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THESIS
2

20%?

This is to certify that the

dissertation entitled

Modeling Yellow Perch (Perca flavescens) Abundance in
Inner Saginaw Bay, Lake Huron, 1971-2001: The
Importance of Density Independent and
Density Dependent Processes

presented by

 

Sarah Ann Thayer

has been accepted towards fulfillment
of the requirements for

 

 

Ph.D. degmin Fish. 5. Wildl.

magma
Major professor

Date November 25, 2002

 

MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

LIBRARY
Michigan State
University

 

 

 

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.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIRC/DateDue.p65«p. 15

 

MODELING YELLOW PERCH (PERCA FLA VESCENS) ABUNDANCE IN INNER
SAGINAW BAY, LAKE HURON, 1971-2001: THE IMPORTANCE OF DENSITY
INDEPENDENT AND DENSITY DEPENDENT PROCESSES
By

Sarah Ann Thayer

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

2002

ABSTRACT
MODELING YELLOW PERCH (PERCA FLA VESCENS) ABUNDANCE IN INNER
SAGINAW BAY, LAKE HURON, 1971-2001: THE IMPORTANCE OF DENSITY
INDEPENDENT AND DENSITY DEPENDENT PROCESSES
By
Sarah Ann Thayer
Yellow perch (Percaflavescens) in Inner Saginaw Bay, Lake Huron, have

exhibited wide variations in total abundance over the last thirty years. Abundance was
low in the 19703, high in the 19803, and low in the 19903. Abundance increased
gradually from the late 19703 to the early 19803, but decreased abruptly at the end of the
19803. Fifteen competing hypotheses involving density independent survival, density
dependent survival, and/or density dependent growth from age 0 through age 5 were
modeled and evaluated using a method that minimized residual summed Squares between
observed and modeled catch per unit effort at age, 1971-2001. Model strength was
measured using Akaike Information Criteria and Akaike weights. The top-ranking
models showed the following processes to be important in determining the abundance of
this yellow perch population: (1) compensatory survival at age 2-4, (2) compensatory
survival at age 0, and (3) compensatory survival at age 1; however, a model with constant
survival and growth rates ranked the highest. Compensatory survival at age 2—4 ranked
second and is supported by other studies; however, compensatory survival at age 1 is best
supported by auxiliary data in this study. In addition, the modeling process itself
indicated that the dramatic drop in abundance in 1991 could only be accurately simulated
if all age groups experienced a Significant reduction in survival, suggesting the shifi in

abundance at this time was caused by a mortality event that affected all age groups. The

top four models also predict total abundance to gradually increase in the future, an

increase that could be hastened with a reduction in fishing mortality.

This work is dedicated to Robert C. Haas, fisheries research biologist at the Michigan
Department of Natural Resources Lake St. Clair Great Lakes Fisheries Research Station,
who has devoted a large part of his career and life to studying yellow perch in Inner
Saginaw Bay.

iv

ACKNOWLEDGMENTS

I would like to acknowledge the following people and organizations for the
financial, academic, professional, and personal support they provided me during the
course of my work. I am thankful to Michigan State University and the Saginaw Bay
Walleye Club, for their financial support. Dr. William W. Taylor, my major advisor, was
a wonderful mentor. He was very supportive of my endeavors and provided me with
wonderful advice and personal support. He was also instrumental in helping me find
financial support. I am also very thankful to Dr. Daniel B. Hayes, who devoted an
enormous amount of time helping me with model formulation and interpretation. Aside
from his beneficial input and suggestions, he always made a point to set aside time to
work with me, and his office was always open to me (and my children). I thank my
committee members, Dr. Gary Mittelbach and Dr. Richard Merritt, for the advice they
provided me while I was in the process of formulating my research goals and objectives.
They steered me in the right direction when they encouraged me to narrow down my
original lofty goals so that I would have the time to produce a thorough analysis, which is
exactly what I did. Outside the University, M.D.N.R. fisheries research biologist, Robert
C. Haas, made enormous contributions to this work by helping me convert the historical
computer data files of the late John Weber into formats that we could use with modern
computer software. I could not have done this without him. I thank him, Mike Thomas,
and the Staff at the M.D.N.R. Lake St. Clair Fisheries Research Station, Alpena Fisheries
Research Station and Bay City District office for the assistance they provided during data
collections, and for all the contributions they made to create the data files I used in these

models. I would also like to acknowledge Saginaw Bay commercial fishermen William

Lentz and Forrest Williams, who allowed me to collect data at their businesses. They
were always very accommodating, very knowledgeable of Saginaw Bay yellow perch,
and always concerned about the fiiture of Saginaw Bay yellow perch. Finally, I would
like to acknowledge my family and friends for the personal support they provided me
during the course of my work. My husband, George Thayer, provided me with loving
encouragement. I was happy to have my son, George, join me during many field data
collections. My daughter, Erika, and son, Reeve, who were born during the course of my
work, had the unique experience of sharing my thoughts and experiences in utero. I
thank my doctor, Dr. Bartol Biocic, for taking such good care of me during my
pregnancies and deliveries so that I was able to maintain my stamina and complete this
work. I thank my wonderful babysitters, Faye Pearce and Korine Shuler, for taking such
good care of my children when I was at school. And, last, but not least, I thank my
neighbors and friends, the Tesners, for listening to me talk about fish “models” for many
years, and often letting me use their kitchen as an office (i.e., I could never do “serious”

work in my own house).

vi

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vii
LIST OF FIGURES ................................................................................ ix
INTRODUCTION ................................................................................... 1
METHODS
Observed Data ................................................................................. 4
Hypothesis Formulation ............................................................................. 7
Population Model and Parameter Estimation ............................................... 9
The Population Model ................................................................... 9
Survival Rate Parameter Estimates ................................................... 12
Growth Rate Parameter Estimates .................................................... 15
Model Selection ............................................................................... 18
Auxiliary Data .................................................................................. 20
RESULTS AND DISCUSSION
Observed Abundance ......................................................................... 21
Model Evaluation .............................................................................. 26
The Null Hypothesis ......................................................................... 27
Age 0 Survival Hypotheses ............................................................ 27
Age 1 Hypotheses ...................................................................... 41
Age 2-4 Hypotheses .................................................................... 50
A Multi-age Hypothesis ................................................................ 54
Hypothesis Ranking ........................................................................... 55
Auxiliary Data ................................................................................. 59
DISCUSSION ........................................................................................ 70
APPENDICES ....................................................................................... 80
BIBLIOGRAPHY ................................................................................... 87

vii

Table 1.

Table 2.

Table 3.

Table 4.

LIST OF TABLES

Alternative hypotheses, by age, of the processes that may have

impacted overall abundance of yellow perch in Inner Saginaw Bay,

Lake Huron, 1971-2001. Each hypothesis is incorporated in one or

more models, indicated by model number ....................................... 8

Constant values used in models, as calculated from observed
female yellow perch data, 1971-2001 ........................................... 11

Models and their fitness components, for which the smallest RSS,
Like Comp, Par Comp, and AICc = the best fitting model .................. 28

The ten best models based on AICc criteria (A) and residual
Sum of squares (B) ................................................................ 57

viii

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

LIST OF FIGURES

Inner and Outer Saginaw Bay, Lake Huron, separated by an imaginary
line between Point Lookout and Sand Point .................................... 5

Temporal changes in total CPUE and age 1 yellow perch in Inner
Saginaw Bay, Lake Huron. ...................................................... 23

Temporal changes in CPUE of age 2, age 3 and age 4 yellow perch in
Inner Saginaw Bay, Lake Huron. ............................................... 24

Temporal changes in CPUE of age 5 and age 6 yellow perch in
Inner Saginaw Bay, Lake Huron ................................................ 25

Temporal changes in CPUE of age 7 and age 8 yellow perch in
Inner Saginaw Bay, Lake Huron ................................................ 26

Model 1: Constant age 0 survival ............................................... 29

Model 2: Variable age 0 survival (A) and survival rate from age 0 to
age 1 versus egg number (B) ..................................................... 30

Model 3: Free fitting age 0 survival every year with a
perturbation (A) and Model 3 survival rates versus egg density (B) ....... 33

Model 4: Constant age 0 survival with a perturbation in 1991 ............. 34

Density dependent survival of age 0 yellow perch without log
transformation of egg number (A), Model 5: density dependent survival
of age 0 yellow perch, based on modeled, log transformed egg number,
and Model 6: density dependent survival of age 0 yellow perch, based

on observed, log transformed egg number (C) ................................ 36

Model 7: density dependent survival of age 0 yellow perch based on
modeled egg number, with a transformation (A) and Model 8: density
dependent survival of age 0 yellow perch based on observed egg

number, with a perturbation (B) .................................................. 37

Model 9: Compensatory growth of age 0 yellow perch based on

modeled density (A), Model 10: Compensatory grth of age 0

yellow perch based on modeled density with a perturbation (B), and
comparison of modeled and observed mean length at age from

Model 9 ............................................................................. 39

ix

Figure 13.

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Figure 24.

Figure 25.

Figure 26.

Figure 27.

Model 11: Age 0 compensatory survival and growth, with a
perturbation ......................................................................... 40

Model 12: Age 0 depensatory survival based on observed density (A),
and Model 13: Age 0 depensatory survival based on observed density
with a perturbation (B) ............................................................ 42

Model 14 and 15: Age 1 compensatory survival based on modeled
density (A), and observed density (B) .......................................... 43

Model 16 and 17: Age 1 compensatory survival with a 3 year time
delay based on modeled density (A) and observed density (B) ............. 45

Model 18 and 19: Age 1 compensatory survival based on observed
density with a 5 year time delay (A) and a 7 year time delay (B) .......... 46

Model 20: Age 1 compensatory survival, based on observed density,
with a 5-year time delay and a perturbation .................................... 47

Model 21: Age 1 compensatory growth based on modeled density (A)
and Model 22: Age 1 compensatory growth based on modeled density

with a perturbation (B) ............................................................ 48
Model 23: Age 1 compensatory survival, with a 5-year time delay,

combined with compensatory growth and a perturbation .................... 49
Model 24: Depensatory survival at age 1 with a perturbation .............. 50

Model 25: Age 2-4- compensatory survival based on modeled density
with a perturbation (A) and Model 26: Age 2-4 compensatory survival
based on modeled density with a perturbation and 5-year time

delay (B) ............................................................................ 52
Model 27: Age 2-4 compensatory grth based on observed density,

with a perturbation ................................................................. 53
Model 28: Age 2-4 compensatory grth and survival with a

perturbation ......................................................................... 54
Model 29: Age 1 compensatory survival with a 5-year time delay
combined with age 2-4 compensatory survival and a perturbation. . . . . ....55
Age 1 and age 2 CPUE from top four models ................................. 60
Age 3 and age 4 CPUE from top four models ................................. 61

Figure 28. Relative instantaneous mortality rate (Z) versus CPUE of age 1 (A)
and age 2 (B) female yellow perch from the observed data. Age 1

CPUE = both sexes*0.5 ........................................................... 62
Figure 29. Relative instantaneous mortality rate (Z) versus CPUE of age 3 (A)

and age 4 (B) female yellow perch from observed data ...................... 63
Figure 30. Decadal changes in length increments of age 1-6 yellow perch (B)

from Saginaw Bay .................................................................. 64
Figure 31. Annual changes in grth increment of female yellow perch, age 0

to age 1 (A) and age 1 to age (B) ................................................ 66
Figure 32. Annual changes in grth increment of female yellow perch, age 2

to age 3 (A), age 3 to age 4 (B), and age 4 to 5 (C) ........................... 67
Figure 33. Observed female length increments from age 0 to age 1 versus egg

estimates (A) and age 1 to age 2 versus observed female CPUE (B) ...... 68

Figure 34. Observed female length increments from age 2 to age 3 (A), age 3
to age 4 (B), and age 4 to age 5 (C) versus observed female CPUE ....... 69

Figure 35. Estimated age 0 survival rates from the free-fitting model with a
perturbation (Model 3) ............................................................ 75
Figure 36. Projected Inner Saginaw Bay yellow perch abundance based on

Model 4 (Constant age 0 survival rate with a perturbation) and

Model 8 (Age 0 compensatory survival with a perturbation) (A), and
Model 25 (Age 2-4 compensatory survival with a perturbation) and

Model 20 (Age 1 compensatory survival with a 5-year time delay

and a perturbation) (B) ............................................................ 78

xi

INTRODUCTION

Fisheries management seeks to produce sustainable, optimal yields of fish
(Roedel 1975, Levin 1993). To do this, it is often necessary to understand how changes
in birth, survival, and individual grth rates lead to observed fish abundance. Birth,
survival, and grth rates affect fish abundance by changing how many fish are born
each year, how many survive, and how well they grow (which affect birth and/or survival
rates). These rates may or may not be affected by fish density. Rate changes are
independent of fish density if they are caused by mechanisms such as weather and
physical lake processes that are not casually related to fish density. In this case, the rates
may change randomly within some upper and lower bounds. Rate changes are dependent
on fish density if they are caused by mechanisms such as competition, predation, and
mating success, to name a few (Goodyear 1980, Hilborn and Walters 1992, Rose et al.
2001) that vary predictably with fish density. These are the types of mechanisms that are
thought to “regulate” a population around some “equilibrium density”, thus preventing
fish density from growing without an upper bound or inevitably declining to zero. Birth,
survival and growth rates can be compensatory (e. g., survival rate increases as density
decreases) or depensatory (e.g., survival rate increases as density increases) (Goodyear
1980, Hilbom and Walters 1992, Rose et al. 2001). Compensatory rates change
according to fish density by either causing the population to grow or shrink towards some
upper and lower bounds. Recognition of how changes in density independent and
density dependent rates can impact population size can help fisheries managers determine
the ways in which a fish population can be enhanced, as well as make better predictions

of future yield.

In general, most animal populations are influenced by both density independent
and density dependent processes (Milne 1962, Wolda 1989); therefore it is difficult to
separate the relative impacts that they have on population size (Murdoch 1994). In
addition, density dependent processes can occur among separate age classes. One way to
evaluate the relative importance of these processes is to test them within separate
hypotheses, or models (Franklin et al. 2001). This approach, however, presents some
challenges. First, these separate models, individually, may not be representative of the
real world, where it is unlikely that only one or two rates will vary, while all other rates
remain constant. However, the results of these separate models are still useful because
the effect of a single rate change on population abundance can still be indicative of its
influence on population size, particularly if the impact is large. Second, the separate
models represent a single hypotheses; therefore, this type of testing does not fit the
traditional “Frequentist” framework of testing a single null and a single alternative
hypothesis (Royall 1997). An alternative approach to null hypothesis testing is that of
“statistical model selection” in which competing hypotheses (models) are tested relative
to one another. A discussion of this approach can be found in Franklin et a1. (2001 ).

Yellow perch in Inner Saginaw Bay, Lake Huron, have exhibited wide
fluctuations in abundance during the 20th century (Hile and Jobes 1941, El-Zarka 1959,
Eschenroder 1977, Haas and Schaeffer 1992). Some studies suggest compensatory
growth has been an important factor regulating population size (El-Zarka 1959, Diana
and Salz 1991, Haas and Schaeffer 1992, Schaeffer et al. 2000), while others suggest
density independent influences, such as spring water temperatures (Eshenroder 1977), are

important. El-Zarka (1959) reported increased abundance through 1955, accompanied by

slow individual grth rates. In the 19805, yellow perch abundance increased and
growth rates decreased, and this has been hypothesized to be due to the density dependent
impacts of food limitation (Diana and Salz 1991, Haas and Schaeffer 1992).

There are multiple hypotheses about the density independent and density
dependent mechanisms that affect yellow perch population size in Inner Saginaw Bay.
For example, recruitment may be influenced by weather-related events (density
independent mechanisms), or young-of-the-year competition for food (density dependent
mechanisms). Furthermore, these multiple hypotheses are not limited to age 0 yellow
perch (i.e., recruitment). Evidence in Haas and Schaeffer (1992) and Diana and Salz
(1991) suggest the Saginaw Bay yellow perch population may be regulated at older ages
(e.g. age 2-4) due to food limitation. The goal of this study, therefore, was to determine
the importance of density independent and density dependent rates (birth, survival and
growth) in influencing the abundance of yellow perch in Inner Saginaw Bay, Lake
Huron. My primary objectives were to determine the likelihood of various models that
describe how the abundance of yellow perch has changed. This was done by (1)
compiling a dataset of observed abundance and growth of yellow perch, (2) developing a
population modeling framework capable of representing competing hypotheses, (3)
determining which hypothesis(es) are most consistent with changes in yellow perch

abundance, and (4) using auxiliary data to lend further evidence to top ranking models.

METHODS
Observed data

Observed data were used to model abundance of yellow perch in the Inner portion
of Saginaw Bay, and to calculate survival and growth increments that could be compared
to model output. Observed data were based on yellow perch trawl survey catch statistics
gathered by the Michigan Department of Natural Resources (MDNR) Fisheries Division
from Inner Saginaw Bay, Lake Michigan. Detailed descriptions of these methods can be
found in Haas and Schaeffer (1992) and Thomas and Haas (1994). The Inner Bay is
separated from the Outer Bay by a shallow area that extends from Point Lookout in the
Northwest to Sand Point in the Southeast (Figure 1). Yellow perch were collected with
bottom trawls from a variety of index sites in September or October of each year, 1971-
2001, except data from 1985 were unavailable. Data from individual sites were usually
collected from three consecutive ten-minute tows. In the 19903, randomly selected sites
were added to the index site design.

Catch statistics were used to calculate age specific relative abundance and size
composition. Catch-per-unit-effort (CPUE) at age was used as an index of abundance,
and mean total length at age represented size at age. CPUE was equivalent to actual
abundance and gear selectivity; however, selectivity was not parameterized in these
models because ( 1) this would have introduced too many parameters into some of the
models, therefore making it difficult to estimate survival and growth rate parameters, and
(2) it was not necessary to determine absolute abundance for this study since relative

comparisons among models could be made. It was, however, assumed that catchability at

/

 

Inner ’
Saginaw Bay .

 

\ I
\‘~ a
.. ”N. N
x. :25:
\v . O 5 10
Kilometers
Figure 1. Inner and Outer Saginaw Bay, Lake Huron, separated by an imaginary line

between Point Lookout and Sand Point.

age remained constant over time. Calculations of CPUE, with and without random sites,
indicated that the addition of random sites did not significantly change CPUE or length
values, but added greater precision to measures of relative abundance. There were no
data for 1985. To ensure a continuous time series, CPUE and mean length for this year
were interpolated from the CPUE (or mean length) of the previous age group in the
previous year and the CPUE (or mean length) of the following age group in the following
year. The interpolation assumed CPUE decreased exponentially, and length at age
increased exponentially.

The CPUE for each age group, age 1-8, for every year, 1971-2001, was used to fit
the models. I assumed no change in abundance or grth from October to April, the
following year, and applied all data from fall collections to the spring of the following
year. This adjustment was needed to simulate changes in abundance within the
population models, which apply birth rates in the spring. For example, young-of-the-year
yellow perch collected in the fall were assumed to be age 1 yellow perch in April of the
following year. This increased the “assigned” ages of observed fish by one year, and
changed the dataset years to 1971-2001. In the models, age 0 yellow perch were assumed
to be egg number, rather than young-of-the-year. Observed data consisted of data on
females only, because the population model included fecundity rates that only involved
females. Age 1 yellow perch were not separated by sex in the observed datasets;
therefore, female CPUE of this age group was calculated as one half the CPUE of both

sexes, assuming there was a 1:1 ratio of females to males.

Hypothesis formulation

The hypotheses in this study (Table 1) were related to population processes
(survival, growth, and fecundity), rather than specific mechanisms (e.g., water
temperature, predators, food) that affect the population processes. Hypothesis
formulation was based on (1) the biological likelihood of the hypothesis, (2) the
limitations of the population model (e. g., the models used could not simultaneously test
density independent and density dependent rates within the same age group), and (3) the
number of parameters (i.e., some hypotheses were overparameterized). Age 2-4 yellow
perch were grouped together because they represented the benthivorous feeding stage of
yellow perch in Inner Saginaw Bay (Tharratt 1959, Haas and Schaeffer 1992,
Synnestvedt 1996). Hypotheses based on compensatory rates at age 5+ were not included
in this study because preliminary model testing showed these rates had very weak
relationships with abundance. Compensatory growth impacts on fecundity were
hypothesized for all age groups, including immature yellow perch. Immature fish do not
produce eggs; however, it was assumed that their grth would influence their size and

fecundity when they became mature.

MODEL

 

 

 

 

 

 

 

HYPOTHESES NO.
Null Hypothesis
1. Constant survival and grth rates at all ages 1,4
Age 0 (recruitment) Hypotheses
2. Density independent survival rates at age 0 2,3
3. Compensatory survival rates at age 0 5,6,7,8
4. Compensatory individual growth rates (impacting fecundity) at age 0 9,10
5. Compensatory individual growth rates (impacting fecundity) and 11
compensatory survival rates, at age 0
6. Depensatory survival rates at age 0 due to the "predator satiation" 12,13
hypothesis
Age 1 Hypotheses
7. Compensatory survival rates at age 1 14,15
8. Compensatory individual grth rates (impacting fecundity) at age 1 21,22
9. Compensatory individual growth rates (impacting fecundity) and 23

survival rates, at age 1
10. The above processes occur with time delays because age 1 yellow 16-20,23
perch respond to the impacts of previous generations

 

 

 

11. Depensatory survival rates at age 1 due to the "predator satiation" 24
hypothesis

Age 1+ Hypotheses

12. Compensatory survival rates at age 2-4 25

13. Compensatory individual grth rates (impacting fecundity) at 26
age 2-4

14. Compensatory individual grth rates (impacting fecundity) and 27
survival rates, at age 2-4

15. The above processes may occur with time delays because age 2-4 28
yellow perch respond to the impacts of previous generations

16. Compensatory survival rates at more than one age group 29

 

 

Table 1. Alternative hypotheses, by age, of the processes that may have impacted
overall abundance of yellow perch in Inner Saginaw Bay, Lake Huron, 1971-2001.
Each hypothesis is incorporated into one or more models, indicated by model
number.

Population model and parameter estimation
The population model
Observed changes in abundance of yellow perch were simulated with the

following population model:

Population abundance at age “a” and time “t” = Na;

8
_ * * *
N0,t- ZlNa,t-1 ma,t—1 fa,t-l p,where
a:

m = percent mature, f = fecundity, and p = sex ratio = 0.5.
_ *
N(a > 0,t) ‘ N(a — 1,: — 1) S(a —l,t —1)""’here

S(a-1,t_1) = survival from a-1,t-1 to a,t.

Modeled Na,t was fit to observed Na; through an optimization process that

minimized the residual sum of squares (RSS) between the modeled and observed N.
Model predictions of N, at every age, (a = 1-8), in every year (t = 1971-2001), were
compared to observed N, at every age and year. The differences between them were

squared and summed, so as to produce a RSS. Parameters contained within components

of the model were selected based on the values that minimized the RSS. N(0,t) was not

included in the minimization; but was needed to calculate NUJ)’ and to model those

hypotheses that involved age 0 survival or grth rates. It was not included in the

minimization because there were no observed measures of egg number. The ability of

the model to fit observed data was assessed by the value of the RSS (i.e., a lower RSS
indicates a better fit), as well as through the model selection process described in the
following section. All models were tested with different parameter starting values to
increase the likelihood that all global minima would be found. Those starting values that
produced the most realistic model with the lowest RSS were used.

Parameters within the population models were estimated based on the closest fit
of modeled abundance to observed abundance as described above. Estimated parameters
were either representative of a rate (e.g., density independent age 0 survival rate), or they
were contained within a function that represented a rate (e. g., density dependent age 0
survival rate = a — (b*density), where a and b are the parameters). Parameters were
automatically scaled during the optimization procedure because they ofien differed in
magnitude. Models were always evaluated to determine if survival and growth rates were
realistic. “Realistic” rates were considered to be those growth and survival rates that fell
within a range of values found in other studies that measured growth or mortality of
yellow perch (Schneider 1973, Brazo et a1. 1975, Ney and Smith 1975, Weber and Les
1982, Pycha and Smith 1984, Henderson 1985, Henderson and Nepszy 1989, Hayes
1990, Haas and Schaeffer 1992). I used constraints when the model output was not
realistic, but only after many attempts at searching for the parameters that minimized the
sum of squared deviations. All models presented in the results have realistic survival and
growth rates. Growth and survival rates remained constant when they were not
parameterized within a model.

In models where survival rates remained constant, they were based on a mean

survival rate (S), at age (a), averaged across years (t) (Table 2). These survival rates were

10

actually “relative” survival rates because they were not adjusted for catchability. They

were calculated from observed data as:

2%,,

S =
a ZNa—l,t—l

 

Age 0 survival rate was not known; therefore it was estimated in every model, except
where noted.
In models where length remained constant, length was based on the mean length

at age of yellow perch, summed and averaged across all years, 1971-2001 (Table 2).

 

 

 

 

 

 

 

 

Mean Length Percent Fecundity
Age Survival Rate at Age (mm) Mature (eggs/female)

0 Estimated 0 0.000 0

1 0.653 81 0.000 0

2 0.571 134 0.384 2,138
3 0.388 165 0.890 9,482
4 0.302 187 0.980 15,600
5 0.302 205 0.997 21,246
6 0.401 229 1.000 30,558
7 0.450 251 1.000 40,573
8 0.000 260 1.000 45,732

Table 2. Constant values used in models, as calculated from observed female

yellow perch data, 1971-2001.

11

Percent maturity was determined from the mean maturity of all yellow perch in the
observed dataset, 1971-2001 (Table 2). Fecundity was calculated using an equation for

yellow perch in Lake Erie from Henderson and Nepszy (1989):

Log (fecundity) = 3.1795 * log (length) — 3.0202

Survival rate parameter estimates

In models based on hypotheses of density independent and compensatory
survival, the survival rate involved parameter estimates. The parameter estimates were
equivalent to the survival rates themselves in density independent models. One
parameter was estimated in density independent models with a constant survival rate over
time. Thirty parameters were estimated in density independent models with variable
survival rates over time, one for each year. Density independent models were actually
considered to be “free-fitting” models because the survival rate estimates were selected
independently, but were not necessarily unrelated to density. To determine if they were
related to density, I used linear regression to compare modeled survival rates with log-
transformed modeled density, and based the significance of this relationship on analysis
of variance.

In models based on hypotheses of density dependent “compensatory” survival,
survival rate for each year was based on the relationship the instantaneous mortality rate

(Z) had with the density of fish; therefore, the parameter estimates were contained within

12

this relationship. The mathematical relationship was derived from a Ricker function

(Ricker 1954, 1975), and parameter estimates were contained within the solution for Z:

Zage = a + b(Nage), where a and b = parameter estimates

Zage was converted to Sage because an annual survival rate was needed in the

population model:

Age 0 and age 1 instantaneous mortality rates were dependent on the density of
one age group (i.e., age 0 or age 1). In contrast, I modeled one instantaneous mortality
rate per year for age 2-4 yellow perch because hypotheses were based on a combination
of these age groups. This single mortality rate was dependent on the combined density of
all three age groups in the current year and was applied to each of the three age groups in
the model. All survival rates were constrained to be greater than 0.0 and less than 1.0.

Some of the density dependent compensatory survival rate models included a time

delay, in which survival rate of an age group was based on the density of that age group

13

from a number of previous years (three, five or seven), thus simulating the time delayed

effect that a previous population had on the current population:

3,5,or7
— *
Sa,t —a+(b Z Na,t—x)

x=l

Time delays were used in age 1 and age 2-4 models because I hypothesized that these age
groups may have been impacted by the abundance of the same age group from many
previous years, rather than just the current year. This could occur if prey required more
than a year to rebound from heavy predation by yellow perch. For example, a dense
population of age 1 (or age 2-4) yellow perch can reduce their food base for many years
if the invertebrates take many years to recover, thereby having a potential impact on age
1 (or age 2-4) yellow perch a few years later.

In all of the above density dependent models, variation in survival rates was
explored in two ways: (1) based on the modeled abundance (one version of the same
model), and (2) based on the observed abundance (another version of the same model).
Models based on observed abundance were developed because a density dependent
model that incorporates deterministic feedback based on its own egg production may be
incapable of simulating the stochastic population variability that occurs in nature.

In density dependent “depensatory” models, survival rate was based on a
depensatory relationship from Liermann and Hilbom (2001) that assumes depensation
occurs as a result of predator satiation. When yellow perch are at low abundance, their

survival rate is very low (predators are not satiated) relative to higher abundance

14

(predators become satiated), although they eventually become abundant enough so that
food limitation occurs, leading to compensatory survival rates (lower survival rates at
very high abundance).

Unlike the above models, in which parameters were contained within the solution

for the survival rate, parameters in these models were contained within the solution for

N(a), for which a = 0 (Model 12 and 13) or a = 1 (Model 24):

S mS
N = —*——————
(0) p5 exP£g h2 + s2]

In the above equation, p = productivity, S = spawning stock, m = maximum level of
predation, g = maximum spawner-recruit curve in the absence of predation, and h =
number of spawners at which predation = '/2 the maximum level of predation (m). I
assumed S = egg number. All depensatory models were based on observed density of age
0 yellow perch because models based on modeled density had the potential to
continuously increase or lower yellow perch density rather than simulate variability in the

production of eggs.

Growth rate parameter estimates
In models that did not parameterize growth, fish length for every year was based

on an overall mean length at age calculated from the observed data (Table 2). In

15

contrast, models based on hypotheses of compensatory growth (as it impacts fecundity),

fish lengths were based on the following equations:

Length(a,t) = length(a_1,t-1) + length increment

Length increment = intercept - [slope*length(a-1,t_1)], where

the slope = a parameter estimate and 0.4<=slope>=0. 1.

Length(a,t) was based on the length of a fish from the previous age and year, plus

a length increment. Rather than arbitrarily estimating the length increment, it was
estimated from the linear relationship it had with length from the previous year and age;
therefore accommodating the prior grth history of the fish. A negative linear
relationship was used because increments were assumed to linearly decrease as fish
length increased, which parallels how length changes in 3 von Bertalanffy grth
equation (von Bertalanffy 1938). I had the option to incorporate a density impact on the
intercept or the slope of the increment equation. In addition, the relationship between
density and the intercept (or slope) could be based on a multitude of equations. I tested
several types of equations for the intercept and slope, and found a modified inverse

equation to be the best:

a
Intercept = , where

+N(a—1,t—l)

 

a and b = parameter estimates, a >=15000 and 40<=b<=200.

l6

When other equations were used, fish length was often too low or equal to zero when fish
density was high. The modified inverse equation prevented these unrealistic low values
from occurring because it forced the length increment to be greater than zero. The
equation worked equally well for the intercept or the slope, so I also applied the equation
to the intercept of the length increment equation. The constraints were needed in order to
obtain realistic fish lengths. As with survival parameterization, grth increments could
be dependent on modeled abundance or observed abundance; therefore both types of
formulations were tested. Time delays were not included in these models because
preliminary model testing indicated growth models fit the observed data worse when time
delays were included.

In models that did not parameterize growth, fish lengths for every age were
equivalent to the mean lengths in Table 2. In the grth models, grth was
parameterized for one age group as described above. The mean lengths in Table 2 were
applied to age groups younger than the parameterized age group; however, length-at-age
of fish that were older than the parameterized age group were based the length of the fish

that were parameterized using the following equation:

Length(a,t) = a - [b*length(a-1,t-1)], where

a and b = parameter estimates, a = 75 mm, and O.4<=b<=0.1

As with the younger fish in these growth models, it was also necessary to constrain the
intercept and slope to obtain realistic lengths for older fish. Finally, it was necessary to
set age 0 survival at a rate of 0.0002, rather than parameterize this rate. The value of this

rate was estimated using a method by Vaughan and Salia (1976) that uses a Leslie matrix

17

population model (Leslie 1945) to determine the rate based on an observed age
distribution of fish. When age 0 survival rate was parameterized in these models, it was
not possible to obtain realistic growth rates because the age 0 survival rate would be
estimated at a value that accounted for changes in abundance regardless of fish growth.
Fish would grow poorly, and produce fewer eggs, but good survival at age 0 would

compensate for the reduced fecundity.

Model selection

In addition to RSS, models were also evaluated with Akaike’s Information

Criterion (AIC) (Akaike 1973):

AIC = -210ge.t'(parameters I data) + 2(number of parameters)

AIC is another measure of model fit. It is different from RSS because it penalizes
models for the number of parameters, thus incorporating the principle of parsimony
(Bumham and Anderson 1992). AIC is an estimate for samples in which n/K >= 40,
where n = sample size, and K = number of parameters. Even though each year of the
data used in these models was based on CPUE of several age groups from several

collections and index sites per year, sample size was set at 31 because measures of CPUE

18

were “nested” within each year. This required the use of a modified AIC (Hurvich and

Tsai 1989):

[2K(K +1)]

n-K—

AICc = AIC + , where

K = parameter number and n = sample size

Inspection of the residuals between modeled and observed CPUE from several
models indicated most models were normally distributed; therefore the likelihood
equation could be based on the residual summed squares (RSS) obtained from the
optimization methods described above. I used the following equation adopted from

Seber and Wild (1989) to calculate log likelihood:
loge L(parameters|data) = {g} * loge(RSS)

I did not include any of the constant values often found within the above likelihood
equation because they do not affect the equation outcome (Seber and Wild 1989). The
value of AICc was used to rank the models, with the best-ranking model equivalent to the

lowest AICc. Neither RSS or AIC provided measures of the relative importance of each

19

model; therefore, Akaike weights (Buckland et al. 1997) were used to compare the

models with each other:

 

M = total number of models, and A1 = difference in AICc of an individual model

compared to the best model.

Auxiliary data

Several types of auxiliary data were used to support or refute the top-ranking
models. First, age-specific survival and grth rates were estimated for each year
directly from the observed data. This estimation process differed from model estimation,
which estimated survival and growth parameters using an optimization process that
minimized the sum of squared deviations between the observed and modeled abundance
for each age and year. The observed data estimates were compared to fish density each
year to determine if density dependent survival and grth occurred. Linear regression
was used to compare the relationship between observed female survival (or grth
increment) and observed female fish density, and significance was determined from
ANOVA.

Observed survival rates were determined by:

loge (N j

S = + l
a loge Na

20

Observed growth was evaluated from length increments. Length increments were

calculated as:

[ncrementa = L

—>a+l a+l--La’Where

L = mean length-at-age obtained from the observed data

Growth rate evaluations from observed data served as auxiliary evidence for (1)
compensatory growth, as it impacts fecundity, and (2) compensatory survival models,
since compensatory growth can act as a mechanism of compensatory survival.
Additional auxiliary data were obtained by comparing observed temporal trends
in grth increments (based on the above estimates) to temporal trends in abundance,
1971-2001. Furthermore, temporal changes in grth increments were examined for the
entire 20th Century using data from Hile and Jobes (1941), El-Zarka (1959), Eschenroder

(1977), and data from this study.

RESULTS AND DISCUSSION

Observed abundance

Although highly variable, estimated yellow perch abundance (CPUE) in Inner
Saginaw Bay generally was low in the early 1970s, high in the 1980s, and low again in
the 19905 (Figure 2A). Overall abundance appeared to have declined abruptly in 1983
and 1991, although it remained consistently low afier 1991. Total abundance appeared to

be most variable in the 19803. Similar to total abundance, age 1 yellow perch were most

21

abundant in the 19803 (Figure 2B). Age 2-4 yellow perch were most abundant from the
late 19703 through the mid 19803 (Figure 3A-C). By age 5, yellow perch became less
abundant, likely due to a combination of (1) lower vulnerability to gear and (2) lower
survivorship at older ages. Age 5 yellow perch were most abundant from the late 19703
through the mid 19903 (Figure 4A). Age 6 yellow perch began to show more distinct
peaks in the late 19703 and late 19803 (Figure 48). Age 7 and 8 yellow perch were most
abundant in the 19903 (Figure 5A and B). CPUE of a cohort could be easily tracked
through subsequent year and age groups, implying CPUE measures were reasonably
precise.

Overall abundance declined abruptly in 1991. This event, however, was actually
observed in the previous year. This discrepancy occurs because observed data were
applied to spring of the following year, when reproduction occurs. To avoid any
confilsion over the actual timing of the change, the change will always be called “the
change in 1991” (later referred to as “the perturbation in 1991”); however, it still
represents the change that occurred between Fall, 1989, and Fall, 1990, rather than
between 1990 and 1991. Graphical representations of the data will also show this abrupt
change occurring in 1991, rather than 1990. It is important to be aware of this because
the mechanism that may have caused the abrupt change in yellow perch abundance would

have occurred between Fall, 1989, and Fall, 1990, rather than between 1990 and 1991.

22

600 -~

 

 

500 ~! Total
I
400 l
m I
E 300 “l
U l
200 t
100 l
O ~L—l"_l_"f_ l” ' "I ’l 'l" 1” 1T" "7 ‘T' T” 1.1 ‘l' T
v--‘ m 0\ M l\ v- V) O\
l\ I\ I\ 00 00 ON 0\ O\
9: a 9: a _. a a a
Year
600 ~
500 4 Age 1
400 -
LU
753 300 -
U l
200 l
100
0 _ ,
—* In O\ m l\ v— V3 ON
[\ (\ [\ oo 00 ON ON 0‘
9: a: a 2: a a 52 a:
Year
Figure 2. Temporal changes in total CPUE and age 1 yellow perch in Inner Saginaw

Bay, Lake Huron.

23

 

 

 

LL]
:3
9..
U L
_' fl
.— VN ON M l\ —-1 tn ON
I\ t\ [\ 00 00 ON ON ON
ON ON ON ON ON ON ON ON
.—1 '— v— p—t '— '— v—t v—t
250| Age3
Ill
.2)
a.
U
.— IIN ON M [\ '— tn ON
I\ r\ I\ oo 00 ON ON ox
°~ 0* °~ °~ e: a a e:
2501 Age4
200i
m -.
a 150
o 1001,
50
O‘l Ifll I’T—TT—‘I—T'W I; l T‘I‘l‘.’
v-n In ON «3 l\ v-‘ V) ON
I\ l\ l\ 00 00 ON ON ON
9: e 9: a: 2: 2: e 2:
Year

Figure 3. Temporal changes in CPUE of age 2, age 3 and age 4 yellow perch in
Inner Saginaw Bay, Lake Huron.

24

’ 1

go _

Age 5

33

I' Val T ‘1’ 7 Fri

32

59

$3

TI 2 7..

3.3

mug

of TTIVTTI‘H‘T

illllllllll~8_

O 5 0 5 0

2 1 1

mDmU

Year

6
e
g
A
,llllli-llll.:2
0 5 0 5 0
2 1 1
BED

 

Year

Temporal changes in CPUE of age 5 and age 6 yellow perch in Inner
25

Saginaw Bay, Lake Huron.

Figure 4.

Age 7

CPUE
O H N b) # kl! ON

 

1 F T'ITI I I T TFIATTT‘T Fifi I ['77 I‘IT‘TTI ‘ T l
v— V) ON ('1 l\ v— V) ON
I\ l\ I\ 00 00 ON ON ON
ON ON ON ON ON ON ON ON
'— v—d F4 —1 F1 I—fl '— —‘
Year
Age 8

CPUE

O .—
a__. _- _
-.
.
a
s
_
a
e-fi
q
a
_.
l
A
—1
a
—4

 

r ’Tfi
v—4 V) ON m l\ '-‘ V3 ON
l\ I\ I\ 00 00 ON ON ON
ON ON ON ON ON ON Q ON
F-fl v-1 '— v—i '— '— P-l F“
Year

Figure 5. Temporal changes in CPUE of age 7 and age 8 yellow perch in Inner
Saginaw Bay, Lake Huron.

Model evaluation

The following models represent the hypotheses listed in Table 1. Some
hypotheses involve more than one model. Generally speaking, hypotheses are presented
in the same order as listed in the table. Parameter number and estimates are listed in

26

Appendix A. In this section, models are evaluated based on (1) their “visual” fits to the
observed data, and (2) the residual summed squares (RSS) that were minimized during
the optimization procedure. Residual summed squares are listed in Table 3. In the
following section, the same models are ranked and weighted according to Akaike’s

Information Criteria (AIC).

The null hypothesis
Model 1. A simple exponential model

This is the simplest model, in which one parameter, age 0 survival rate, was
estimated and treated as a constant over time. It was not based on fish density, but was
estimated based on a rate that would provide the best overall fit of modeled CPUE to
observed CPUE.

Although a constant survival rate of age 0 yellow perch likely does not occur, this
model represents the null hypothesis, in which all rates remain constant over time, and
are not dependent on fish density. In this model, survival rate was estimated as 0.0003,
and abundance increased exponentially because constant survival rates caused the
population to increase continuously (Figure 7). Visually, the model did not match the

data well because it was incapable of modeling annual fluctuations in density.

Age 0 survival hypotheses

Model 2. Free fitting age 0 survival rate for every year
In this model, age 0 survival rate was estimated for each year, while all other

survival and growth rates in the model were held constant. This model was

27

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28

500 ‘1 l: +Modeled
m 400 .1 \ _ ~I~— Observed
D J "- R
63 300 l , \ ’H'! \ I"! '/

 
   

EE’: i \' .gf‘I'lll'
0 —l T—“T'T T‘l‘TTT TTT'”. TT‘FT‘T‘T‘T—T—T‘fl—lfl-FTI'TI I_Tl_T—T"“T"’l '1
'— lf) C’\ M [\ v—t V) O\
l\ t\ l\ 00 00 O\ a ON
a: a: a: e 9.: a a 9:
Year
Figure 6. Model 1: Constant age 0 survival.

more flexible than Model 1 because it fit thirty parameters (one age 0 survival rate for
each year), and because each survival estimate was not dependent on any value, such as
fish density. This does not, however, mean that the survival rates were not related to
density; hence the model was considered to be “free fitting”, rather than density
independent.

In this model, abundance closely matched the observed data (Figure 7A). As
indicated by the RSS, the model provided a good fit to the observed data (Table 3). This
close fit was possible because (1) age 0 survival rates in fish populations can have a
strong impact on the total abundance in subsequent years, and (2) age 0 survival rate is fit
independently for each year, allowing for tremendous flexibility of parameter values.
Age 0 survival rate estimates varied widely (0.000001 to 0.00011, mean = 0.00018),
with the lowest rates occurring between 1992-1994. These low rates were the model’s

attempt to match the large drop in observed abundance that occurred after 1991. When

29

estimated survival rates were compared to CPUE, there was significant evidence that they

may be density dependent (Figure 7B).

 

 

A
600 1
500 ; +Modeled
400 i "F ObSfiI'VCd
LI-J
D .l
a. l
U 1
: W 2155033 8:
a 2: <2 2 a 9: a a
Year
8.0E—04 ! B
Io . °
6-013'04 E Fm: 11.07 P=0.0002
5 : 2 _
P; 4,013.04 3 r ‘0-23
Cd
.2 !
E 2.0E-04 i
m l
l " o
0.0E+00 l - *9?” f °—~ H 4.7———~or—— "1
0 200000 400000 600000 800000 10000001200000
-2.0E—04 »
Egg number
Figure 7. Model 2: Variable age 0 survival (A) and survival rate from age 0 to age

1 versus egg number (B).

30

Unlike the last model, this model was capable of modeling annual fluctuations in
yellow perch abundance. It was capable of matching low abundance in the 19915, but it
could not simulate the “sudden” drop in abundance that occurred in 1991. Despite the
influence age 0 survival rate can have on a population and the “free fitting” nature of this
model, the model could not simulate this sudden change. One way to model this change
is to reduce the survival of yellow perch across all age groups, thereby simulating a

perturbation to the entire population.

Model 3. Free fitting age 0 survival rate every year with a perturbation

Model 3 was identical to Model 2, except a perturbation was added in 1991. This
perturbation consisted of a reduction in CPUE that was applied to all age groups in 1991.
In this model, as well as all subsequent models with a perturbation, CPUE at age was not
modeled for 1991, but was fixed at an abundance that was approximately 10 times less
than CPUE at age for models without perturbations. The same reduced CPUE values
were used in all models with a perturbation, rather than “fitting” the perturbation for each
model. This was done to maintain consistency among models with the perturbation, and
to reduce additional model complexity. The amount of the reduction was determined in
preliminary models by testing several factors and selecting the one that best simulated the
sudden change in overall abundance that occurred in 1991. The perturbation was applied
to a model prior to optimization and essentially acted as if the model contained an
additional starting point (i.e., 1971 CPUE was the first starting point and 1991 CPUE was

a second starting point).

31

Visually, this model (Figure 8A) matched the observed data slightly better than
Model 2 (Figure 7), and the RSS also indicated a better fit (Table 3). This closer fit was
possible because the perturbation enabled the model to match the drop in abundance in
1991. Age 0 survival rates varied widely across a similar range as Model 2. As with
Model 2, when estimated survival rates were compared to CPUE (Figure SB), there is an
indication of density dependence, although the overall relationship was not significant. I
also fit a nonlinear regression line to the data and found the linear relationship had a

better fit.

Model 4. Model 1 with a perturbation

The above perturbation appeared to be a necessary feature of the free-fitting
models; therefore, it was added to Model 1, creating a model with two parameters. The
new model (Model 4) provided a better match to the observed data (Figure 9) than Model
1 (Figure 6). Based on RSS, it also had a dramatically better fit than without the

perturbation (Table 3).

Model 5 — Model 8. Compensatory survival of age 0 yellow perch

In this suite of models, the age 0 survival rate (survival from egg to age 1) for
each year was dependent on the density of age 0 yellow perch (eggs) from the previous
year. All of these models estimated at least two parameters (intercept and slope of the

linear relationship between age 0 survival rate and egg density).

32

Survival rate

Figure 8.

A

+ Modeled
+ Observed

CPUE

 

 

 

v—~ tn 0\ M '— 10 O\

1\ t\ 1\ oo oo Ox ox GN

51‘ 3‘ 51‘ 9‘. 2: 5’: 93 2:

Year
0'0016 ' F 3 46 P 0 073
l,28 = - = -
0.0014 .. . 12:011
0.0012 1 0 '
0.001
0.0008 l .
0
0.0006 . .
0.0004 . .
0.0002 - , ’ 0 ° 9
0 ‘4‘ :.£ i"_ 7*" .“TT _" .— TT— ‘ fl
0 200000 400000 600000
Egg density

Model 3: Free fitting age 0 survival every year with a perturbation (A)
and Model 3 survival rates versus egg density (B).

33

. + Modeled
/\ ~I-~ Observed

CPUE

 

1 , ; H
-—-4 m C\ m [\ " m G
(\ I\ oo 00 O\ o O\
a a: 9: a: 2: 2: a a
Year
Figure 9. Model 4: Constant age 0 survival with a perturbation in 1991.

In these models, age 0 survival rate was dependent on either egg density predicted
within the model (one version of the model), or egg density estimated from the observed
data (another version of the model). In all of the age 0 models, observed egg number, as
determined from the fecundity and abundance of mature females from the previous year,
was variable. This variability produced models with visible cycles in abundance (Figure
10A), which may be indicative of compensatory survival of age 0 yellow perch; however,
the models were very unstable. Age 0 survival rates were either zero, or too high;
therefore, egg number was log-transformed prior to estimating the parameters of the
survival rate equations. This produced a model (e. g., Model 5, Figure 10B) with
reasonable age 0 survival rates, but the variability in egg number was greatly reduced.

In Model 5, age 0 survival rate was dependent on egg density predicted within the
model. In Model 6, age 0 survival rate was dependent on egg density estimated from the

observed data. Both Model 5 and Model 6 (Figure 10A and Figure 10B) were visibly

34

similar to the null model (Model 1; Figure 7). Based on RSS, Model 5 fit the observed
data slightly better than Model 1 (Table 3). Model 5 also had a better fit to the observed
data than Model 6 (Table 3) because it was able to come closer to some of the data points
in the earlier years. For this pair of models, the use of observed egg density to simulate
greater annual variability in egg density did not appear to benefit model fit.

Survival rates were distributed differently between the two models. They
decreased over time from 0.00046 to 0.00023 in Model 5; but remained virtually constant
in Model 6 at 0.00027. A slowly decreasing density of total yellow perch in the model-
based formulation (Model 5) caused the survival rate to slowly increase, although it did
not fluctuate. In the observed-based formulation (Model 6), a virtually constant value
was fit because changes in age 0 density over time likely forced the model to fit a value

that kept the model between extremes; therefore, it was very similar to Model 1.

35

A

1 +Modeled
400 . Ill — I— Observed

  
  

CPUE

 

 

        

 

   

i
.—n W O\ M [\ fl V3 0‘
[\ [\ (\ OO 00 O\ 0‘ 0‘
e: 2 a: a: e a: a a
B
600 —;
500 l g +Modeled
g 400 4‘ —-I— Observed
a, 300 x
0 200 1 ,
100 J. '
Oiit'fl1t1rrrr1—r tTrrrthl. 31.1.1.1 1
F‘ W 0‘ m [\ v—I V5 0‘
[\ [\ [\ 00 00 ON 0‘ o‘
a: a 52 9: 9: :2: 93 93
C -
:83 1 +Modeled
d
g 400 1 A +Observc
E3 300 7 f \
200 1
100 *
O i fl 1 T T—T T l l l l l l l l I I I l l T l
_ V3 0‘ M [\ '— V3 ON
.\ .\ I\ oo 00 as a o
9: 9: a 2: e a a a
Year

Figure 10. Density dependent survival of age 0 yellow perch without log
transformation of egg number (A), Model 5: density dependent survival of
age 0 yellow perch, based on modeled, log transformed egg number (B),
and Model 6: density dependent survival of age 0 yellow perch, based on
observed, log transformed egg number (C).

36

Model 5 and Model 6 were incapable of fitting the sudden change in abundance that

occurred in 1991. When the perturbation (an additional parameter) was added to Model 5

(Model 7) and Model 6 (Model 8), they appeared to fit the observed data better (Figure

11A and Figure 11B), and RSS was lowered (Table 3); however, the model-based

Figure 11.

CPUE

CPUE

 
   

600 1 A
500 1
400 1 +Modebd
300 1 ll +Observed
200 1 . ..
J V ‘ l
100 _l , 1‘
0 imTTT‘” j r 1’: .‘ 1T 1_r“—'7*. L.
'— V3 0\ M l\ "-‘ V3 0‘
l\ l\ l\ 00 OO 0\ O\ O\
a a: e a: 2: a a 2:
Year
B
600 ‘1
500 1
400 l
I +Modeled
300 l 4. Observed
200 ~
100
0 t‘ '* I 1 1 r1" 1
.— ln 0\ M l\ .— m 0
l\ l“ l\ 00 00 O\ O\ Ox
2 e: a: 9: 9.: 9: a: 9:
Year

Model 7: density dependent survival of age 0 yellow perch based on
modeled egg number, with a transformation (A) and Model 8: density
dependent survival of age 0 yellow perch based on observed egg number,
with a perturbation (B).

37

formulation could no longer fit the data as well as observed-based formulation (Table 3).
The observed-based formulation probably fit the observed data better because the
perturbation enabled the model to use slightly higher age 0 survival rates and provide a

slightly better fit to the data points that followed the perturbation.

Model 9 — Model 10: Compensatory growth of age 0 yellow perch, as it impacts future

fecundity.

In these models, at least seven parameters were estimated, consisting of intercepts
and slopes found within the incremental grth equation. Compensatory growth models
based on “modeled” yellow perch abundance provided the best fit to the observed data
for these models; therefore, only modeled-based formulations are presented. It was not
initially clear that a perturbation was needed.

The model of compensatory growth of age 0 yellow perch (Model 9) did not fit
the observed data well (Figure 12A; Table 3). When a perturbation was added (Model
10), the model matched the observed data better (Figure 12B), and, according to RSS, the
fit was better (Table 3). As with previous models, the perturbation added an additional
parameter to the model, but was needed to accommodate the sudden change in abundance
in 1991. Model 10 also had a steeper incline in abundance over the first two decades
(Figure 12B).

In all growth models, I examined yellow perch mean length at age (averaged
across all years) to determine if it was similar to observed mean length at age. In Model

10, mean length at age was similar to observed mean length at age at young ages;

38

Figure 12.

   
  

   
  

600 '1
500
Lu 400 (I +Modeled
a 300 l +Observed
U 2001
100 'l
0 fi’fTerVVFIT Ti l'_l 17leerka
'—* V3 ON m K\ F‘ V) ON
I\ l\ l\ 00 00 ex as as
1 1 1‘ 1‘ 1 1‘ 1‘ 1‘.
600 l B
500
1.1.1 400 { +Modeled
a 300 J —I— Observed
U 200 I
100 l gr
0 lfl fl TT 7<17l‘l j i T—rlfl lfi 1“? l l'Tl rl'1
v-1 11') O\ m [\ .— ln 0\
;\ [\ |\ 00 00 ex as O\
1‘ 1‘ 1‘ 1 1 1‘ 1‘. 1‘
Year
E
5
3’0
(6 7 fi_.i ,
a 11:1 Modeled]
a lIQbsgrvgctl
.13
5
(D
2

   

Model 9: Compensatory grth of age 0 yellow perch based on
modeled density (A), Model 10: Compensatory growth of age 0 yellow
perch based on modeled density with a perturbation (B), and comparison
of modeled and observed mean length at age from Model 9.

39

however, modeled mean lengths remained lower than observed mean lengths at older

ages (Figure 12). All remaining growth models showed similar results.

Model 11. Compensatory survival and growth of age 0 yellow perch.

I combined the best-fitting model of compensatory survival of age 0 yellow
perch (Model 8) with the best-fitting model of compensatory growth of age 0 yellow
perch, as it impacts fecundity, (Model 10). This combined model (Model 11) had a better
match to the observed data (Figure 13) and, according to RSS, fit the observed data better
than the individual models (Table 3). The number of parameters increased to 8. The
combined model was similar to the individual models, except that the predictions had a

slightly better fit to the observed abundance points due to the added effects of

   

compensatory growth.
600 i
500 1
400 J "1 +Modeled
L” V ,_ —-— Ob d
a 300 l . serve
U
200 1
100 1 1'1
01'; — . -1 421V, Tl.-
t\ (\ I\ oo oo 0\ O\ ox
e a a: 9: a: e a a
Year

Figure 13. Model 11: Age 0 compensatory survival and growth, with a
perturbation.

40

Depensatory survival of age 0: Model 12-Model 13

When depensation occurred at age 0, (Model 12), the model fluctuated in density
similar to observed density, but the fluctuations did not match the observed changes
(Figure 14A); therefore, according to RSS, the model did not have a good fit to the
observed data (Table 3). Interestingly, this was the first model that was able to produce a
significant drop in density in the 1991, yet the drop was immediately followed by a sharp
rise in abundance. When a perturbation (additional parameter) was added to the model
(Model 13), the sudden drop was modeled more accurately (Figure 14B); however, RSS

showed little improvement (Table 3).

Age 1 hypotheses

Model 14 - Model 20. Compensatory survival of age 1 yellow perch

In this suite of models, age 1 survival rate (survival from age 1 to age 2) was
based on the density of age 1 yellow perch in the previous year. These models fit at least
three parameters. Two of the parameters were the intercept and slope of the linear
equation that described the relationship between survival rate and density of age 1 yellow
perch. The third parameter was a constant age 0 survival rate. In Model 14 and Model
15, survival rates were dependent on the density of age 1 yellow perch from the model
(Model 14) or from the observed data (Model 15).

As with the age 0 models, the model-based formulation (Model 14; Figure 15A),
matched the observed data similarly as Model 1 (Figure 7). Age 1 survival rates
decreased over time from 0.985 to 0.355 (Mean = 0.451). Although these survival rates

were initially high for this age group, the average rate was realistic. In contrast, the

41

600 -‘
500
400
300
200
100

CPUE

CPUE

Figure 14.

1971 1f“ .__._ ____ “r *4“ _

A

+ Modeled

- ._ .- —»-— Observed

' '7 1’" 1" F1 1’ 1'1 '- 1—1'1'1’ ’

In ON (*3 l\ '— V3 0\

l\ l\ 00 00 ON ON 0\

ON 0\ 0\ ON ON 0\ O\

—i v—II v—I v-4 —d v—fl '—
Year

 

 

Model 12: Age 0 depensatory survival based on observed density (A), and

Model 13: Age 0 depensatory survival based on observed density with a
perturbation (B).

42

CPUE

CPUE

Figure 15.

observed-based formulation (Model 15; Figure ISB) exhibited more reversals in

abundance and, according to RSS, was a better fit to the observed data (Table 3). Age 1

600 1 A

 
 
 

 

 

  

500 1 1
‘ 1.
400 l \ . + Modeled
,. Q It
300 1 ./ \ ~ —-I-- Observed
200 1 / ‘
100 l ,, .
O 'lH'V—IW Tv‘T‘I—‘T—THT'T—T'T'Tfi 71‘ THT“T_I'T”F“>l-1K'T' ”7‘1 ’7— 'x'1
v-' V) C\ M l\ '— V) O\
l\ I\ [\ 00 00 O\ O\ O\
a: 9: 9; a a 9: a 9;
Year
600 W
500 1 B
1
400 l, l I.
i F '1
300 *l f l // + Modeled
I \ ' "n " --II—- Observed
200 1 Fl
0 4h» l‘Tjfil T—‘T’T l 71.1 l l —‘ l _" ’lj TT‘F‘T I‘l.llf‘:l ' 1’7":-l
v-‘ In ON M I\ 1—1 V) O
[\ .\ I\ oo oo Ox ox Ox
2 a: a a: e a a 9.:
Year
Model 14 and 15: Age 1 compensatory survival based on modeled density

(A), and observed density (B).

43

survival rates did not steadily decline like they did in Model 14, but increased and
decreased densities (range = 0.013 - 0.749; mean = 0.475) based on observed.

When a 3 year time delay was added to Model 14 (Model 16), RSS indicated that
it fit the observed data better (Figure 16A; Table 3), so much so, that it had a better match
to the observed data and a lower RSS than when a 3 year time delay was added to Model
15 (Model 16B; Figure 16; Table 3). Longer time delays (5 year and 7 year) also
performed better on Model 14, rather than Model 15; therefore, longer time delays are
only presented for the latter. Five and seven year time delays (Model 18 and 19)
impacted Model 14 similarly as the 3-year delay (Figure 17A and 17B). The 7-year time
delay provided the best fit, as indicated by RSS (Table 3). The time delays improved the
fit of the above models because modeled data points matched the variability in the
observed data points much better than previous models; however, these models could not
simulate high density in the late 19803, as well as the sudden drop in 1991.

When the perturbation (additional parameter) was added to the time delay models,
the time delay models had a better RSS. A 5-year time delay model with a perturbation
(Model 20) had the best match to the observed data (Figure 18) and the lowest RSS
(Table 3). This model was now able to fit some of the higher data points in the late

19805, and incorporate the sudden drop in 1991.

44

CPUE

CPUE

Figure 16.

600 1

 

1 A
500 1
400 1 + Modeled
300 .. + Observed
200 - ‘
100 « ,, ‘ "
O Th—F—‘l—fi’W—FTVTI' '7‘1"_7—T T T‘T TW—‘1—1- "‘1
1— In 0\ m l\ —4 1n Ch
1\ 1\ I\ oo 00 O\ a ON
a a: a a: e a e 9.:
Year

+ Modebd

 

 

1 1 ' a
v-1 In O\ M l\ .— 1n O\
l\ l\ l\ 00 00 O\ O\ O\
0\ ON 0\ O\ O\ 0\ ON 0\
v—1 1— v— I-i —‘ —1 F—‘ —I
Year

Model 16 and 17: Age 1 compensatory survival with a 3 year time delay
based on modeled density (A) and observed density (B).

45

CPUE

CPUE

Figure 17.

   

   

' A
500 p f
400 l +Modeled
l
300 l ' —-I-- Observed
1
200
100 1 _
i i t“?
0 ‘l T ‘1 1‘7 T T 1"" 1 1' l I 1’ m' T l 1 T lfil—T"? 1'1
v-t V) O\ M l\ —‘ V3 ON
I\ l\ l\ 00 00 O\ O\ O\
a 9:. 9: 2 2: a: a: a:
Year
600 B
500 1 A
400 ; , ' +Modeled
300 a l —I-— Observed
200
100 4'
0 + 1 T‘UT 1 ‘ r": 1 1 r“ '1‘ '1" r—1 1.:lir—1‘1’1'nr’1
u—d V) O\ M l\ —‘ V5 Ox
(\ l\ [\ OO 00 O\ O\ O\
9: a a a: a 9: a 9:
Year
Model 18 and 19: Age 1 compensatory survival based on observed

density with a 5 year time delay (A) and a 7 year time delay (B).

46

   

500 1
g , —I~— Observed
a, 300 1
C) .
200
100
._ * .n .. .. h ._ sir
I\ l\ 00 00 0\ ON 0\
1 1 1 1 1 1 1 1
Year
Figure 18. Model 20: Age 1 compensatory survival, based on observed density, with

a 5-year time delay and a perturbation.

Model 21 - Model 22: Compensatory growth of age 1 yellow perch, as it impacts future
fecundity.

Compensatory grth models based on “modeled” yellow perch abundance
provided the best fit (i.e. lowest RSS) to the observed data for these models; therefore,
only model-based formulations are presented. The model of compensatory growth of age
1 yellow perch (Model 21) did not match the observed data well (Figure 19A).
According to RSS, the fit was not good, either (Table 3). When a perturbation (additional
parameter) was added to Model 21, this new model (Model 22) had a better match to the
observed data (Figure 19B). The model now had a steeper incline in abundance over the
first two decades. According to RSS, the perturbation also greatly improved the fit of the
age 1 growth model and this model also fit the observed data better than age 0 grth

models (Table 3).

47

 

  
  

1
500 1| l A
m 400 ; 1 _ +Modeled
- Q /.
D g 11 . —I—-— Observed
200 -7
100
0 ill 7' ' . 1 1‘1 "
H V3 0\ M l'\ .—-4 W ON
l\ l\ 00 00 O\ O\ O\
a a 9: a: e 9: e a
Year
600 ‘1 B
500 "
1.1.1 400 i +Modeled
a 300 1 +ObSCI'VCd
U 1
200 1
1
100 1 :1 $1 .
O l l. 1_’T I CI‘T 1' I I T 1”1 1 ITI‘E.p '1" ilTT‘T 1!1
'— V) O\ M l\ —* V3 O\
l\ l\ l\ 00 OO O\ O\ ON
9.: a: 2: 9: a 9: a a

Figure 19. Model 21: Age 1 compensatory grth based on modeled density (A) and
Model 22: Age 1 compensatory growth based on modeled density with a
perturbation (B).

Model 23 .' Compensatory survival and growth of age 1 yellow perch.
When I combined the best-fitting model for compensatory survival of age 1

yellow perch (Model 15) with the best-fitting model for compensatory growth of age 1

48

yellow perch impacting fecundity (Model 22) this combined model (Model 23) matched
the observed data slightly better, and had a lower RSS than each individual model (Figure
20; Table 3). The combined model was almost identical to Model 22 (Figure 19B),
except that its modeled abundance had a slightly better fit to the observed points due to

the added effects of compensatory growth.

600
500 7 1

400 g + Modeled

 

— I— Observed

L5] .
a, 300 -‘
U .
200 -'

100 l g

0 ~ 1 I ‘1 r T 7‘1 1
8 8
2 a

Figure 20. Model 23: Age 1 compensatory survival, with a 5-year time delay,
combined with compensatory growth and a perturbation.
Model 24 .' Depensatory survival of age 1 yellow perch with a perturbation.

When depensation occurred at age 1 (Model 24), the model did not match the
observed data well (Figure 21). According to RSS, it also did not provide a good fit
(Table 3). As with the age 0 depensatory survival model (Model 12), this model was also
not able to simulate the drop in density that occurred in 1991; therefore a perturbation

was necessary.

49

600 1
500 '
400

   
  

+ Modeled
+ Observed

J
l
1
300 l

m
D
a.
U l
2001
100
Oiifi
'— V3 0\ M (\ v— V) O\
l\ l\ 00 G 0\ ON
a: 2 e a a: a: a 9:
Year

Figure 21. Model 24: Depensatory survival at age 1 with a perturbation.

Age 2-4 hypotheses

Model 25 and 26: Compensatory survival of benthivorous (age 2-4) yellow perch

In these models, age 2-4 survival rates were based on the summed density of age
2-4 yellow perch in the current year. These models had three parameters. Two of the
parameters were the intercept and slope of the linear relationship that described the
relationship between survival rate and the summed density of age 2-4 yellow perch).
The third parameter was the perturbation, which was added to all remaining models
because it consistently improved the fit of all previous models. When age 2-4
compensatory survival models were optimized, they produced unrealistic survival rates
for older yellow perch. If age 0 survival was fixed at a value of 0.00008, the model

provided realistic survival rates of age 24 yellow perch; therefore, age 0 survival rate

50

was not freely estimated in these models. Age 2-4 compensatory survival models also fit
best (lowest RSS) when compensatory survival rates were based on modeled densities,
rather than observed densities; therefore, only modeled densities are presented.

This model (Model 25) provided a relatively good match to the observed data
(Figure 22A), and, according to RSS, a fairly good fit (Table 3). Survival rates of age 2-
4 yellow perch in this model were quite high (Range = 0.504 — 0.902; mean = 0.705) as
compared to estimates made from the observed data (Table 3). In addition, abundance
increased in the 19913 much more quickly than what was observed. Nonetheless, the
rates varied over time in a way that was consistent with the hypothesis that survival rates
were lowest in the 19803 when yellow perch abundance was high. There was no benefit
to adding a time delay to this model (Model 26; Figure 228); however, the model still

had a reasonable match and fit to the observed data (Table 3).

51

CPUE

CPUE

Figure 22.

600 ~

500
400
300
200
100

  

   
  

*1

1 A

1 1‘

1 1'1

1 1 + Modeled

1 I I 1 P.
/1 —I-- Observed
1 1‘ ‘ '

1
1. I 1141‘

. I

I

‘l’ T_T_T'— — T l 1 l T—’T' F‘d’ T‘ ‘T fiT"’j‘T—T T—' a
.—. In ON M l\ ‘— tn ON

I\ l’\ I\ 00 00 ON ON ON

3‘. 3} 3? 2 E 2 3‘ 3‘.

Year
B
T
11
I. - 1 + Modeled
1' —I— Observed

+_ ._- N. .4, “--__. __l_,._. ..J__ __
1
.

fir:

'—l T - 1 ITT T TWIN I' l
—-1 tn ON «1 [\ —- to ON
[\ l\ [\ OO 00 ON ON ON
ON ON ON ON ON ON ON ON
F‘ '— v—l '— v—d '— F-I v—I

Model 25: Age 2-4 compensatory survival based on modeled density with
a perturbation (A) and Model 26: Age 2-4 compensatory survival based on
modeled density with a perturbation and 5 year time delay (B).

52

Model 27: Compensatory growth of age 2-4 yellow perch, as it impacts fecundity

This model was based on observed abundance and included the perturbation.
Visually, this model (Figure 23) did not match the data better than previous
compensatory growth models that included a perturbation (Model 8 and Model 22), nor
did the RSS differ greatly (Table 3). Of all the grth models, compensatory growth of
age 1 yellow perch, as it impacts fecundity (Model 22), had the lowest RSS (Table 3).
This age 2-4 growth model fit similarly as the age 1 compensatory growth model (Model
22), except that it fit a lower peak in 1991, and tended to increase at the end of the 19905,

like the age 2-4 compensatory survival model.

   
  
  

600 1
50° 1 '1 +Modeled
m 400 —-I-— Observed
q
E 300 1 ‘
U 1
200 1
100 1 u. -« fix
0 41d. T‘j—TT T—T V 1"‘1—“1' l ' "IT—7' f' T’Al—‘T'I F_lhl'r l 1: lfi—Tfl.'i
'— In ON M l\ H V) ON
I\ l\ 00 00 ON ON ON
a: 2: a: a: a: a: 9: a:
Year

Figure 23. Model 27: Age 2-4 compensatory growth based on observed density, with
a perturbation.
Model 28. Age 2-4 compensatory survival and growth
I combined the model of compensatory survival of age 2-4 yellow perch with the

lowest RSS (Model 25) with the model of compensatory growth of age 2-4 yellow perch
53

impacting fecundity with the lowest RSS (Model 27), increasing the number of
parameters to 9. This combined model (Model 28; Figure 24) did not match the observed
data better than the individual models. In contrast to previous survival and growth
combination models (Model 11 and Model 23), an age 2-4 combination did not lower the
RSS because it could not model the variability in the observed data as well as the other

combined models.

   

600
1
500 1 1!1 +M0deled
400 1
m 1 -——I— Observed
E 300 1
U 1
200 1
100 1
0 _1_ TT'TJTWI-di-TT—l l 7771 1 Til. 1 7—7 1' #1 177—111 1 ir-T T—r—T’fiml
-~ In ON ('3 l\ '— 10 ON
I\ l\ l\ 00 00 ON
3‘. E} 3‘ 2 51‘ ‘33 9.3 E?
Year

Figure 24. Model 28: Age 2-4 compensatory growth and survival with a perturbation.

A multi-age hypothesis
Model 29: Compensatory survival of two age groups

Of all the compensatory survival models, Model 15 (age 1) and Model 24 (age 2-
4) had the lowest RSS. A combination of these two models (Model 28) simulating
compensatory survival at more than one age group increased the number of parameters

but did not show an improved match with the observed data (Figure 25), and did not

54

lower RSS (Table 3). In fact, Model 24 fit similarly as Model 15, suggesting the
constraints on Model 15 overrode any impacts from another model. Multi-age models
were not applied to compensatory growth models because the number of constraints

became too high, and a solution could not be found.

600 .
500 1
400 .

   
  

+ Modeled
4~ Observed

LL]
E3 300
L) .
200 »
100
O E* 1 1 ' r '
"-1 If) ON M l\ v-t V) G
I\ l\ l\ 00 00 ON ON ON
ON ON ON ON ON ON ON ON
F—‘ —4 r—fl v—I I—fl '— v—i ’—
Year

Figure 25. Model 29: Age 1 compensatory survival with a 5-year time delay
combined with age 2-4 compensatory survival and a perturbation.
Hypothesis ranking
Table 3 presents several measures used in the process of selecting and ranking
models. These measures include ( 1) a likelihood “component” of the Akaike equation,
(2) the likelihood component with the parameter penalization (complete Akaike
equation), and (3) a correction for small sample size (corrected Akaike). The likelihood
component values tracked the information provided by the residual summed squares, with
lower RSS being associated with lower likelihood values, indicating a more likely model.

The likelihood values were penalized for parameter number by increasing their value

55

within the Akaike equation, and further penalized because there was a small sample size
relative to parameter number, resulting in the corrected Akaike equation. Once the
models were ranked, they were compared to the best model and weighted accordingly.

The ten best models, based on Akaike rank are listed in Table 4, alongside the ten
best models based on RSS criterion. Some of the models were listed in both tables, thus
signifying the strength of their RSS value in combination with a low parameter number.

According to RSS criteria, the best-fitting models were those that best account for
the variability in the observed data, and therefore “visually” fit the observed data the best.
The best-fitting model was Model 3, where age 0 survival rate is freely fit for each and
every year, and a perturbation was included.

According to the Akaike criteria, the best models had a low RSS relative to the
amount of parameters. Models that ranked well, according to RSS, often were not
included in this list because the RSS was not low enough to compensate for the high
parameter number (e. g., Model 3). The models in this list were considered to be the most
likely models and were, therefore, explored more thoroughly.

Models that manipulated age 0 and age 1 survival rates dominated the top ten
models, with the exception of the second best model, which involved compensatory
survival of age 2-4 yellow perch. All involved constant or compensatory survival rates.
None of them involved compensatory grth rates (as they impact fecundity), and most
involved a perturbation. The best model was not necessarily the dominant model (i.e. its
weight value was not 80-99%). In fact, the top models all have relatively high weights,

yet they model the observed data differently, as discussed below.

56

(A)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AICc Model Like Akaike
Rank No. Model Description Pl RSS Comp3 AICc4 weightd
1 4* Constant age 0 surv5 + pert6 1 207604 379.55 383.97 0.388
2 25* Age 2-4 comp7 surv (mods) + pert 2 196549 377.85 384.74 0.265
3 8* Age 0 comp surv (obs9) + pert 2 204770 379.12 386.01 0.140
4 20* Age 1 comp surv (obs), 5 yr delay 2 196596 377.86 387.39 0.070
+ pert
5 19 Age 1 comp surv (obs), 7 yr delay 3 226446 382.24 389.13 0.030
6 18 Age 1 comp surv (obs), 5 yr delay 3 226459 382.24 389.13 0.029
7 7 Age 0 comp surv (mod) + pert 3 231190 382.88 389.77 0.021
8 17 Age 1 comp surv (obs), 3 yr delay 3 234884 383.37 390.26 0.017
9 26 Age 2-4 comp surv (mod), 5 yr 3 242189 384.32 391.21 0.010
delay + pert
10 6 Age 0 comp surv (obs) 3 271632 387.88 392.31 0.006
(B)
RSS Model
rank No. Model Description P RSS
1 3 Variable age 0 surv + pert 30 105411
2 2 Variable age 0 surv 30 124881
3 29 Age 1 (mod/5 yr delay) + 6 194997 Not
age 2-4 (mod) comp surv + pert
4 23 Age 1 surv (mod), 5 yr delay + 10 195688 Applicable
age 1 growth (obs) + pert
5 25* Age 2-4 comp surv (mod), + pert 2 196549
6 20* Age 1 comp surv (obs), 5 yr delay 3 196596
7 11 Age 0 surv (obs) + 10 197127
age 0 growth (mod) + pert
8 8* Age 0 comp surv (obs) + pert 2 204770
9 4* Constant age 0 surv + pert 1 207604
10 28 Age 2-4 surv (mod) + 9 210058
age 2-4 grth (obs) + pert

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lparameter number, 2residual sum of squares, 3 likelihood component of AICc,

4corrected Akaike Information Criteria, 5 survival, 6perturbation, 7compensatory,

8using modeled density, 9using observed density.

*Model present in both (A) and (B).

Table 4.

The ten best models based on AICc criteria (A) and residual sum of

squares (B).

57

 

 

The top-ranking model was one of the simplest, containing all constant survival
and growth rates, including age 0 survival, but with the added perturbation in 1991. The
model that ranked second was a model of compensatory survival rate of age 2-4 yellow
perch, based on modeled abundance, and with a perturbation in 1991. The third model
included compensatory survival rates of age 0 yellow perch, based on observed
abundances, with a perturbation in 1991. This model (and most of the age 0
compensatory survival rate models) did not differ much from the constant model because
modeled survival rates varied little over time. The fourth best model included
compensatory survival rate of age 1 yellow perch, based on observed abundances, with a
five-year time delay, and a perturbation in 1991. All of the best models exhibited an
increase in total abundance up until the perturbation, and a very slow increase in
abundance following the perturbation, with the exception of the second best model, in
which abundance increased more rapidly in the 19915. The first (Model 4) and the third
(Model 8) best models were similar to each other: both exhibited a steady exponential
increase in total abundance until the perturbation. Without the perturbation, total
abundance increases exponentially without a clear carrying capacity. In contrast, the
second (Model 25) and third (Model 20) best models exhibited a logistic increase in
abundance, suggesting the population is regulated by a more apparent carrying capacity.
These model pairs also showed some similarities to each other when changes in
abundance were examined age-specifically (Figure 26 and 27). Qualitatively speaking, it
was difficult to determine which model(s) matched age 1 and age 2 the best; however, all
of the models matched the overall change in CPUE at age 1, and none of them could

match the decreasing abundance of age 2 prior to the perturbation in 1991 (Figure 26).

58

Model 25 appears to match the overall trend in CPUE at age 3 and at age 4 the better than
the other models (Figure 27). Overall, Model 25 matched age-specific changes in

abundance the best, although the differences among models were not dramatic.

Auxiliary data
Compensatory survival rate

The ten best models involved either a constant survival rate (Model 4), or
compensatory survival rate (remaining top 9 models). Observed data were used to
provide additional evidence that survival rates were compensatory. Female instantaneous
mortality rates for age 0-4 yellow perch were compared to corresponding CPUE. Gear
selectivity influenced the values of the instantaneous mortality rates; therefore they are
considered to be “relative”, and were some times negative in value (Figure 28 and Figure
29).

There was a significant relationship between instantaneous mortality rate from
age 1 to age 2, and age 1 density (Figures 28A); but none for age 2 to age 3 (Figure 28B).
Mortality rate from age 3 to age 4 was not significantly related to density (Figure 29A);
but there was some indication of a relationship between mortality rate from age 4 to 5 and
density (Figure 29B). These results provide additional support for the fourth best model

(Model 20) and the second best model (Model 25).

59

Age 1
+ Observed

+ Model 4
+ Model 25

—:s—— Model 8

—9— Model 20

 

 

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+ Model 4
+ Model 25

+ Observed
—cr— Model 8

—9— Model 20

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Age 1 and age 2 CPUE from top four models.

Figure 26.

60

+ Observed

Age3

 

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Age 2 and 3 CPUE from top four models.
61

Figure 27.

Figure 28.

 

1
1
1 F139 = 10.88 P = 0.003

': . r2=0.27

 

 

 

400
6 1
1 B
1
4; F1,29=2.53 P=O.12
2 110 r2=0.08
O
o
0 W
1‘ .. . O
2 1 O . 0 fi
1
4 ' 1 1 1 1 l
50 100 150 200 25
CPUE

Relative instantaneous mortality rate (Z) versus CPUE of age 1 (A) and
age 2 (B) female yellow perch from the observed data. Age 1 CPUE =
both sexes * 0.5.

62

F1,29=1.87 P=O.18

3:006
M‘
O

 

 

 

 

 

O
l l l
50 100 150
CPUE
l
i 131,29: 3.89 P =0.06
.. 3:012
M ’ o
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i I I I I
20 40 60 80
C?UE

Relative instantaneous mortality rate (Z) versus CPUE of age 3 (A) and

age 4 (B) female yellow perch from observed data.

Compensatory growth

63

Decadal changes in female growth increments over the entire 20th century (Figure
30) suggest compensatory growth may have occurred in age 1-3 yellow perch. Growth
increments exhibited a cyclic pattern that may be indicative of compensatory grth if

changes were related to fish density. Abundance data were not comparable for these time

periods because fish were collected with different gear; however, studies suggest
abundance was relatively higher in 1943-51 (El-zarka 1959), 1968-70 (Eschenroder
1977), and the 19805, as shown in this study. For older yellow perch (age 4-6), grth

increments did not exhibit any cyclic pattern, but decreased at a continuous rate.

 

 

100 I
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o'\ «'3 c': ob ._'. ..'. ._'.
N V \O \O l\ 00 O\
9.} 1‘ 2‘ 2 2 2 2
Yearrange
Figure 30. Decadal changes in length increments of age 1-6 yellow perch in Saginaw
Bay.

64

Within the last thirty years, observed annual changes in growth increments of age
0 to age 1, and age 1 to age 2, female yellow perch (Figure 31A and B) suggest grth
was lower in the 1980s relative to the surrounding years, which is consistent with
changes in CPUE (Figure 2), suggesting a compensatory grth effect on young yellow
perch. Older female yellow perch (age 2-3), however, exhibited a pattern of slowly
increasing grth over the last thirty years (Figure 32A and B), while growth from age 4-
5 decreased recently (Figure 32C). Both of these observations are not consistent with
changes in CPUE (Figure 2). Yellow perch grth from age 2-4, on average, appears to
have been exceptionally poor within the last thirty years, as evidenced by the very small
growth increments in Figure 32A and 328. In fact, length increments from age 4-5 are
actually greater than increments from age 2-4, when it should be less.

When female growth increments from observed data were compared to
corresponding CPUE, there was very little indication that density dependent grth
occurs (Figures 33 and 34). The lack of significant relationships in the observed data for
older yellow perch provides support for the grth models, which were not effective at
modeling yellow perch abundance. There is some evidence; however, that young yellow
perch (age 0-2) do experience compensatory growth, as seen in the temporal trends in
growth increments for recent years, and the temporal trends in mean lengths over the
entire 20th century. This may have had a greater impact on survival rate, rather than
fecundity, since compensatory grth models in this study did not explain observed

CPUE.

65

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Annual changes in growth increment of female yellow perch, age 0 to age

1 (A) and age 1 to age 2 (B).

Figure 31.

66

 

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Annual changes in growth increment of female yellow perch, age 2 to age

3 (A), age 3 to age 4 (B), and age 4 to 5 (C).

Figure 32.

67

100 1 F1‘27=1..10P=O30 A

 

 

 

 

Li I r2=0.04
3 90-1. ’ . o .
15, J ’0 ° 9 o
E 80 M 6 Q
“5 .° * . . 7"“ a
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g 70 ..
t— 60 O
J: 50 A .’ . . v Q
Ea? 40 to ~ .
20..I I A - f 4:444.
0 100 200 300 400
CPUE

Figure 33. Observed female length increments from age 0 to age 1 (A) and age 1 to
age 2 (B) versus egg estimates (A) and observed female CPUE (B).

68

Length Increment (mn

Length increment (mr

Length increment (mr

Figure 34.

F127 = 2.53 P = 0.12
2 =0.08

A

Fm = 0.05 P = 0.82
r2 = 0.00

F1,27 = 0.65 P = 0.43
r2 = 0.03

CPUE

Observed female length increments from age 2 to age 3 (A), age 3 to age 4
(B), and age 4 to age 5 (C) versus observed female CPUE.

69

DISCUSSION

The goal of this study was to determine how density independent and density
dependent processes determine the abundance of yellow perch in Inner Saginaw Bay,
Lake Huron, so that fisheries managers can (I) explain observed abundance and (2)
predict future abundance. My primary objective was to determine the likelihood and
relative weights of various density independent and density dependent models that
describe how the abundance of yellow perch changed over time. I compiled a dataset of
observed abundance and growth of yellow perch, developed population models that
represented competing hypotheses, and determined which models most likely accounted
for changes in yellow perch abundance.

The top ranking models matched the general pattern that occurred in the observed
abundance; however, they did not fit the observed data in a precise way; therefore, it was
somewhat difficult to determine the actual strength of the above processes in influencing
yellow perch abundance. Density dependent processes have been shown to occur in
some fish populations (Elliot 1985, Bromley 1989, Forrester 1995, Ferrari and Taylor
1996 to name a few); however, it is usually very difficult to detect regulation in animal
populations because variability, or “noise”, in the data obscures these processes
(Murdoch 1994, Turchin 1995). Despite historical arguments over the importance of
density dependent processes in determining the abundance of animals (Andrewartha and
Birch 1954, Nicholson 1954, Milne 1962), fisheries scientists generally agree that they
are important and suggest better methods of detection could help clarify their existence
(Rose et a1. 2001). For this study, the importance of density independent and density

dependent processes was determined by evaluating (1) many models of separate

70

processes at various ages and (2) auxiliary data. The models in this study were evaluated
based on (1) their ability to visually match the observed data, (2) the residual summed
squared value, (3) the corrected AIC rank, and (4) the weighted ranks. All of these
methods identified models that best explain the variation in the observed data, but
corrected AIC ranking can be viewed as the best method to account for the trade-off
between model fit (likelihood) and the number of fitted parameters (Franklin et a1. 2001).
The top-ranking models listed in Table 4(A) were selected based on the corrected AIC
ranking. If models were selected based on RSS, which does not account for parameter
number, one would have assumed the annual free-fitting age 0 survival rate model
(Model 2) was the “best” model. This model fit the data well only because it contained
multiple free-fitting parameters. If this model had an exceptional fit to the observed data
(i.e. very low RSS), its likelihood value would have been strong enough to withstand
parameter penalization; however, this was not the case. The modeling process used in
this study showed that yellow perch abundance in Inner Saginaw Bay may be weakly
regulated with compensatory survival; however, it did not separate the relative
importance of density independent and density dependent processes because these twp
types of models often fit the data in a similar fashion (i.e., Model 4 and Model 8).

Based on the top-ranking models in this study, some of the dominant processes
that determine the abundance of yellow perch in Inner Saginaw Bay include: (1)
compensatory survival of age 2-4 yellow perch, (2) compensatory survival of age 0
yellow perch, and (3) compensatory survival of age 1 yellow perch. These are the
processes that appear to be important based on how well these models fit the observed

data, and the model selection process used in this study. Compensatory grth (as it

71

impacts fecundity) and depensation do not appear to be as important in determining
yellow perch abundance in Inner Saginaw Bay.

Despite the above results, the top-ranking model was a model in which all grth
and survival rates remained constant (Model 4). It had the highest weight of all of the
models; however, its weight (0.3 88) was not overwhelmingly higher than the other top-
ranking models, suggesting other processes are also important. Model 4 simulated
changes in yellow perch density with constant rates, including a constant age 0 survival
rate. At first, a constant age 0 survival rate does not appear to be biologically realistic
because fish populations, in general, ofien have highly variable age 0 survival rates
(Hilbom and Walters 1992). The average of these variable rates over time, however,
results in a population that either decreases or increases; therefore, a constant age 0
survival rate may be adequate in describing the overall changes in population abundance.

Model 25 ranked second, with a weight (0.265) that was not much lower than the

top-ranking model. This model suggests the population is regulated by compensatory
survival of the benthivorous age group, age 2-4. Model 8 and Model 20 (third and fourth
best models) suggest compensatory survival at age 0 (Model 8) and age 1 (Model 20) also
play a role in population regulation. It appears, however, that age 0 survival is weakly
compensatory because Model 8, which explicitly included compensatory survival,
simulated abundance almost identically to Model 4, a model with constant rates. The
significant linear relationship between survival rates estimated in the free-fitting model

(Model 2), and age 0 density, suggests age 0 survival is weakly compensatory.

72

Of the top four models, Model 25 visually matches the “age-specific” abundance
best, thereby reinforcing its rank among the top models. Auxiliary data suggest

compensatory survival may be important within this age group; however, these data best
support compensatory survival at age 1. In combination with the results of other studies
(Diana and Salz 1991; Haas and Schaeffer 1991; Schaeffer et a1. 2000), it seems likely
that yellow perch in Inner Saginaw Bay have been regulated to some extent by
compensatory survival at age 2-4 over the last 30 years. It is not as clear, however,
whether compensatory grth causes compensatory survival during that time.

Compensatory survival is commonly controlled by compensatory growth
(Goodyear 1980); however, I could not determine if compensatory grth caused
compensatory survival in this study because I could not model density dependent impacts
on these rates separately within the same model. The auxiliary data in this study,
however, suggest the following. If observed yellow perch grth increments over the last
thirty years (Figure 30) are compared to yellow perch CPUE over the last thirty years
(Figure 2A), the two are not inversely related. In addition, growth of age 3 and 4 yellow
perch has steadily declined over the entire century, and has been very low over the last 30
years (Figure 30), suggesting fish density at these ages has not impacted grth as much
as other factors, such as the disappearance of Hexagenia sp. in the 1960s (Schaeffer et al.
2000). Historical mean length increments of age 1 and age 2 yellow perch suggest fish
size has changed in a cyclic pattern over the whole century (Figure 30); however, it is not
known whether these changes in growth were dependent on fish density. It may be that
the perturbation in 1991 has played the greatest role in determining the abundance of

yellow perch in Inner Saginaw Bay.

73

During the process of modeling yellow perch abundance in Inner Saginaw Bay, I
discovered the necessity of the perturbation in 1991. This sudden change in abundance
was observed between fall of 1989 and fall of 1990. This perturbation improved the fit of
almost every model, and was present in six of the top ten models. It appeared to be a
necessary feature to account for the sudden change in abundance that occurred at this
time. Without the perturbation, I could not model this change accurately. The need for
this process suggests that mortality events affecting the total abundance of yellow perch
occur sporadically. Fisheries managers often target recruitment mechanisms to account
for changes in fish abundance because small changes in recruitment can have significant
impacts on the abundance of fish (Ricker 1975, Cushing 1981). Yet, the need for a
perturbation in these models, in which the survival rate of all ages of yellow perch was
reduced, suggests that poor recruitment alone could not account for the sudden drop in
total abundance that occurred in 1991 in the models. Observed age and year specific
CPUE shows that the abundance of all age group dropped dramatically between 1990 and
1991.

A perturbation suggests that a poor recruitment rate throughout the 1990s may not
explain poor abundance in the 1990s. Instead, abundance was likely low during this time
period because the dramatic reduction in stock size in 1991 made it impossible for the
population to rebuild itself for some time. A relatively simple model with a perturbation
(e.g. Model 25) can adequately explain the changes in abundance that have occurred for
this population, rather than a model with variable recruitment. When recruitment was fit
for every year (Model 3), age 0 survival rates were not consistently low in the 19908

(Figure 35). The cause of the perturbation is not known. It has been suggested that

74

yellow perch of all ages may have collapsed their food base by 1991 because they
became so abundant in the 19808 (R.C. Haas, personal communication). Other possible
mechanisms include disease (Mills and Hurley 1990), winterkill (Cooper and Washbum
1946, Tonn and Paszkowski 1986, Hall and Ehlinger 1989), and rapid temperature
fluctuations (Busch et al. 1975). Yellow perch in Lake Michigan exhibited a similar drop
in abundance around the same time as yellow perch in Inner Saginaw Bay (Makauskas
and Clapp 2000), suggesting a regional density independent event could have caused

widespread reductions

0.0016 1
0.0014 1
0.0012 1
0.001
0.0008 1
0.0006 a
0.0004
0.0002 7

011 ‘I III—1’ Ir‘liiI—I—l lfl—fiI’kl’T—T—VT l l I l I I m

1971 1975 1979 1983 1987 1991 1995 1999

Survival rat:

 

 

Year

Figure 35. Estimated age 0 survival rates from the free-fitting model with a
perturbation (Model 3).
in yellow perch. However, yellow perch populations in Lake St. Clair (Synnestvedt
1996) and in Lake Erie (Tyson and Knight 2001) have flourished in the last decade.
In summary, the results of this study identified the processes that may be

important in determining the abundance of yellow perch in Inner Saginaw Bay, Lake

75

Huron. Compensatory survival at age 2-4 appears to be the dominant regulatory process,
although the regulation does not appear to be strong. A reduction in survival across all
age groups (perturbation) was needed to match the dramatic drop in abundance that
occurred in 1991, suggesting low abundance in the 1990s resulted from a perturbation in
1991, rather than poor recruitment throughout the 1990s. Finally, predictions indicate
yellow perch abundance will slowly increase in the future, a recovery that could be

hastened if fishing mortality is reduced.

Management Implications

Fisheries managers can target management actions at specific processes and age
groups to enhance the population if they know the rates and ages at which regulation
occur. The above discussion suggests yellow perch in Inner Saginaw Bay were
regulated the most by compensatory survival at age 2-4 in the last thirty years. It is not as
clear, however, whether compensatory grth caused compensatory survival during that
time, only that growth of this age group has been very poor. Management may be most
effective if it were to control the abundance of age 2-4 in a way that produced optimal
survival rates. Any management action to improve growth rate, such as habitat
improvement, would improve the quality of fish, but may not be a dominant force in
determining the abundance of fish.

The models in this study are also beneficial to fisheries scientists as predictive
tools. Compensatory impacts can be incorporated into models that predict absolute
abundance of yellow perch (e. g. catch-at-age analyses), or they provide relative

predictions of future abundance on their own. The best model (constant age survival) and

76

third best model (age 0 compensatory survival) predict future abundance to continuously
increase in the near future (Figure 36), yet they do not include an inevitable upper
threshold. The second best model (age 2-4 compensatory survival rate) and third best
model (age 1 compensatory survival rate) predict a more logistic change in abundance, in
which density increases, but eventually levels off at some point (Figure 37).

These predictions are consistent with the different behavior that these model
“pairs” exhibit. In Model 2 and Model 8, there is no clear carrying capacity, and the
population is reduced by a single event that has a strong impact on survival rate (the
perturbation). In contrast, Model 25 and Model 21 exhibit a logistic pattern, suggesting
the population was regulated by a more apparent carrying capacity. In these models, the
population was self-regulating, was interrupted by a catastrophic event, and continued to
self-regulate itself following the perturbation. For yellow perch in Inner Saginaw Bay,
both of these behaviors are likely occurring. When managers know the relative
contributions that density independent and density dependent processes make to fish
abundance, they are able to separate events into those that they can control (density
dependent processes), and those that they cannot control (density independent processes).
The density dependent models in this study did not show strong relationships between
density dependent rates and observed abundance of yellow perch; therefore, managers
may not be able to manage for an Optimal abundance of yellow perch. Since a
perturbation caused a large reduction in abundance of this population, fisheries scientists
may wish to collect information that could identify the mechanism(s) that cause a
perturbation. Although it is not known whether widespread mortality resulting from a

perturbation is more likely to occur when yellow perch abundance is high, increased

77

600 ;
500

400 f

CPUE

200 7

100

600

500 '

400

300

CPUE

200

100

Figure 35.

300 -

 

——Observed
------ Model4
' ' - Model 8

 

 

 

1981 7;
2011 ?
2016 i

 

1 Observed
1 ------ Model 25
- - - Model 20

 

 

 

T l i TT—T—T—T—‘T‘W‘I‘LT l i I I I_T"”_' '

'—
O
O
N

1986
2006
2016 f

1981 -3

Year

Projected Inner Saginaw Bay yellow perch abundance based on Model
4 (Constant age 0 survival rate with a perturbation) and Model 8 (Age
0 compensatory survival with a perturbation) (A), and Model 25 (Age
2-4 compensatory survival with a perturbation) and Model 20 (Age 1
compensatory survival with a 5 year time delay and a perturbation)

(B).

78

abundance of yellow perch prior to the perturbation in this study suggests this may be a
factor. When yellow perch abundance in Inner Saginaw Bay increases to levels similar to
the last period of high abundance in the 19805, fisheries researchers should retain fish for
disease evaluation, and collect detailed information on ice cover, temperature changes,
and food availability.

An immediate management concern for yellow perch in Inner Saginaw Bay is
whether this population has the ability to increase, given that abundance has remained
consistently low in the last ten years. The models suggest these fish are capable of better
survival at low abundance; however, many of the models show an increased abundance
sooner than what was observed in the 1990s. The perturbation in 1991, which affected
all age groups, reduced abundance enough that it may take many years for yellow perch
to be recruited into the fishery, grow and replenish the stock. If recruitment is
consistently poor due to density independent events, this process will be prolonged and it
may take some time for the population to rebound. The population may, however,
rebound more quickly if birth rates increase and/or survival rates increase. If fishing
mortality is reduced, it may be possible for the population to rebound more quickly than
if fishing mortality remains at the current level. Managers need to consider a reduction in

fishing pressure if they want the population to rebound more quickly.

79

APPENDICES

80

Appendix A. List of parameters (and their fitted values) for each model.

Model 1: Constant age 0 surv
1 parameter:
Age 0 survival rate = 0.0003.

Model 2: Variable age 0 surv
30 parameters:

30 age 0 survival rates:
1971 1972 1973 1974 1975 1976 1977 1978
0.0003 0.0006 0.0006 0.0002 0.0007 0.00005 0.0001 0.0002

1979 1980 1981 1982 1983 1984 1985 1986
0.0003 0.00004 0.0001 0.0003 0.0001 0.0001 0.0001 0.0001

1987 1988 1989 1991 1991 1992 1993 1994
0.0001 0.0001 0.0002 0.00002 0.0001 2.8E-06 1.0E-06 7.7E-06

1995 1996 1997 1998 1999 2000
0.0001 0.0001 0.0004 0.0003 0.00004 0.00004

Model 3: Variable age 0 surv + pert
30 parameters:
29 age 0 survival rates:
1971 1972 1973 1974 1975 1976 1977 1978
0.0003 0.0013 0.0012 0.0004 0.0014 0.0001 0.0002 0.0004

1979 1980 1981 1982 1983 1984 1985 1986
0.0005 0.0001 0.0003 0.0006 0.0002 0.0002 0.0002 0.0001

1987 1988 1989 1991 1991 1992 1993 1994
0.0002 0.0003 0.0007 na 0.0004 0.0002 0.00003 0.0001

1995 1996 1997 1998 1999 2000
0.0004 0.0003 0.0013 0.0007 0.0001 0.00005

Perturbation in 1991.

81

Model 4: Constant age 0 surv + pert
2 parameters:

Age 0 survival = 0.000383.
Perturbation in 1991.

Model 5: Age 0 comp surv (mod)
2 parameters:
Age 0 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.0001 and b = 1.5389

Model 6: Age 0 comp surv (obs)
2 parameters:
Age 0 survival = exp(-Z) = exp(-a-(b*N)) where a = 8.2247 and b = 0.0001

Model 7: Age 0 comp surv (mod) + pert

3 parameters:

Age 0 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.0001 and b = 1.4984
Perturbation in 1991.

Model 8: Age 0 comp surv (obs) + pert

3 parameters:

Age 0 survival = exp(-Z) = exp(-a-(b*N)) where a = 7.954 and b = 0.0001.
Perturbation in 1991.

Model 9: Age 0 comp growth (mod)

4 parameters:

Age 0 length increment = a1 - b1*length = a2/(b2+density) - (b1*length) where a2 =
15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.4
(constraint: 0.4<=b l <=0. 1 )

Age 1-8 length increment = 75 - b3*length where b3 = 0.4 (constraint:

0.4<=b3<=0. 1)

Age 0 survival rate = 0.0003

Model 10: Age 0 comp growth (mod) + pert

5 parameters: -

Age 0 length increment = a1 - bl*length = a2/(b2+density) - (b1*1ength) where a2 =
15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.4
(constraint: 0.4<=b1<=0. 1)

Age 1-8 length increment = 75 - b3*length where b3 = 0.2322 (constraint:
O.4<=b3<=0. 1)

Perturbation in 1991.

82

Model 11: Age 0 surv (obs) + age 0 growth (mod) + pert

8 parameters:

Age 0 survival = exp(-Z) = exp(-a-(b*N)) where a = 7.602 and b = 0.002.

Age 0 growth increment = a1 + b1*length = a2/(b2+density) + (bl *length)
where a2 = 15,0000, b2 = 161, and bl = -0.4

Age 1-8 growth increment = a3 + b3*length where a3 = 75 and b3 = -O.4.

Perturbation in 1991.

Model 12: Depensation at age 0 (obs)

4 parameters:

N(l) = pSexp((S/g*mS/h2+S2) where p = 0.00017, g = 1.54E+12, m = 0, and
h = 141,969.

Model 13: Depensation at age 0 (obs) + pert

5 parameters:

N(l) = pSexp((S/g*mS/h2+SZ) where p = 0.00018, g = 1.54E+12, m = 0, and
= 141,969.

Model 14: Age 1 comp surv (mod)

3 parameters:

Age 0 survival rate = 0.00004

Age 1 survival rate = exp(-a-(b*N)) where a = 0.0001 and b =0.0138

Model 15: Age 1 comp surv (obs)

3 parameters:

Age 0 survival = 0.0004

Age 1 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.2876 and b = 0.0129

Model 16: Age 1 comp surv (mod), 3-yr delay

3 parameters:

Age 0 survival rate = 0.0004

Age 1 survival rate = exp(-a—(b*N)) where a = 0.0001 and b =0.0040

Model 17: Age 1 comp surv (obs), 3-yr delay

3 parameters:

Age 0 survival = 0.0004

Age 1 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.0001 and b = 0.0051

83

Model 18: Age 1 comp surv (obs), 5-yr delay

3 parameters:

Age 0 survival = 0.0004

Age 1 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.0001 and b = 0.0031

Model 19: Age 1 comp surv (obs), 7-yr delay

3 parameters:

Age 0 survival rate = 0.00004

Age 1 survival rate = exp(-a-(b*N)) where a = 0.0001 and b =0.0022

Model 20: Age 1 comp surv (obs), S-yr delay + pert

4 parameters:

Age 0 survival = 0.0003

Age 1 survival = exp(-Z) = exp(-a—(b*N)) where a = 0.0001 and b = 0.0015
Perturbation in 1991.

Model 21: Age 1 comp growth (mod)

5 parameters:

Age 1 length increment = a1 - b1*length = a2/(b2+density) - (b1*length) where a2 =
15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.4
(constraint: 0.4<=b l <=0. 1)

Age 2-8 length increment = 75 - b3*length where b3 = 0.4 (constraint:
0.4<=b3<=0.1)

Age 0 survival rate = 0.0004

Model 22: Age 1 comp growth (mod) + pert

6 parameters

Age 1 length increment = a1 - b1*length = a2/(b2+density) - (bl*length) where a2 =
15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.4
(constraint: 0.4<=b1<=0. 1 )

Age 2-8 length increment = 75 - b3*length where b3 = 0.1765 (constraint:
0.4<=b3<=0. 1)

Age 0 survival rate = 0.0003

Perturbation in 1991.

84

Model 23: Age 1 surv (mod), 5-yr delay + age 1 growth (obs) + pert

8 parameters:

Age 1 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.00001 and b = 0.0014

Age 1 length increment = a1 - b1*length = a2/(b2+density) - (b1 *length) where a2 =
15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.4
(constraint: 0.4<=b l <=0. 1)

Age 2-8 length increment = 75 - b3*length where b3 = 0.1607 (constraint:
0.4<=b3<=0. 1)

Age 0 survival rate = 0.0003

Perturbation in 1991.

Model 24: Depensation at age 1 (obs) + pert

6 parameters:

Age 0 survival = 0.0003

N(1) = pSexp((S/g*mS/h2+SZ) where p = 0.6093, g = 1.29E+12, m = 0, and
h = 137,418

Perturbation in 1991 .

Model 25: Age 2-4 comp surv (mod) + pert

Age 0 survival fixed at 0.00008

3 parameters:

Age 2-4 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.0785 and b = 0.0043
Perturbation in 1991.

Model 26: Age 2-4 comp surv (mod), 5 yr delay + pert

Age 0 survival fixed at 0.00008

3 parameters:

Age 2-4 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.1231 and b = 0.0001
Perturbation in 1991.

Model 27: Age 2-4 comp growth (obs) + pert

6 parameters:

Age 2-4 length increment = a1 - b1*length = a2/(b2+density) - (bl *length) where
a2 = 15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.1
(constraint: 0.4<=b1<=0. 1)

Age 5-8 length increment = 75 - b3*length where b3 = 0.2528 (constraint:
0.4<=b3<=0. 1)

Age 0 survival rate = 0.0003

Perturbation in 1991.

85

Model 28: Age 2-4 surv (mod) + age 2-4 growth (obs) + pert

8 parameters:

Age 2-4 survival = exp(-Z) = exp(-a-(b*N)) where a = 0.1182 and b = 0.0025
Age 2-4 length increment = a1 - bl*length = a2/(b2+density) - (bl *length) where
a2 = 15000 and b2 = 200 (constraint: a2>=15000 and 40<=b2<=200) and bl = 0.1
(constraint: 0.4<=b1<=0. l )

Age 5 to 8 length increment = 75 - b3*1ength where b3 = 0.1 (constraint:
0.4<=b3<=0. 1)

Age 0 survival = 0.0001

Perturbation in 1991

Model 29: Age 1 (mod/S-yr delay) and age 2-4 (mod) comp surv + pert

6 parameters:

Age 0 survival rate = 0.0002.

Age 1 survival rate = exp(-Z) = exp(-a—(b*N)) where a = 7.954 and b = 0.0001.
Age 2-4 survival rate = exp(-Z) = exp(-a-(b*N)) where a = 0.0783 and b = 0.0036.
Perturbation in 1991.

86

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