0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709
0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709 0.0709

# slimy sculpin ( avg wt for all ls from trawl data 1965-1997)

0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009
0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009
0.009 0.009 0.009 0.009 0.009 0.009 0.009

# deep water sculpin ( avg wt for all ls from trawl data 1965-1997)

0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009
0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009
0.009 0.009 0.009 0.009 0.009 0.009 0.009

#

# WEIGHT AT AGE FOR PREDATORS- FROM SCOL H

# lake trout (1-11)

0.030437 0.218535 0.615904 1.19584 1.902372 2.676988 3.470269 4.245325 4.977302
6.196222 6.196222

# coho (1-3)

0.030 1.021 2.222603

# chinook (0-5)

0.0045 0.586 3.1756 8.1931 10.3675 13.119 13.119

0.0045 0.586 3.1756 8.1931 10.3675 13.119 13.119

0.0045 0.586 3.1756 8.1931 10.3675 13.119 13.119

0.0045 0.586 3.1756 7.4845 10.3675 13.119 13.119

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.1756 7.4845 10.3675 14.361 14.361

0.0045 0.586 3.0626 7.2001 10.1687 13.8155 13.8155

0.0045 0.586 2.9361 6.7659 9.5928 12.7807 12.7807

0.0045 0.586 2.8096 6.3317 9.017 12.0171 12.071

0.0045 0.586 2.6831 5.8975 8.4412 11.2534 11.2534

0.0045 0.586 2.5566 5.4633 7.8653 10.4898 10.4898

0.0045 0.586 2.4301 5.0291 7.2895 9.7261 9.7261

0.0045 0.586 2.3036 4.5949 6.7137 8.9625 8.9625

0.0045 0.586 1.9891 4.0832 5.9739 7.7668 7.7668

0.0045 0.586 2.0017 3.9099 5.7042 7.9687 7.9687

0.0045 0.586 2.0384 4.2497 5.969 9.1125 9.1125

162

0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586
0.0045 0.586

1.9483 4.2866 6.4195 9.6971 9.6971
2.1218 4.3643 6.7269 10.5566 10.5566
2.0189 4.6794 7.0007 11.2297 11.2297
2.4066 4.8601 7.3379 11.5067 11.5067
2.5202 5.2934 8.7143 15.625 15.625
1.8563 5.5349 8.8163 14.6839 14.6839
1.8129 4.7052 8.0439 11.6902 11.6902
2.3591 4.9034 7.5731 12.1888 12.1888
1.8079 5.0352 7.6961 12.0795 12.0795
1.5112 4.0172 6.4552 8.2757 8.2757
1.5112 4.0172 6.4552 8.2757 8.2757

# rainbow trout (1—6)

0.005 0.680 2.495 3.8102 5.443 5.443
# brown trout (1-6)

0.2 0.8 2.3 3.7 3.7 3.7
#

# LEN GTH-WEIGHT COEFFICIENTS (a b) FROM SIMPLE
# alewife

498 0.33

# weight-length params (ln(alpha) beta) for bloater by year from growth modeling
-14092 3.44337
-l40801 3.44566
-14.0683 3.44794
-14.0521 3.45109
-14.045 3.4524]
-l4.0424 3.45279
-14.0438 3.45233
-14.0441 3.45204
- 14.0481 3.45099
- 14.0458 3.4512

- 14.042 3.4517
-14.0386 3.45212
-14.0341 3.45279
-14.0317 3.45303
-l4.0331 3.45242
-l40387 3.45091
-14.0532 3.44762
-l4.0689 3.44408
-14.0849 3.44048
-l4.0995 3.43708
-l40989 3.43658
-14.0953 3.43669
-14.086 343793
-14.0725 3.44009
-l40657 3.44105
-14.0629 3.44133

163

-l40682 3.4401
-14.0748 3.43876
-14.0841 3.43691
-l4.0982 3.43421
~141144 3.43109
-14.1218 3.42968
-14.1253 3.42901
-14.1247 3.42914
-14.124 3.42926
# rainbow smelt

520.7 0.396

# slimy sculpin

300 0.33

# deep water sculpin
300 0.33

# lake trout

467.2969 0.3195

# coho

481.04 0.31

# chinook

481.7 0.31

# rainbow trout

459.68 0.366

# brown trout

395.65 0.389

164

Appendix C
Methods used to Estimate the Parameters of the Lake Michigan Stock Assessment Model
for Chinook Salmon.

This appendix was written by Jim Bence (Michigan State University) to document
the analysis that he completed to describe the relationship between chinook salmon
growth and mortality events. These models were included in the decision model
described in Chapter 3 and I have included this appendix to provide further details on the
chinook salmon mortality models.

Here we first describe the approach to estimating the parameters of an age
structured stock assessment for chinook salmon in Lake Michigan. This was an updated
version of the model developed by Benjamin and Bence (in press a), which we used to
produce the estimates of abundance and consumption for Madenjian et al. (2002). We
then describe how we used results from that assessment model to explore the relationship
between time-varying natural mortality and individual chinook salmon size.

Estimation of stock assessment parameters

The equations defining the age structured stock assessment model are in Table 19,
and the symbols used in those equations are defined in Table 20. This is the same model
described in Appendix A. Here we have added details and repeated information needed
to understand the process of fitting the model to observed data. We do not repeat the
rationale underlying the model or all sources and values for assumed constants.

The system model
The model recognizes two time periods, with natural mortality operating only

during those periods, and numbers at the beginning of the year (and period 1) are given

165

by eq. T191. Between those two periods fishing mortality and mortality associated with
maturation occur as pulses in that order. Predictions of the number present at the end of
the first period before pulses of mortality (eq. T192), after the fishing pulse (eq. T193)
but before the maturation pulse, and after the maturation pulse at the start of the second
period (eq. T194) are used to either predict observed quantities or in calculations of
consumption (Appendix A).

Natural mortality was modeled as a background rate plus a time varying
component (eq. T195), and the time-varying component was modeled as the product of
an age and year specific effect (eq. T196). The background rates were assumed known.
Prior to 1985 and after 1997, the time varying component was assumed to be zero. For
1985 through 1996, the year specific effects were estimated. Separate age specific effects
were estimated for ages 0, 1, 2. Ages 3, 4 and 5 were assumed to have the same age

effect and this was fixed as a reference baseline (y = 1.0) and not estimated.

The proportion of fish surviving the fishing pulse is exp(-Fa,y) (eq. T197), where

F a.y is the product of age-specific vulnerability and year specific fishing intensity (eq.

T198). Vulnerability was estimated for ages 0, 1 and 2. Vulnerability for ages 3 through
5 was assumed equal and fixed to a reference baseline of 1.0.

For 1985 through 1999, fishing intensity was assumed to be proportional to
observed fishing effort up to an estimated multiplicative error (eq. T198). For 1967 to
1984, fishing intensity was assumed to increase linearly from zero to the 1985 level
estimated by the model.

From 1985 through 1999, proportion of mature chinook salmon at age was

assumed to be follow a logistic function of weight-at-age during the summer- fall (eq.

166

T199) for ages 1 through 4. Proportion mature at age 0 was assumed to be zero and

maturity at age 5 was assumed to be 1.0. For years prior to 1981, a constant maturity

schedule was used based on results reported by Stewart (1980): (Pmfl = 0, 0.12, 0.33, 1.0,

1.0, 1.0 for ages 0 through 5 respectively). From 1981 through 1984, the maturity at each
age was linearly interpolated between the value assumed for 1980 and the value
estimated for 1985.

The initial numbers at age in 1967 and the numbers of fish that recruited each
year at age-0 were assumed known based on stocking records and estimates of the
numbers of wild fish in the system.

The observation model

The model used observed fishery information to estimate the dynamics of the
population. This information included: (1) annual total harvest from 1985-1999, (2) age-
frequency compositions (proportions at age) of the annual total harvest from 1985-1997,
(3) age-frequency compositions of the annual total harvest of mature fish from 1985-
1997, (4) age-frequency compositions of the annual weir harvest of fish captured during
the spawning run from 1985-1996, and (5) annual fishing effort directed at chinook
salmon from 1985-1999. Sources of these data are described by Benjamin and Bence (in
press a) and in Appendix A. A Bayesian approach was used in estimating parameters,
taking the parameters associated with the highest posterior density as point estimates
(Schnute 1994). AD Model Builder Software (version 6.02) was used to estimate
parameters using a quasi-Newton method based on derivatives obtained by automatic

differentiation (see “The objective function” below).

167

Fishery catch-at-age was predicted by eq. T1911, and total annual harvest and
proportions at age were calculated from these, for comparison with observed quantities.
The catch-at-age of mature fish in the fishery harvest is given by eq. T1912, and
associated proportions were calculated from these for comparison with observations. The
numbers at age in the spawning run were predicted by eq. T1913, and proportions at age
calculated from these were compared with observed age-frequency compositions of the
annual weir harvest for 1985 through 1996.

Strictly speaking, the observed effort was not treated as data to which predictions
were compared. Instead fishing intensity was assumed proportional to observed effort up
to multiplicative error and these errors were estimated. These errors derive both from
measurement error associated with observations of effort and process error because

catchability will actually vary over time. Deviations of fishing intensity from direct

proportionality were penalized during the model fitting process. Thus, in effect, qu is a

prior estimate for fy (see “The objective function” below).

The objective function

Above we defined a system and observation model and we adjust the estimated
parameters of these models to obtain the best fit to the data. This best fit is defined by the
the minimum value of the negative log posterior density, which is our objective function.

The adjustable parameters are the fishery catchability (q), fishery vulnerabilities (S0, S 1,

S 2), fishery effort deviations ( f), for years 1985 through 1999), parameters associated

with time-varying natural mortality including age-effects (xla for ages 0 through 3) and

168

year effects (y), for years 1985 through 1996), and parameters associated with maturity at
age (,60‘“ and 6’0,“ for ages 1 through 4).

The posterior density function is proportional to the product of the likelihood
given the data and the prior density of the parameters. We minimized the negative log of
this density, (which we sometimes call the negative log-likelihood), which can be
expressed as sum of components (eq. T1914). In our application all the priors were
bounded uniforms on a log-scale, except for those associated with the effort deviations
(which were lognormal priors). As a result we drop the priors for those parameters with
uniform prior from the likelihood equations, since these priors are constant within the
bounds. The priors are still implicit, and were implemented by using bounded estimation
for these parameters in the Ad Model Builder software. All parameters were estimated
on a log-scale. Bounds were set widely enough apart so they had little influence on the
solution when parameters converged to a solution within the bounds. Bounds were used
to ensure a proper posterior density function and to avoid the model becoming stuck at
implausible solutions that did not maximize the posterior density during the fitting
process.

Total annual harvest was assumed to follow a lognormal distribution with an
assumed known dispersion parameter (eq. T1915). The proportions-at-age were
assumed to behave as though calculated from multinornial samples (eq. T1916) with
effective sample sizes equal to either the actual number of age determinations or data
type specific maximum number intended to avoid over-weightin g observed age

compositions based on large samples (Foumier and Archibald 1982). The deviations

169

about the assumed direct proportionality between fishing intensity and effort were
assumed to be lognormal with a known dispersion parameter (eq. T1917).

The effective sample sizes (the n’s in equation T1916) and the dispersion

parameters (0% and 0;) act to weight how important the various data and priors are

considered during the fitting process. These were set at 0.08 and 0.04 for 0'62. and 0'; and

100, 50 and 50 for the effective sample sizes for the harvest, mature harvest and weir
harvest. The dispersion parameter for the harvest was set based on the average of
coefficient of variation estimates for the Michigan creel surveys. The other values were
chosen after a sequence of repeated fits of the model in an attempt to avoid strong
patterns in the residuals, and achieve a match between the specified dispersion values for
the effort deviations and for annual recreational catch, and ones calculated based on the
estimated effort deviations or log-scale differences between observed and predicted
harvest, after the model had converged to a solution.
The mortality versus individual size models

The assessment model produced estimates of “age effects” and “year effects” for
time varying natural mortality. The age-effects were quite small for ages 0 and 1 (about
0.007), and quite high for age-2 (12.9) relative to the reference value for age-3 and older
(1.0). The year effect was estimated to be quite small in 1985 and for this analysis we
assume that year effects on a log-scale were at the lower bound used during estimation
for 1983 and 1984.

The implied trends in estimated natural mortality together with weight of age-3
fish at annulus formation are given in Figure 30. While the relationship between natural

mortality and size is not clear cut in this figure or in other exploratory ones using

170

different measures of fish size, it is clear that mortality increased after a period of low
growth and recovered after a period when growth had increased above that during the
slowest growth period. Another way of looking at these data is to consider how changes
in mortality are related to size achieved by the start of age-3 (Figure 31). This suggests
there is some inertia to mortality (it tends to be similar to that observed in the previous
year), and that it tends to decrease when larger sizes are in the process of being achieved.
The subsequent analysis of the relationship between natural mortality and growth
conditions is based on the year effects, and takes into account the above observations.
We developed two closely related models that attempt to describe the growth
effect and “inertia”. In the first of these models the year effect is assumed to be a
deterministic function of the year effect observed at the previous time step and size

attained at annulus formation by age-3:

“IO/y) : 'u+pln0/y-1)_flln(WANN,3,y+l) (1)

The second of these models differs only in allowing for process error:

ln(yy) = ,u + ,oln(yy_1) — ,Bln(WANN,3,y+1) + (By (2)

These two models were fit to the “observed” year effects (estimated previously in the
stock assessment model). In the first of these models, all variations between predicted
and observed were treated as measurement errors. In the second, all deviations between
observed and predicted were treated as process errors. Errors on the log-scale of eqs. 1
and 2 were assumed normally distributed and point estimates were obtained by

minimizing the negative concentrated likelihood by altering u, lnp, and lnB:

3
— log(conc) = 51042 (In 7), — 1n yy )2] ( )
y

171

where k is the number of years involved in the comparison of observed and predicted
values. Both p and B were assumed to only take positive values and were estimated on
the log-scale. All three parameters were restricted within bounds: u (-50, 50), lnp (-
23,0), and lnB (-10,10). By specifying bounds we are assuming a prior uniform
distribution for these parameters on their respective scales, so that all values within the
bounds are assumed a priori equally likely and all values outside the bounds are assumed
impossible. Using such bounds ensures that when a Bayesian posterior distribution is
determined it will be a proper one. The bounds for u and lnB were arbitrary chosen as
large and small values below or above which parameter values would be quite unlikely.
The bounds for lnp correspond to values of 0.1 and 1.0 for p. Initially a much smaller
value for the lower bound for lnp was used, but in the case of the process error model, the
resulting MCMC chains were quite sticky. When low values for p were visited the chain
remained in the vicinity of these low values for long sequences even though these
combinations of parameters had low likelihood values. The point estimate for 02 (which

is concentrated out of the likelihood) was obtained as:

02:20:17,011ny (4)
k

 

Here (52 represents either the measurement error variance or the process error variance
depending on whether the model of eq. 1 or eq. 2 is used. Point estimates for all the

parameter estimates are in Table 21.

172

Note that the measurement error model essentially assumes that next year’s time-
varying mortality will be the same as this year’s with an adjustment for growth. The
process error model led to very uncertain estimates of effects due to history and growth
and random year to year variation plays a substantial role. We decided that the
measurement error model was implausible, since it seemed unlikely to us that most of the
variation between observed and predicted mortality came about because of poor estimates
of mortality. The process error model was therefore pursued further.

Posterior distributions for the parameters of the process error model were
estimated by Markov Chain Monte Carlo (MCMC) methods. A chain of 20 million was
run, with output saved every 2000 steps, leading to 10,000 saved samples. Because of
the use of concentrated likelihood the output of the chain for the process error variance
had to be post-processed. This was done following the same two-stage procedure used in
Chapter 2 for variance about a stock-recruitment model. The estimate of variance
originally calculated for each set of parameters is the most likely variance estimate given
those parameters. This was replaced in each sample in the chain by a single draw from a
scaled inverse chi-square distribution, with degrees of freedom equal to v-3, where v is
the number of observations used in fitting the mortality model. Calling the result of eq. 4

for the 1"” sample MSEi:

2 V ' M SE1 (5)
0} = ——

Xi

where X, was generated from a Chi-square distribution with v degrees of freedom.

In general the resulting chains were well behaved and appeared to provide

reasonable estimates of the posterior distributions. Autocorrelations fell to near zero after

less than 50 steps for all four parameters, and all effective sample sizes exceeded 500.

173

The trace plots did not reveal long-term correlations. See Chapter 2 for further
explanation of these diagnostics.

The marginal posterior distributions for p, B, and o are shown in Figures 32
through 34. These distributions indicate that there is a substantial probability that growth
in fact has relatively little impact on mortality and that the observed patterns could reflect
chance events associated with process errors, combined with inertia in the system.

Our above analysis does not fully take into account observed patterns in growth
and mortality. In particular there is at least anecdotal information suggesting that natural
mortality did not become high, as it was during the late 1980s, from 1967 through 1982,
and during those years chinook size at age was generally larger (e.g., Stewart et al. 1981).
We developed two alternative mortality models based on this information and making
stronger assumptions regarding the relationship between mortality and individual size and
for inertia with respect to mortality. The premise of these models is that the system
consists of two states for each age, a high natural mortality state and a low natural
mortality state. In the first of these alternative models it is assumed that after the high
mortality state is first entered the system must stay in that state for five years before
further transitions are possible. When such transitions are possible, the transition
probability from a high mortality state to a low mortality state increases with increases in

weight at annulus formation at the next age in the next year following a logistic function:

1 (6)
1+ CXp(—b1,a(wann,a+l,y — bzfl)

 

Pa,y,H—>L :

174

The transition probability from a low mortality state to a high mortality state is the
complement of that given by equation 6. The parameter values were chosen by trial and
error so that in Monte Carlo simulations using the observed/assumed weight-at-age data
from 1980 through 1999, an average of about seven high mortality years occurred. These
parameters produce a substantial increase in the probability of making the transition from
high to low over the range of observed weight-at-age, as is illustrated for age 3 (Figure
35).

The second alternative model resembles the first, in that the transition from low to
high mortality is given by equation 6. However, to mimic patterns like the observed
mortality sequence, the probability of transitions from the low to high mortality state can
no longer be the complement of the probability of a transition from the high to low
mortality states. In this model transition from high to low mortality occurs
deterministically when “growth is good enough”. Thus the transition probability is 1.0

when weight at annulus formation for the next age in the next year exceeded a threshold

War The same values of b] and b2 were used as in the first alternative model and Wax

was again chosen by trial and error to produce on average of about 7 years of high
mortality in Monte Carlo simulations using the observed sequence of weight-at-age. For
these parameters there is a non-negligible probability of transition to high mortality even
when above the threshold, but not a certain transition to bad even when growth is
substantially below the threshold. This allows the mortality to stay low even following
the occasional poor growth year as we have seen. The model even can allow no mortality
event to occur given the observed growth sequence, although this is relatively rare given

the current parameter values.

175

We note that these two alternative models are not statistically derived models, and

we do not provide uncertainty estimates for b 1’s and bz’s (or for the second alternative

Wags). These models cannot attempt to mimic year-to-year quantitative variation in

mortality rates, but do reproduce periods of high mortality following periods of slow
chinook salmon growth. In applying these models to produce actual mortality rates, it
would be reasonable to assume a zero or near zero year effect for low mortality state
years and a value equal to the average for 1986-1992 for high mortality years.
Calculating this average on a log-scale yields 7 = 0.0986, which corresponds to a MTVMJJ
= 1.27.

It is possible given these models for salmon to achieve unreasonably low sizes
even when mortality has not become very high. This is rare for the two alternative
models but is not infrequent for the model described by eq. 2. To ameliorate this, all

three models are augmented by the rule that when ln(y), as determined by any of the

above models is less than 5.0, ln(y) is set to 5.0 with probability:

 

(7)
P 1

. . 5 =
a y 7—9 1+ exp(-b1(b2 " WHAR,3 ))

The constants b, and 122 were set to 6 and 4.15 respectively as follows. As mean weight
in fall for age-3 falls below 4.9, if there is no ongoing mortality event, we assumed that
the probability of a mortality event increases. We assumed that the probability of having
this mortality event (when otherwise the model did not have one) was near zero for

weights just below 4.9, but increased to near 1.0 as the mean length approached a value

176

for which 95% of the age-3 fish in the fall were larger than in every year. We looked at
distributions of Michigan creel caught chinook salmon of age-3 for months starting in

August. by year. The lowest 5th percentile was 3.4kg (in 1986).

177

Table 19. Equations defining the Lake Michigan stock assessment model.

 

System Dynamics Model

-M
Na+l,y+l : Na.ye a,y(1_ PF,a,y)(l" Pm,a,y)

-7/12M .

+ __ a,
Na,y.l ‘ Na.ye )
N‘ —N+ 1 P

a.y.2_ a,y.l( _ F.a.y)
N0.ya2 = Na,y»2(1_ Pm'a’y)

Ma,y : Ma + MTVM,a,y

MTVM,a,y : lolly

PM”, =1— exp(—Fa.).)
Far 2 Safy

f, -—- (15.4,. 6 ~ Lil/(0,02 )

l
Pmfl.) " 1+ e-(fl0,a+fl1,aWHAR,a,y)

 

Observation Model
Cm, = N;,,,,, (1 — exp(—Fa,y))
cm“, = Nfa,},(1—exp(—Fa’y))Pm,a’y
Nsww : szwpmflay

(T191)

(T192)

(T193)

(T194)

(T195)

(T196)

(T197)

(T198)

(T199)

(T1910)

(T1911)

(T1912)

(T1913)

 

178

Table 19. cont.

 

Objective Function

5
— LOgL = ,2 — L,- + Ignored Constant (T19- 14)

(=1

L1 = —1-2-2(1n 5‘, —In C).)2 (T1915)
20C “

For i =2, 3 and 4 (age compositions for harvest, mature harvested fish, weir return fish),

with subscripts to distinguish the three data types dropped on the right hand side:

L,- =Zny2fia,yln(pa,y) (T1916)
y a

1 2 (T1917)
= ——Zln
L5 20? (9‘)

 

179

Table 20. Definition of symbols used in equations for chinook salmon stock assessment

 

model.
Symbol Description
N (Ly Abundance-at-age at start of year
N+ Abundance-at-age at end of first period
a, y,1
N — Abundance-at-age after fishing pulse
a, y.2
N Abundance-at-age after spawning pulse (start of 2nd period)
a. y,2
M Instantaneous natural mortality rate
a . V
Ma Constant age-specific portion of natural mortality
M Time varying component of natural mortality
TVM ,a,y
4 Age effect for time varying natural mortality component
(1
7 Year effect for time varying natural mortality component
y
Proportion of fish that die during fishing pulse
PF ,a,y
F Fishing “rate” determining the proportion dieing during
dry fishing pulse
Fishery vulnerability
SCI
f Fishing intensity
y
q Fishery catchability
Observed fishing effort
EY
é: Fishery effort deviation
y
P Proportion of fish that mature
m,a, y
fl Intercept parameter for age-specific logistic maturation
0’” function
fl Slope parameter for age-specific logistic maturation
l’a function
W Observed weight in late summer/fall (time of harvest and
HAR,a J maturation)
Cm Predicted catch by age and year
C Predicted catch of mature fish by age and year
"1.0 , V

 

180

Table 20, cont.

Symbol

may

Oz

Pa.)

pa,y

[‘1‘

a...

Q q
WNQN

.3

Description

Predicted number of mature fish in
spawning runs by age and year
Predicted annual recreational harvest

Observed annual recreational harvest

Predicted proportion (for the year) of catch.
These quantities are defined for
recreational harvest, mature recreational
harvest, and weir catch
Observed proportion (for the year) of catch.
These
quantities are defined for recreational
harvest, mature recreational harvest, and
weir catch
Total posterior density

Log of likelihood or prior component
Dispersion parameter for recreational

harvest
Dispersion parameter for effort deviations

Effective sample size (one defined for each
year for recreational, mature recreational
and weir caught fish).

18]

Table 21. Parameter estimates and their asymptotic standard errors for the models

described by equations 1 (measurement error model) and 2 (process error model).

 

Parameter estimates for the measurement error model

Asymptotic correlation matrix

parameter estimate asymptotic lnp Inf} )1 02
SE
lnp -9.07E-08 1 .375-04 1
Int?» 2.56E+OO 1.73501 0.0019 1
(1 2.00501 3.37500 0.0021 0.9998 1
62 4.10500 3.38504 -1 0.0019 0.0021 1

 

 

Parameter estimates for the process error model

Asymptotic correlation matrix

parameter estimate asymptotic SE lnp mp 11 02

lnp -2.62E-01 2.76501 1

Int) 6.38E-01 3.40500 0.6001 1

,1 1.85E+OO 1.045+01 0.656 0.9961 1

02 3.78E+00 4.48507 0.7995 0.123 0.1685 1

 

182

 

._...vv._|p.nn

 

 

 

 

 

 

 

715 2
7 . 1+ 18 A
1+ 1.6 E,
'3 Eis‘ «~-L4 g;
3' 6‘ 1J2j§
j§ 555 ~1 g
j; 5 hi}: ii
0 - . h
3“15 »(14 i;
4 ~t12
305 I F V 0
1983 1988 1993
Year

Figure 30. Temporal patterns in weight at annulus formation for age 3 (W, diamonds)

and model estimated natural mortality rate at age-2 (M, squares).

183

2.5 -

 

 

2- .
5 :9. 1.5-
8:; 1-
,E 0
g 3 04 O . o O o o
5 i -0.5- . Q
-11 . Q
'15 U I I I I
1.3 1.4 1.5 1.6 1.7 1.8

Log Weight at annulus formation age~3 in year y+1

Figure 31. Relationship between changes in mortality at age-2 from year y-l to year y

and weight at annulus formation in year y+l.

184

 

 

 

 

2 .
A
1.5 -
O.
v 1 .
q—
O.5 ‘
0 I U I I I
O 0.2 0.4 0.6 0.8 1

 

 

 

Figure 32. Estimated posterior density for p, which is the effect of last year’s natural

mortality year effect on this year’s year effect (eq. 2).

185

 

 

0.1 -

.09 -
0.0 -
0.07 ..
0.06 a
0.05 -
0.04 -
0.03 '-
0.02 -
0.01 -

fUOQB)

 

 

 

I09 B

 

 

 

 

Figure 33. Estimated posterior density for B, the effect of age-3 weight at time of annulus

formation in year y+l on the year effect for natural mortality in year y (eq. 2).

186

 

0.25 -

 

 

 

0.21
A
N 0.15 -
U
z: 0.1 -
0.05 -
O I I f I U
0 2 4 6 8 10
! 02

Figure 34. The estimated posterior density for 02, the variance for the process errors (8)

in equation 2.

187

 

.0
A
a

 

Transition probability (high to low)
9 o
N O)

 

O

 

T i V U

4.5 5 5.5 6
Weight at age 3

9°
01
.p.

 

 

Figure 35. Probability of transition from high to low mortality regime for age-2 predicted

for a given weight at age 3 (annulus formation) by equation 6, with b, = 3.0 and b2 = 4.0

188

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