UALG
Statistical catch at age models
Einar Hjörleifsson
2
The data: Catches at age (here million of fish)
Age effect
Cohort
effect
Year effect
Year
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Age -->
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
3
4
9
9
30
14
10
3
7
11
16
29
9
15
8
3
4
3
14
17
8
5
2
6
3
4
32
17
54
48
40
122
71
51
21
48
38
62
50
18
29
31
13
34
12
51
36
29
34
9
21
5
55
55
50
91
76
73
144
130
65
38
65
42
50
49
23
50
62
37
68
25
65
53
58
64
18
6
88
57
55
67
111
59
66
123
135
68
38
50
25
47
51
27
72
85
37
66
25
64
58
58
67
7
60
71
35
43
53
74
32
31
79
95
44
18
22
21
46
40
31
61
53
27
42
23
42
37
45
8
41
46
43
27
27
26
35
13
17
36
46
17
8
10
13
38
30
18
34
26
14
22
15
24
23
9
77
14
20
19
13
11
10
7
5
7
14
20
5
2
6
8
23
13
9
15
11
7
12
8
12
10
17
22
5
11
7
5
4
2
3
2
2
10
5
2
2
4
5
7
5
4
6
3
4
5
3
Total
380
298
282
323
365
390
378
363
335
308
265
251
178
169
181
203
244
260
235
234
208
208
227
213
196
Recall the lecture of structure fisheries data
3
The model in math
  a  a fu ll 
e  L
for a  a full
sa  
 a  a
2
 Rfu ll
for a  a full
e
2
Na, y
Fay  sa Fy
 Ra , y

( F
M
)
  N a 1, y 1e a1, y1 a1, y1

 ( Fa1, y 1  M a1, y 1 )
( F
M
)
 N a , y 1e a , y1 a , y1
 N a 1, y 1e
Cˆ ay 
Fa , y
Fa , y  M ay
1  e
 ( Fa , y  M a y )
N
a  1 or y  1

1  a  a plus

a  a plus  11
ay
 p S 1  Fay  M ay 
S1
S1 S1
S1
ˆ
U ay  qa N ay where N ay  N ay e
 p S 2  Fay  M ay 
Uˆ ayS 2  qaS 2 N ayS 2 where N ayS 2  N ay e

ˆ
min SSE  C  a ln C ay  ln Cay
y
a

2

 S1  aS1 ln U ayS1  ln Uˆ ayS1
y
a

2

  S 2   aS 2 ln U ayS 2  ln Uˆ ayS 2
y
a

2
The model in words
4

Make a separable model having:

A fixed (constant through time) selection pattern (sa) for each age, assume
selection pattern follows double-half Gaussian





Fishing mortality (for some reference age) for each year (Fy)
Numbers of fish that enter the stock each year (year class size, recruitment, N1,y)
and in the first year (Na,1)
A plus group: Catches of the oldest age groups are summed - Needs to be taken
into account in the model
Calculate:


The number of fish caught each year and age by the fishermen (Cay-hat). This is
the modeled Cay number.
The number of fish caught each year and age by the scientist (Uay-hat). This is
the modeled Uay number.


Fixed selectivity with time is commonly referred to as a separable model.
Assume the relationship between stock size and survey indices as: Uay = qNb
Set up an objective function (minimizing SS):

Constrain the model such that we minimize the squared difference between
observed values (Cay and Uay) and predicted values (Cay-hat and Uay-hat)
5
Can you disentangle this?
The model as a map
pattern
Year Age ----->
|
| 4a. Population
V
Year Age ----->
|
| 5. Predicted
V
Year Age ----->
|
| Observed
V
numbers
Year Age ----->
|
| 3. Fishing
V
mortality
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Year Age ----->
|
| 6a. Predicted
V
catch
survey 1
indices
catch at age
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
squared
squared
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
Objective functions
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| Natural mortality
V
Measurements
1. Parameters
Year Age ----->
|
| 2. Selection
V
Population and observation model
9. Solver
6
The separable part
7


Selectivity describes the relative fishing mortality within
each age group.
In this simplest model setup we assume that the
selectivity is the same in all years. Fishing mortality by
age and year can thus be described by:
Fay  sa Fy



Fay: Fishing mortality of age a year y
sa: Selectivity of age a
Fy: Fishing mortality (of some reference age) in year y

Note: The separability assumption reduces the number fishing
mortality parameters from:
n = (#age groups x #years) to
n = (#age groups + #years)
Modelling selectivity
8

The selection pattern is a function of size/age

sa = f(age)





Logistic
Gaussian
…
So instead of having independent values of s1, s2, .. sa
we could use a function to describe the selection
pattern as a function of age/size
we will use the normal distribution here for illustrative
purpose, but that is not quite often applicable in
practice

Assume double half-Gaussian
9

Lets make a further assumption here by letting selectivity follow:

 
L
e
for a  a full
sa  
 a  a fu ll 2
 R
for a  a full
e
 a  a fu ll 2






The
afull: age at full selectivity
R: Shape factor (standard deviation) for right hand curve
L: Shape factor (standard deviation) for left side of curve
Note: by using this selection function we reduce the number of parameters from
whatever number of age groups we have, to only 3 parameters.
But could just as well just estimate each Sa without resorting to a particular
function.
R, L and Afull are parameters that we estimate
Selectivity - double half-Gaussian
10
Selectivity
Note asymmetry
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
L=5, 10, 15
afull = 8
R=10000, 100, 15
0
5
10
Age
The map
9. Solver
11
Year Age ----->
|
| 2. Selection
V
Calculate sa here
Year Age ----->
|
| 4a. Population
V
Store Afull, L
and R here
1. Parameters
pattern
Year Age ----->
|
| 5. Predicted
V
Year Age ----->
|
| Observed
V
numbers
Year Age ----->
|
| 3. Fishing
V
mortality
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Year Age ----->
|
| 6a. Predicted
V
catch
survey 1
indices
catch at age
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
squared
squared
Year Age ----->
|
| Natural mortality
V
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
12
The map: selection pattern (say)
Parameters
Param.
L
R
afull
Fy
..
..
Fy+8
Nay
Nay+1
Nay+2
Nay+3
Nay+4
...
y/a
y
y+1
y+2
y+3
y+4
y+5
y+6
y+7
y+8
a
Sa
Sa
Sa
Sa
Sa
Sa
Sa
Sa
Sa
a+1
Sa+1
Sa+1
Sa+1
Sa+1
Sa+1
Sa+1
Sa+1
Sa+1
Sa+1
a+2
Sa+2
Sa+2
Sa+2
Sa+2
Sa+2
Sa+2
Sa+2
Sa+2
Sa+2
a+3
Sa+3
Sa+3
Sa+3
Sa+3
Sa+3
Sa+3
Sa+3
Sa+3
Sa+3
a+4
Sa+4
Sa+4
Sa+4
Sa+4
Sa+4
Sa+4
Sa+4
Sa+4
Sa+4
a+5
Sa+5
Sa+5
Sa+5
Sa+5
Sa+5
Sa+5
Sa+5
Sa+5
Sa+5
  a  a fu ll 
e  L
for a  a full
sa  
 a  a fu ll 2
 R
for a  a full
e
2
Excel speak: =exp(-((a-afull)^2/if(a=<afull;sL;sR)))
A word on nomenclature
13

Often make the following distinction:

Selectivity: The probability of catching an individual of
a given age scaled to the maximum probability over
all ages, given that all animals are available to be
caught by a certain gear in a certain plaice.



This is what gear technologist study at lengths when they are
studying the properties of various gears.
Availability: The relative probability, as a function of
age, of being in the area in which catching occurs.
Vulnerability: The combination of selectivity and
availability.

Thus should really refer to vulnerability but lets stick with the more
ambiguous word selectivity, the reason being its wide usage.
Setting up Fy and calculating Fay
14


The fishing mortality each year (Fy) are
parameters of the model that we want to
estimate.
Since we already calculated sa we can
calculate fishing mortality by age and year
from:

Fay = saFy
The map
Year Age ----->
|
| 2. Selection
V
1. Parameters
9. Solver
15
pattern
Year Age ----->
|
| 4a. Population
V
F1
F2
..
..
Fy
Year Age ----->
|
| 5. Predicted
V
Year Age ----->
|
| Observed
V
numbers
Year Age ----->
|
| 3. Fishing
V
mortality
Calculate Fay here
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Year Age ----->
|
| 6a. Predicted
V
catch
survey 1
indices
catch at age
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
squared
squared
Year Age ----->
|
| Natural mortality
V
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
Setting up Ninit and calculating Nay
16

The number of fish entering the system in first
year and in the first age (Ninit) are parameters of
the model that we want to estimate. Need:



The number of fish in each age group in the first year
(Na,1)
The number of recruits entering each year (N1,y)
Given the above we can then fill in the
abundance matrix by the conventional stock
equation
Na 1, y 1  Nay e
 ( Fay  M ay )
The map
9. Solver
17
Year Age ----->
|
| 2. Selection
V
pattern
Year Age ----->
|
| 4a. Population
V
numbers
Calculate Nay here
1. Parameters
Store Ninit
here
Year Age ----->
|
| 5. Predicted
V
Year Age ----->
|
| Observed
V
Year Age ----->
|
| 3. Fishing
V
mortality
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Year Age ----->
|
| 6a. Predicted
V
catch
survey 1
indices
catch at age
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
squared
squared
Year Age ----->
|
| Natural mortality
V
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
18
The map: Population numbers details
Pastel green area: Estimated parameters
Param.
L
R
afull
Fy
..
..
Fy+8
Nay
Nay+1
Nay+2
Nay+3
Nay+4
...
y/a
y
y+1
y+2
y+3
y+4
y+5
y+6
y+7
y+8
a
Na,y
Na,y+1
Na,y+2
Na,y+3
Na,y+4
Na,y+5
Na,y+6
Na,y+7
Na,y+8
a+1
Na+1,y
Na+1,y+1
Na+1,y+2
Na+1,y+3
Na+1,y+4
Na+1,y+5
Na+1,y+6
Na+1,y+7
Na+1,y+8
a+2
Na+2,y
Na+2,y+1
Na+2,y+2
Na+2,y+3
Na+2,y+4
Na+2,y+5
Na+2,y+6
Na+2,y+7
Na+2,y+8
a+3
Na+3,y
Na+3,y+1
Na+3,y+2
Na+3,y+3
Na+3,y+4
Na+3,y+5
Na+3,y+6
Na+3,y+7
Na+3,y+8
Na 1, y 1  Nay e
a+4
Na+4,y
Na+4,y+1
Na+4,y+2
Na+4,y+3
Na+4,y+4
Na+4,y+5
Na+4,y+6
Na+4,y+7
Na+4,y+8
 ( Fay  M ay )
a+5
Na+5,y
Na+5,y+1
Na+5,y+2
Na+5,y+3
Na+5,y+4
Na+5,y+5
Na+5,y+6
Na+5,y+7
Na+5,y+8
Predicting catch: Cay-hat
19

Once the population matrix is calculated it is
simple to calculate the predicted catch (Cay-hat)
according to the catch equation:
Cˆ ay 

Fa , y
Fa , y  M ay
1  e
 ( Fa , y  M a y )
N
ay
The C-hats are values that we will later “confront”
with the measurements that we have.
The map
1. Parameters
9. Solver
20
Year Age ----->
|
| 2. Selection
V
pattern
Year Age ----->
|
| 4a. Population
V
Year Age ----->
|
| 5. Predicted
V
numbers
catch
Predicted Cay-hat
Year Age ----->
|
| Observed
V
catch at age
Year Age ----->
|
| 3. Fishing
V
mortality
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Year Age ----->
|
| 6a. Predicted
V
indices
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
squared
survey 1
squared
Year Age ----->
|
| Natural mortality
V
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
Confronting the model with data
21

Until now we have only set up equations that follow the
progression of each year class and calculated catch.

This is more or less a population simulator.




If we let recruitment be a function of SSB and we add some noise
to recruitment we have a closed system and thus almost a
medium/long term simulator
This is also more or less the same thing as we do when we do a
short term projection.
Or for that matter in a yield per recruit analysis, except that
there we focus only on one cohort (here one diagonal line).
At present we are only interested in fitting the model to
observations (measurements). Need thus some kind of
objective function (minimizing sums of squares).
The objective function in words
22

Find the value of the parameters:





fishing pattern by age, sa (controlled by L, R and Afull)
yearly fishing mortality (Fy)
population number in the first year (Na,1)
recruitment (N1,y) in each year
that minimize the squared deviation of estimated catch
(Cay-hat) and measured catch (Cay)

ˆ
SSE

ln
C

ln
C

min
C
ay
ay
y


2
a
Note that here we assume a log-normal error distribution. Could
easily be replace with other type of error structure.
The map
23
Year Age ----->
|
| 2. Selection
V
1. Parameters
9. Solver
The sum
to minimize
pattern
Year Age ----->
|
| 4a. Population
V
Year Age ----->
|
| 5. Predicted
V
Year Age ----->
|
| Observed
V
numbers
Year Age ----->
|
| 3. Fishing
V
mortality
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Year Age ----->
|
| 6a. Predicted
V
catch
survey 1
indices
catch at age
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
(obs-pre)
squared
(obs-pre)^2
squared
Year Age ----->
|
| Natural mortality
V
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
1 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
Different weights by age
24

The catch of different age groups are often
measured with different accuracy. Thus often set
different weights to the residuals, so that the
information from age groups that are measured
with the most accuracy weigh more in the
objective function:

ˆ
SSE


ln
C

ln
C
 a
min
C
ay
ay
y
a
 is inversely related to the variance

2
If we only have Cay
25


If there are no other available data for a stock
than catch at age one could attempt to fit the
model to catches alone.
May need a extra “stabilizer”: The brave one
may assume that fishing mortality does not
change much between consecutive years:
min
SSE  SSEC  SSEF
SSEC   ln Fy  ln Fy 1 
2
y
Tuning with survey indices
26

If additional information are available it is
relatively easy to add them to the model. If agebased survey indices are available one may use:
Uˆ ay  qa N ay

where qa is a parameter (catchability). The
minimization is by (again assuming log-normal
errors):

ˆ
SSE


ln
U

ln
U
 a
min
U
ay
ay
y
a

2
27
iCod: Age based survey indices
Age -->
Year
0
1982
1983
1984
0
1985
0
1986
0
1987
0
1988
0
1989
0
1990
0
1991
0
1992
0
1993
0
1994
0
1995
0
1996
0
1997
0
1998
0
1999
0
2000
0
2001
0
2002
0
2003
0
2004
0
2005
0
2006
0
2007
0
2008
1
2
3
4
5
6
7
8
9
10
17
15
4
3
4
6
4
1
4
14
1
4
1
8
7
19
12
1
11
7
3
9
6
111
61
29
7
16
12
16
17
5
15
29
5
22
6
33
28
22
38
4
25
15
7
18
35
96
103
72
22
26
18
33
31
9
25
43
14
30
7
55
37
41
46
8
39
23
9
48
22
82
102
78
14
30
19
36
27
9
29
56
16
42
7
38
40
39
62
10
38
21
64
21
21
67
68
27
15
16
13
22
24
13
29
62
13
30
5
36
32
35
43
11
28
23
26
12
8
34
32
18
7
10
6
18
15
9
28
24
8
15
7
19
25
23
28
9
15
7
12
6
4
14
21
6
2
4
4
14
9
7
11
8
3
8
4
14
11
10
10
5
2
3
6
1
2
4
5
2
1
2
4
7
5
2
4
2
1
5
3
6
4
5
3
1
1
1
1
1
1
1
1
0
0
1
1
3
1
1
1
1
1
3
1
1
2
2
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
1
Population numbers at survey time
28

If survey time is not in the beginning of the year
we need to take that into account by:
N  N ay e
'
ay
 p ( Fay  May )
'
ˆ
U ay  qa N ay

Where


N’ is the population size at survey time
p is the fraction of the year when the survey takes
place.
The map
9. Solver
29
The sum
1. Parameters
Store qa
here
Year Age ----->
|
| 2. Selection
V
pattern
Year Age ----->
|
| 4a. Population
V
Year Age ----->
|
| 5. Predicted
V
Year Age ----->
|
| Observed
V
numbers
Year Age ----->
|
| 3. Fishing
V
mortality
Year Age ----->
|
| 4b. Nay at
V
survey 1 time
Calculate N’ay
Year Age ----->
|
| 6a. Predicted
V
catch
survey 1
indices
Predict Uay-hat
catch at age
Year Age ----->
|
| Observed
V
survey 1 indices
Year Age ----->
|
| 7. Catch residuals
V
Year Age ----->
|
| 8a. Survey
V
Color code
Population model
Observation model
Year Age ----->
Measurements |
Objectives
| 7. Catch residuals
Parameters
V
Year Age ----->
|
| 8a. Survey
V
1 residuals
obs-pre
squared
1 residuals
(obs-pre)^2
squared
Year Age ----->
|
| Natural mortality
V
Year Age ----->
|
| 4c. Nay at survey
V
Year Age ----->
|
| 6b. Predicted
V
2 time
survey 2
indices
Year Age ----->
|
| Observed survey
V
2 indices
Year Age ----->
|
| 8b. Survey
V
2 residuals
Year Age ----->
|
| 8b. Survey
V
2 residuals
squared
Objective function I
30

Simple to combine the two objective functions:
min
SSE  SSEC  SSEU

    ln U

 ln Uˆ 
    a ln Cay  ln Cˆ ay
y
a
a
y
2
2
ay
ay
a
Lets not worry about pa for now
31
Getting it all together
THE MINIMIZATION STUFF
Sum of squares
C@A
U@A - Survey 1
U@A - Survey 2
SSE total
Lambda
48.419
1
53.900
1
49.084
1
151.4023
PARAMETERS
Name
Ln Afull
Ln L
Ln R
Ln(parameter) Switches Parameter
2.3900
10.91
1.4481
4.26
5.0000
148.41
The heart of the
setup lies in the
left side of the spreadsheet. There we have the
the objective functions
(minimize SS) in one place.
The only thing left
is to setup the solver
such that it minimizes
the total SSE by
changing the parameters. And this you
were introduced
already at day 2 of the
course.
The output: Historical development of the stock
33
Recruitment
SSB
400
350
300
F
1000
0.90
900
0.80
800
0.70
700
250
600
200
500
150
400
0.60
0.50
0.40
300
0.30
200
0.20
50
100
0.10
0
0
100
1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924
0.00
1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924
250
Yield
300
Survey
200
250
1 numbers
100
Survey
90
80
70
60
50
40
30
20
10
0
200
150
150
100
100
50
50
0
0
1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924
1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924
Survey
1 residuals
12
16
Catch
residuals
1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924
1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924
Survey 2 residuals
12
14
10
10
10
8
8
8
6
6
6
4
4
2
2
12
4
2
0
0
1895
1900
1905
1910
1915
1920
1925
1930
1895
2 numbers
0
1900
1905
1910
1915
1920
1925
1930
1895
1900
1905
1910
1915
1920
1925
1930
Age/size structured models
34

Advantages


Populations do have age/size structure
Basic biological processes are age/size specific







Growth
Mortality
Fecundity
The process of fishing is age/size specific
Relatively simple to construct mathematically
Model assumption not as “strict” as in e.g. logistic models
Disadvantages






Sample intensive
Data often not available
Mostly limited to areas where species diversity is low
Have to have knowledge of natural mortality
For long term management strategies have to make model assumptions
about the relationship between stock and recruitment
Often not needed to address the question at hand
UALG
Some addition points on Y/R and then
on reference points in light of model
uncertainty
36
37
Atlantic mackerel
38
NEA cod
39
NEA cod
40
NEA cod
41
NEA haddock
Find the Fmsy and the Bmsy!