SIUS
ABSTRACT
A GENERAL NUMERICAL MODEL
FOR EVALUATING SIZE LIMIT REGULATIONS
WITH APPLICATION TO MICHIGAN BLUEGILL
(LEPOMIS MACROCHIRUS RAFINESQUE)
BY
Kelley David Smith
OI
S
A general numerical model was developed which can be
used to study the biological response of a fishery to a
variety of size limit restrictions. Effects of minimum,
maximum (inverted), or slot size limits, or a catch-and-
release regulation can be studied using this model. Fishing
and hooking mortality are adjustable for simulating effects
of different gear types and restrictions. Density-dependent
growth can be used and seasonal fluctuations of growth may
be assessed. It is also possible to model
fluctuation in fishing mortality, including shifts in season
length or time periods.
Effects of a 7.0 inch maximum size limit (i.e., all
fish 7.0 inches and longer have to be returned to the water)
were analyzed for slow, average, and fast growing bluegill
(Lepomis macrochirus Rafinesque) populations in Michigan.
Density-dependent mortality was used to estimate the number
of fry surviving to age I. Density-dependent growth was
simulated using a relationship between number of fry
produced and total initial mean length.
Equations were developed to simulate the processes of
mortality (natural, fishing, and hooking), growth, and
recruitment. Number of fish in a population, number
harvested, number caught and released, number lost to
hooking mortality, and number of natural deaths were
calculated using these equations and length-frequency
information. Yield was calculated for harvested and caught
a
S
and released fish using a length-weight regression.
Model simulations demonstrated that a 7.0 inch maximum
size limit restriction was not effective in controlling
bluegill populations. Variable and constant recruitment
were modeled separately, and in neither case did the size
limit regulation increase the number of bluegills 7.0 inches
and larger, nor did mean length at each age change
appreciably as compared to the same populations simulated
under a 5.0 inch minimum size limit.
The greatest impact was observed in the fast growing
population (using constant recruitment). Equilibrium
numbers of 7.0 inch plus bluegills increased from 589 fish
under a 5.0 inch minimum to 724 fish under the 7.0 inch
maximum restriction, at a fishing mortality rate (m) of 0.40
for both. Total catch remained about equal - 2,038 and
2,089 fish per year for a 5.0 inch minimum and a 7.0 inch
maximum size limit respectively, but legal harvest dropped
from 1,530 under the existing conditions to 1,332 fish per
year for the special regulation. As the conditional fishing
mortality rate decreased, the population characteristics
became virtually equal for the 5.0 inch minimum and the 7.0
0
e
U10
a
MO
a
e
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inch maximum restrictions.
A GENERAL NUMERICAL MODEL
FOR EVALUATING SIZE LIMIT REGULATIONS
WITH APPLICATION TO MI CHIGAN BLUEGILL
(LEPOMIS MACROCHIRUS RAFINESQUE)
van
'- --
by Kelley David Smith
A thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science
(Natural Resources)
in the University of Michigan
1981
Master's Committee:
Dr. Richard L. Patterson, Chairman
Dr. Alvin L. Jensen
Dr. William C. Latta
For Mom and Dad
With Love
::
-
ACKNOWLEDGEMENTS
I would first like to thank Dr. Richard L. Patterson"
not only for chairing my thesis committee, but also for
advising me throughout my entire Master's program. His .-
eagerness and willingness to help in any matter went far and
above any necessary duties. With his generous help, my
years at the University of Michigan have been a wonderful
experience. I would also like to thank Dr. Alvin L. Jensen
for his critical review of this manuscript. His expertise
in the field of fisheries modeling was extremely helpful
during the development of this research.
I also want to express my thanks to the employees of
the Institute for Fisheries Research who gave of their time
willingly. Dr. William C. Latta reviewed the thesis and was
very helpful in the data collection and analysis. Mercer
H. Patriarche, Percy W. Laarman, James C. Schneider, and
Roger N. Lockwood also gave much of their time and knowledge
in the analysis of the biological data published for
bluegills. James R. Ryckman helped in some of the
statistical analysis as did Dr. Gary W. Fowler from the
School of Natural Resources at the University of Michigan.
Dr Latta obtained funding for the project from the
Michigan Department of Natural Resources, which was greatly
appreciated.
I am also indebted to Richard D. Clark, Jr. who allowed
me to work on and modify his trout model. His willingness
U
ii
to help and the long hours he devoted to discussing and
interpreting some of the theoretical aspects of the model
were greatly appreciated and very useful in the development
of this study. Without his generosity and inspiration, this
study may never have been undertaken.
Finally, I would like to thank my mother and father,
Hadley and Maureen Smith for their love and understanding
during my time here at the University. Without their help,
this reality may have forever remained a dream.
iii
TABLE OF CONTENTS
Page
DEDICATION
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ACKNOWLEDGEMENTS ..
LIST OF FIGURES. ...
LIST OF TABLES ....
ABLES . . . .
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vi
INTRODUCTION ......
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MODEL ASSUMPTIONS. i . i
Mortality . . . . . .
Growth. . . . . . . .
Recruitment, . . . . .
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MODEL DEVELOPMENT. . . . . . . . . . . . . . . .
Model Variables ............
Mortality . . . . . . . . .
Growth. . . . . . .
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RecrUI Cuent . . . . . . . . . . . . . . . . . .
Combining Mortality, Growth, and Recruitment..
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NNN
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mm m d
•
MODEL APPLICATION.
Population Characteristics. . . . . . . . .
Recruitment . . . . . . . . . . . . . . . . .
Fishing Mortality . . . . . . . . . . . . . .
Hooking Mortality . . . . . . . . . . . . . . .
Seasonal Distribution of Natural Mortality,
Growth, and Fishing Mortality . . . . . . . .
a ono A A
51
RESULTS AND DISCUSSION . . . . . . . . . . . . .
Results Using Variable Recruitment. . . . . . .
Results Using Constant Recruitment. . . . . . .
66
SUMMARY.
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APPENDIX
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BIBLIOGRAPHY ...........
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104
iv
LIST OF FIGURES
Page
1. Flow diagram depicting input strategy and
output objectives to be analyzed using the
population model for warmwater inland lake
fisheries .
. . . . . . . . .
.
----
-----...da
A graphical representation depicting the
interactions between competing processes of
loss acting in a fish population. . . . . . . .
Hypothetical length-frequency distribution of a
fish cohort depicting a slot size limit.
regulation. . . . . . . . . . . . . . . . .
4.
A graphical representation depicting the
interactions between competing processes of
removal acting in a fish population which has
been divided into two independent groups. ....
. 5.
Fit of a stock-recruitment curve to observed
fry densities. · · ... · · · · · · · · · · · ·
6. Fit of a length-density regression equation to
observed mean lengths of fry. . . . . . . . .
7.
Possible hypothetical functions relating fry
survival to the density of 7.0 inch and larger
bluegills . . . . . . . . . . . . . . . . .
LIST OF TABLES
Page
Strata (mean total length in inches) used to
classify slow, medium, and fast growing.
bluegill populations in Michigan. .......
34
2.
----
.
---
-
--
Estimated mean total length in inches (1),
standard deviation of length (s), and mean
weight in pounds (w) of slow, medium, and fast
growing bluegill populations in Michigan. ...
36
Initial population (assuming a 100 acre lake
and 100 fish per acre) and associated annual
conditional natural mortality rates (n) for
slow, medium, and fast growing bluegill
populations in Michigan . . . . . . . . . . . .
37
Parameter values (and coefficients of
determination) estimated for Ford's growth
equation and the length-weight relationship for
slow, medium, and fast growing bluegill
populations in Michigan i... . . . . . . .
The number of weeks allotted to each month, and
monthly percentage distribution of natural
mortality (%n), growth (%), and fishing
mortality (%m). . . . . . . . . . . . . . . .
48
Predicted equilibrium number and attained mean
length (1) by age group with variable
recruitment for two size limit regulations and
three annual conditional fishing mortality
rates (m) for a slow growing bluegili
population in Michigan. . . . . . . . . . . . .
54
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.40 for a slow growing bluegill
population in Michigan. . . . . . . . . . . . .
55
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming. variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.20 for a slow growing bluegill
population in Michigan. . . . . . . . . . . . .
50
vi
.
.
9.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
an annual conditional fishing mortality rate
(m) of 0.10 for a slow growing bluegill
population in Michigan. . . . . . . . . . . . .
57
Predicted equilibrium number and attained mean
length (1) by age group with variable
recruitment for two size limit regulations and
three annual conditional fishing mortality
rates (m) for a medium growing bluegill
population in Michigan. . . . . . . . . . . . .
------.. je...
-.-
58
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
· regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.40 for a medium growing bluegill
population in Michigan. . . . . . . . . . . . .
59
12.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.20 for a medium growing bluegill.
population in Michigan. . . . . . . . . . . . .
60
13.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.10 for a medium growing bluegill
61
14.
Predicted equilibrium number and attained mean
length (ī) by age group with variable
recruitment for two size limit regulations and
three annual conditional fishing mortality
rates (m) for a fast growing bluegill
population in Michigan. . . . . . . . . . . . .
62
15.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.40 for a fast growing bluegill
population in Michigan. . . . . . . . . . . . .
63
vii
16.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.20 for a fast growing bluegill
population in Michigan. . . . . . . . . . . . .
64...
17.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming variable recruitment and
an annual conditional fishing mortality rate
(m) of 0.10 for a fast growing bluegill
population in Michigan. . . . . . . . . . . . .
-
-•-.
-
--
--
65
18.
Predicted equilibrium number and attained mean
length (1) by age group with constant
recruitment for two size limit regulations and
three annual conditional fishing mortality
rates (m) for a slow growing bluegill
population in Michigan. . . . . . . . . . . . .
68
19.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.40 for a slow growing bluegill
population in Michigan. . . . . . . . . . . . .
20.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.20 for a slow growing bluegill
population in Michigan. . . . . . . . . . . . .
70
21.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.10 for a slow growing bluegill
population in Michigan. . . . . . . . . . . .
71
22.
Predicted equilibrium number and attained mean
length (1) by age group with constant
recruitment for two size limit regulations and
three annual conditional fishing mortality
rates (m) for a medium growing bluegili
population in Michigan. ............
72
viii
23.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.40 for a medium growing bluegill
population in Michigan. . . . . . . . . . . . .
73
24.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.20 for a medium growing bluegill
population in Michigan. . . . . . . . . . . . .
--. ----.
...
.
.
74
25.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.10 for a medium growing bluegill
population in Michigan. . . . . . . . . . . . .
75
26. Predicted equilibrium number and attained mean
length (1) by age group with constant
recruitment for two size limit regulations and
three annual conditional fishing mortality
rates (m) for a fast growing bluegill
population in Michigan. . . . . . . . . . .
76
27.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.40 for a fast growing bluegill
population in Michigan. . . . . . . . . . . . .
77
28. Predicted equilibrium levels of density, catch,
• and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.20 for a fast growing bluegill
population in Michigan. . . . . . . . . . . . .
78
29.
Predicted equilibrium levels of density, catch,
and yield by length group for two size limit
regulations, assuming constant recruitment and
an annual conditional fishing mortality rate
(m) of 0.10 for a fast growing bluegill
population in Michigan. ..........
79
INTRODUCTION
Management techniques developed for the purpose of
controlling and improving stunted or slow growing bluegill
(Lepomis macrochirus Rafinesque) populations are many and
often controversial. Techniques such as partial or complete
poisoning and restocking of lakes, poisoning of spawning
beds, encouragement of predatory fishes, or lowering of
water levels to expose fingerlings to predation have either
failed or only worked for short periods of time (Snow et al.
1960; Hooper et al. 1964; Beyerle and williams 1967 and
1972; Schneider 1973b; Becker 1976; Beyerle 1977; Novinger
and Legler 1978). These methods were usually discontinued
because funds needed for reapplication or use on a widescale
basis were not available.
New types of management techniques, which are not
limited by money or manpower, have become the focus of
recent studies. Regulations aimed at controlling a
particular species have been imposed on lakes and streams.
Minimum, maximum (inverted), or slot size limits, catch-and-
release, and gear restrictions are being used to optimize
survival, growth rate, and ultimately the yield of a fishery
(Patriarche 1968; Schneider 1973a and 1978; Clark et al.
1979 and 1980).
Optimum yield can be defined as total harvest, "trophy"
harvest, or total number of fish caught and released,
depending on anglers' preference. Because fishermen define
"quality fishing" in a number of ways, many sociological
problems arise when implementing such complex regulations
(Gulland 1968; Anderson 1975; Weithman and Anderson 1978).
Although angler reaction and behavior can cause any fishery
restriction to fall short of expected goals, biological'...
response of the population is also of importance.
Management techniques should not be employed until a careful
theoretical analysis of the biological response by the
population has been made using a variety of possible
environmental conditions. With the widespread use of
computers, numerical modeling is growing as a feasible and
useful tool. in fishery management (Clark and Lackey 1976;
Clark et al. 1977). Biological response of fish
populations to size limit regulations may be simulated and
studied in detail. This allows selection and use of the
best size restriction for achieving management objectives,
hopefully eliminating most of the field oriented trial and
error process.
The purpose of this study was twofold. The major
objective was to develop a general numerical model which
would aid in assessing the biological response of fish
populations to a variety of size limit regulations. This
model was patterned after one developed for trout fisheries
by Clark et al. (1980). Five basic improvements were made
on Clark's model to make it more applicable for warmwater
inland lake fisheries (e.eu, bass, bluegill, and similar
species). These modifications included the use of density-
dependent growth, a normal distribution of lengths at each
-
-
3
n
age, a more explicit definition of the annual conditional
hooking mortality rate, a breakdown of the annual
conditional fishing and hooking mortality rates into a
probability of capture and a probability of death given
capture, and finally, distributing the different types of
losses that are interacting in a fish population as a
percentage of the competing mortality rates. The advantages
of these improvements will be discussed as each is
developed.
The second step of this research was to apply the model
to a study of Michigan bluegill. Computer simulation was
used to evaluate the effects of a 7.0 inch maximum limit on
slow, average, and fast growing bluegill populations in
Michigan. A FORTRAN computer program, Size Limit Regulation
Analysis (S.L.R.A.), was written to perform the model
simulations. A' listing of the program appears in the
appendix.
MODEL ASSUMPTIONS
it
Perfect simulation of real world processes is neither
feasible nor possible. All numerical models must be built
using basic assumptions as the starting foundation. If the
assumptions are good approximations of the situation being
simulated, then model performance is usually adequate. If
too many assumptions are made or many are unrealistic, model
results can be disastrous. The assumptions used to build
this model are discussed in the following sections.
Mortality
Mortality includes many subdivisions which may be
treated separately. Natural mortality at each age was
assumed to remain constant from one year to the next. It
was distributed by percentage throughout each year.
Survival of fry was assumed to be density-dependent. A
regression equation relating number of eggs to survival rate
was used to estimate the number of fry reaching age 1.
Fishing and hooking mortality were developed as two
independent sources of death. Fishing mortality was applied
only to legal fish in the population ("legal" refers to fish
that can be harvested, and "illegal" to fish that must be
released). The annual conditional fishing mortality rate
(m) was constant for all age and size groups from one year
to the next. This rate was divided into two components, a
probability of capture (p') and a probability of death given
capture (d'). The latter was assumed to be constant for all
legal fish regardless of age or size.
Only illegal fish were susceptible to hooking
mortality. The annual conditional hooking mortality rate
(h) was constant for all ages and size groups from one year
to the next, and all illegal fish caught were released. -- --
This assumption excluded the effects of poaching which were
considered to be negligible. Although this could lead to
erroneous results, especially for certain species or size
limits, such an assumption was necessary because the amount
of poaching is basically unknown.
Hooking mortality was divided into two components, a
probability of capture ('') and a probability of death
after being caught and released (d''). The latter was
assumed to be constant for all illegal fish regardless of
age or size.
Growth
A basic assumption used in many fishery models is that
growth remains constant in the population (Patriarche 1968;
Clark et al. 1980). However, constant growth is not always
observed in fish populations. Changes in growth are caused
by density-dependent factors (Goodyear 1980).
Observed changes in mean length of a population over
time may be simulated if the actual growth rate in a
population remains constant but the initial mean length of a
cohort varies with changes in density of fry (Gerking 1967;
on.
--.-
.
-
-
-:
Goodyear 1980). Therefore, although each new cohort
experiences the same rate of size increase, their initial
mean length may be different than that of other previous
cohorts. This gives different lengths at each age than
previously observed for the population.
The assumption of density-dependent growth was used in
developing this model. The actual growth rate was constant
for all cohorts, while their initial mean length depended on
the number of fry hatching. Use of an expression developed
empirically by Ford (1933) and a density-dependent
regression relating number of fry to length allowed changes
in growth over time to be a function of initial length
rather than changes in the actual growth rate.
Recruitment
Many assumptions must be made in any fishery model when
dealing with a process as complex as recruitment.
Recruitment is a function of number of spawning adults,
number of eggs produced, survivorship of fry, and growth
(Goodyear 1980). Both variable and constant recruitment
were studied in separate simulation runs.
A major problem observed in many fisheries is year
class dominance caused by many density related factors
(e.g., food or available spawning areas). With variable
recruitment, it is possible to model and study this
phenomenon. By using density-dependent relationships for
fry survival, number of eggs produced, and growth, variable
-
---
-
recruitment was incorporated as a density-dependent
function. Two basic assumptions were necessary to perform
this task. First, the sex ratio was assumed to be 1:1 when
calculating the number of females in each age-length group.
Second, spawning was allowed during only one time period
each year. Although many warmwater lake species may spawn
two or more times during a summer season, a spawning "peak"
is usually observed. It was assumed that the majority of
spawning activity was accomplished during this period.
Constant recruitment was also simulated using these
same basic assumptions. Although the number of fry
surviving to age I remained constant, the number of eggs
produced and the actual fry survival were still predicted
each year to allow use of the density-dependent growth
function.
MODEL DEVELOPMENT
. A general numerical model based on a modification of
Ricker's (1975) method was developed to simulate the
processes of natural mortality, fishing mortality, hooking
mortality, growth, and recruitment. This model may be used
to study minimum, maximum (inverted), or slot size limits,..
or catch-and-release regulations by subdividing a population
into length groups of illegal and legal size fish. Model
predictions include estimates by age and length groups of
number and weight of legal fish harvested and illegal fish
caught and released, hooking deaths, and natural deaths.
Such a breakdown of population dynamics is more useful in
evaluating size limit restrictions on sport fisheries than
is one of the older models (e.q., surplus production model)
which estimate total weight of harvested fish.
Model. Variables
The variables used in a simulation model may be
categorized into three major groups. The important input,
state, and output variables used in this model are
summarized in the following section. A flow diagram showing
input strategy and output objectives is found in Figure 1.
Input:
A)
Size limit regulation.
B) Fishing pressure (seasonal distribution).
c) Density-dependent growth relationship.
9
D) Variable (constant) recruitment.
State Variables:
A) Nondynamic state variables.
1) Natural, fishing, and
hooking
mortality
rates.
2) Seasonal distribution of mortality (natural,
: fishing, and hooking) and growth.
Dynamic state variables.
1) Population numbers (weight).
2) Total catch and total catch-and-release
(numbers and weight).
3) Harvest and hooking deaths (numbers and
weight).
Natural deaths (numbers).
5) Mean length and standard deviation by age
group.
Output:
A) Population structure.
1) By age group.
2) By length-frequency distribution.
Yield.
1) Total catch and harvest.
2) Catch-and-release and hooking deaths.
C) Mean length.
10
'
'
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.
.
--
MANAGEMENT
STRATEGY
STRATEGY
EFFECT

-
SLOT LIMITS
(MAXIMUM LIMIT)
NUMBERS 7 INCHES
AND LARGER
POPULATION
THREE FISHING
MORTALITY RATES
(m = 0.40, 0.20,
AND 0.10)
SIZE DISTRIBUTION
OF THE POPULATION
MODEL
YIELD (CATCH PLUS
CATCH-AND-RELEASE)
Figure 1.
Flow diagram depicting input strategy and output
objectives to be analyzed using the population
model for warmwater inland lake fisheries.
.
...
.
.
11
Mortality
Ricker (1975) showed a method for separating mortality
(2) operating in a fish population into two major
instantaneous fishing mortality (F). Relationships between
these parameters is described by the following equations:
-*
-
.
- ...
Instantaneous total mortality rate:
Z = F
+ M
Annual expectation of death:
A = 1 - e-Zt
Annual survival rate:
S = e-Zt
Conditional mortality rates may be defined as:
Annual conditional natural mortality rate:
n = 1- e-Mt
Annual conditional fishing mortality rate:
m = 1 - e-Ft - pidi
(5).
· where:
p - probability of capture for legal fish.
d' - probability of harvest after being caught
(constant over time and age).
The use of the variable d' is an improvement of the model
developed by Clark et al. (1980). Clark's method assumed
that all legal size fish caught were harvested. The model
in this paper relaxes that assumption and allows appropriate
adjustment of the total harvest if necessary. This may be
useful in fisheries where fishermen voluntarily release many
of the fish they catch, regardless of size of the
regulations currently in effect.
By combining equation's (4) and (5), a new expression
for the annual expectation of death (A) may be written as a
sum of the conditional rates:
A = 1 -e-Zt = m + n - mn
conditional
rate
is the fraction
of the
population which would have been killed by a certain type of
mortality if no other source of death was operating in the
population. Because both natural and fishing mortality
compete for the same fish, the interaction term in equation
(6) must be subtracted to prevent a fish from being counted
as lost to more than one type of mortality.
Using this same logic, hooking mortality may be added
as a third component of total mortality by redefining
equations (1) and (6) and including a conditional hooking
mortality rate:
13
Instantaneous total mortality rate:
Z
=
F
+
M
+ H
(7)
where:
H - instantaneous hooking mortality.
Annual conditional hooking mortality rate:
h = 1 - e-Ht - p'id"
(8)
where:
D'' - probability of capture for illegal fish.
d'' - probability of a fish dying after being
caught and released (constant over time
and age).
N
This explicit definition of the annual conditional hooking
mortality rate was not used by Clark et al. (1980). They
used the same variable to describe the conditional fishing
mortality rate for legal fish and the catch rate of illegal
fish. The probability of capture (p'') used in this model
allows hooking mortality to be defined in a way that is
consistent with the definitions of the natural (n) and
fishing (m) mortality rates. This explicit definiton is
also advantageous in that separate probabilities of capture
(and separate probabilities of death given capture) may be
assigned to both legal and illegal size fish.
Using equations (4), (5), and (8), equation (6) can be
redefined as:
A = 1 - e 46 = m + n + h - m
- mh - nh + mnh
(9)
14
Including the interaction terms in equation (9), although
they may be insignificant, is necessary from both an
analytical and a biological standpoint. First, the terms
are theoretically necessary to satisfy the numerical
relationships between parameters. Second, these terms
adjust for the combined effect of the three conditional
mortality rates. Since these rates are acting concurrently,
some fish already lost to one source of mortality may also
be counted as lost to another. The Venn diagram in Figure 2
depicts this problem. Thus, from a biological viewpoint,
the interactions are necessary to prevent losing the same
fish to more than one type of mortality.
A change in the number of fish in the population from
one time period to the next is described by the equation
(Ricker 1975):
Nt+1 = Nt - NA1
(10)
Because of the assumptions made concerning the
different types of mortality, it is reasonable to subdivide
a cohort into five length groups as (Clark et al. 1980):
Nt = Pt + Rt + Tt + vt + Vt
(11)
where:
Interval Pc represents fish affected only by
natural mortality.
15
POPULATION

m - Fishing mortality
n - Natural mortality
h - Hooking mortality
Figure 2.
A graphical representation depicting the
interactions between competing processes
of loss acting in a fish population.
:
16
:
Intervals R and U, represent fish of illegal
size.
Intervals T. and Vr represent fish of legal
(harvestable) size.
This subdivision (Figure 3) allows simulation of slot limits...
directly, and other limits with slight modification.
By combining equations (9), (10), and (11), a change in
the number of fish from one time period to the next becomes:
Nt+1 = Nt - N4", - Těmi - Vm2 - Ruhy - uchy + .
Těm,+ Věm,nı + Run,hı + Utnythy - Omyhı +
Omynı hı
.
(12)
Note that the interaction terms m, h, and m, n, h, do not apply
to any part of the cohort. This is necessary to conform to
the assumptions concerning mortality. There can be no
interaction between fishing and hooking mortality because
they apply to two independent groups of the cohort as
depicted by the Venn diagram in Figure 4. The interactions
shown in Figure 4 are not between the conditional fishing
and hooking mortality rates explicitly, but between the
respective probabilities of capture as necessitated by the
variables defined above. This implies that a fish must be
caught before it can die from harvest or hooking.

-
-
-
NUMBER
-
-
-
-
-
-
-
H
X
PP Re Tu U V
- X, X₂ X₃ X4 Xs
LENGTH
Pe - Uncatchable range
Re - First illegal range
Te - First legal range
Uç - Second illegal range
Ve - Second legal range
Figure 3. Hypothetical length-frequency distribution of ai?
fish cohort depicting a slot size limit
regulation.
::
.-
18
LEGAL
FISH
ILLEGAL
FISH

al :
.
.
.
p'
D
p"
- Probability of capture for legal fish
- Natural mortality
- Probability of capture for illegal fish
re 4.
u
A graphical representation depicting the
interactions between competing processes of
removal acting in a fish population which has
been divided into two independent groups.
19
A further breakdown of N++, may be written as (Clark et
al. 1980):
N7+1 = Nt - Ct - Dt - Ht
(13)
where:
- number of legal fish harvested.
Di - number of fish dying natural deaths.
H- number of fish lost due to hooking
mortality.
Each of the three losses in equation (13) may be expressed
as a combination of terms from equation (12) as:
Legal Catch:
C+ = Tem, + Vem, - {nz/1p'1 + ny)}T4","1 -
{ny/10/2+ ny)}v_mını
(14)
Natural Deaths:
Dt = Nény - {p''1/(ny + p''])}Runghi - {p''i'
(ny + p''])}u4Nyhy - {p']/(p'+ n,)}T4 min, - .
{p'/(p'1 + n,)}V4m,
(15)
Hooking Deaths:
H+ = Ruhy + uthy - {ny/(ny + p''])}R¢n, h1 -
{ny/(ny + p''])}U+n7",
(16)
20
The interaction terms which apply to two sources of
mortality have been distributed using a ratio of the
mortality rate to the total combined effect as a weighting
factor. This seems reasonable as the number of fish
involved in an interaction term should be divided between
the two sources of death according to the respective sizes
of the mortality rates in question. This is a more accurate
division than used by Clark et al. (1980), who divided the
effects of the interaction terms equally.
Two final quantities, numbers of legal and illegal fish
caught and released that lived, may be calculated as:
JLt = TEP'i + vtP'1 - {ny/p'i + n,)}T+"yP'i -
{n/p'. + n ) }verP'1 = c/d'
.
-
.
(17)
JIE - RUP''1 + U1P''1 - {ny/1n, + p''])}R_''
i {ny/In, + p''])}U",0''; = H[/d"
(18)
Growth
. The numerical model of mortality developed in the
previous section requires the approximation of a length-
frequency distribution for a cohort to determine the number
of fish in each of the areas defined in Figure 3. Any
unimodal distribution which adequately describes the length
distribution may be used for this purpose. Clark and Lackey
(1976) demonstrated a method for approximating a length
21
frequency distribution using a three parameter Weibull
probability density function. Clark et al. (1980) used
this method in estimating length-frequency distributions for
trout. Another distribution that may be used for this
purpose is the normal (Jones 1958; Ricker 1969), with the
probability density function:
-00
<
x
<
+00
(19)
f(x) = (1/V200 )e-(x - x)2/202
where:
x - random variable (length).
H - mean of the distribution.
o - standard deviation of the distribution.
The cumulative distribution function is:
g(x) = (1/V270)
-( -.)2/202 dx
(20)
(20)
This distribution is very versatile as any normal
distribution may be standardized using the transformation:
2 = (x - 1)/0
(21)
giving:
P(X x) = P({X - } /0 S (x - H} /0 ) =
P(Z < {x - x}/0)
(22)
22
thus:
g(x)
=
f(x) dx = P(Z
<
{x
- u}/0)
(23)
If the mean and standard deviation of the distribution are
known, g(x) may be evaluated directly from a standard normal...
table.
i
Using equation (23), the number of fish in each area
defined in Figure 3 may be expressed as:
(24)
(25)
Pt = NƯ9.(X2)
R¢ = N{g(xy) - g(x2)
Tt = N{g(x3) - g(x2)}
Ut - N{g(x4] - g(x3)}
V= = N_{1 - g(x4)}
(26)
(27)
(28)
length
Using equation (23), g(x) may be defined for each
interval in Figure 3:
Vi
g(xq) = 0; = P(X s xq) = P(Z'S {xq - Mi,t}/01,t? (29)
g(x) = @z = P(X S xg) = P(Z = {xq - Wint}/01,+) (30)
g(x3) = @z = P(x = xg) = P(Z = {xz - Mi,t}/01,t) (31)
g(x4) = 24 = P(X 5 XA) = P(Z = {x4 - Hint}/01,t> (32)
23
where:
subscript "i" denotes the age of a cohort.
Mint = lint - mean length at time "t".
Substituting respective values of g(x) from equations (29)
through (32) into equations (24) through (28) gives:
(33)
(34)
Pi,t = Ni,tes
Ri,t = Ni,t(ez - ;
Ti,t = Ni,t(@z - @z?
up,t = Nint(04-03)
Vi,t • vị, (1 - x)
(35)
(36)
(37)
To simulate a fishery, the length distributions for
each cohort must be moved through time as the fish grow
larger. The fraction (G) of annual growth experienced in
some small time interval (1.0/t') was used to move
distributions through time, with:
Š G+ - 1.0
(38)
where:
t' - number of time intervals in one year.
24
The use of G allows modeling of seasonal patterns in growth
(and thus in recruitment of fish into and out of legal
ranges) to be simulated more accurately within a year.
Changes in mean length for a given cohort within a year
were expressed as:
-..-
.-
.
11,t+1 = 11,t + (GL) (11+1,1 - 11,1
(39)
Clark et al. (1980) used constant growth in their
trout model. However, this assumption is not realistic for
inland lake fisheries where growth may be affected by
density, food availability, and other environmental factors.
The assumption of density-dependent growth used in this
model necessitated the use of a function to relate the mean
length at age "i+1" to the mean length at age "i", while
keeping the mean annual growth rate constant. Such an
equation was developed by Ford (1933) of the form:
11+1,1 = Log (1.0 - k) + kli,1
(40)
where:
k - Ford's growth coefficient.
Logo - mean asymptotic length.
Ti, - mean length at age "i".
The parameters "K"
and "L" may be estimated from actual
(VONB) developed by Allen
data using a computer program
(1966 and 1967).
25
By keeping. "k" and "L" constant for all cohorts, it
can be seen from equation (40) that the mean length at any
age "i" depends on the initial length at age 1. Thus, it is
possible to observe different lengths at some age "i" for
each new cohort (depending on 1,,) even though the actual
growth rate remains unchanged.
A regression relating mean length of fry to number of
fry was used to estimate 1, for each new cohort. The form
of the equation used was:
-
-
-
-
11,1 = a + blog (FRY)
(41)
where:
a - intercept.
b - slope.
ī, , - mean length in time period i (age 1).
FRY - number of fry produced.
The mean length estimated for fry from equation (41) was
used as the mean length of age-1 fish at the start of each
year. Ford's equation (40) was used to calculate mean
lengths for all other ages using the initial mean length
estimated by equation (41).
A ratio of mean length to standard deviation for a
previous cohort at each age“ was used to calulate the
standard deviations of new cohorts at each age. The
relationship used was:
26
Oq',i,1 = (Ig',i,1)(0g,i,1)/([q,i,1'
q'>q
(42)
where:
q - denotes each cohort.
Thus, the ratio of mean length to standard deviation for all
cohorts is the same at each age "i".
The reasoning used to develop equation (42) was based
on the regression used in equation (41). As the number of
fry increases, the initial mean length of the cohort
decreases. A decrease in length should cause a
corresponding drop in the standard deviation. With
increasing density, competition for food and space would be
spread out more evenly, thus decreasing the chances for any
fish to gain or lose a competitive edge (Goodyear 1980).
Therefore, length and associated variation should both
decrease. This is often observed in lakes containing
stunted populations which have very large numbers of fish of
roughly the same length.
The opposite holds true if the number of fry decreases.
Mean length and associated deviation increase as fish have a
greater chance to gain or lose a competitive edge depending
on their ability to survive.
Standard deviations of length were moved through time
analogous to the method used for mean lengths. Using.
equations (38) and (39), this was expressed as:
.
=
ºi,t+1 = 'i,t + (GL) (0i+1,1 - 01,1)
(43)
27
Equation (43) allows changes in standard deviation through
time to be proportional to corresponding changes in mean
length.
Recruitment
Recruitment of fish depends upon the number of young.--.
produced during the spawning season and their ability to
survive to harvestable length. The total number of eggs
produced in one season was calculated as (Clark et al.
1980):
EGGS - Š
Š (FEM; ;) (FMAT;;) (EC;)
(44)
1
..
where:
.
- number of age groups.
y - number of length groups.
FEM;; - number of females in each age-length
group.
FMAT;; - percent females mature in each age-
EC
length group.
- mean egg content of females in each
length group.
Mean egg content was determined using a regression relating
length to number of eggs of the form:
EC; = a + bly
(45)
28
where:
a - intercept.
b - slope.
ly - length group.
When the total number of eggs was calculated from
equation (44), a stock-recruitment curve (Ricker 1975)
relating number of eggs produced and number of fry was used
to estimate the number of fry surviving to age 1. The form
of the equation used was:
Sp = a(EGGS) e-b(EGGS)
(46)
where:
a - intercept.
b - slope.
Se - number of fry surviving to age I.
The number of fry calculated from equation (46) was used as
the initial number of age-I fish at the start of the next
year.
Combining Mortality, Growth, and Recruitment
The numerical model thus far developed allows the
population processes of interest to be described as single.
equations.
29
Population Numbers:
Changes in number of fish
for each age group was
expressed by combining equations (12) and (33) through (37):
Ni,t+1 = Ni,t(1.0 - ,1)(1.0 - mi,11.0.- @z + @z -
04] + hi,1{@z - @z + 03 - 04}} (47)
(47)
Catch:
Total catch of legal fish was calculated using
equations (17), (35), and (37):
Ji,t = Ni,tP'1,1(1.0 - {ni,1/(p'1,1 + ni,1)}ni,1)
11.0 - ©2 + 03 - 04)
(48)
---
using equations (18), (33), (34), and (36):
.JIi,t = Ni, to''1,1({ni,1/(ni,1 + p'' 1,1}}ni,1 -
1.0/10 - 02 + 03 -).
(49)
Harvestable catch (legal size fish) was expressed as a
combination of equations (14), (35), and (37):
Ci,t = Ni,t",1(1.0 - {ni,1/(p'i,1 + ni,1)}ni,1)
(1.0 - ©2 + 03 - 04)
(50)
Hooking Deaths:
The number of fish lost to hooking mortality was
expressed using equations (16), (33), (34), and (36):
Hi,t = Ni,thi,1({ni,1/(ni,1 + p''i,1)}ni,1 - 1.0)
--.
.
lei - 02 + 03 - 04)
(51)
Natural Deaths:
Number of fish lost to natural deaths was defined as a
combination of equations (15) and (33) through (37):
Di,t = Ni,t":,161.0 + {p''1,1/(n1,1 + p''1,1)}h1,110, -
@z + 03 -04) - {p'i,1/(P'1,1 + 11,1 )}m,1
(1.0 - ©2 + 03 -0))
(52)
Yield:
Yield in weight may also be calculated using the model
equations developed for estimating catch. Length and weight
for fish of a given age were related using the regression
(Ricker 1975):
loge (ū;) = a + blogelī;)
(53)
where:
a
b
- intercept.
- slope.
31
Wi - mean weight of fish at age "i".
īj - mean length of fish at age "i".
The catch equation (14), which represents a sum of the
total harvest in two length classes (i.e., intervals Tit
and Vit in Figure 3), may be used to calculate yield in
weight as:
Fint - WT1 ,t":,1 + ūvi, t"i ,1 - {n},1!
(p'1,1 + ":,1)}ā?i, t":,1"1,1 -
{n} ,1/(p'1,1 + 11,1)Būvi,t":,191,1
(54)
where:
Wr - mean weight of a fish in interval Ti.
Wy - mean weight of a fish in interval Vit.
f
Mean weight in each interval was calculated using equation
(53) and the corresponding mean lengths in each interval (In
The mean lengths were calculated as: :
Iqxf(x) dx/(@z - Oz!
(55)
1.82
1.0
|
xf(x) dx/11.0 - 04)
.
(56)
.: 32
where:
f(x) is the normal probability density
function.
Using equations (35), (37), and (53) through (56), harvested
yield was expressed as:
71,t - Ni,t":,161.0 - {n},1/(p':,1 + 11, 2)}n: , 1'
let. 9220,1 +692103) - 24:00))
(57)
.::
where:
02 = a + blogelig
04 = a + blogelīy)
Similarly, using equation (18), yield in weight of
illegal fish caught and released was expressed as: .
YJ ,t = WpRE, EP"'1,1 + Wyli, tP''1,1 - {n},1'
(ni,1 + p''1,1) }WR Ri, t'i,10''1,1 -
{n}.,1/(ni, 1 + p''1,1)Wyi ,t":,10''1,1
(58)
.
..where:
We - mean weight of a fish in interval Ri..
Wag - mean weight of a fish in interval Ui.:
Mean lengths were calculated as:
p^2
IR = 1, xf(x) dx/(ez - @)
IR
(59)
J 81
Iy =
xf(x) dx/(- )
(60)
183
.
..
..
.
Combining equations (33), (34), (36), (53), and (58) through
(60) gave a final form for the yield in caught and released
fish:
YJi,t = Ni,tP''1,1({ni,1/(ni,1 + p''1,1)}ni,1 - 1.0)
To 92100) - 09710,1 +29710,1 - **(0,1). (61)
where:
Q1 = a + bloge (IR)
03 = a + bloge(īy)
.
MODEL APPLICATION
The model presented above was coded into a FORTRAN
program (S.L.R.A.) and used to simulate population responses
by Michigan bluegill to a 7.0 inch maximum limit. Necessary
input data were collected from a variety of sources and are
summarized in this section.
Population Characteristics
Bluegill populations were classified into three
specific groups to cover a wide range of the existing
conditions found in Michigan, Slow, medium, and fast
growing populations were defined according to the mean
length of a cohort at a given age. The stratification used
is summarized in Table 1.
Table 1.
Strata (mean total length in inches) used to
classify slow, medium, and fast growing bluegill
populations in Michigan. .
Age
Slow
Medium
Fast
2.9 - 3.9
-
•
II
III
IV
52.8
53.8
54.9
55.8
56.4
56.9
57.3
58.0
00v van WN
ano ano
24.0
25.0
26.1
27.0
27.6
28.1
28.5
29.2
VI
VII
. VIII
7.5
- 8.0
- 8.4
- 9.1
Data published by Laarman (1963) were grouped using the
above classes, and mean length with associated standard
35
deviation and mean weight were calculated for each
population at each age (Table 2).
Annual conditional natural mortality rates for each
population were obtained from data published by Schneider
(1973b) for Mill Lake in Washtenaw County, Michigan.
Natural mortality was assumed to be size- rather than age-
specific (Table 3). Plots of mortality against mean length
from Schneider's data served as a guideline for determining
the rates used in this study. These rates were allowed to
increase as growth increased and resulted in a natural
mortality rate of 55% per year for bluegills larger than 6.0
inches (Schneider 1973b).
The initial population size was chosen to be 10,000
fish and a lake size of 100 acres was used for all three
populations. The initial age structures were calculated
using the esimated natural mortality rates to determine the
number of fish at each age (Table 3).
Ford's growth equation (40) was fit to the length data
in Table 2 for each population using a computer program
(VONB) developed by Allen (1966 and 1967). The calculated
coefficients are presented in Table 4.
The regression equation (53) relating mean weight to
mean length was fit for each population using least squares.
A single regression was desired for all three populations,
but tests for equal slopes and equal intercepts were both
found to be highly significant (p<0.001). Therefore,
separate regression equations were used for the slow,
36
.

Table 2. Estimated mean total length_in inches (1), standard deviation of length (s),
and mean weight in pounds (w) of slow, medium, and fast growing bluegill
populations in Michigan.
Slow
Medium
Fast
Age
1
S
S
:
I
II
III
2.5
3.5
4.4
5.1
5.6
6.1
6.7
7.3
3.5
4.4
5.4
6.4
0.237
0.275
0.403
0.464
0.496
0.519
0.581
0.689
IV
0.009
0.024
0.050
0.080
0.113
0.148
0.190
0.244
0.027
0.054
0.106
0.175
0.222
0.274
0.330
0.336
0.306
0.314
0.243
0.326
0.391
4.5
5.7
6.7
7.5
8.1
18.9
9.3
19.7
0.576
0.615
0.697
0.573
0.582
0.593
0.447
0.463
0.065
0.132
0.218
0.306
0.401
0.542
0.668
0.776
6.9
:
is . VI
VII
VIII
7.4
7.8
0.323
0.438
....
'
L
37
Table 3. Initial population (assuming a 100 acre lake and 100 fish per acre) and
associated annual conditional natural mortality rates (n) for slow, medium,
and fast growing bluegill populations in Michigan,

Slow
Medium
Fast
Fish
per
. acre
Fish
per
acre
Fish
per
acre
Age
n
:
0.44
II
III
IV
59.04
12.99
8.83
6.80
5.17
3.72
2.38
1.07
0.78
0.32
0.23
0.24
0.28
0.36
0.55
0.81
40.01
22.41
17.03
12.26
6.01
1.92
0.33
0.03
0.24
0.28
0.51
0.68
0.83
0.91
0.98
49.15
23.10
14.78
8.72
3.31
0.83
0.10
0.01
0.53
0.36
0.41
0.62
0.75
0.88
0.95
0.98
VI
VII
VIII
Total
100.0
100,0
100.0
38
.
The
regression
medium, and fast growing populations.
coefficients are summarized in Table 4.
Table 4.
Parameter values (and coefficients of
determination) estimated for Ford's growth
equation and the length-weight relationship for
slow, medium, and fast growing bluegill
populations in Michigan.
-
--
-

Growth
L.
Slow
Medium
Fast
0.8861
0.8793
0.8428
10.6659
11.9500
11.9720
0.99383
0.98866
0.99830
Length-weight
Slope (b)
Intercept (a)
Slow
Medium
Fast
3.1275
3.0815
3.2118
-7.6057
-7.4670
-7.6063
0.99922
0.99972
0.99783
Recruitment
Three additional assumptions were necessary to
incorporate variable recruitment into the model. First,
spawning activity was assumed to peak in the third week of
June. Choice of this week was based on studies by Carbine
(1939) who reported that spawning peaked in late June,
Karvelis (1952) who reported mid-June, and Snow et al.
(1960), Breder and Rosen (1966), and Becker (1976) who
reported early to mid-June ranges. This week also
39
corresponded to "to" in the model and it was at this time
that fish were moved to the next age group. This was done
not only because spawning was assumed to occur at this time,
but also because the mean lengths reported by Laarman (1963)
were considered early to mid-summer estimates. Second, a
sex ratio of 1:1 was assumed for all ages capable of ™
spawning (Beckman 1946; Fabian 1954; Parker 1958). Third,
it was necessary to assume some minimum length for mature
females. Ulrey et al. (1938) reported a minimum of 5.2
inches in Indiana lakes, Mayhew (1956) showed 4.3 inches in
an Iowa lake, Snow et al. (1960) found the minimum to be
4.5 inches in Wisconsin lakes, and Scott and Crossman (1973)
reported 5.4 inches for Canadian lakes. An average of these
estimates was used in this study. A value of 4.8 inches was
chosen as the minimum length of mature females.
: A regression relating the mean egg content of females
(EC) to mean length in inches (ī) was reported by Latta and
Merna (1976). This equation was:
EC = -50,154.78 + 10,697.64(ī)
This equation shows that the average 4.8 inch female
bluegill would contain 1,194 eggs and an average 6.0 inch
female would have 14,031 eggs. These figures correspond
well with estimates reported by Ulrey et al. (1938), Mayhew
(1956), and Snow. et al. (1960). This equation also agrees
well with the choice of a 4.8 inch minimum length for mature
8
40:
to
females. Equation (44) was then used to calculate the total
number of eggs produced (EGGS) for the year.
In this model, fish were considered fry from the time
of hatching until the following second week in June when
they recruited into the age-I group. To use the assumption
of density-dependent survival of fry, it was necessary to
relate the number of eggs produced to number of fry
surviving. Such a relationship for bluegills was developed
by Latta and Merna (1976). A stock-recruitment curve
(equation (46)) was fit to their observed data of the form:
S. = 0.74647 (EGGS) e-(1.776 x 10 °) (EGGS)
This curve (Figure 5) results in a maximum number of fry
surviving (154,000. per acre) when the egg production is
about 563,000 per acre. Because data were lacking beyond
this peak area of the curve, an assumption was made that the
curve to the right of the peak should decrease. This
implies that as egg production increases beyond 563,000 per
acre, the number of fry decreases. This seems reasonable
because, as density increases, available spawning sites may
be used up causing many fish to spawn in areas unsuitable
for successful hatching, or suppression of spawning
altogether (Snow et al. 1960). Swingle and Smith (1943)
reported that at high densities bluegills will eat many or
all of their own eggs which further supports this
assumption.
216.00
:
180.00
007
THOUSAND FRY PER ACRE
108.00
72.00
36.00
R = 0.72500
908.00
THOUSAND EGES COPRODUCIDO PER
668.00
Figure 5.. Fit of a stock-recruitment curve to observed fry
densities.
42
Another regression developed by Latta and Merna (1976)
related fry density to mean length of fry... Use of this
regression allowed density-dependent growth to be
incorporated into the model. Equation (41) was fit to these
data giving:
14,i = 4.4396 - 0.28716(10g. (FRY))
The resulting curve (Figure 6) shows decreasing length with
increasing density. This phenomenon was also reported by
Karvelis (1952), Anderson (1959), and Novinger and Legler
(1978). Goodyear (1980) also supported such a relationship
in his comprehensive report on compensation in fish
populations
The regressions developed by Latta and Merna (1976)
were obtained from experiments done in ponds at the Saline
Fisheries Research Station in Michigan. These regressions
needed to be adjusted for use with the slow, medium, and
fast growing populations previously described for two
reasons. First, the predicted number of fry surviving and
their estimated mean length were probably higher in the
ponds than would be found in natural lakes. This would be
caused by the fact that no predation existed in the ponds,
except from the few older bluegills stocked to produced the
fry. However, the effect of their predation was assumed
negligible. Second, the populations studied at Saline
showed some characteristics. seen in slow growing
14.00

R = 0.95273
.
*
3.00
MEAN LENGTH (INCHES)
2.00
*
*
1.00
*
.
.
.
In
36.00 THOUSAZOO FRY PIES COACRE 14.00
180.00
Figure 6. Fit of a length-density regression equation to
observed mean lengths of fry.
..
.-
,..
44
-
-
populations. The regressions predicted survival and initial
length better for the slow growing model populations than
for the other two. However, these regressions were adjusted
for use with each of the three populations.
Another problem existed in that the regressions
predicted number of fry per acre and mean length occurring
in the fall. Since the model "to" was the third week in
June, mortality rates were assigned for fry from the fall to
the second week in June of 0.99915033, 0.99968773, and
0.99968186 in slow, medium, and fast growing populations
respectively. Although these may seem high, it must be
remembered that the predicted estimates of fry per acre in
the fall were also high due to lack of predation. These
rates were chosen so that the populations to be simulated
were in an equilibrium state when no fishing was allowed.
Because fry dynamics were not modeled in any detail, the
only important value was number of fry recruiting into the
age-1 group. The initial lengths predicted in the fall were
also increased by one season of growth to keep them in phase
with the model "year".
The constant mortality rate of fry from the fall to the
following spring did not allow the population to compensate
for any change in the number of large bluegills (7.0 inches
plus), the conditional fishing mortality rate, or the size
of previous year classes. Thus, population characteristics
were also simulated using constant recruitment each year).
The number of fry entering the age-I group each year was set
::
1.
.
.
-
45
equal to the number of age-I fish found in the unexploited
populations, 5,904, 4,001, and 4,915 fish for slow, medium,
and fast growing populations respectively (Table 3).
However, the actual number of fry produced each season was
calculated using the fecundity relationships so that the
density-dependent growth relationship could still be used in
the model.
Fishing Mortality
Three annual conditional fishing mortality rates (m)
were chosen for use in this model. The values picked were
0.40, 0.20, and 0.10, each rate being applied to all three
populations of bluegills during separate simulation runs. A
consideration in determining what fishing rates to choose
was that special regulations have often caused fishing
pressure to drop on those lakes included in the regulations
(Schneider and Lockwood 1979). Therefore, an upper level of
0.40 was chosen and then reduced substantially to cover a
fairly wide range of possible fishing mortality rates.
The probability of death (harvest) given capture for
legal fish (d') was set equal to 1.0, meaning that all legal
fish caught were harvested. Although this may not be true
especially for a 7.0 inch maximum limit on bluegill, Bennett
(1962) showed that most fishermen would keep bluegills that
were 5.0 inches or larger. Thus, a lower bound on the legal
(harvestable) range was set at 5.0 inches, supporting the
assumption that all legal fish caught would be harvested.
46
Because the probability of harvest given capture was set to
1.0, the capture probability for legal fish (p') was equal
in magnitude to the conditional fishing rates chosen,
depending on which was being used in the simulation process.
Hooking Mortality
An annual conditonal hooking mortality rate (h) was
chosen corresponding to each fishing rate. The probability
of death given capture for illegal fish (a'!) was set equal
to 0.20, and the probability of capture of illegal fish
(p'') was set equal to the probability of capture for legal
fish (p'). This resulted in hooking mortality rates of
0.08, 0.04, and 0.02 for fishing rates of 0.40, 0.20, and
0.10 respectively (h = pild'').
The assumption of equal catchability for legal and
illegal fish seems reasonable because, as Snow et al.
(1960) and Becker (1976) stated in their studies, "bluegills
are always ready and willing to take a hook". The
assumption of a probability of death given capture for
illegal fish equal to 0.20. also seems reasonable since it
applies to bluegills over 7.0 inches with no restrictions on
the types of fishing gear or bait. used. Many bluegills
often swallow hooks when fishermen are still-fishing with
live bait (the most popular way to fish for bluegill) and
will probably die when released. Although no experiments
have been done on hooking mortality for bluegill, this
.
.
"..
.
47
figure is probably representative according to P. W. Laarman
(personal communication).
Seasonal Distribution of Natural Mortality,
Growth, and Fishing Mortality
-
-
,
The numerical model developed here allows a simulated
year to be broken down into as many discrete time periods as
seems reasonable. Since the dynamics of fish populations
are all continuous processes, it follows that as the number
of discrete time periods within a year increases (i.e., as
t'-), better estimates of the population characteristics
will be calculated. Ricker (1975) pointed out that
intervals as small as a day were probably unnecessary and
that any period less than a day was unreasonable because
diurnal fluctuations in predation, etc., would. invalidate
many results unless a calculus of finite differences was
employed in the model.
3. A discrete time period of one week was assumed to give
accurate results, and any smaller interval would not be
justified by the increase in precision of the population
estimates. Based upon this reasoning, time intervals of one
week (t'=52) were used. Each month was assigned a specific
number of weeks (Table 5), allowing natural and fishing
mortality rates and growth to be spread throughout a year.
Pátriarche (1968) published seasonal natural mortality
rates for two lakes in Michigan. He assigned 7% to the
spring (May 1 to June 7), 81% to the summer (June 7 to
September 1), 12% to the fall period (September 2 to
48
Table 5. The number of weeks allotted to each month, and
monthly percentage distribution of natural
mortality (%), growth (%), and fishing
mortality (%).
Month
Number
of weeks
n.
.
g
%m
0.0
OOO
4.0
4.0
1.0
Jan.
Feb.
Mar.
Apr.
May
June
July
Aug.
Sep.
Oct.
Nov.
Dec.
0.0
5,6
21.6
0.0
.0.0
0.0
16.0
29.0
25.0
12.0
14.0
0
33.8
.0
26.0
21.0
11.0
هه سا
3.7
0.0
December 1), and no natural mortality during the winter.
(December 2 to April 30). Schneider (1973a) used this same
type of distribution in his model for Mill Lake in Washtenaw
each month was computed as a ratio of the number of weeks in
each month to the total number of weeks attributed to each.
season (Table 5).
Anderson (1959) showed a monthly percentage breakdown
of total annual growth in his study of Third Sister Lake in
Washtenaw County, Michigan. Both field data and laboratory
experiments performed by Anderson gave similar results.
Karvelis (1952) and Fabian (1954) reported approximately the
same distribution for Ford Lake in Otsego County, Michigan
as did Snow et al. (1960) for Wisconsin lakes. Schneider
49
(1973a) also used a similar pattern of growth in his Mill
Lake. model. Anderson's figures were used in this model
(Table 5).
:: The distribution of fishing mortality was calculated
from angler census data collected on Bear Lake (1952-1953)
in Hillsdale County, Michigan (Schneider and Lockwood,
1979). Fishing pressure on this lake of 117 acres in size
was assumed to be representative of pressure received on
small lakes in the 100 acre range as used in this study. On
the basis of these data, 62% of the total fishing mortality
was assigned to the summer months (June 24 to September 15),
6% to the fall (September 16 to December 1), 10% to the
winter (December 2 to March 30), and 22% to the spring
(April 1 to June 23). This pattern was slightly modified
(as recommended by M. H. Patriarche and P. W. Laarman,
personal communication) and a final distribution of percent
fishing mortality was estimated to be (same seasonal dates):
57.5% in the summer, 7.5% in the fall, 10% in the winter,
and 25% in the spring. Christensen (1953) showed a
distribution of percent fishing mortality for six lakes in
Michigan (averaged over a five year period) that was very
similar to this latter distribution. These seasonal values
were then spread over the corresponding months (Table 5)
using the same method as described earlier for natural:
mortality.
The monthly percentages of natural mortality, growth;
and fishing mortality were spread uniformly over the weeks
.
50
in each month. Each week was given an equal amount of the
total value for the month by dividing the number of weeks in
a month into the monthly percent estimate.
..
.
RESULTS AND DISCUSSION
The characteristics of slow, medium, and fast growing
bluegill populations were simulated using the S.L.R.A.
computer program. Simulation was continued until the
populations reached equilibrium for the fishing mortality
rate being used. The initial populations were also
subjected to a 5.0 inch minimum size limit regulation, used
as a control to determine the impact of the 7.0 inch maximum
restriction. Although no regulation was in effect for
bluegills, a 5.0 inch minimum was assumed to be
representative of conditions found in Michigan at this time.
Two separate sets of output were generated, one using
variable recruitment and the other constant recruitment.
Results Using Variable Recruitment
Annual fishery statistics of the three bluegill
populations at equilibrium (assuming variable recruitment)
are summarized in Tables 6-17. Number of fish (per 100
acres) and attained mean length in inches at each age are
found in Tables 6, 10, and 14 for slow, medium, and fast
growing bluegill populations respectively. These results
show no change in length (at a given fishing mortality rate)
between a 5.0 inch minimum and a 7.0 inch' maximum
regulation, and a large decrease in the total number of fish
as the fishing mortality rate increases, especially in the
slow growing population. This suggests that the density-
dependent recruitment relationships used in the simulation
..
51
52
were not adequate in describing the processes of fry
survival and recruitment into the age-I group. This could
be attributed to many factors including the assumptions of
only one spawning period per year and/or a 4.8 inch minimum
length of mature females. Both of these assumptions could
cause the number of eggs produced to be far less than seen
in the field under the same conditions. Multiple spawning
periods were not modeled because of the lack of data
concerning this phenomenon and the complexity involved in
simulating such a process. The effect of a 4.8 inch minimum
length of mature females was not as important in the medium
or fast growing populations where this length was attained
by age I or II. However, in the slow growing population,
this length was not reached until age III or IV and the
spawning stock had been greatly depleted by then because of
fishing. This caused the large reduction in total number of
fish as the fishing mortality rate increased.
It was hoped that the use of a density-dependent growth
relationship would offset these problems. However, the
regression used to predict the mean length of fry was not
sensitive enough to changes in the density of fry. Although
the mean length at each age increased as density decreased,
the change in length was not enough to offset the loss of
fish over 4.8 inches.
Another important factor was the assumption of a
constant mortality of fry from the fall to the followingii
spring. This assumption did not allow any compensation by
53
the population for changes in the number of large bluegills
(7.0 inches plus), the density of the previous year class,
or the fishing rate. Better feedback mechanisms are
necessary to model this compensation in any greater detail.
The fishery statistics are recorded by length group in
Tables 7-9 for slow growing, Tables 11-13 for medium
growing, and Tables 15-17 for fast growing populations. The
largest impact of the special regulation was seen in the
fast growing population at a fishing mortality rate of 0.40
(Table 15). Numbers of 7.0 inch plus fish increased from
398 under existing conditions to 590 using the special
regulation. Total catch was greater under the 7.0 inch
maximum restriction (1,617 fish versus 1,363 fish), while
harvest was essentially equal (1,024 and 1,020 fish per year
for a 5.0 inch minimum and a 7.0 inch maximum respectively).
The special regulation had little effect on controlling the
slow and medium growing populations and the fishery
statistics were virtually the same for the 5.0 inch minimum
and 7.0 inch maximum restrictions.' Changing from a 5.0 inch
minimum to a 7.0 inch inverted regulation had little effect
because less than 5% of the fish in these populations were
over 7.0 inches (i.e., the same fish were still harvested).
Any change between the existing conditions and the 7.0 inch
maximum in all three populations diminished as the fishing
mortality rate decreased.
mn
54

Table 6. Predicted equilibrium number and attained mean length (l) by age group
with variable recruitment for two size limit regulations and three annual
conditional fishing mortality rates (m) for a slow growing bluegill population
in Michigan.
m
= 0.40
m
= 0.20
m
= 0.10
5" minimum
7" maximum
5" minimum
7" maximum
5" minimum
7" maximum
Age
Number
1
Number 1
Number 1
Number i Number 1
Number 1
II
11
IV
721 3.0
163 4.0
104 4.9
62 5.6
33 6.1
18 6.4
8 7.0
2 7.5
751 3.0
175 4.0
115 4.9
68 5.6
38 6.0
19 6.4
9.7.0
.4 7.5
2,546 2.7
571 3.7
388 4.6
275 5.3
180 5.8
109 6.3
59 6.9
22 7.4
2,670 2.7
599 3.7
401 4.6
285 5.3
186 5.8
114 6.3
64 6.9
27 7.4
4,156 2.6
922 3.6
622 4.5
465 5.2
330 5.7
219 6.2
129 6.8
54 7.3
4,252 2.6
946 3.6
637 4.5
475 5.2
337 5.7
224 6.2
134 6.8
57 7.3
VI
VII
VIII
Total
1,lll
1,179
4,150
4,346
6,897
7,062
55
Table 7. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.40 for a slow growing bluegill population in Michigan.'
}
Simulated fishery statistics*
Fishing regulation
YJ
C+ u
Y + YJ
-
1.0- to 2.9-inch fish
3.0- to 4.9-inch fish
98
104
4.2
4.6
5.0- to 7.0-inch fish
62
7.7
8.3
69
over 7.0-inch fish
5 inch minimum
7 inch maximum
09
All fish
8
.4
.
164
5 inch minimum
7 inch maximum
361
376
98
104. 4.6
4.2
5 inch minimum
7 inch maximum
592
627
5 inch minimum
7 inch maximum
148
165
7.7
8.3
0.0
0.0
66
69
5 inch minimum
7 inch maximum
1,111
1, 179
69
98
107
8:.
4.2
5.5
109
9:3
196
2.8
:8.3
13.8
22
*N = number of fish in the population; C = annual number of legal-size bluegills harvested: Y = annual
yield in pounds of harvest; J = annual. number of illegal-size bluegilis caught and released; Y = annual
yield in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills
dying from hook ing mortality.
56
Table 8. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.20 for a slow growing bluegill population in Michigan.
_
_
Fishing regulation
N
Simulated fishery statistics
V J Yu C#
J
Y
+ YJ
1.0- to 2.9-inch fish
2,352
3.0- to 4.9-inch fish
5.0- to 7.0-inch fish
110
I
13.7
13.7
13.8
115
13.8
Over 7.0-inch fish
5 inch minimum
7 inch max i mum
2.0
2.6
All fish
2.244
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
1,280
1,337
146
153
146
153
6.5
6.9
5 inch minimum
7 inch maximum
573
596
5 inch minimum
7 inch maximum
4, 150
4,346
120
115
15.7.
13.8
146
163
266
278
22.2
23.3
9.5
*N = number of fish in the population; C: annual number of legal-size bluegills harvested; Y annual
yield in pounds of harvest; v = annual number of illegal-size blueg111s caught and released; YJ = annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
57
Table 9. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.10 for a slow growing bluegill population in Michigan.
Simulated fishery statistics*
Fishing regulation
N
Yu.
Ct u
V + YU
1.0- to 2.9-inch fish
3.0- to 4.9-inch fish
23
5 inch minimum
7 inch maximum
1,860
1.907
114
117
5.2
5.4
24
114
5.2
117
5.4
5.0- to 7.0-inch fish
981
1.005
11.3
11.2
0.0
0.0
89
11.3
11.2
92
Over 7.0-inch fish
99
1.8
103
OO
1.8
0.0
0.0
2.2
2.2
All fish
:
13.1
•
212
5 inch minimum
7 inch maximum
3,957
4,047
0.0
0.0
0.0
5 inch minimum
7 inch maximum
5 inch minimum
7 inch max i mum
5 inch minimum
7 inch maximum
6,897
7,062
114
126
5.2
7.6
92
18.3
18.8
23
2
11.2
218
26
*N = number of fish in the population; C annual number of legal-size bluegills harvested; Y = annual
yield in pounds of harvest; v = annual number of illegal-size bluegills caught and released: Yu : annual yield
in pounds of illegal-size bluegills caught and released; H & annual number of illegal-size bluegills dying
from hook ing mortality.

Table 10. Predicted equilibrium number and attained mean length (1) by age group
with variable recruitment for two size limit regulations and three annual
conditional fishing mortality rates (m) for a medium growing bluegill
population in Michigan.
m = 0.40
m = 0.20
m = 0.10
5" minimum 7" maximum 15" minimum 7". maximum 5" minimum 7" maximum
Number 1 Number 1 | Number Ī Number 1 | Number 1 Number 1
Age
I
II
|
III
IV
1,076 3.9
569 4.8
315 5.8
145 6.7
46 7.1
10 7.6
1 8.0
0...
1,229 3.9
639 4.8
349 5.8
165 6.7
60 7.1
17 7.6
3 8.0
0...
2,506 3.6
1,363 4.5
933 5.5
552 6.5
222 7.0
59 7.5
8 7.9
1 8.8
2,606 3.6
1,416 4.5
966 5.5
571 6.5
237 7.0
71 7.5
12 7.9
1 8.8
3,195 3.5
1,766 4.4
1,286 5.4
845 6.4
377 6.9
110 7.4
17 7.8
18.7
3,244 3.5
1,795 4.4
1,305 5.4
857 6.4
387 6.9
119 7.4
20 7.8
2 8.7
VI
VII.
VIII
Total
2,162
2,462
5,644
5,880
7,597
7,729
Table 11. Predicted equilibrium levels of density, catch, and yield by length, group for two size limit:
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.40 for a medium growing bluegill population in Michigan.
:
Simulated fishery statistics
Fishing regulation
Fishing regulation
N
C +
J
Y
+ YJ
H
1.0- to 2.9-inch fish
3.0- to 4.9-inch fish
17.4
1.483
330
376
330
376
17.4
19.8
66
75
19.8
5.0- to 7.0-inch fish
.
36,9
41.1
Over 7.0-inch fish
65
24
0.0
7.1
5.5
7.1
86
27
27
All fish
629
66
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
1,686
5 inch minimum
7 inch max i mum
613
688
275
311
36.9
41.1
275
311
5 inch minimum
7 inch max i mum
5 inch minimum
7 inch maximum
2,162
2,462
299
311
42.4
41.1
330
403
17.4
26.9
59.8
68.0
714
80
*N = number of fish in the population; C = annual number of legal-size bluegills harvested: Y = annual
yield in pounds of harvesti u = annual number of illegal-size bluegills caught and released; YJ = annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
60
.
.
Table 12. Predicted equilibrium levels of density, catch, and yield by length group for two size limit.
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.20 for a medium growing bluegill population in Michigan
i
Simulated fishery statistics*
Yu C+j
Fishing regulation
N
V+YJ
1.0- to 2.9-inch fish
3.0- to 4.9-inch fish
5.0- to 7.0-inch fish
Over 7.0-inch fish
35
0.0
35
8.4
0.0
8.4
9.6
36
9.6
36
All fish
85
96
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
3,801
3,951
426
442
20.8
21.7
426
442
20.8
21.7
85
89
0.0
5 inch minimum
7 inch maximum
1,595
1,655
309
322
43.1
44.0
309
322
43.1
44.0
0.0
5 inch minimum
7 inch maximum
207
232
·
5 inch minimum
7 inch maximum
5,644
5,880
344
322
51.5
44.0
426
478
20.8
31.3
770
800
72.3
75.3
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual
yield in pounds of harvesti J + annual number of illegal-size bluegills caught and released; Y J = annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
Table 13. Predicted equilibrium levels of density. catch, and yield by length group for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
..:of 0.10 for a medium growing bluegill population in Michigan.
:
.
.
Simulated fishery statistics*
Y u Yu . C+jS
Fishing regulation
N
.
C
c.
+ Yu
1.0- to 2.9-inch fish
109
3.0- to 4.9-inch fish
5.0- to 7.0-inch fish.
0:0
203
5 inch minimum
7 inch maximum
2,261
2,298
28.6
28.6
28.6
0.0
0.0
203
7
206
206
28.6
28.6
206
0
Over 7.0-inch fish
5 inch minimum
7 inch maximum
1 10
5 inch minimum
7 inch maximum
4,942
5,020
274
278
13.2
13.4
274
278
13.2
13.4
5 inch minimum
7 inch maximum
285
301
5.4
0.0
5.4
6.0
.
All fish
.
54
5 inch minimum
7 inch maximum
1,597
7,729
226
206
34.0
28.6
274
301
13.2
19,4
500
507
47.2
48.0
60
'
.
I
*N = number of fish in the population; C = annual number of legal -size bluegills harvested; Y = annual
yield in pounds of harvest; J = annual number of illegal-size bluegilis caught and released; Yu - annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
.
-
.
Table 14. Predicted equilibrium number and attained mean length (1) by age group
with variable recruitment for two size limit regulations and three annual
conditional fishing mortality rates (m) for a fast growing bluegill population
in Michigan.

m = 0.40
m = 0.20
m = 0.10
5" minimum
7" maximum
5" minimum
7" maximum
5" minimum
7" maximum
Age
Number
1
Number
Number
Number
Number
1
Number
1
I
II
III
IV.
3,281 4.7
1,123 5.8
457 6.8
169 7.6
40 8.2
6 9.0
3,837 4.7
1,314 5.81
564 6.8
263 7.6
90 8.2
21 9.0
2 9.4
O ...
4,443 4.5
1,841 5.7
966 6.7
463 7.5
143 8.1
29 8.9
3 9.3
4,597 4.5
1,906 5.7
1,019 6.7
536 7.5
193 8.1
46 8.9
5 9.3
0 ...
4,791 4.5
2,121 5,7
1,236 6.7
661 7.5
228 8.1
52 8.9
6 9.3
4,830 4.5
2,138 5.7
1,257 6.7
702 7.5
260 8.1
64 8.9
8 9.3
VI
VII
VIII
0
...
Ó
...
0
...
O
...
O
...
Total
5,076
6,091
7,888
8,302
9,095
9,259
63
Table 15. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.40 for a fast growing bluegill population in Michigan...
Simulated fishery statistics ::
-Fi
Fishing regulation
YJ
..
C+ u
.
Y + VJ
1.0- to 2.8-inch fish
3.0- to 4.9-inch fish
.
21.7
27.6
68
79
5.0- to 7.0-inch fish
.
OO
Over 7.0-inch fish
164
48.6
164
200
46.7
68.9
0.0
68.9
39
200. 68:9
All fish
5 inch minimum
7 inch maximum
5 inch minimum
7 inch max i mum
2,374
2,777
339
397
21.7
27.6
339
397
5 inch minimum
7 inch maximum
2,296
2.715
860
1,020
131.4
147.2
860
1,020
131.4
147.2
0.0
5 inch minimum
7 inch maximum
398
590
5 inch minimum
7 inch maximum
5,076
6,091
1,024
1,020
178.1
147.2
339
597
21.7
96.5
1,363
1,617
199.8
243.7
'68
118
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual yield in
pounds of harvest; J = annual number of illegal-size bluegills caught and released; YJ = annual yield in pounds of
illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from hook ing mortality.
-
:
..
Table 16. Predicted equilibrium levels of density, catch, and yield by length group. for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.20 for a fast growing bluegill population in Michigan.
m
Simulated fishery statistics
YU
Fishing regulation
C +
J
Y
+ YJ
H
1.0- to 2.9-inch fish
3.0- to 4.9-inch fish
289
20.2
5.0- to 7.0-inch fish
289
299
18.1
20.2
Over 7.0-inch fish
900
1.044
160
0.0
59. 1
160
170
47.7
59. 1
0.0
170
All fish
:
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
3,808
3,941
18.1
299
5 inch minimum
7 inch maximum
3, 160
3,296
577
602
91.7
88.8
0.0
0.0
.:
577
602
91.7
88.8
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
7.888
8,302
737
: 602
139.4
88.8
.
289
469
18.1
79.3
1:02
1.026
1,071
157.5.
168.1
57
94
*N = number of fish in the population; C = annual number of legal-size bluegilis harvested; Y = annual yield
in pounds of harvest; J = annual number of illegal-size bluegills caught and released; YJ = annual yield in pounds
of illegal-size bluegilis caught and released; H = annual number of illegal-size bluegilis dying from hook ing
mortality.
Table 17. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming variable recruitment and an annual conditional fishing mortality rate (m)
of 0.10 for a fast growing bluegill population in Michigan... .. .
Simulated fishery statistics*
V .
YJ . C
Fishing regulation
.
Y + YJ.
:
1.0- to 2.9-inch fish
.
.
3.0- to 4.9-inch fish
5 inch minimum
7 inch maximum
4, 125
4, 159
156
157
9.8
10.8
156
157
9.8
10.8
31
5.0- to 7.0-inch fish
3.685
3,730
320
325
52.0
48.4
320
325
52.0
48.4
on
Over 7.0-inch fish
.
106
32.4
0.0
0.0
37.9
106
109
32.4
37.9
109
A11 fish
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
5 inch maximum
7 inch maximum
1.263
1,349
5 inch minimum
7 inch maximum
9,095
9,259
426
325
84.4
48.4
156
266
9.8
48.7
582
591
94.2
97.1
*N = number of fish in the population; C = annual number of legal-size bluegills harvested: Y = annual
yield in pounds of harvest; J = annual number of illegal-size bluegills caught and released; YJ : annual yield
in pounds of illegal-size bluegilis caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
66
Results Using Constant Recruitment
Annual fishery statistics of the three bluegill
populations at equilibrium (assuming constant recruitment)
are summarized in Tables 18-29. Number of fish (per 100
acres) and attained mean length in inches at each age are
found in Tables 18, 22, and 26 for slow, medium, and fast
growing populations respectively. As in the results using
variable recruitment, no change in length (at a given
fishing rate) was observed between a 5.0 inch minimum and a
7.0 inch maximum. However, because constant recruitment
allows some compensation in the recruitment of age-I fish,
the total number of fish in the population is much higher
than seen in the earlier results for variable recruitment.
The population structures seem much more realistic and are
no longer greatly altered by the choice of a fishing
mortality rate. The largest impact from using constant
recruitment was observed in the slow growing population at a
fishing mortality rate of 0.40. Total number increased from
1,111 fish (assuming variable recruitment) to 8,880 fish
(assuming constant recruitment). This was expected because
of the slow growth and the 4.8 inch minimum length of mature
females. This impact decreases from medium to fast growth,
but a significant change in total numbers was still observed
between the variable and constant recruitment estimates.
Although the assumption of constant recruitment did
increase total numbers, the use of a 7.0 inch maximum
regulation was still not effective in controlling the
i.
bluegill populations. Yield was much greater over that seen
in the previous results, but this was expected because of
the increase in total number of fish in the populations.
Simulated fishery statistics by length group are found in
Tables 19-21 for slow growing, Tables 23-25 for medium
growing, and Tables 27-29 for fast growing bluegill
populations. The greatest change between existing
conditions and the special regulation was again seen in the
fast growing population using a conditional fishing
mortality rate of 0.40 (Table 27). Number of fish over 7.0
inches increased from 589 (5.0 inch minimum) to 724 fish for
a 7.0 inch maximum regulation. Harvest decreased slightly
from 1,530 to 1,332 fish per year when the special
regulation was applied. Total catch was virtually the same,
2,038 under existing conditions and 2,089 fish per year
using a 7.0 inch maximum restriction. The statistics for
the slow and medium growing populations showed much smaller
changes at a fishing mortality rate of 0.40 when the special
regulation was applied, than observed for the fast growing
population. The number of fish over 7.0 inches, harvest,
and total catch became essentially equal for a 5.0 inch
minimum and a 7.0 inch maximum with a decrease in the
fishing mortality rate, as observed in the simulation
results using variable recruitment.
68
Table 18. Predicted equilibrium number and attained mean length (1) by age group
with constant recruitment for two size limit regulations and three annual .
conditional fishing mortality rates (m) for a slow growing bluegill population
in Michigan.

m
= 0.40
m = 0.20
m
= 0,10
7" maximum
5" minimum
7" maximum
5" minimum
Number 1
7" maximum 15" minimum
Number 1 | Number 1
Age
Number
1 | Number 1.
Number
1
II
III
IV
5,904 2.7
'1,281 3.7
803 4.6
483 5.3
243 5.8
110 6.3
44 6.9
12 7.4
5,904 2,7
1,281 3.7
803 4.6
483 5.3
243 5.8
112 6.3
49 6.9
17 7.4
5,904 2.6
1,293 3.6
848 4.5
593 5.2
379 5.7
223 6.2
116 6.8
42 7.3
5,904 2.6
1,293 3.6
848 4.5
593 5.2
379 5.7
224 6.2
120 6.8
48 7.3
5,904 2.6
1,296 3.6
866 4.5
637 5.2
446 5.7
292 6.2
170 6.8
69 7.3
5,904 2.6
1,296 3.6
866 4.5
637 5.2
446 5.7
293 6.2
173 6.8
74 7.3
VI
VII
VIII
Total
8,880
8,892
9,398
9,409
9,680
9,689
__
69
.
Table 19. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m)
of 0.40 for a slow growing bluegill population in Michigan.
Simulated fishery statistics
Fishing regulation
Yu
C+
J
Y
+ YJ
rYUH
1.0- to 2.9-inch fish
0.0
.
0
0.0.
0
3.0- to 4.9-inch fish
.
127
28.1
28.3
127
5.0- to 7.0-inch fish
844
372
43.5
43.0
372
374
43.5
43.0
850
374
Over 7.0-inch fish
3.4
All fish
5 inch minimum
7 inch maximum
5,202
5, 202
5 inch minimum
7 inch maximum
2,793
2,793
639
639
28.1
28.3
639
639
5 inch minimum
7 inch maximum
5 inch minimum
7 inch max imum
5 inch minimum
7 inch maximum
8.880
8,892
389
374
46.9
43.0
639
656
28.1
32.4
1,028
1,030
75.0
75.4
127
130
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; y = annual yield
in pounds of harvest; u : annual number of illegal-size bluegills caught and released; YJ : annual yield in pounds
of illegal-size bluegills caught and released; H = annual number of illegal-size bluegilis dying from hook ing
mortality.
.
70
.
-
: Table
). Predicted equilibrium levels of density. catch, and yield by length group for two size i imit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m)
of 0.20 for a slow growing bluegill population in Michigan.
1
Simulated fishery statistics
NCY.
Yoctu. V + YUH
Fishing regulation
1.0- to 2.9-inch fish
3.0- to 4.8-inch fish
317
:
317
14.2
14.4.
64
64
5.0- to 7.0-inch fish
1, 128
1,133
220
221
26.9
26.2
220
221
26.9
26.2.
.
0.0
Over 7.0-inch fish
5 inch minimum
7 inch maximum
5.619
5,619
5 inch minimum
7 inch maximum
2,564
2.564
317
317
14.2
14:4
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximuin
87
93
16
: 0
0
3.2
.0
3.2
4.0
16
4.0
All fish
:
5 inch minimum
7 inch maximum
..9.398
9,409
236
221
30.1
26.2
317
333
14.2
18.4
553
554
44.3
44.6
-
64
67
::
-
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual
yield in pounds of harvest; .: annual number of illegal-size bluegills caught and released; YJ = annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
Table 21. Predicted equilibrium levels of density, catch, and yield by length group for two size limit.
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m)
of 0.10 for a slow growing bluegill population in Michigan. :
_
_
Simulated fishery statistics*
: J .You C +
Fishing regulation
N
C
Y
Y + YJ
H
1.0- to 2.9-inch fish
5,619
5.619
oo
3.0- to 4.9-inch fish
0
.0
7.2
160
160
7.2
7.3
160
160
ooo
5.0- to 7.0-inch fish
.
.
15.4
14.9
oo
.
Over 7.0-inch fish
0
0.0
11. 2.9
2.3
2.3
2.9
no
All fish
293
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
2,604
2.604
5 inch minimum
7 inch maximum
1,328
1.331
122
123
15.4
14.9
122
123
0.0
5 inch minimum
7 inch maximum
129
135
5 inch minimum
7 inch maximum
9,680
9,689
133
123 .
17.7
14.9
160
171
7.2
10.2
24.9.
25.1
32
34
294
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual
yield in pounds of harvest; J = annual number of illegal-size bluegills caught and released; YJ : annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality...

Table 22,
Predicted equilibrium number and attained mean length (1) by age group
with constant recruitment for two size limit regulations and three annual
conditional fishing mortality rates (m) for a medium growing bluegill.
population in Michigan..
m = 0.40
5" minimum 7" maximum
Number 1 Number 1
m = 0.20
5" minimum 7" maximum
Number 1 Number 1
m = 0.10
5" minimum 7" maximum
Age
Number
Number
I
II
III
IV
ON
4,001 3.6
2,083 4.5
1,224 5.5
551 6.5
168 7.0
33 7.5
4 7.9
4,001 3.6
2,083 4.5
1,224 5.5
552 6.5
183 7.0
49 7.5
8 7.9
1 8.8
4,001 3.5
2,168 4.4
1,492 5.4
876 6.4
349 6.9
91 7.4
13 7.8
1 8.7
4,001 3.5
2,168 4.4
1,492 5.4
876 6.4
357 6.9
104 7.4
17 7.8
2 8.7
4,001 3.5
2,204 4.4
1,597 5.4
1,044 6.4
464 6.9
135 7.4
21 7.8
28.7
4,001 3.5
2,204 4.4
1,597 5.4
1,044 6.4
469 6.9
143 7.4
. 24 7.8
2 8.7
VI
VII.
VIII
0
...
Total
8,064
.
8,101
8,991
9,017:
9,468
9,484
73
.
.
Table 23. Predicted equilibrium levels of density. catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m).
of 0.40 for a medium growing bluegill population in Michigan.;
:
Simulated fishery statistics*
_
_
latice cou
voru
Fishing regulation
YU
CH J
V + YJ
1.0- to 2.9-inch fish
67
00
67
3.0- to 4.8-inch fish
270
:
00
1,348
1,348
65.5
65.5
1,348
1,348
65.5
65.5
270
5.0- to 7.0-inch fish
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
5,962
5,962
5 inch minimum
7 inch maximum
1,886
1,895
846
850
111.9
111.1
850
Over 7.0-inch fish
149
58
5 inch minimum
7 inch maximum
58
13.6
0.0
0.0
. 15.0
177
13.6
15.0
57
:57
11
All fish
5 inch minimum
7 inch maximum
8,064
8, 101
904
850
: 125.5
111.1
1,348
1,405
65.5
80.5
2, 252
2.255
191.0
191.6
270
281 .
:
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual yield in
pounds of harvest; J = annual number of illegal-size bluegills caught and released; YU : annual yield in pounds of
illegal-size bluegilis caught and released; H = annual number of illegal-size bluegills dying from hook ing mortality.
74 .
i
Table 24. Predicted equilibrium 'levels of density, catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate. (m)
:: of 0.20 for a medium growing bluegill population in Michigan.
ini
.
Simulated fishery statistics*
1 -
-
u
Yu
Fishing regulation
C + J
óru
V + YJ
vo vd
I
1.0- to 2.9-inch fish
5 inch minimum
7 inch maximum
.
0.0
136
136
...oo
3.0- to 4.9-inch fish
32.7
6, 132
6,132
0.0
0.0
683
683
137
137
683
32.7
683
32.8
5.0- to 7.0-inch fish
2,470
2,475
47
64.9
64.1
0.0
0.0
470
471
64.9
64.1
.
· Over 7.0-inch fish
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
253
274
10. 1
0.0
0.0
11.1
.
:
10.1
11.1
8
.
:
A11 fish
8.991
513
5 inch minimum
7 inch maximum
75.0
64.1
683
725
32.7
43.9
1, 196
1,196
107.7
108.0
9.017
137
145
471
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual yield
in pounds of harvest; J = annual number of illegal-size bluegills caught and released: YJ : annual yield in pounds
of illegal-size bluegills caught and released; H = annual number of illegal-size bluegilis dying from hook ing
mortality.
: 75
Table 25. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m)
of 0. 10. for a medium growing bluegill population in Michigan.
Simulated fishery statistics*
Fishing regulation
v + YJ
uyucu
1.0- to 2.9-inch fish .
130
K16:
3.0- to 4.9-inch fish
.
342
5 inch minimum
7 inch maximum
.6.179
6, 179
342
342
16.5
16.5
342
16.5
16.5
5.0- to 7.0-inch fish
252
35.5
0,0
0.0
252
252
35.5
34.9
252
Over 7.0-inch fish
0.0
28
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
2,801
2,805
5 inch minimum
7 inch max imum
6.7
0 . 0
352
364
28
7.3
28
7.3
5 inch minimum
All fish
342 16.5
9.468
9.484
220
42
280
252
42.2
34.9
342
370
16.5
23.8
622
622
58.7
58.7
69
74
*N = number of fish in the population; C = annual number of legal-size bluegilis harvested; Y = annual
yield in' pounds of harvest; u = annual number of illegal-size bluegills caught and released; YJ : annual yield
in pounds of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
76
Table 26. Predicted equilibrium number and attained mean length (i) by age group
with constant recruitment for two size limit regulations and three annual
conditional fishing mortality rates (m) for a fast growing bluegill population
in Michigan.

m
= 0.40
_
m = 0.20
5" minimum 7" maximum
5" minimum
7" maximum
m = 0.10
5" minimum 7" maximum
| Number 1 Number 1
Age
Number
1
Number
1
Number
1
: Number
1
II
III
IV
V
VI
VII
VIII :
4,915 4.7
1,678 5.8
678 6.8
249 7.6
59 8.2
9. 9.0
1 9.4
0...
4,915 4,7
1,741 5.7
720 6.8
337 7.5
115 8.2
26 8.9
3 9.4
0. ...
4,915 4.5
2,037 5.7
1,068 6.7
512 7.5
158 8,1
32 8.9
3 9.3
0 ...
4,915 4.5
2,037 5.7
1,089 6.7
573 7.5
206 8.1
50 8.9
6 9.3
4,915 4.5
2,176 5.7
1,268 6.7
678 7.5.
234 8.1
53 8.9
69.3
4,915 4.5
2,176 5.7
1,279 6.7
:-714 7.5
264 8.1
65 8.9
8 9.3
0 ...
:
0
...
.
0
...
Total
7,589
7,857
8,725
8,876
9,330
9,421
77
Table 27.
Predicted equilibrium levels of density. catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m).
of 0.40 for a fast growing bluegill population in Michigan...
Simulated fishery statistics*
.
Fishing regulation
YJ
c + J
Y + YJ
.
1.0- to 2.8-inch fish
00
0.0
3.0- to 4.8-inch fish
101
. 508
514
00
32.5
35.9
:
508
514
32.5
35.9
5.0- to 7.0-inch fish
3,511
1,286
1.332
196.3
190.8
1,286
1.332
196.3
190.8
.
Over 7.0-inch fish
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
3,557
3.610
3,431
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
589
724
1
244
69.2
0.0
244
243
69.2
82.8
243
49
0.0
243 82.8
All fish
508
32.5
: 5 inch, minimum
7 inch max imum
7.589
7.857
1,530
1,332
265.5
190.8
757
2,038
2,089
298.0
309.5
101
152
118.7
*N = number of fish in the population; C = annual number of legal-size bluegills harvested: Y = annual yield in
pounds of harvest; J = annual number of illegal-size bluegills caught and released; YJ & annual yield in pounds of
illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from hook ing mortality.
..
Table 28Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m)
of 0.20 for a fast growing bluegill population in Michigan.
Simulated fishery statistics*
Fishing regulation
с
C+JO
Y + YU
1.0- to 2.9-inch fish
OO
0.0
3.0- to 4.9-inch fish
20.0
.: oo
320
320
20.0
21.6
64
64
5.0- to 7.0-inch fish
0.0
3,496
3,523
638
644
101.4
94.9
638
644
101.4
94.9
Over 7.0-inch fish
.
177
O
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
4,213
4,213
320
320
21.6
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
993
1, 117
52.7
0.0
0.0
63.2
177
183
52.7
63.2
4.3
A11 fish
.
5 inch minimum
7 inch maximum
8,725
8,876
815
644
154. 15. 320
94.9 : 503
20,0
84.8
1,135
1, 147
174.1
179.7
64
64
101
*N = number of fish in the population; c = annual number of legal-size bluegills harvested; Y = annual yield
in pounds of harvesti u = annual number of illegal-size bluegilis caught and released; YJ : annual yield in pounds
of illegal-size bluegills caught and released; H = annual number of illegal-size bluegills dying from hook ing
mortality.
79
Table 29. Predicted equilibrium levels of density, catch, and yield by length group for two size limit
regulations, assuming constant recruitment and an annual conditional fishing mortality rate (m)
of 0.10 for a fast growing bluegill population in Michigan.
Simulated fishery statistics*
Fishing regulation
No
c
Y
J
YJ
c + c
Y + Y
H.
1.0- to 2.9-inch fish.
NN
3.0- to 4.9-inch fish
161
33
O
10.1
11.0
161
33
161
10.1
161
11.0
5.0- to 7.0-inch fish
O
53.3
49.2
0.0
0.0
329
330
53.3
49.2
.
Over 7.0-inch fish
5 inch minimum
7 inch maximum
5 inch minimum
7 inch maximum
4,232
4,233
5 inch minimum
7 inch maximum
3,781
3,796
329
330
5 inch minimum
7 inch maximum
1,294
1,369
108
33:2
0.0
38.5
108
111
33.2
38.5
0.0
111
All fish
86.5
5 inch minimum
7 inch maximum
161
9.330
9,421
437
330
86.5
.92
10.1
49.5
OS
.
598
602
96.6
98.7
33
. 55
*N = number of fish in the population; C = annual number of legal-size bluegills harvested; Y = annual
yield in pounds of harvest; v = annual number of illegal-size bluegilis caught and released; YJ : annual yield
in pounds of illegal-size bluegilis caught and released; H = annual number of illegal-size bluegills dying from
hook ing mortality.
SUMMARY
The simulations using the model developed in this
report demonstrated two important points. First, a 7.0 inch
maximum size limit regulation can not improve and control
bluegill populations. The best results were obtained for a
population showing characteristics of fast growth, and which
was subjected to heavy fishing mortality even after
application of the special regulation. However, these
results depicted only a nominal improvement of the
population. Any gains realized by restricting the size of
harvestable bluegills could be easily offset or even
reversed by poaching or any natural disaster.
Further analysis reveals that a closed season on
harvest in May, June, and some or all of July might be
beneficial (along with the special size restriction) for two
reasons. One, 55% of the annual fishing mortality occurs
during these months. Large bluegills are very susceptible
to angling at this time because they come into shallow water
to spawn. Closing the fishing season during these months
would protect the larger fish and possibly cut down on
poaching. Second, 70% of the annual growth occurs during
this three month period. A closed season would allow many
more of the fish to recruit into the illegal size range
(i.e., over 7.0 inches) and thus not be harvested. This
would increase the number of large fish and possibly make
the 7.0 inch special regulation much more effective as a
management strategy for improving poor bluegill populations.
.
80
...Hooking mortality was essentially insignificant using
0.20 as the probability of a hooking death for a fish caught
and released. This implies that special restrictions on
gear (e.g., artificial lures only) would not help improve
the populations and are thus unnecessary if the probability
of a hooking death is 0.20 or less. However, if the
probability of a hooking death is much higher than 0.20,
gear restrictions may be necessary to prevent large numbers
of fish from dying after being caught and released.
The second point demonstrated by this model is the need
for further research regarding bluegill spawning, survival.
of fry, and recruitment into the age-I group. It is
necessary to determine the factors, dependent and/or
independent, which influence these processes and allow the
population to compensate for environmental and man-made
stresses. The assumptions in the model of variable
recruitment, one spawning period per year, a 4.8 inch
minimum length of mature females, and a constant mortality
of fry from the fall to the following spring were not
sufficient to produce a bluegill population with realistic
characteristics. Constant recruitment was then assumed, and
although this did improve the population characteristics, it
was still somewhat inadequate in producing the population
dynamics associated with spawning and recruitment. A
further density-dependent feedback mechanism may improve
model performance. One hypothesis is cannibalism by large ...
bluegills, illustrated in Figure 7. A variety of curves
82

SURVIVAL OF FRY
DENSITY OF BLUEGILLS 7 INCHES AND LARGER
Possible hypothetical functions relating fry
survival to the density of 7.0 inch and larger
bluegills.
83
relating fry survival to the number of bluegills 7.0 inches
.
n
.
line is the assumption used in this model along with the
assumption of variable recruitment (i.e., high constant
mortality regardless of the density of large fish).
However, one of the other curves may be more realistic, the
use of which might possibly enhance model performance
depending on the sensitivity of the relationship.
The model presented in this paper is very useful as a
management tool. Its general applicability makes possible
the simulation of a wide variety of size limit regulations,
gear and season restrictions, and studies of seasonal growth
and natural mortality. It may be applied to any inland
warmwater fishery if the necessary data are available for
accurate simulation. Application of this model to Michigan
bluegill populations showed that more research is still
needed before a successful solution for managing bluegill
populations can be determined. It is much easier to find a
cure if the symptoms are known rather than to use over and
over a trial and error process.
APPENDIX
Necessary data input for running program S.L.R.A.
FORTRAN
SYMBOL
TYPE
FORMAT
1)
Titte for output:
TITLE(20)
INTEGER
2014
2)
REG
INTEGER
A4
Type of reguition:
MIN - Minimum Size Limit
MAX - Maximum Size Limit
KST - Kill Slot with Trophy Size Limit
KS - K111 Siot Size Limit
RS - Release Slot Size limit
CR - Catch and Release Only
UN - Unexploited Population
3) Units for output:
Population
Weight
Length
Length-frequency distribution
Year starting date
PUNIT(20)
WUNIT(20)
LUNIT (20)
FMTUNI
YRSTRT (20)
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
2014
2014
2014
A4
2018
4) Population Characteristics:
Number of age groups to be entered
oldest expected age of any fish
Number of time periods per year
Number of years to be simulated
Time per iod when spawning occurs
Minimum length for mature females
Capture rate of legal sized fish
Probability of death given capture
for legal sized fish
Capture rate of Pllegal sized fish
Probability of death given capture
for illegal sized fish
0
NUMAGE
MAXAGE
NUMTIM
NYRSIM
SPWNTM
MLMAT
PPRIM
INTEGER
INTEGER
INTEGER
INTEGER
INTEGER
REAL
REAL
FREE
FREE
FREE
FREE
FREE
FREE
FREE
O DO DO
mmmmmmm
mmmmmmm
DPRIM
PDPRIM
REAL
REAL
FREE
FREE
DOPRIM
REAL
FREE
REAL
FREE
5) Fry characteristics and regression
cofficients:
Natural mortality of fry
NFRY
Length-fry regression calibration
factor
LENDIV
Intercept for egg-length regression EALPHA
Siope for egg-length regression
EBETA
Intercept for fry-egg regression
FALPHA
Siope for fry-egg regression
FBETA
Intercept for length-fry regression LALPHA
Slope for length-fry regression
LBETA
Intercept for weight-length regression WALPHA
Slope for weight-length regression WBETA
REAL
REAL
REAL
REAL
REAL
REAL
REAL
REAL
REAL
FREE
FREE
FREE
FREE
FREE
FREE
FREE
FREE
FREE
|
6) Annual conditional natural mortality by age
group:
LNRD ( 15 )
REAL
FREE
7)
Spread of annual conditional natural mor-
tality within a year:
Number of time periods
Percent to be spread uniformly over
: the time periods
NUMPRD
INTEGER
FREE
• PERC
REAL
FREE
8)
Spread of annual conditional fishing (and
nook ing if present) mortality within a year:
Number of time per tods
NUMPRD
Percent to be spread uniformiy over
the time periods
PERC
INTEGER
FREE
PERC
REAL
FREE
84
85
Necessary data input for running program S.L.R.A.*
FORTRAN
SYMBOL
TYPE
FORMAT
9)
Growth equation coefficients:
Brody's growth coefficient
Mean asymptotic growth coefficient
Maximum length possibly attained
LK
LINF
MAXLEN
REAL
REAL
INTEGER
FREE
FREE
FREE
10) Mean length and associated standard devi-
ation by age group: .
Mean length at each age
Standard deviation for each length
LEN( 16 )
LENDEV( 16 )
REAL
REAL
FREE
FREE
11)
NUMPRD
INTEGER
Spread of growth within a year:
Number of time periods
Percent to be spread uniformly over
the time periods
FREE
PERC
REAL
FREE
12)
Correction for lengths predicted by the
growth equation (each age):
CORLN( 16 )
REAL
FREE
13)
size regulation limits (zero if not used):
Lower bound on first illegal range
Lower bound on first legal range
Lower bound on second illegal range
Lower boundt on second legal range
REAL
REAL
REAL
REAL
FREE
FREE
FREE
FREE
X3
х4
14)
Initial starting population:
POPNUM( 15 )
REAL
FREE
. *Fortran symbols also show dimensions if present. All real type variables are ..
double precision. Free format may be changed to specific formatting if desired.
86
军事中事事第串串星来华事事本案当事事多多来来来来事半来来来来来​,事事事事事事事事家事事年年事事为多系本多多多多多多
​事事都事事事事事非事事事事有事​,事事事第米老本事事中事事本事事事事事本来事事事事军军事军事基军事基本事出事基本事事事
​单车事本等等等等等等第 ​本集客串第第第革本事事法律事事都非常多多多多多多多多多多多多多多年来本事事军事军事基
​}
i2345676
Size Limit Regulation Analysis
(S.L.R.A.)
事集第
​}
}
}
本基本
​This program simulates the effects of imposing any
基本事 ​of six size limit regulations on sport fisheries.
Changes in seasonal distribution of natural, fishing, and
hooking mortality may also be studied. Fry survival and
** growth are modeled as density-dependent relationships.
. The program will handle up to 15 age groups. A year ***
*** may be broken up into as many as 52 discrete time periods. ***
All calculations are done in double precision aritn- ***
metic. Execution is terminated when the number of years
to be simulated has been reached. Unit 5 is assigned to
input. Unit 6 is assigned to output.
16
17
INPUT:
Data input is performed using free format. However,
specific formatting may be used if necessary. For a list ***
of the FORTRAN variables used and data file setup, see
attached documentation.
非軍事
​22222222。
ot 2345678
*本中
​本东南
​出事事
​PROCEDURE:
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc.
T 2 4456
本事事
​## #
事本事
​The model (Smith, 1981 - Master's thesis, University *
of Michigan) is a modification of Clark's method (Clark *
et. al., 1980 - Trans. Amer. Fish. Soc.). Computed in a **
time period are - population numbers, mean length, stand- ***
ard deviation about the mean length, total catch of legal
fish, legal harvest, total catch and release of illegal
pish, nooking deaths, natural deaths, and a length-
frequency distribution of the catch data. The catch data 事事事
​also includes the weight for each component.
mo
43
OUTPUT:
事事
​事事事​,
事事事
​本事事
​Output is in two parts. First, ali pertinent data
input is printed along with a table defining symbols used
事事事 ​in the output. Second, beginning and ending population
事事事 ​characteristics for the year simulated are printed, along **
with a length-frequency distribution of the starting pop-
*** ulation. If fishing and/or hook ing mortality are present, ***
本事事 ​a length-frequency distribution of the catch data is also
printed
本事第
​Catch and natural death data may be printed for each
time period if desired.
来来来
​Ctes
at dds. 5s 5555
7896i 2345678
事事非事事事事​事事事律事第4集第3年半军事军军部落事​》第第第第第第第第表
​本集第第第第拿第易第2集第
​事事律事事军军事军事事事事基本事事事第第第第第第第第骨事事事事事
​59
星期事来来来来單集第第第第革军事革是靠本事来表其事本事事本本来来来来第第第
​事事事事本军军事
​60
61
Declare variable types and dimensions ...
6666 66
2345678
IMPLICIT REAL*8 (A-2)
INTEGER TITLE(20), PUNIT(20), WUNIT(20), LUNIT(20), FMTUNI. REG..
+ REGCHK(7),TYPE(7,10),r.J,K, FLAGX, YEAR, NY RSIM, NUMAGE,
+ MAXAGE, MX AGE, NUMTIM, SPWNTM,AGEPRN(15), MAXLEN, INCH, INCHM1,
+ INITK, INITK2, LL, UL. NUMPRD, LINES ( 18), YRSTRT ( 20 )
LOGICAL SKPRES, NOLAST
LOGICAL*1 FREE( 1)/'*'/
DIMENSION LARD(15), LN( 15.52), LM(52),LH(52),LPP(52), LPDP(52),
12
.
+ PERLN(52), PERLM( 52), LEN( 16), LENDEV( 16), PERLEN( 52);
* LENDIF( 15), DEVDIF(15), RATLOV(16), CORLN( 16), POPNUM( 15),
# WP(15), D(15,52), LHVHD( 15, 52,2), TCLTCR( 15, 52,2),
+ WLHWHD ( 15,52,2), WCLWCR ( 15.52,2). LHV(15), TCL ( 15), WLH( 15).
+ WCL (15), HD( 15), TCR(15), WCR (15), WHO ( 15). DSMAG( 15 )
+ LFREQ(30,2), LFRSUM(2), CFREQ.( 30,2), CFRSUM(2); WFREQ(30,2),
+ WFRSUM(2), CFRQ(30,2), CFRSM(2). WFRQ(30,2), WFRSM(2),
* CT4 (15,52), CT5( 15, 52), CT6( 15,52), CT7( 15,52), CT9(15),
+ CT 10( 15 )
COMMON X1, X2, X3, X4
:
...
Initialize variables
.
:
DATA POPNUM, LFREQ, LFRSUM, CFREQ, CFRSUM, WFREQ. WFRSUM, CFRQ,
* CFRSM, WERQ, WFR$M, THETD1, THETD2, THETD3, THETD4, CT4,
+ CT5, CT6 CT7. CT9, CT 10/325*0.ODO, 4*1.ODO, 3150*0.ODO/
DATA REGCHK, YEAR, LINES/'MIN '. 'MAX'. 'KST ','KS :.'RS ',
'CR ','UN !,1, 18-'n-o-'/
DATA TYPE/MINI', 'MAXI, 2* KILL', 'RELE','CATC', 'UNEX, 2* MUM ,
* 2*, SLO', 'ASE: .,'HAN','PLOI', 'SIZE', (INV', 'T WI','T SI',
'SLOT!,'D RE', 'TED'' LIM', 'ERTE', 'TH T', 'ZEL, SIZ', 'LEA
POPU: 'IT ... 'D) s', 'ROPH', 'IMIT', 'E LI','E ON','LATI',
* 'IZE "i'Y CA'.'. ''MIT ":'LY ''ON , ' ' LIMI TCH 1
SIZE , 6* ',! LIM!,6*' '. 'IT .4*
DATA AGEPRN/' ','II ','III : , ' IV ', 'V ','VI '. 'VII '.
VIII, IX , X .; XI . XII. XIII', 'XIV , 'XVI
93
94
95
97
98
X
io. Begin data input
99
100
101
102
READ (5, 1) (TITLE(1),1-1,20).
1 FORMAT (20A4)
READ. (5,2) REG
2 FORMAT (A4)
103
104
105
106
... Determine type of regulation being simulated and value of
FLAGX
107
108
109
110
111
DO 31*1.7
FLAGX=1
IF (REG. EQ. REGCHK( I )) GO TO 4
CONTINUE
112
113
114
115
116
117
... Continue data input ...
4 READ (5.1) (PUNIT(I),I=1,20), (WUNIT(I),I=1,20), (LUNIT(I).1-1.20)
READ (5,2) FMTUNI
READ (5, 1) (YRSTRT(I),I=1,20)
READ (5, FREE) NUMAGE, MAXAGE, NUMTIM, NYRSIM, SPWNTM, MLMAT,
+ PPRIM, OPRIM, POPRIM, DOPRIM
READ (5. FREE) NFRY, LEND IV, EALPHA, EBETA, FALPHA, FBETA. LALPHA,
+ LBETA, WALPHA, WBETA
MXAGE=MAXAGE # 1
READ (5, FREE) (LNRD(I).1*1. MAXAGE)
Calculate percent of annual conditional natural mortality
occuring in each time period within the year ....
:.
UL=0
5 READ (5, FREE) NUMPRD, PERC
LLSUL # 1
UL-UL * NUMPRD
DO 6 J*LL, UL
PERLN(j) PERC/DBLE(FLOAT(NUMPRD))..
6 CONTINUE
IF (UL.NE. NUMTIM) GO TO 5
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
:: ; . Calculate percent of annual conditional fishing (and
hook ing if present) mortality occuring in each time period
within the year ...
143
UL:0
7 READ (5. FREE) NUMPRD, PERC
144
LLEUL 4.1
ULEUL # NUMPRD
DO 8 JELL, UL
PERLM(J)=PERC/DBLE (FLOAT(NUMPRD ) )
CONTINUE
IF (UL.NE .NUMTIM) GO TO 7
ono
...:
Input growth data
...
145
146:
147
148
149
150
151
152
153
154
155
156
157
158
159
READ (5, FREE) LK, LINF, MAXLEN
READ (5, FREE) (LEN(I),1*1, MXAGE), (LENDEV(I),1=1, MXAGE)
uovo
. Calculate percent of annual growth occuring in each time
period within the year ...
:.
160
ULo
9 READ (5. FREE) NUMPRD, PERC
LL=UL + 1
ULEUL + NUMPRD
DO 10 JELL,UL
PERLEN(J)=PERC/DBLE (FLOAT( NUMPRD))
10 CONTINUE
IF (UL.NE. NUMTIM) GO TO 9
... Continue data input one
READ (5, FREE) (CORLN(I), 131.MXAGE)
(5, FREE) X1, X2, X3, X4
READ (5. FREE) (POPNUM(I).I=1, MAXAGE)
Calculate annual conditional fishing and nooking
mortalities ...
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
nnnn
LMRDOPPRIMOPRIM.
LHRD-PDPRIM*DOPRIM
-
nonn
... Calculate constant term in Ford's growth equation and
initialize summation variables to zero ...
182
CTLNEQ=LINF*(1.ODO - LK)
POPSUM=0.ODO
WPSUM=0.000
:
183
184
185
186
187
188
189
190
191
.. noon
Calculate annual conditional natural, fishing, and hooking
mortalities for each time period within the year. Calculate
length to deviation ratios and starting population weight by
age ...
192
-
-
.
193
194
195
196
197
198
..
199
MSAVEDLOG( 1.ODO - LMRD)
HSAVODLOG(1.0DO - LHRD )
DO 14 1:1, MAXAGE
RATLDV () LEN(I)/LENDEV(I)
NSAVEDLOG(1.ODO - LNRD(I))
IF (.GT. NUMAGE) GO TO 17
WP ( 1.) =DBLE(FLOAT( IFIX (SNGL (DEXP (WALPHA + WBETA*DLOG(LEN(I)))*
C POPNUM( I )* 10000.00 + 0.5DO))))/ 10000. ODO
POPSUM=POPSUM + POPNUM(I)
WP SUM=WPSUM + WP (I)
11. DO 13 ja 1. NUMTIM
LN(1,)=1.000 - DEXP (NSAV*PERLNU))
IF (PERLM(U).LE.O. ODO) GO TO 13
IF (I.NE. 1). GO TO 12
LM(J): 1. ODO - DEXP ( MSAV*PERLM(J))
LPP (J)=PPRIM*PERLM(J)
IF (DPRIM: GT.O.ODO) LPP (U)=LM(U)/DPRIM
LH()1. ODO • DEXP (HSAV*PERLM(J))
LPDP (U)*POPRIM*PERLM(U)
: IF (DOPRIM.GT.O.ODO) LPDP (J) LHIJ)/DDPRIM
200
201
202
203
204
205
206
207
208
209
210
211
212
:
.
1
89
213
.. Calculate common terms used in more than one population
equation ...
214
215
2:16
217
218
219
220
12 CT 1-LPP (U) om LN(I,J)
CT2-LPDP(u) LN(I, U)
CT3-LN(I, U)**2
IF (CT1.GT.O.ODO) CT4(I, U)=LPP (U)*(1.ODO - CT3/CT 1)
IF (CT2.GT.O.ODO) CT5(1,0)=LPDP (J)*(CT3/CT2 - 1.000)
IF (CT2.GT.0. ODO) CT6(1,0) LN(I,J)*LH(U)*LPDP (U)/CT2
IF (CT 1.GT.O.ODO) CT7(1.0)=LN(I.J)*LM(J)*LPP(J)/CT 1
13 CONTINUE
14 CONTINUE
RATLDV (MXAGE )=LEN(MXAGE)/LENDEV(MXAGE)
29
225
226
227
Determine initial values to be used in calculating
the frequency distributions .....
228
229
230
231
232
233
234
235
IF (FLAGX. EQ.7) GO TO 15
INITX-X2
INITK1
INITK2*2
IF (FLAGX. EQ.6) INITXEDBLE(FLOAT (MAXLEN))
IF (DABS(X2 - X1).GT, 1.00-05) GO TO 15
INITX=X3
INITKO2
INITK2=1
IF (FLAGX.EQ.1) INITX=OBLE(FLOAT(MAXLEN))
236
237
238
239
240
241
... Output of pertinent data
...
242
...
.
246
247
248
: 249
250
251
252
253
254
255
256
257
258
259
260
15 WRITE (6, 16) (TITLE(I),1=1,20), (TYPE(FLAGX. I),1=1,10)
16 FORMAT ('tillli'-SIZE LIMIT REGULATION ANALYSIS ... 2014.1.
*'-TYPE OF REGULATION - ', 10A4)
IF (FLAGX. EQ.7) GO TO 30
WRITE (6, 17) X1, FMTUNI
17 FORMAT (OMÍNIMUM LENGTH AT WHICH FISH ARE SUSCEPTIBLE TO ,
* CAPTURE: ..F5.1, 1X, A4)
GO TO ( 18, 20, 22, 24, 26, 28, 30), FLAGX
18 WRITE (6, 19), X2, FMTUNI
19 FORMAT ('OLENGTH FOR LEGAL HARVEST: ".F5.1, 1X, A4, 'OR LARGER')
GO TO 30
20 WRITE (6:21) X2, X3. FMTUNI
21 FORMAT ( OLENGTH FOR LEGAL HARVEST: ', F5.1,' TO',F5. 1, 1X, A4)
GO TO 30
22 WRITE (6,23 ) X2, X3, X4, FMTUNI
23 FORMAT (OLENGTH SLOTS FOR LEGAL HARVEST: ',F5.1,' TO.55.1,
+.AND ',F5.1, 1X, A4, 'OR LARGER')
GO TO 30
24 WRITE (6,25 ) X2, X3, FMTUNI
25 FORMAT ( OLENGTH SLOT FOR LEGAL HARVEST: ',F5.1.'. TO ',F5.1,
*.1X, A4)
GO TO 30
26 WRITE (6,27) X2, X3, X4, FMTUNI
27 FORMAT (OLENGTH SLOTS FOR LEGAL HARVEST: : ',F5.1,' TO ,F5.1,
* ' AND ', F5.1, 1X, A4, 'OR LARGER').
GO TO 30
28 WRITE (6.29) X1, FMTUNI
29 FORMAT ('OLENGTH FOR CATCH AND RELEASE: ..F5.1, 1X,A4, OR LARGER')
.30 WRITE (6,31) (YRSTRT(I),1-1,20), NYRSIM. NUMTIM, MAXAGE, SPWNTM,
...+ MLMAT :
31 FORMAT 111,'-YEAR STARTING: '.2014./, 'ONUMBER OF YEARS TO BE '.
* SIMULATED : ',12,1,' NUMBER OF TIME PERIODS PER YEAR.; '.12.
" OLDEST EXPECTED AGE OF ANY FISH : '.12./,' 'TIME PERIOD WHEN',
.+! SPAWNING OCCURS : '.12,/,' MINIMUM LENGTH OF MATURE FEMALES:',
* , F5.1)
WRITE (6,32) NFRY, LENDIV. EBETA, EALPHA, FBETA, FALPHA, LBETA.
+ LALPHA, WBETA. WALPHA
32 FORMAT 1 OFRY MORTALITY: '.F10.8,/,' LENGTH - FRY REGRESSION ',
CALIBRATION FACTOR: '.F4.2,///,'-REGRESSION COEFFICIENTS:'.1.
.+ 0,23%, 'SLOPE', 8X, 'INTERCEPT',/ , 'OEGG • LENGTH'.8X, F11.2.5x, .
+ F9.2,/,' FRY - EGG', 10X, E 12.5, 5X, F9.5,/,' LENGTH - FRY'.8X,F11.5,
261
262
263
264
265
266
267
268
269
270
274
272
273
274
275
276
:
277
278
279
280
281
282
283
284
.
90
-...
-
285
286
287
288
289
290
291
292
293
+ 5X,F9.5./.WEIGHT - LENGTH' , 5X,F11.4,5x, F9.4)
WRITE (6,33) LK, LINF, (PERLEN(U).va 1, NUMTIM),
33 FORMAT (11,'- FORD'S GROWTH COEFFICIENT : '.F6.4,/,' MEAN ASYMP! .
+ TOTIC GROWTH (L • INFINITY): ', F8.4,/,'-PERCENT CHANGE IN ';.
+ LENGTH FOR EACH TIME PERIOD: './/.(1X, 10(F7.5, 3x)))
WRITE (6,34) LMRD, PPRIM, OPRIM, LHRD, PDPRIM, DOPRIM
34 FORMAT ('t',117,'-ANNUAL CONDITIONAL FISHING MORTALITY (SMALL M)',
+ '; ,F7.5,/,' PROBABILITY OF CAPTURE , 8H(Pi): ,F7.5,1.' PRO',
+ 'BABILITY OF HARVEST GIVEN CAPTURE ', 8H(D ): F7.5.1, 'OANNUAL',
+ " CONDITIONAL HOOKING MORTALITY (SMALL H): '.F7.5.1. PROBABIL.',
+ 'ITY OF CAPTURE !,8H(D''): ,F7.5,/,' PROBABILITY OF DEATH',
*.GIVEN CAPTURE 1,8H(D"): 77.5)
IF (FLAGX.NE, 7. AND.FLAGX.NE.6) WRITE (6,35) (PERLM(J), =1, NUMTIM)
35 FORMAT ('-PERCENT CHANGE IN ANNUAL CONDITIONAL FISHING (AND ';
+ 'HOOKING IF PRESENT) MORTALITY. FOR EACH TIME PERIOD:', /1,
+ (1x, 10(77.5,3X)))
IF (FLAGX. EQ.6) WRITE(6, 36) (PERLM(u),*1, NUMTIM)-
36 FORMAT ('-PERCENT CHANGE IN PROBABLILITY OF CAPTURE FOR EACH ',
'TIME PERIOD:'.11. (1X, 10(F7.5,3x)))
:WRITE (6,37) (AGEPRN(I),1-1, MAXAGE)
37 FORMAT (1, -ANNUAL CONDITIONAL NATURAL MORTALITY (SMALL N) BY',
+ ' AGE:',,'0'; 1X, A4,6(3X, A4), 2X, A4,4X, A4,3(3x, A4).2x, A4,4X, A4,
+ 3X, A4)
WRITE (6, 38) (LINES(1),1=1.MAXAGE)
38 FORMAT (1X, 15( A4,3x))
WRITE (6,39) (LNRD(I),1-4, MAXAGE)
39 FORMAT (1X, 15(F4.2, 3x)):
WRITE (6.40) (PERLN(U).= , NUMTIM)
40 FORMAT ("-PERCENT CHANGE IN CONDITIONAL NATURAL MORTALITY FOR '.
the EACH TIME PERIOD :',//. (1x, 10(F7.5, 3x))).
WRITE (6,41) (PUNIT(I),1-1,20), (WUNIT(I),131,20),
* (LUNIT(I),1=1,20), (PUNIT(I).1-1,20), (WUNIT(I),1-1,20)
(PUNIT(I),1-1,20), (WUNIT(I),1*1,20), (PUNIT(I),1-1,20)
+ (WUNIT(I),1-1,20), (PUNIT(I),1=1,20), (WUNIT(I),1-1,20),
* (PUNIT(I), 1=1,20)
41 FORMAT ('..//.'-DEFINITIONS OF SYMBOLS USED IN THIS OUTPUT (UN'
+ ITS IN PARENTHESES):',7,' op • POPULATION NUMBERS ', 2014,/,'O'.
+ 'WP - POPULATION WEIGHT , 2014./, 'OL - MEAN LENGTH.', 2014, 1.
ODV - STANDARD DEVIATION ABOUT THE MEAN LENGTH";/, 'OTCL • '.
* CATCH OF LEGAL FISH , 2014,/ 'OWCL - WEIGHT OF LEGAL CATCH '.
+ 2014./..OLH - HARVEST OF LEGAL FISH '', 2014,/ , 'OWLH - WEIGHT '.
+ OF HARVESTED CATCH ., 2014./ , 'OTCR - CATCH AND RELEASE OF ILL'.
+ EGAL SIZED FISH., 2014./. 'OWCR - WEIGHT OF CAUGHT AND RELEAS',
+ ED FISH '.2014./ . 'OHD - HOOKING DEATHS OF ILLEGAL SIZED FIS'.
+'H!, 2014., 'OWHD - WEIGHT OF HOOKING DEATHS '. 2014.. 'OD.-',
+ ' NATURAL DEATHS ', 2014)
294
295
296
297
298
299
300
301
302
303
304
305
306
307
:
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
UUUU
Calculate inverse of square root of 2 times Pi used in
calculating conditional means for length ....
ISQ2PI=1.000/DSORT (2.000*3.1415900)
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
ii.
Initialize summation by age variables ...
42 DO 44 1*2.NUMAGE
OSMAG( I - 1)-0.ODO
TCL(I - 1) =(.ODO
TCR(I - 1)-0.ODO
LHVCI - 1)=0.ODO
HD(I • 1) =0.ODO
WCL(I • 1 )=0.000
WCR(I - 1)=O. ODO
WLHI - 1) =O. ODO
WHD ( I - 1)-O. ODO
KNUMAGE + 2 • I
349
onnn
Bump population numbers and lengths to next age group
unless YEAR equals 1 ...
350
351
352
353
354
355
356
IF (YEAR.EQ.1) GO TO 43
POPNUM(K) =POPNUM(K - 1)
WP (K)=WP (K - 1)
LEN(K) = LEN(K - 1)
LENDEV(K) LENDEV(K - 1)
43 IF (POPNUM(K).LE.O.ODO) GO TO 44
onon :
357
358
359
360
361
362
Calculate lengths and deviations to be reached by end
of year
LNGTH=OBLE (FLOAT( IFIX( SNGL((CTLNEQ = LEN(K)*LK)*10.ODO + 0.5DO
C))))/10.ODO + CORLN(K + 1)
LNDV=DBLE (FLOAT(IFIX (SNGL (LNGTH/RATLOV(K + 1))*1000.ODO + 0.500
C))))/ 1000. ODO. .
LEND IF(K)=LNGTH - LEN(K)
DEVDIF(K)=LNDV - LENDEV(K)
44 CONTINUE
IF. (NUMAGE. EQ.1) GO TO 45
DSMAG( NUMAGE ) -0.ODO
TCL (NUMAGE ) =0.ODO
TCR (NUMAGE) -0.0DO
LHV ( NUMAGE)-0.ODO
HD (NUMAGE) =0.ODO
WCL (NUMAGE)*0.000
WCR (NUMAGE ) =0.ODO
WLH( NUMAGE ) -0.ODO
: WHD (NUMAGE)-0.ODO
IF (YEAR. EQ.1) GO TO 45
POPNUM( 1 )-FRY
WP(1) -WFRY
LEN(1) INLEN
LENDEV( 1 )INDEV
45 IF (POPNUM( 1).LE.O.ODO) GO TO 46 :
LNGTHODBLE (FLOAT (IFIX (SNGL( (CTLNEQ + LEN( 1 )*LK)*10.0DO + 0.500
C))))/10.ODO + CORLN(2)
LNDV=OBLE (FLOAT(IFIX(SNGL ((ENGTH/RATLDV ( 2 ) )*1000.ODO + 0.5DO
C))))/1000.ODO
LENDIF(1)=LNGTH. - LEN(1)
DEVDIF(1) LNDV - LENDEV( 1)
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
.... Output of starting population characteristics
....
388
389
390
391
392
393
394
395
396
397
398
399
400
46 WRITE (6,47) YEAR, (LINES(I),1=1,7). (AGEPRN(I), POPNUM(I),
: WP (I), LEN(I), LENDEV(I),1=1, NUMAGE).
47 FORMAT ('9'./ .'-STARTING POPULATION – YEAR',12,' SIMULATION:'./,
'-AGE', 13X, 'p', 12x, .WP', 10X, 'L'.8X, 'DV' ,/. 1x, A4,6X,
2 (2A4.'-'.5x).A4;'o'.5X, A4, '--',l. (1X.A4,6X, F9.2,5X,
+ F9.4,5X, F5:1,5X, F6,3))
WRITE (6.48) POPSUM, WPSUM
48 FORMAT ('OTOTALS', 3X, F10.2,4X,F 10.4)
401
nnnnn
Call subroutine LNFREQ for length (weight) - frequency
distribution of the starting population and then print
these distributions
402
403
404
405
406
407
408
409
410
411
412
4.13
414
415
416
417
418
CALL LNFREQ (POPNUM, WP, NUMAGE, MAXLEN, LEN, LENDEV.LFREQ, INITX, FLAGX)
WRITE (6,49) FMTUNI, (LINES(I),I=1.6)
49 FORMAT 111,'-LENGTH (WEIGHT.) - FREQUENCY DISTRIBUTION OF THE '.
+ 'STARTING POPULATION: ' ,/,'O' ,A4,' GROUP', 11X, 'p', 12X, 'WP',
+ 1, 1X, 2A4,'--'.2(5X, 244,'-'))
DO 51 INCHE 1, MAXLEN
INCHM 1 = INCH - 1
LFREQ( INCH, 1 ) =OBLE(FLOAT(IFIX(SNGL (LFREQ( INCH, 1)*100.ODO + 0.500)
C )))/100. ODO
LFREQ( INCH, 2 ) =DBLE(FLOAT( IFIX(SNGL (LFREQ( INCH, 2)* 10000.ODO + 0.5DO
C))))/ 10000. ODO
WRITE (6,50) INCHM1, INCH, LFREQ(INCH, 1), LFREQ(INCH, 2)
50 FORMAT (1X, 13.' TO', 13, 5X, F9.2,5X, F9.4).
LFRSUM( 1 )=LFRSUM( 1 ) + LFREQ( INCH, 1)
LFRSUM( 2 )*LFRSUMI 2) + LFREQ( INCH, 2)
LFREQ( INCH, 1 ):0.ODO
LFREQ(INCH, 2)-0.ODO
51 CONTINUE
WRITE. (6,52) (LFRSUM(I), 1=1,2)
52 FORMAT ('OTOTALS, 8X,F10.2,4X,F10.4)
419
420
421
422
423
424
425
426
427
OU AWNO
428
92
LFRSUM( 1 )=0.ODO
LFRSUM( 2 ):0.ODO
000
429
430
431
432
433
...
Initialize yearly summation variables
434
POP SUMO. ODO
WPSUM*O.ODO
LHVSUM=0.ODO
TCLSUM=O. ODO
HD SUM=O. ODO
TCRSUM O.ODO
DSUM-O. ODO
WLHSUM=O.ODO
WCRSUM-O. ODO
WCLSUM=O. ODO
WHO SUM=O.ODO
...
იიიი იიი
.... Begin simulation for a year
DO 66 v= 1. NUMTIM
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
... call subroutine SPWNSB if spawning occurs during this
time period ...
IF (D.EQ. SPWNTM) CALL SPWNSB (POPNUM, NUMAGE, LEN, LENDEV, CTLNEQ, LK,
* MLMAT, ISQ2PI, EALPHA, EBETA, FALPHA , FBETA, LALPHA , LBETA, LENDIV,
+ RATLDV( 1), WALPHA, WBETA, NFRY, EGGS, FRY, INLEN, INDEV, WFRY)
...
UN
იიიი
Begin age loop. Initialize variables for keeping catch
and natural death data within a year ...
00
459
460
DO 63 11, NUMAGE
LHVHD(I.J,1)=0.ODO
LHVHD(I,J,2)=0.000
TCLTCR(I.V,1)=0.ODO
TCLTCR(I,J,2)=0.000
WLHWHD(I,1,1)=0.ODO
WLHWHD(1,1,2)=0.ODO
WCLWCR(I.,1)=0.ODO
WCLWCR(I, 0,2)=0.ODO
D(I, U)=0.ODO
461
462
463
464
465
466
467
468
469
470
471
472
473
....
... If the age group has no fish, go to end of age loop
IF (POPNUM( I ).LE.O.ODO) GO TO 63
474
იიი იიიი იიი
...
If no fishing or hook ing mortality, go to calculation of
natural deaths and new population estimate ...
IF (PERLM(U).LE.O.ODO) GO TO 62
... Initialize mean weight of each length interval too...
EPHI 1-0. ODO
EPHI 2-0. ODO
EPHI3=0.ODO
EPHI4O. ODO
475
476
477
478
479
480
48.1
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
... Calculate 2 times the length standard deviation squared
used in calculating conditional means for length ...
იიი იიიი
DSQT2=2.ODO*LENDEV( I ) **2
...
Calculate areas between each length limit ...
THETD 1 3 I NORM((x1 - LEN(I))/LENDEV(I))
IF (FLAGX. EQ.6) GO TO 54
THETD2= I NORM( (X2 - LEN(I))/LENDEV(I))
IF (FLAGX. EQ.1) GO TO 53
THETD3= I NORM( (X3 - LEN(I))/LENDEV(I))
IF (FLAGX. EQ.2. OR. FLAGX. EQ.4) GO TO 53
THETD4+ I NORM( (X4 - LEN(I))/LENDEV(I))
499
500
93
:
501
Calculate conditional mean length and mean weight for areas
between each length limit ...
502
503
504
505
506
507
508
509
510
511
512
513
514
515
16
517
518
OLENCIA THE MOST?! LENGD.ODO
519
520
521
522
523
53 IF (THETD2 - THETD 1.LE. 1.0D-40) GO TO 55
MEAN=(ISQQPI*LENDEV( I )*(DEXP(- (x1 - LEN( I ))**2/DSQT2) -
C DEXP(- (x2 - LEN(I))**2/DSQT2)))/(THETD2 - THETD 1) + LEN(I)
EPHI 1=DEXP (WALPHA + WBETA*DLOG(DBLE(FLOAT(IFIX(SNGL (MEAN* 10.000
C + 0.500))))/ 10.ODO))
GO TO 55
54 IF (1.ODO - THETD 1.LE.1.OD-40) GO TO 60
MEAN=1 SQ2PI*LENDEV( I )*DEXP(- (x1 - LEN(I) )**2/DSQT2)/(1.000 -
C THETD 1). + LEN(I)
EPHI **DEXP(WALPHA + WBETADLOG(DBLE(FLOAT(IFIX( SNGL (MEAN*10.ODO
C + 0.500))))/10.ODO))
GO TO 60
55 IF (FLAGX. EQ.1) GO TO 56
IF (THETD3 - THETD2.LE. 1.0D-40) GO TO 57
MEAN=(ISO2PI *LENDEV(I)*(DEXP(- (x2 - LEN(I))**2/0SQT2) -
C DEXP(- (x3 = LEN(I))**2/DSOT2)))/(THETD3 - THETD2) + LEN(I)
EPHI 2=DEXP (WALPHA + WBETA*DLOG(DBLE(FLOAT(IFIX( SNGL (MEAN*10. ODO
C + 0.500))))/10. ODO))
GO TO 57
56 IF (1.ODO - THETD2.LE.1.0D-40) GO TO 60
MEAN=ISO2P1 *LENDEV( I ) *DEXP(- (x2 - LEN(I) )**2/DSQT2)/(1.000 -
C THETD2) + LEN(I)
EPHI 2 DEXP(WALPHA + WBETA-DLOG(DBLE(FLOAT(IFIX(SNGL (MEAN= 10.ODO
C + 0.5DO))))/10. ODO)).
GO TO 60
57 IF (FLAGX. EQ.2. OR. FLAGX. EQ.4) GO TO 58
IF (THETD4 - THETO3. LE.1.00-40) GO TO 59
MEAN (ISO2PIELENDEV(I)*(DEXP(- (x3 - LEN(I))**2/DSOT2) -
C DEXP(- (X4 - LEN(I))**2/DSQT2)))/(THETD4 – THETD3) + LEN(I)
EPHI3=DEXP(WALPHA + WBETA*DLOG(DBLE(FLOAT(IFIX(SNGL (MEAN*10.000
C+ 0.5DO))))/10.ODO))
GO TO 59
58 IF (1.ODO - THETD3.LE. 1.0D-40) GO TO 60
MEAN ISQ2PI*LENDEV(I)*DEXP(- (x3 - LEN(I) )**2/DSQT2)/(1.000 -
C THETD3) + LEN(I)
EPHI3DEXP(WALPHA + WBETA DLOG(DBLE (FLOAT(IFIX(SNGL (MEAN*10.ODO
C + 0.500))))/10.ODO)).
GO TO. 60
59 IF (1.ODO - THETD4.LE. 1.00-40) GO TO 60
LE:1.0074)GOTIESO S OVA
MEAN=ISO2PI*LENDEV( I ) *DEXP(- (X4 - LEN(I))**2/DSQT2)/(1.ODO •
.
C THETD4) + LEN(I)
EPHI 4*DEXP (WALPHA + WBETA*DLOG(DBLE(FLOAT(IFIX(SNGL (MEAN*10.ODO
C + 0.500))))/10.ODO))
524
525
526
527
528
529
OUW
530
531
532
533
534
535
1
AWN
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
იიიი
... Calculate common terms used in more than one population
equation ...
60 CT8=THETD2 - THETD3 + THETD4
CT9(1):THETD1 - CT8
CT 10( 1 )=1.000 - CT8
CT 11 = POPNUM( I ) *CT4(I,J)
CT 12POPNUM(I)*CT5(1,1)
CT 13*EPHI4 - EPHI2*THETD2 + EPHI2*THETD3 - EPHI4*THETD4
CT 14-EPHI 1*THETD 1 - EPHI 1 *THETD2 + EPHI3* THETD3 - EPHI3*THETD4
...
If catch and release only regulation, skip calculation
of legal harvest ...
იიიი იიიი
560
561
562
563
564
565
566
567
IF (FLAGX.EQ.6) GO TO 64
... Calculate total legal fish caught and legal harvest
(number and weight) ...
TCLTCR(I,J,2)=CT 11 *CT 10(1)
TCL(I)-TCL (I) + TCLTCR(I,J. 2)
WCLWCR(I.J. 2) =CT 11 *CT 13
WCL(I) =WCL(I) + WCLWCR(I,J. 2)
LHVHD(I,J, 2) =TCLTCR(I,J, 2) *DPRIM
570
571
572
94
LHV(I) LHV(I) + LHVHD(1,1,2)
WLHWHD(I..2)=WCLWCR(I,1,2) *DPRIM
WLH(I) =WLH(I) + WLHWHD(I,J,2)
იიიი
... Calculate total illegal fish caught and released and
nook ing deaths (number and weight) ...
61 TCLTCR(I,1,1)=CT 12*CT9(I)
TCR(I) TCR (I) + TCLTCR(I, 1, 1)
WCLWCR(I,1,1) -CT 12*CT 14
WCR(I) =WCR(I) + WCLWCR(I,, 1)
IF (DOPRIM.LE.O.ODO) GO TO 62
LHVHD( 1, 0, 1) TCLTCR(I,J,1) *DOPRIM
HD(I)«HD(I) + LHVHD(I,J,1)
WLHWHD(I,U, 1)-WCLWCR(I, 0, 1) *DOPRIM
WHD ( 1 )=WHD(I) + WLHWHD(I,J, 1)
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
იიიი
... Calculate natural deaths and new estimate of the
remaining population ...
D(1,0)=POPNUM( I )*(LN(I,J) + CT6(1.)*CT9(I) - CT7(1,1) *CT10(I))
DSMAG(1) EDSMAG(I) + D(I,J)
POPNUM( I )=POPNUM(I) • D(I,J) · LHVHD(I,J,2) · LHVHD(I,1,1)
... If at end of year, truncate all variables for later
output ...
595
იიიი
596
597
598
599
600
601
602
603
604
IF (D.NE. NUMTIM) GO TO 63
POPNUM( I )-DBLE(FLOAT(IFIX(SNGL (POPNUM( 1 )*100.ODO + 0.5500))))/
C 100.ODO
LHV( I )-DBLE(FLOAT(IFIX( SNGL (LHV( I )*100.ODO + 0.500
C))))/100.ODO.
TCL(I)-OBLE(FLOAT( IFIX(SNGL (TCL(I)*100.000 + 0.500
C))))/100.ODO
WLH(1)-DBLE (FLOAT(IFIX(SNGL (WLH( I )* 10000.ODO + 0.500
C))))/ 10000. ODO,
WCL(I) =DBLE (FLOAT(IFIX( SNGL (WCL( 1 ) * 10000.ODO + 0.500
C))))/10000.ODO
HD(I ) DBLE(FLOAT( IFIX (SNGL (HD( 1 )*100.ODO + 0.5DO
C))))/100. ODO
TCR ( 1 )-DBLE (FLOAT(IFIX(SNGL(TCR(I)*100.ODO + 0.5DO
C))))/100.ODO .
WHD ( 1 )-DBLE (FLOAT(IFIX (SNGL (WHD ( 1 )* 10000.ODO + 0.500
C))))/10000. ODO
WCR (I) =DBLE (FLOAT( IFIX(SNGL (WCR( I )* 10000.ODO + 0.5DO
c))))/ 10000.000
DSMAG( I )DBLE(FLOAT(IFIX (SNGL (DSMAG(I)*100.ODO + 0.5DO
C))))/100.ODO
605
606
607
608
609
610.
611
612
613
614
6.18
იიი
...
Sum components by age group
...
622
623
624
AWN
LHVSUM LHVSUM + LHV(I)
TCL SUM=TCLSUM + TCL(I)
WLHSUM=WLHSUM + WLH(I)
WCLSUMEWCLSUM + WCL(I)
HDSUM«HD SUM + HD(I)
TCRSUM=TCRSUM + TCR(I)
WHD SUM WHOSUM + WHD(I)
WCR SUM=WCRSUM + WCR(I)
DSUMEDSUM + DSMAG(I)
628
629
630
631
62
633
...
End of age loop
...
63 CONTINUE
იიიიი იიი
...
Call subroutine CRFREQ for length (weight) • frequency
distribution of the catch data (unless an unexploited
population is being studied)
637
638
639
640
641
642
643
644
NOLAST = TRUE.
IF (U.EQ.NUMTIM) NOLAST=.FALSE.
SKPRES= . TRUE.
.
95
.
C
IF (PERLMOJ).LE.O.ODO) GO TO 64
CALL CRFREQ (NUMAGE, TCLTCR , LEN, LENDEV.MAXLEN, FLAGX ,
......+ INITK, INITK2, INITX, CFREQ, U, CT 10, CT9, POPNUM, WP, SKPRES, NOLAST)
IF (FLAGX. EQ.6. AND . DDPRIM.LE.O.ODO) SKPRES=. FALSE.
CALL CRFREQ (NUMAGE , WCLWCR , LEN, LENDEV, MAXLEN, FLAGX,
+ INITK, INITK2, INITX, WFREQ, U, CT 10, CT9, POPNUM,WP, SKPRES, NOLAST)
IF (FLAGX.EQ.6. AND.DDPRIM.LE.O.ODO) GO TO 64
CALL CRFREQ (NUMAGE , LHVHD, LEN, LENDEV, MAXLEN, FLAGX,
+ INITK, INITK2, INITX, CFRQ.0, CT 10, CT9, POPNUM, WP, SKPRES, NOLAST).
CALL CRFREQ (NUMAGE, WLHWHD, LEN, LENDEV, MAXLEN, FLAGX,
+ INITK, INI TK2. INITX, WFRQ,J,CT 10, CT9, POPNUM, WP, FALSE. NOLAST)
... Bump lengths and deviations for the next time period -
truncate for output if at end of year io.
DO 65 1=1, NUMAGE
IF (POPNUM(I).LE.O. ODO) GO TO 65
LEN(I) = LEN( I ) PERLEN(J)*LENDIF(I)
LENDEV(1) LENDEV(I) + PERLEN(J) *DEVDIF(I)
IF (J.NE. NUMTIM) GO TO 65
LEN(I)-DBLE(FLOAT(IFIX(SNGL(LEN(I)* 10.ODO * 0.5DO))))/ 10.ODO
LENDEV(I) •OBLE (FLOAT(IFIX (SNGL (LENDEV( I )* 1000.ODO + 0.500))))
C / 1000.ODO
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
...
...
...
Sum population and weight at end of year ...
669
670
671
672
IF(I. EQ. MAXAGE) GO TO 65
POPSUM POPSUM * POPNUM(I)
WP(1) -OBLE(FLOAT(IFIX (SNGL (DEXP (WALPHA + WBETA*DLOG(LEN(I)))*
C POPNUM( I )* 10000.00 + 0.5DO))))/10000.ODO
WPSUMEWPSUM + WP(I)
65 CONTINUE
Add any fish rema ining in POPNUM ( MAXAGE) to natural
deaths and reset too ...
IF (J.NENUMTIM) GO TO 66
LEN(MAXAGE ) +0.ODO
LENDEV( MAXAGE ) =0.000 :
D(MAXAGE, J) D(MAXAGE.J) + POPNUM( MAXAGE)
DSMAG ( MAXAGE) -DSMAG(MAXAGE) + POPNUM( MAXAGE)
DSUM DSUM + POPNUM ( MAXAGE)
POPNUM( MAXAGE) =0.ODO
WP (MAXAGE ) -0.0DO
673
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685
686
687
688
689
690
i. End of year simulation loop
66 CONTINUE
691
692
693
694
ion Add in number of fry and their weight to the current
population ...
.
-
oond. 000 000
POPSUM=POPSUM # FRY
WP SUM=WPSUM + WFRY
TOTCAT TCL SUM + TCRSUM
TOTWT.C*WCL SUM * WCRSUM
695
696
697
698
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703
iii Write out end of year summary
...
704
WRITE (6,67') YEAR, (LINES(I),1=1, 15)
67 FORMAT ('q..1.-END OF YEAR ',12,' SUMMARY:'./,'-AGE'. 13X, P., 12x,
to :'WP., 10X, 'L'.8%'DU' , 11%, 'TCL , 11, 'WCL', 12X, LH', 11X. 'WLH'.J
+ 1X, A4,5X.2A4,'--',5X. 244.'.', 5X, A4,'.', 5X, A4,'.n'.4(5X, 244,'-'))
WRITE (6.68 ) EGGS, FRY, WFRY, INLEN, INDEV, (AGEPRN(I.),
+ POPNUM( I ). WP(I), LEN(I), LENDEV(I), TCL(I), WCL(I), LHV(I).
+ WLH(I), 1*1 , NUMAGE )
68 FORMAT. ( EGGS', 5x, F10.0,/,' FRY'. ?X.F9.2.5X, F9.4, 5X, F5.1,
+ 5X.F6.3.7. (1X, A4,6X, F9.2, 5X, F9.4.5X, F5.1,5X, F6.3.2(5X, F9.2.5X,
+ F9.4)))
WRITE (6,69 ) POPSUM, WPSUM, TCLSUM, WCLSUM, LHVSUM, WLHSUM
69. FORMAT ( OTOTALS',3X,F10.2, 4X, F10.4,25x, 2(F10.2,4x, F10.4,4x))
705
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707
708
709
710
711
712
713
714
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·
96
716
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718
719
720
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722
WRITE (6.70) (LINES(I),1*1,11). (AGEPRN(I), TCR(I).. WCR(I), HD(I).
+ WHD(I), DSMAG(I),1=1, NUMAGE)
70 FORMAT (11,'-AGE', 11%, 'TCR , 11%, 'WCR', 12x, HD, 11X, 'WHD. , 13X,
+ 'D' .1, 1X, A4,6X, 5( 244,'-',5x),/, (1XA4,6X, 2(F9.2,5X, F9.4.5x),
+ F9.2))
WRITE (6,71) TCRSUM, WCRSUM, HDSUM, WHOSUM, DSUM
71 FORMAT ('OTOTALS'.3x, F10.2,2(4X, F10.4,4X, F10.2))
WRITE (6,72) TOTCAT, TOTWTC
72 FORMAT (1.'-TOTAL CATCH 2',F10.2,1, OTOTAL WEIGHT OF CATCH ::.
* F 10.4)
IF (FLAGX. EQ.7) GO TO 78
WRITE (6,73) FMTUNI, (LINES(I),1*1, 18)
73 FORMAT ('1',L,'-LENGTH (WEIGHT) • FREQUENCY DISTRIBUTION OF THE '.
+ CATCH DATA:'.1.'0',A4,' GROUP: .9x,'TCL., 11'WCL', 12X, 'LH' , :
+ 11X, 'WLH'. 11%, 'TCR', 11%, 'WCR', 12X, 'HO', 11x. 'WHD'./. 1X, 244,
* '--'.8(5X, 244,'-'))
onnn
... Truncate and output length (weight) - frequency
distributions for the catch data ...
DO 75 INCH« 1, MAXLEN
INCHM 1 = INCH - 1
CFREQ( INCH. 2)-DBLE (FLOAT(IFIX(SNGL (CFREQ(INCH, 2)*100.ODO + 0.500
C))))/ 100.000
CFREQ(INCH, 1 )-DBLE (FLOAT(IFIX( SNGL (CFREQ( INCH, 1)* 100.ODO + 0.500
C))))/100.ODO
WFREQ( INCH,2)=OBLE (FLOAT( IFIX( SNGL (WFREQ( INCH, 2)*10000.ODO + 0.5DO
C))))/ 10000.ODO
724
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C 0.500))))/ 10000.ODO
CFRQ(INCH, 2) DBLE(FLOAT(IFIX (SNGL (CFRQ( INCH, 2)*100.ODO + 0.5DO
C))))/100.ODO
CFRQ(INCH, 1) =OBLE (FLOAT( IFIX(SNGL (CFRQ(INCH, 1)*100.ODO + 0.5DO
C))))/100.ODO
WFRO(INCH, 2 ) =DBLE(FLOAT(IFIX(SNGL (WFRQ(INCH, 2)* 10000.ODO + 0.5DO
C))))/ 10000.ODO
WFRO(INCH, 1)-DBLE(FLOAT(IFIX(SNGL ( WFRO( INCH, 1)* 10000.ODO + 0.5DO
C))))/10000.ODO
.WRITE (6,74) INCHM1, INCH, CFREQ(INCH, 2), WFREQ(INCH, 2),
+ CFRQ( INCH.2), WFRO(INCH, 2), CFREQ( INCH, 1), WFREQ(INCH, 1),
* CFRQ( INCH, 1), WFRQ(INCH, 1)
74 FORMAT (1X,13,' TO :,134(5X, F9.2, 5X,F9.4)).
CFRSUM( 2)=CFRSUM (2) CFREQ(INCH, 2)
CFRSUM( 1 ) *CFRSUM( 1) + CFREQ( INCH, 1)
WFRSUM(2)=WFRSUM(2) + WFREQ( INCH, 2)
WFRSUM( 1) =WFRSUM(+) + WFREQ(INCH, 1).
CFRSM(2)=CFRSM(2) + CFRQ(INCH, 2)
CFRSM( 1 )-CFRSM(1) + CFRQ( INCH, 1)
WFRSM(2) *WFRSM(2) + WFRO( INCH, 2)
WFRSM( 1 ) =WFRSM(1) + WFRQ(INCH, 1)
CFREQ(INCH.2)*0. ODO
WFREQ(INCH, 2)=0.ODO
CFRQ(INCH, 2)-0.ODO
WFRO( INCH, 2)-0.ODO
CFREQ(INCH, 1) =0.ODO
WFREQ(INCH, 1)=0.ODO
CFRQ(INCH, 1)*0.ODO
WFRO(INCH, 1)*0.ODO
75 CONTINUE
WRITE (6,76) CFRSUM(2), WFRSUM(2), CFRSM(2), WFRSM(2).
+ CFRSUM( 1), WFRSUM( 1), CFRSM( 1), WFRSM( 1)
76 FORMAT OTOTALS', 8X,F10.2.3(4x,F10.4.4x,F10.2), 4X.F10.4)
DO 77 1=1,2
CFRSUM(I):0. ODO
WFRSUM( I ) =0.ODO
CFRSM(I):0. ODO
WFRSM( I )=0.ODO
77 CONTINUE
749
750
751
752
753
754
755
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757
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759
760
761
762
763
764
765
766
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
97
Bump year 1. check if reached number of years to
be simulated ...
784
785
786
787
788
789
78
onno. 0000
YEAR:YEAR 1
IF (YEAR. GT.NYRSIM) GO TO 79
790
...
Add 1 to the number of age groups unless the maximum
has already been attained
IF (NUMAGE.LT. MAXAGE) NUMAGE=NUMAGE + 1
GO TO 42
79 WRITE (6.80)
80 FORMAT ('1!)
STOP
END
791
792
793
794
795
796
797
798
799
-
.
.
99
單军事某事事事本事集第基本来自某事某事某事某事非常重要事某事某事来事事​,事事原来第車事本事事多事事事有本事者三事
​本事来表事事事事事来事事為事本事​,事事事事事事本本来由本军事律事業集第基军军事事業者基本事事事事事集第5军事军事军事军事
​事事争等来事半年事事不是基本非事事非事事事非事事非事事基本法事事非事事事中事事事军事军来来来华事事非事事基法第第第第第华
​იიიიიიიიიიიი
Subroutine LNFREQ - calculates length (weight) -
frequency distributions of the population.
**
本事来
​本事本事事都事事本無事​,事事​,事事都事為本来事事事事某事某事事非事事事本事中事事第律事和基本事​,事事事客串串串串串事
​800
801
802
803
804
805
806
807
808
809
80
811
812
813
814
事事事事事事事非事事事事事军事军事军事事事事事本事军事军事基本常事​,事事系军事革事事都事事多第串串串串串串串串串串串
​多年来事事等事律事者年事事法律事事事​,事事事事非事事弟弟弟弟弟弟弟常喜事事事求事事半军事罪名来非常非常多多多多多
​SUBROUTINE LNFREQ (P, WP, NUMAGE , MAXLEN, LEN, LENDEV, LFREQ, INITX,
+ FLAGX)
ccc
... Declare var table types and dimensions
...
815
816
817
IMPLICIT REAL*8 (A-2)
INTEGER NUMAGE, MAXLEN, K, I. INCH, LL, UL, FLAGX
LOGICAL FLAG
DIMENSION P(15), WP(15), LEN(16),LENDEV(16). LFREQ(30,2),
+ THETA1(5)
COMMON X1,X2,X3, X4
FLAGE. TRUE.
LLv1
UL MAXLEN
818
819
820
821
822
823
824
825
826
827
828
... Loop over age groups
...
829
1 DO 3 1=1, NUMAGE
IF (P(I). LE.O.ODO) GO TO 3
IF(FLAG) THETA1(1)0.000
... Loop over length groups - calculate number (weight) in each
.:
group
grow
...
oooo
DO 2 INCHELL, UL :
THETA2+INORMI (DBLE(FLOAT( INCH)) - LEN(I))/LENDEV(I))
CT1=( THETA2 - THETAI(I))
LFREQ( INCH,1)=LFREQ(INCH, 1) + P(I)*CT 1
LFREQ(INCH,2)=LFREQ( INCH, 2) + WP() *CT 1
THETA 1(I) - THETA2
2 CONTINUE
3 CONTINUE
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
SAS
850
oooo
If all population numbers accounted for, RETURN. Otherwise,
bump MAXLEN and calculate next group ..:
IF (LFREQ(MAXLEN, 1).LE.4.00-03) RETURN
MAXLEN-MAXLEN +1
IF (FLAGX, EQ.6.OR. FLAGX. EQ.1.AND. DABS(X2 • X1).LE. 1.OD-05)
+ INITX=0BLE(FLOAT(MAXLEN))
LL-MAXLEN
UL MAXLEN
FLAG:. FALSE.
GO TO 1
END
8888888
15 5s
o123
dse
99
出来事事事本军事基本事事事事法律事事​,事事基本事事未来基本参军军事基本事事為事出事業基法第基本法多多多多多多多多集第1集
​事事事事事事事事事事非事事事法律事本本事事多事本其事本军事基本事事事為事事事事事事事事事来家里事事非事事法律事等事本
​事事本事​,事事事事事未来 ​基本指名率第基本本基本事喜事事事非事事者弟弟来是来害本事事第基本
​多事​,
oooooo
得得得得得得得​,
| Subroutino CRFREQ - calculates length(weight).
frequency distribution of the catch data.
857
858
859
860
861
862
863
864
365
***
多太多事事都事事事非事事基本事来本军军事学事实本事多年来基本事事​多事来来来事本案基本需事事事来来来来来来来来来事本
​来来来军事事非事事非事事事事事基本常事事非事事事事非事事事事者本事事事事事事本事事事非事事事事非事事事基本事当事事事事事事基本
​军事事事事兼第 ​拿出事本本军事基本事事事本事事事事事​,事事事未来事事事事本事事主事来事事事事等事事事亲事事事事都事本事本本本事
​866
: SUBROUTINE CRFREQ (NUMAGE, CW, LEN, LENDEV, MAXLEN, FLAGX, INITK,
* INITK2, INITX, CWFREQ, U, LEGPRC, ILLPRC, POPNUM, WP, SKPRES, NOLAST)
... Daciaro variable types and dimensions
...
IMPLICIT REAL*8 (A-Z)
INTEGER NUMAGE, MAXLEN, FLAGX, INITK, INITK2, I, J, KR. .
+ KR 1 SAV, KR2SAV, INCH, INCHSV, KOUN
LOGICAL FLAG, SKPRES, NOLAST
DIMENSION Cu(15,52,2),LEN(16), LENDEV(16), CHEREO(30,2)..
+ PERC(2),LEGPRC(15), ILL PRC(15), POPNUM(15),WP(15)
COMMON X1,X2,XS,X4
FLAG, FALSE,
FMAXLNEOBLE(FLOAT( MAXLEN))
INCHSVIFIX (SNGL (X1))
FINSVEDBLE(FLOAT(INCHSV))
867
868
869
870
971
872
873
374
875
876
877
378
879
880
881
882
883
884
cccccccccc
... Loop over age groups
DO 11 1:1, NUMAGE
IF (LENDEV(I).LE.O.ODO) GO TO 11
... 'Set KOUN for first length interval of the regulation.
X1 equals X2 then reset KOUN to 2 ...
If
885
886
887
888
889
390
891
892
893
894
895
896
KOUN=1
TF (DABS(X2 · X1). LE.1.00-05) KOUNKOUN + 1
... Initialize percent in legal and 111egal ranges
897
898
899
PERC(2)=LEGPRC(I).
PERC(1): – ILLPRC(I)
900
... Set lower bound on area in the fishable range
THETA 1= I NORM((x1 - LEN(I))/LENDEV(I))
KREINITK
KR2SAVOINITK2
X INITX
INCHEINCHSV
FINCH=FINSV
901
902
903
904
905
906
907
.
08.
909
910
Check if at full increment value. If not - calculate number
(weight) in partial increment group and sum ...
IF (DABS(FINCH - X1).LE. 1.00-05 ) GO TO 1
INCH INCH +1
FINCHEOBLE(FLOAT(INCH))
THETA2- I NORMI (FINCH • LEN(I))/LENDEV(I))
IF (PERC(KR).LE.O. ODO) GO TO 1
CWFREQ(INCH, KR ) =CWFREQ(INCH, KR) + CW(I,J,KR) *(THETA2 - THETA 1)/
C PERC(KR)
12345
99ssssssssss9999
t222222
9012345
... Update THETA 1 - begin loop over all jength groups
..
1 THETA 1=THETA2
2 INCH2 INCH + 1
100
...
Check if reached maximum length interval
i.
IF (INCH. GE.MAXLEN) GO TO 10.
FINCHEDBLE(FLOAT(INCH))
on 000 000
926
927
928
929
930
931
932
933
934
935
936
... Check of reached next length bound of the regulation ...
IF (DABS(FINCH - X).LE. 1.00-05.OR.FINCH - X.GE.5.OD-02) GO TO 4
Caiculate number (weight) in next length group and
sum ...
...
937
THETA2+INORM((FINCH - LEN(I))/LENDEV(I))
IF (PERC(KR).LE.0.0DO) GO TO 3
CWFREQ(INCH , KR) -CWFREQ(INCH,KR) + CW(I,J,KR)*( THETA2 - THETA 1)/
C PERC(KR)
3 THETA 1=THETA2
GO TO 2
938
939
940
941
942
943
944
იიიი
... Calculate number (weight) at upper bound of current
Tength interval of the regulation and sum ...
4 THETA2+I NORM((x • LEN(I))/LENDEV(I))
IF (PERC(KR).LE.O. ODO) GO TO 5
CWFREQ(INCH, KR) -CWFREQ(INCH,KR) + CW(I,J.KR) *(THETA2 - THETA 1)/
C PERC(KR)
5 THETA 1-THETA2
945
946
947
948
949
950
951
952
953
954
ono
955
... Check if at full increment value. If not, update limit and
switch to next range (legal or illegal) ...
956
957
IF (DABS(FINCH - X).LE.1.00-05.OR.FLAG) GO TO 6
X=FINCH
KR 1 SAVOKR
KROKRASAV
KR2SAVEKR 1SAV
FLAG. TRUE.
GO TO 4
nnnn
...
Switch to next range unless FLAG: . TRUE. - meaning, switch
done already ...
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
6 IF (FLAG) GO TO 7
KR 1SAVOKR
KROKR2SAV
KR2SAVEKR 1SAV
ono
...
Reset FLAG, update KOUN
...
7 FLAG: FALSE.
KOUN KOUN + 1
X+FMAXLN
979
...
onnn
"Go to" for specific regulation • reset upper limit to
calculate length (weight) frequencies in ...
980
981
982
983
984
985
GO TO (2.8.9.8.9), FLAGX
8 IF (KOUN. EQ:2) X=X3
GO TO 2
9 IF (KOUN. EQ.2) X3X3
IF (KOUN. EQ.3) X*X4
GO TO 2
i
Calculate number (weight) in last length interval and
non
sum
986
987
988
989
990
991
992
993
994
995
996
10 THETA2= I NORM( (FMAXLN - LEN(I))/LENDEV(I))
IF (PERC(KR).LE.O.ODO) GO TO 11
CWFREQI INCH , KR ) CWFREQ( INCHKR) + CW(I,J,KR) *(THETA2. THETA 1)/
C PERC(KR)
101
nonn
... If age group is 0. reset length, deviation, and weight
to O ...
IF (POPNUM(I).GT.O.ODO.OR. SKPRES.OR.NOLAST) GO TO 11
LEN(I)-0.ODO
LENDEV(I)=0.ODO
WP(I)*0.ODO
11 CONTINUE
RETURN
END
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
102
事基本常事名单靠本事拿者為未来事奉事事事事基本是第1事事第第第第第第第表第革本事事事有本事事第三事業集第4集
​事事非常多事事第基本事事事事事事事事軍事事事集第串串串串串事本来是军事家事法第第第第第第草本生集​_军事表
​事事基本常事事事事事事事本事来军事军事军事军事军事等事事本無事事事事事亲弟弟弟弟弟出事原来来来来来来事事未来集第
​Subroutine SPWNSB - Calculates number of eggs pro-
duced, number of fry produced, number of fry remaining at ***
the end of the year, mean length (and deviation) of the
fry, mean. 1ength (and deviation) of fry at the end of the
year, and weight of the fry at the end of the year.
იიი? იიიიიიიიიიიიიიი
"事事事非事事非事事事​,事事都事事事都需事事事事单单单单单靠本举来来来来来来事事本来来来来来事事事事事​#本军事革考生基本第
​事事事事事事有本事事事事事非事事事非事事事事事本来事事事本事事事非事事事事都事事非事实本来事事事非事事家事事事事事事
​事事事非事事非事事事事事事事無事​,事事事事事事事非事事非华军事革等多事事求事事非事事本事案等事来华军事基本島本军事革第害事本军事
​SUBROUTINE SPWNSB (POPNUM, NUMAGE, LEN, LENDEV, CTLNEQ, LK.MLMAT,
+ ISQ2PI, EALPHA , EBETA, FALPHA, FBETA, LALPHA , LBETA, LENDIV, RATLOV,
+ WALPHA , WBETA, NFRY, EGGS, FRY, INLEN, INDEV, WFRY)
... Declare variable types and dimensions ...
IMPLICIT REAL*8 (A-Z)
INTEGER NUMAGE, I, THETA 1
DIMENSION POPNUM( 15), LEN( 16), LENDEV( 16)
oooo
Set EGGS to 0, loop over ages to calculate number of females
mature (1:1 sex ratio) and number of eggs produced ..
EGGS=0.000
DO 2 I 1, NUMAGE
IF (POPNUM( I ).LE.O.ODO) GO TO 2
DSQT2-2.ODO*LENDEV( 1 )**2
THETD 1=1 .ODO - INORM( (MLMAT - LEN(I))/LENDEV(I))
THETA 1-IFIX (SNGL (THETD 1* 10000.ODO + 0.5DO))
IF (THETA1.LE.O) GO TO 2
PRCNT1,000
XBROLEN(I)
IF (THETA1.GE.10000) G0 T01
XBR ISO2PI*LENDEV(I)*(DEXP(- (MLMAT - LEN(I) ) **2/DSOT2)/THETD1)
C + LEN(I)
XBR-OBLE(FLOAT(IFIX( SNGL (XBR*10.ODO + 0.5DO))))/ 10.ODO
PRCNT •DBLE (FLOAT(THETAT))/ 10000.ODO
1 EGCALC08LE(FLOAT(TFIX(SNGL((EALPHA + EBETA * XBR)+0.500))))
| POP-DBLE(FLOAT(TFTX(SNGL(POPNUM(I)*100.000/2.000 +0.5500))))/
c 100,000
EGCALC-DBLE(FLOAT(IFIX(SNGL(EGCALC-PRENT +0.500))))
EGGSEGGS + OBLE(FLOAT(ITFIX(SNGL(EGCALC*POP))))
2 CONTINUE
1008
1009
1010
101
1012
1013
1014
1015
1016
017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
083
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
046
1047
1048
1049
1050
1051
1052
1053
1054
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1057
1053
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1062
1069
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
000
... Calculate number of fry - if less than o, set too ...
FRYR-DBLE(FLOAT(IFIX(SNGL(FALPHA * EGGS DEXP(FBETA EGGS)))))
FRY=OBLE(FLOAT(IFIX( SNGL((FRYR - FRYR*NFRY) * 100.ODO + 0.5DO))))
c./100,000
IF (FRY.LT.O.ODO) FRYEO. ODO
INLEN-O. ODO
INDEV-O. ODO
WFRY=0.ODO
იიი იიი
... If no fry - RETURN ...
IF (FRY.LE.O.ODO) RETURN
... Calculate mean length of fry ...
INLEN-08LE(FLOAT(TFIX(SNGL((LALPHA + LBETA * DLOG(FRYR))* 10.000
C + 0.5DO))))/ 10.000
INLEN=OBLE(FLOAT(IFIX( SNGL( (CTLNEQ + INLEN*LK)*10.ODO + 0.500))))
c / 10.000
INLEN=DBLE(FLOAT(IFIX(SNGL (INLEN/LENDIV* 10.ODO + 0.500))))/10.ODO
103
...
Calculate standard deviation of the mean length for fry
...
იიი იიი
INDEVEDBLE (FLOAT(IFIX( SNGL (INLEN/RATLDV*1000.ODO + 0.500))))
C/1000. ODO
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
.. Calculate weight of fry
...
WFRY ODBLE (FLOAT(IFIX (SNGL (DEXP (WALPHA + WBETA*OLOG( INLEN) ) *
RETURN
END
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