The exercises are numbered according to the numbers of the relevant sections of the manual.
Exercise 2.1 Mean value and variance
In this exercise we use part of the lengthfrequency data of the coral trout (Plectropomus leopardus) presented in Fig. 3.4.0.2, namely those in the length interval 2329 cm. These fish are assumed to belong to one cohort. The lengthfrequencies are presented in Fig. 17.2.1.
Tasks:
Read the frequencies, F(j) from Fig. 17.2.1 and complete the worksheet. Calculate mean, variance and standard deviation.
Worksheet 2.1
j 
L(j)  L(j) + dL 
F(j) 
_{}(j) 
F(j) * _{}(j) 
_{}(j)  _{} 
F(j) * (_{}(j)  _{})^{2} 
1 
 



2.968 

2 
 



2.468 

3 
 



1.968 

4 
 



1.468 

5 
 



0.968 

6 
 



0.468 

7 
 



0.032 

8 
26.527.0 
6 
26.75 
160.50 
0.532 
1.698 
9 
27.027.5 
2 

54.50 
1.032 
2.130 
10 
27.528.0 
2 

55. 50 
1.532 
4.694 
11 
28.529.0 
2 

56.50 
2.032 
8.258 
12 

1 

28.75 
2.532 
6.411 
sums 

S F(j) 

_{} 

_{} 
_{} = 
s^{2} = 
s = 
Fig. 17.2.1 Lengthfrequency sample
Exercise 2.2 The normal distribution
This exercise consists of fitting a normal distribution to the lengthfrequency sample of Exercise 2.1, by using the expression:
_{} (Eq. 2.2.1)
for a sufficient number of xvalues allowing you to draw the bellshaped curve.
For your convenience introduce the auxiliary symbols:
_{}
so that the formula above can be written
_{}
Since A and B do not depend on L and as they are going to be used many times, it is convenient to calculate them separately beforehand.
Tasks:
1) Calculate A and B_{}B = 1/(2s^{2}) =
2) Calculate Fc(x) for the following values of x:
Worksheet 2.2
x 
Fc(x) 
x 
Fc(x) 
22.0 

26.0 

22.5 

26.5 

23.0 

27.0 

23.5 

27.5 

24.0 

28.0 

24.5 

28.5 

25.0 

29.0 

25.5 

29.5 

3) Draw the bellshaped curve on Fig. 17.2.1
Exercise 2.3 Confidence limits
Tasks:
Calculate the 95% confidence interval for the mean value estimated in Exercise 2.1.
Exercise 2.4 Ordinary linear regression analysis
It is often observed that the more boats participate in a fishery the lower the catch per boat will be. This is not surprising when one considers the fish stock as a limited resource which all boats have to share. In Chapter 9 we shall deal with the fisheries theory behind this model.
The data shown below in the worksheet are from the Pakistan shrimp fishery (Van Zalinge and Sparre, 1986).
Tasks:
1) Draw the scatter diagram.
2) Calculate intercept and slope (use the worksheet).
3) Draw the regression line in the scatter diagram.
4) Calculate the 95% confidence limits of a and b.
Worksheet 2.4


number of boats 

catch per boat per year 


year 
i 
x(i) 
x(i)^{2} 
y(i) 
y(i)^{2} 
x(i) * y(i) 
1971 
1 
456 

43.5 

19836.0 
1972 
2 
536 

44.6 

23905.6 
1973 
3 
554 

38.4 

21273.6 
1974 
4 
675 

23.8 

16065.0 
1975 
5 
702 

25.2 

17690.4 
1976 
6 
730 
532900 
30.5 
930.25 

1977 
7 
750 
562500 
27.4 
750.76 

1978 
8 
918 
842724 
21.1 
445.21 

1979 
9 
928 
861184 
26.1 
681.21 

1980 
10 
897 
804609 
28.9 
835.21 

Total 

7146 

309.5 

211099.5 
_{} = 
_{} = 

_{} = 
_{} 

_{} = 
_{} 

_{} 

_{} 
sx = 

_{} 
sy = 

_{} 

slope: _{} 
intercept: _{} = 

variance of b: 

_{} 
sb = 

variance of a: 

_{} 
sa = 

Student's distribution: t_{n2} = 
Exercise 2.5 The correlation coefficient
Refer to Exercise 2.4. Does the correlation coefficient make sense in the example of catch per boat regressed on number of boats? Consider which of the variables is the natural candidate as independent variable. Can we (in principle) decide in advance on the values of one of them?
Tasks:
Irrespective of your findings in the first part of the exercise carry out the calculation of the 95% confidence limits of r.
Exercise 2.6 Linear transformations of normal distributions, used as a tool to separate two overlapping normal distributions (the Bhattacharya method)
Fig. 17.2.6A shows a frequency distribution which is the result of two overlapping normal distributions "a" and "b". We assume that the lengthfrequencies presented in Fig. 17.2.6B are also a combination of two normal distributions. The aim of the exercise is to separate these two components. The total sample size is 398. Assume that each component has 50% of the total or 199. Further assume that the frequencies at the left somewhat below the top are fully representative for component "a", while those at the bottom of the right side are fully representative for component "b".
Fig. 17.2.6A Combined distribution of two overlapping normal distributions
Fig. 17.2.6B Lengthfrequency sample (assumed to consist of two normal distributions
Tasks:
1) Complete Worksheet 2.6a.2) Plot D ln F(z) = y' against x + dL/2 = z and decide which points lie on straight lines with negative slopes (see Fig. 2.6.5).
3) On the basis of the plot select the points to be used for the linear regressions. (Avoid the area of overlap and points based on very few observations). Do the two linear regressions and determine a and b.
4) Calculate _{}, s^{2} = 1/b and s = Ö s^{2} for each component.
5) Draw the two plots which represent each distribution in linear form.
6) We now want to convert the straight lines into the corresponding theoretical (calculated) normal distributions. Using Eq. 2.2.1 calculate Fc(x) for both normal distributions for a sufficient number of xvalues to allow you to draw the two bellshaped curves superimposed on Fig. 17.2.6B. Assume n = 199 for both components. (Use the same method as presented in Exercise 2.2). Complete Worksheet 2.6b.
Worksheet 2.6a
interval 
x 
F(x) 
ln F(x) 
D ln F(z) 
z = x + dL/2 
45 
4.5 
2 
0.693 






0.916 
5 
56 
5.5 
5 
1.609 






0.875 
6 
67 
6.5 
12 








7 
78 
7.5 
24 









89 
8.5 
35 









910 
9.5 
42 









1011 
10. 5 
42 









1112 
11.5 
46 









1213 
12.5 
56 









1314 
13.5 
58 









1415 
14.5 
45 









1516 
15.5 
22 
3.091 






1.145 
16 
1617 
16.5 
7 
1.946 






1.253 
17 
1718 
17.5 
2 
0.693 


Worksheet 2.6b
First component 
Second component 

_{} 
_{} 

B = 
B = 

_{} 
_{} 

_{} 

x 
Fc(x) 
Fc(x) 
x 
Fc(x) 
Fc(x) 
1.5 


11.5 


2.5 


12.5 


3.5 


13.5 


4.5 


14.5 


5.5 


15.5 


6.5 


16.5 


7.5 


17.5 


8.5 


18.5 


9.5 


19.5 


10.5 


20.5 


Exercise 3.1 The von Bertalanffy growth equation
The growth parameters of the Malabar blood snapper (Lutjanus malabaricus) in the Arafura Sea were reported by Edwards (1985) as:
K = 0.168 per year
L_{¥ } = 70.7 cm (standard length)
t_{0} = 0.418 years
Edwards also estimated the standard length/weight relationship for Lutjanus malabaricus:
w = 0.041 * L^{2.842} (weight in g and standard length in cm)
as well as the relationship between standard length (S.L.) and total length (T.L.):
T.L. = 0.21 + 1.18 * S.L.
Tasks:
Complete the worksheet and draw the following three curves:1) Standard length as a function of age
2) Total length as a function of age
3) Weight as a function of age
Worksheet 3.1
age 
standard length 
total length 
body weight 
age 
standard length 
total length 
body weight 
years 
cm 
cm 
g 
years 
cm 
cm 
g 
0.5 



8 



1.0 



9 



1.5 



10 



2 



12 



3 



14 



4 



16 



5 



(do not use ages above 16 in the graph) 

6 







7 



20 







50 



Exercise 3.1.2 The weightbased von Bertalanffy growth equation
Pauly (1980) determined the following parameters for the pony fish or slipmouth (Leiognathus splendens) from western Indonesia:
L_{¥ } = 14 cm
q = 0.02332
K = 1.0 per year
t_{0} = 0.2 year
Tasks:
Complete the worksheet and draw the length and the weightconverted von Bertalanffy growth curves.
Worksheet 3.1.2
age 
length 
weight 
age 
length 
weight 
0 


0.9 


0.1 


1.0 


0.2 


1.2 


0.3 


1.4 


0.4 


1.6 


0.5 


1.8 


0.6 


2.0 


0.7 


2.5 


0.8 


3.0 


Exercise 3.2.1 Data from age readings and length compositions (age/length key)
Consider Table 3.2.1.1 (age/length key) and suppose we caught a total of 2400 fish of the species in question during the cruise from which this age/length key was obtained and that only 439 specimens of Table 3.2.1.1 were aged. The remaining fish were all measured for length. To reduce the computational work of the exercise only a part (386 fish) of this lengthfrequency sample is used. This part is shown in the worksheet.
Tasks:
Estimate how many of these 386 fish belonged to each of the four cohorts listed in Table 3.2.1.1, by completing the worksheet.
Worksheet 3.2.1
cohort 
1982 
1981 
1981 
1980 

1982 
1981 
1981 
1980 
length interval 
key 
number in length sample 
numbers per cohort 

3536 
0.800 
0.200 
0 
0 
53 
42.4 
10.6 
0 
0 
3637 
0.636 
0.273 
0.091 
0 
61 
38.8 
16.7 
5.6 
0 
3738 




49 




3839 




52 




3940 




70 




4041 




52 




4142 
0.222 
0.444 
0.222 
0.111 
49 
10.9 
21.8 
10.0 
5.4 




total 
386 
187.2. 
133.8 


Exercise 3.3.1 The Gulland and Holt plot
Randall (1962) tagged, released and recaptured ocean surgeon fish (Acanthurus bahianus) near the Virgin Islands. Data of 11 of the recaptured fish are shown in the worksheet, in the form of their length at release (column B) and at recapture (column C) and the length of the time between release and recapture (column D).
Tasks:
1) Estimate K and L for the ocean surgeon fish using the Gulland and Holt plot.
2) Calculate the 95% confidence limits of the estimate of K.
Worksheet 3.3.1
A 
B 
C 
D 
E 
F 
fish 
L(t) 
L(t + D t) 
D t 
_{} 
_{} 

cm 
cm 
days 
cm/year 
cm 




(y) 
(x) 
1 
9.7 
10.2 
53 


2 
10.5 
10.9 
33 


3 
10.9 
11.8 
108 


4 
11.1 
12.0 
102 


5 
12.4 
15.5 
272 


6 
12.8 
13.6 
48 


7 
14.0 
14.3 
53 


8 
16.1 
16.4 
73 


9 
16.3 
16.5 
63 


10 
17.0 
17.2 
106 


11 
17.7 
18.0 
111 


a (intercept) = 
b (slope) = 

K = 
L_{¥ } = 

_{} 

sb = 
t_{n2} = 

confidence interval of K = 
Exercise 3.3.2 The FordWalford plot and Chapman's method
Postel (1955) reports the following length/age relationship for Atlantic yellowfin tuna (Thunnus albacares) off Senegal:
age 
fork length 
1 
35 
2 
55 
3 
75 
4 
90 
5 
105 
6 
115 
Tasks:
Estimate K and L_{¥ } using the FordWalford plot and Chapman's method.
Worksheet 3.3.2
Plot 
FORDWALFORD 
CHAPMAN 

t 
L(t) 
L(t + D t) 
L(t) 
L(t + D t)  L(t) 
1 




2 




3 




4 




5 




a (intercept) 




b (slope) 




_{} 




t_{n2} 




confidence limits of b 




K 




L_{¥ } 




Exercise 3.3.3 The von Bertalanffy plot
Cassie (1954) presented the lengthfrequency sample of 256 seabreams (Chrysophrys auratus) shown in the figure. He resolved this sample into normally distributed components (similar to Fig. 3.2.2.2) using the Cassie method (cf. Section 3.4.3) and found the following mean lengths for four age groups (cf. Fig. 17.3.3.3):
A 
B 
C 
D 
age group 
mean length 
D L/D t 
_{} 
0 
3.22 




2.11 
4.28 
1 
5.33 




2.29 
6.48 
2 
7.62 




2.12 
8.68 
3 
9.74 


Note: a Gulland and Holt plot gives (cf. Columns C and D): K = 0.002 and L_{¥ } = 950 inches, which makes no sense whatsoever.
Tasks:
1) Estimate K from the von Bertalanffy plot.
2) Why does it not make sense to ask you to estimate t_{0}?
Exercise 3.4.1 Bhattacharya's method
Weber and Jothy (1977) presented the lengthfrequency sample of 1069 threadfin breams (Nemipterus nematophorus) shown in Fig. 17.3.4.1A. These fish were caught during a survey from 29 March to 1 May 1972, in the South China Sea bordering Sarawak. The lengths measured are total lengths from the snout to the tip of the lower lobe of the caudal fin.
Figs. 17.3.4.1B and 17.3.4.1C show the Bhattacharya plots for the data in Fig. 17.3.4.1A, where B is based on the original data in 5 mm length intervals and C on the same data regrouped in 1 cm intervals. You should proceed with Fig. C for two reasons: 1) because it appears easier to see a structure in Fig. C than in Fig. B and 2) because the number of calculations is much lower.
Tasks:
1) Resolve the lengthfrequency sample (1 cm groups, Fig. C) into normally distributed components and estimate thereby mean length and standard deviations for each component. Use the four worksheets and plot the regression lines.2) Estimate L_{¥ } and K using a Gulland and Holt plot. Draw the plot.
3) Do you think the analysis could have been improved by using Fig. B (5 mm length groups) instead of Fig. C (1 cm groups)?
Fig. 17.3.4.1A Lengthfrequency sample of threadfin breams. Data source: Weber and Jothy, 1977
Worksheet 3.4.1a
A 
B 
C 
D 
E 
F 
G 
H 
I 
length interval 
N1+ 
ln N1+ 
D ln N1+ 
L 
D ln N1 
ln N1 
N1 
N2+ 
5.756.75 
1 
0 
 
 
 
 
1 
0 
6.757.75 
26 
3.258 
(3.258) 
6.75 
1.262 
 
26 
0 
7.758.75 
42# 
3.738# 
0.480 
7.75 
0.354 
3.738# 
42# 
0 
8.759.75 
19 
2.944 
0.793 
8.75 
0.554 
3.183 
19 
0 
9.7510.75 
5 


9.75 




10.7511.75 
15 


10.75 




11.7512.75 
41 


11.75 




12.7513.75 
125 


12.75 




13.7514.75 
135 


13.75 




14.7515.75 
102 


14.75 




15.7516.75 
131 


15.75 




16.7517.75 
106 


16.75 




17.7518.75 
86 


17.75 




18.7519.75 
59 


18.75 




19.7520.75 
43 


19.75 




20.7521.75 
45 


20.75 




21.7522.75 
56 


21.75 




22.7523.75 
20 


22.75 




23.7524.75 
8 


23.75 




24.7525.75 
3 


24.75 




25.7526.75 
1 


25.75 




Total 
1069 







a (intercept) = 
b (slope) =  
_{} 
_{} 
Worksheet 3.4.1b
A 
B 
C 
D 
E 
F 
G 
H 
I 
interval 
N2+ 
ln N2+ 
D ln N2+ 
L 
D ln N2 
ln N2 
N2 
N3+ 
5.756.75 








6.757.75 



6.75 




7.758.75 



7. 75 




8.759.75 



8.75 




9.7510.75 



9.75 




10.7511.75 



10.75 




11.7512.75 



11.75 




12.7513.75 



12.75 




13.7514.75 



13.75 




14.7515.75 



14.75 




15.7516.75 



15.75 




16.7517.75 



16.75 




17.7518.75 



17.75 




18.7519.75 



18.75 




19.7520.75 



19.75 




20.7521.75 



20.75 




21.7522.75 



21.75 




22.7523.75 



22.75 




23.7524.75 



23.75 




24.7525.75 



24.75 




25.7526.75 



25.75 




Total 








a (intercept) = 
b (slope) =  
_{} 
_{} 
Worksheet 3.4.1c
A 
B 
C 
D 
E 
P 
G 
H 
I 
interval 
N3+ 
ln N3+ 
D ln N3+ 
L 
D ln N3 
ln N3 
N3 
N4+ 
5.756.75 








6.757.75 



6.75 




7.758.75 



7.75 




8.759.75 



8.75 




9.7510.75 



9.75 




10.7511.75 



10.75 




11.7512.75 



11.75 




12.7513.75 



12.75 




13.7514.75 



13.75 




14.7515.75 



14.75 




15.7516.75 



15.75 




16.7517.75 



16.75 




17.7518.75 



17.75 




18.7519.75 



18.75 




19.7520.75 



19.75 




20.7521.75 



20.75 




21.7522.75 



21.75 




22.7523.75 



22.75 




23.7524.75 



23.75 




24.7525.75 



24.75 




25.7526.75 



25.75 




Total 








a (intercept) = 
b (slope) =  
_{} 
_{} 
Worksheet 3.4.1d
A 
B 
C 
D 
E 
F 
G 
H 
I 
interval 
N4+ 
ln N4+ 
D ln N4+ 
L 
D ln N4 
ln N4 
N4 
N5+ 
5.756.75 



 




6.757.75 



6.75 




7.758.75 



7.75 




8.759.75 



8.75 




9.7510.75 



9.75 




10.7511.75 



10.75 




11.7512.75 



11.75 




12.7513.75 



12.75 




13.7514.75 



13.75 




14.7515.75 



14.75 




15.7516.75 



15.75 




16.7517.75 



16.75 




17.7518.75 



17.75 




18.7519.75 



18.75 




19.7520.75 



19.75 




20.7521.75 



20.75 




21.7522.75 



21.75 




22.7523.75 



22.75 




23.7524.75 



23.75 




24.7525.75 



24.75 




25.7526.75 



25.75 




Total 








a (intercept) = 
b (slope) =  
_{} 
_{} = 
Exercise 3.4.2 Modal progression analysis
Fig. 17.3.4.2A shows a time series over twelve months of ponyfish (Leiognathus splendens) from Manila Bay, Philippines, 195758. (Data from Tiews and CacesBorja, 1965; figure redrawn from Ingles and Pauly, 1984). The numbers at the right hand side of the bar diagram indicate the sample sizes, while the height of the bars represents the percentages of the total number per length group.
Fig. 17.3.4.2B shows a time series of six samples of mackerel, (Rastrelliger kanagurta) from Palawan, Philippines, 1965. (Data from Research Division, BFAR, Manila; figure redrawn from Ingles and Pauly, 1984).
Tasks:
1) Fit by eye growth curves to these two time series, trying to follow the modal progression (as was done in Fig. 3.4.2.6). Start by fitting a straight line and then add some curvature to it, but do not be too particular about it. (Actually one should have carried out a Bhattacharya or similar analysis for each sample, but because of the amount of work involved in that approach, we take the easier, but less dependable, eyefit. This exercise aims at illustrating only the principles of modal progression analysis  not the exact procedure).2) Read from the eyefitted growth curves pairs of (t, L) = (time of sampling, length), and use the Gulland and Holt plot to estimate K and L_{¥ }. Assume that the samples were taken on the first day of the month. Read for Leiognathus splendens only the length for the samples indicated by "*" in Fig. A, as the figure is too small for a precise reading of each month. Use the worksheet.
3) Use the von Bertalanffy plot to estimate t_{0}.
Worksheet 3.4.2
A. Leiognathus splendens:


GULLAND AND HOLT PLOT 
VON BERTALANFFY PLOT 

time of sampling 
L(t) 
D L/D t 
_{} 
t 
 ln (1  L/L_{¥ }) 
1 June 





1 Sep. 





1 Dec. 





1 March 





a (intercept) 



(slope, K or K) 
L_{¥ } =  a/b = 
t_{0} =  a/b = 

L(t) = ___________ [1  exp ( _______ (t  _________ ))] 
Fig. 17.3.4.2A Time series of lengthfrequencies of ponyfish. Data source: Tiews and CacesBorja, 1965
B. Rastrelliger kanagurta:


GULLAND AND HOLT PLOT 
VON BERTALANFFY PLOT 

time of sampling 
L(t) 
D L/D t 
_{} 
t 
 ln (1  L/L_{¥ }) 
1 Feb 





1 March 





1 May 





1 June 





1 July 





1 August 





a (intercept) 



(slope, K or K) 
L_{¥ } =  a/b = 
t_{0} =  a/b = 

L(t) = ___________ [1  exp ( _______ (t  _________ ))] 
Fig. 17.3.4.2B Time series of lengthfrequencies of Indian mackerel. Data source: BFAR, Manila
Exercise 3.5.1 ELEFAN I
This exercise aims at explaining the details of the lengthfrequency restructuring process. Fig. 17.3.5.1A shows a (hypothetical) lengthfrequency sample, where the line shows the moving average. The worksheet table shows the calculation procedure and some results. Further explanations are given below for each step of the procedure.
Tasks:
1) Fill in the missing figures in the worksheet table.
2) Draw the bar diagram of the restructured data on the worksheet figure (B).
Worksheet 3.5.1
RESTRUCTURING OF LENGTH FREQUENCY SAMPLE  


STEP 
STEP 
STEP 
STEP 
STEP 
STEP 
STEP 
midlength 
orig. freq. 
MA (L) 
FRQ/MA 
_{} 
zeroes 
deemphasized 
points 
highest positive points 
5 
4 
4.6 a) 
0.870 
 0.197 h) 
2 
0.197 
0.109 p) 

10 
13 
4.6 


2 
0.966 k) 

0.966 s) 
15 
6 
4.8 b) 
1.250 e) 

1 
0.123 l) 
0.123 

20 
0 
4.0 
0 
1.000 
1 

0 

25 
1 

0.714 
0.341 i) 
3 
0.341 
0.188 

30 
0 
0.4 
0 
1.000 
2 



35 
0 
1.0 c) 
0 f) 

1 
1.000 


40 
1 

1.000 
0.077 
2 
0.077 


45 
3 


1.770 j) 
2 
1.062 m) 
1.062 

50 
1 



1 

0.127 q) 

55 
0 

0 
1.000 
1 
1.000 
0 r) 

60 
1 
0.4 d) 


3 
0.523 n) 


S = 
SP = 
 
(S /12) = M = 1.083 g) 
SN = 
ASP =  

 SP/SN = R = 0.552 o) 

Fig. 17.3.5.1A Hypothetical lengthfrequency sample. Line indicates moving average over 5 neighbours
Step 1: Calculate the moving average, MA(L) over 5 neighbours.
Examples: (see Fig. 17.3.5.1 A and worksheet table)MA (5) = (0 + 0 + 4 + 13 + 6)/5 = 4.6 a)
(two zeroes added at start of the sample)MA (15) = (4 + 13 + 6 + 0 + 1)/5 = 4.8 b)
MA (35) = (1 + 0 + 0 + 1 + 3)/5 = 1.0 c)
MA (60) = (1 + 0 + 1 + 0 + 0)/5 = 0.4 d)
Step 2: Divide the original frequencies, FRQ(L), by the moving average (MA) and calculate their mean value, M:
Examples:6/4.8 = 1.25 e)
0/1 = 0 f)_{}
_{}
(12 = number of length intervals)
Step 3: Divide FRQ/MA by M and subtract 1
Examples:0.870/1.083  1 = 0.197 h)
0.714/1.083  1 = 0.341 i)
3.000/1.083  1 = 1.770 j)
Step 4a: Count numbers of "zero neighbours" among the four neighbours (two zeroes added to each end of the sample).
Step 4b: Deemphasize positive isolated values: For each "zeroneighbour" the isolated point is reduced by 20%:
_{}
and if there are "zeroneighbours" then multiply this value by [1  0.2 * (no. of zeroes)]Examples:
1.610 * (1  0.2 * 2) = 0.966 k)
0.154 * (1  0.2 * 1) = 0.123 l)
1.770 * (1  0.2 * 2) = 1.062 m)
1.308 * (1  0.2 * 3) = 0.523 n)
Note: In the most recent version (Gayanilo, Soriano and Pauly, 1988) the deemphasizing has been made more pronounced by using the factor:
_{}
Step 4c: Calculate sum, SP, of positive (restructured) FRQs and calculate sum, SN, of negative (restructured) FRQs and calculate the ratio R =  SP/SN
Example:SP = 0.966 + 0.123 + 1.062 + 0.523 = 2.674
SN = 0.197  1  0.340  1  1  0.076  0.230  1 = 4.845
R =  SP/SN = 2.674/4.845 = 0.552 o)
_{}
then multiply this value by R. Values > 0 are not changed.Examples:
0.197 * 0.552 = 0.109 p)
0.231 * 0.552 = 0.123 q)
FRQ (55) = 0 r)
Plot the points in the diagram (Fig. 17.3.5.1B).
Step 6: Calculate ASP (available sum of peaks). Identify the highest point in each sequence of intervals with positive points (a "sequence" may consist of a single interval)
Examples:0.966 is the highest point in the positive sequence 1015 cm s)
1.062 is the highest point in the positive sequence 4545 cm
0.523 is the highest point in the positive sequence 6060 cmASP = 0.966 + 1.062 + 0.523 = 2.551
Fig. 17.3.5.1B Diagram for plotting points obtained after Step 5 (see text)
Exercise 3.5.1a ELEFAN I, continued
This exercise aims at illustrating the importance of the choice of the size of the length interval (cf. Exercise 3.4.1).
Fig. 17.3.5.1C1 shows a lengthfrequency sample (from Macdonald and Pitcher, 1979) of 523 pike from Heming Lake, Canada, grouped in 2 cm length intervals. There are five cohorts, determined on the basis of age reading of scales with the mean lengths shown in the following table:
age 
mean length 
standard deviation 
1 
23.3 
2.44 
2 
33.1 
3.00 
3 
41.3 
4.27 
4 
51.2 
5.08 
5 
61.3 
7.07 
These data put us in a position to test ELEFAN I.
Fig. 17.3.5.1C2 shows the normally distributed components derived from scale readings, and Fig. C3 shows the restructured data.
Except for the largest fish ELEFAN I manages to place the ASPs (indicated by arrows) close to where the "true" mean lengths of the cohorts are, but like all other methods ELEFAN I has difficulties in handling the largest (oldest) fish.
Tasks:
Repeat the restructuring using Worksheet 3.5.1a on the basis of 4 cm intervals (see worksheet figure) instead of 2 cm intervals. Compare the results with those presented in Figs. 17.3.5.1C1 and C2.
Fig. 17.3.5.1D Regrouped lengthfrequency data, 4 cm length intervals (see Fig. 17.3.5.1C)
Worksheet 3.5.1a
RESTRUCTURING OF LENGTH FREQUENCY SAMPLE  


STEP 
STEP 
STEP 
STEP 
STEP 
STEP 
STEP 
midlength 
orig. freq. 
MA(L) 
FRQ/MA 
_{} 
zeroes 
deemphasized 
points 
highest positive points 
20 
14 







24 
32 







28 
45 







32 
109 







36 
115 







40 
78 







44 
45 







48 
29 







52 
23 







56 
11 







60 
12 







64 
5 







68 
2 







72 
1 







76 
2 







S = 
SP = 
 
(S /15) = M = 
SN = 
ASP =  

SP/SN = R = 

Fig. 17.3.5.1E Diagram for plotting points obtained after Step 5 using data from Fig. 17.3.5.1D
Exercise 4.2 The dynamics of a cohort (exponential decay model with variable Z)
Consider a cohort of a demersal fish species recruiting at an age t, which is arbitrarily put to zero. Recruitment is N (0) = 10000.
Tasks:
1) Calculate, using the worksheet, for the first ten half year periods the number of survivors at the beginning of each period and the numbers caught when mortality rates are as shown below:
age group 
natural mortality 
fishing mortality 
Comments 
t1  t2 
M 
F 

0.00.5 
2.0 
0.0 
Cohort still on the nursery ground and exposed to heavy predation due to small size 
0.51.0 
1.5 
0.0 

1.01.5 
0.5 
0.2 
Cohort under migration to fishing ground. Some fish escape through meshes 
1.52.0 
0.3 
0.4 

2.02.5 
0.3 
0.6 
Cohort under full exploitation 
2.53.0 
0.3 
0.6 

3.03.5 
0.3 
0.6 

3.54.0 
0.3 
0.6 
Predation pressure reduced 
4.04.5 
0.3 
0.6 

4.55.0 
0.3 
0.6 

Recruitment: N (0) = 10000 
2) Give a graphical presentation of the results.
Worksheet 4.2
t1  t2 
M 
F 
Z 
e^{0.5Z} 
N(t1) 
N(t2) 
N(t1)  N(t2) 
F/Z 
C(t1, t2) 
0.00.5 
2.0 
0.0 







0.51.0 
1.5 
0.0 







1.01.5 
0.5 
0.2 







1.52.0 
0.3 
0.4 







2.02.5 
0.3 
0.6 







2.53.0 
0.3 
0.6 







3.03.5 
0.3 
0.6 







3.54.0 
0.3 
0.6 







4.04.5 
0.3 
0.6 







4.55.0 
0.3 
0.6 







Exercise 4.2a The dynamics of a cohort (the formula for average number of survivors, Eq. 4.2.9)
Tasks:
Calculate the average number of survivors during the last 3 years for the cohort dealt with in Exercise 4.2 using the exact expression (Eq. 4.2.9) and the approximation demonstrated in Fig. 4.2.3, i.e. calculate N(2.0, 5.0).
Exercise 4.3 Estimation of Z from CPUE data
Assume that in Table 3.2.1.2 the numbers observed are the numbers caught of each cohort per hour trawling on 15 October 1983.
Tasks:
Estimate the total mortality for the stock under the assumption of constant recruitment, using Eq. 4.3.0.3:_{}
Worksheet 4.3


cohort 
1982 A 
1982 S 
1981 A 
1981 S 
1980 A 1) 
age t2 
1.14 
1.64 
2.14 
2.64 
3.14 

CPUE 
111 
67 
40 
24 
15 

cohort 
age t1 
CPUE 





1983 S 
0.64 
182 





1982 A 
1.14 
111 
 




1982 S 
1.64 
67 
 
 



1981 A 
2.14 
40 
 
 
 


1981 S 
2.64 
24 
 
 
 
 

1) A = autumn, S = spring 
Exercise 4.4.3 The linearized catch curve based on age composition data
Use the data presented in Table 4.4.3.1 of North Sea whiting (19741980).
Tasks:
Estimate Z from the catches of the 1974cohort after plotting the catch curve. Calculate the confidence limits of the estimate of Z.
Worksheet 4.4.3
age 
year 
C(y, t, t+1) 
ln C(y, t, t+1) 
remarks 
(x) 


(y) 

0 




1 




2 




3 




4 




5 




6 




7 
1981 
 
 

slope: b = 
sb^{2} = [(sy/sx)^{2}  b^{2}]/(n2) =  
sb = 
sb * t_{n2} = ________________ z = _______ ± _______ 
Exercise 4.4.5 The linearized catch curve based on length composition data
Lengthfrequency data from Ziegler (1979) for the threadfin bream (Nemipterus japonicus) from Manila Bay are given in the worksheet below, L_{¥ } = 29.2 cm, K = 0.607 per year.
Tasks:
1) Carry out the lengthconverted catch curve analysis, using the worksheet.
2) Draw the catch curve.
3) Calculate the confidence limits for each estimate of Z.
Worksheet 4.4.5
L1  L2 
C (L1, L2) 
t(L1) 
D t 
_{} 
_{} 
z 
remarks 


a) 
b) 
c) 
(y) 


78 
11 





not used, not under full exploitation 
89 
69 






910 
187 






1011 
133 





? 
1112 
114 





? 
1213 
261 





? 
1314 
386 





? 
1415 
445 





? 
1516 
535 





? 
1617 
407 





? 
1718 
428 





? 
1819 
338 





? 
1920 
184 





? 
2021 
73 





? 
2122 
37 





? 
2223 
21 





? 
2324 
19 





? 
2425 
8 





? 
2526 
7 





too close to L_{¥ } 
2627 
2 






Formulas to be used:
a) Eq. 3.3.3.2
b) Eq. 4.4.5.1
c) Eq. 4.4.5.2
Details of the regression analyses:
length group 
slope 
number of obs. 
Student's distrib. 
variance of slope 
stand. dev. of slope 
confidence limits of Z 
L1  L2 
Z 
n 
t_{n2} 
sb^{2} 
sb 
Z ± t_{n2} * sb 





















Exercise 4.4.6 The cumulated catch curve based on length composition data (Jones and van Zalinge method)
Lengthfrequency data from Ziegler (1979) for the threadfin bream (Nemipterus japonicus) from Manila Bay are given in the worksheet below,
L_{¥ } = 29.2 cm, K = 0.607 per year.
Tasks:
1) Determine Z/K by the Jones and van Zalinge method, using the worksheet. (Start cumulation at largest length group).2) Plot the "catch curve".
3) Calculate the 95% confidence limits for each estimate of Z (worksheet).
Worksheet 4.4.6
L1  L2 
C(L1, L2) 
S C (L1, L_{¥ }) cumulated 
ln S C (L1, L_{¥ }) 
ln (L_{¥ }  L1) 
Z/K 
remarks 



(y) 
(x) 
(slope) 

78 
11 




not used, not under full exploitation 
89 
69 





910 
187 





1011 
133 




? 
1112 
114 




? 
1213 
261 




? 
1314 
386 




? 
1415 
445 




? 
1516 
535 




? 
1617 
407 




? 
1718 
428 




? 
1819 
338 




? 
1920 
184 




? 
2021 
73 




? 
2122 
37 




? 
2223 
21 




? 
2324 
19 




? 
2425 
8 




? 
2526 
7 




too close to L_{¥ } 
2627 
2 





Details of the regression analyses
length group 
slope 
number of obs. 
Student's distrib. 
variance of slope 
stand. dev. of slope 
confidence limits of Z 
L1  L2 
Z 
n 
t_{n2} 
sb^{2} 
sb 
Z ± K * t_{n2 }* sb 



































Exercise 4.4.6a The Jones and van Zalinge method applied to shrimp
Carapace lengthfrequency data for female shrimp (Penaeus semisulcatus) from Kuwait waters, 19741975, from Jones and van Zalinge (1981), are presented in the worksheet below. L_{¥ } = 47.5 mm (carapace length). Input data are total landings in millions of shrimps per year by the Kuwait industrial shrimp fishery.
Note: In this case the length intervals have different sizes, because the length groups have been derived from commercial size groups, which are given in number of tails per pound (1 kg = 2.2 pounds).
Tasks:
1) Determine Z/K by the Jones and van Zalinge method using the worksheet.
2) Plot the "catch curve".
3) Calculate the 95 % confidence limits for each estimate of Z/K.
Worksheet 4.4.6a
carapace length 
numbers landed/year 
cumulated numbers/year 



remarks 
L1  L2 
C(L1, L2) 
S C(L1, L_{¥ }) 
ln S C(L1, L_{¥ }) 
ln (L_{¥ }  L1) 
Z/K 




(y) 
(x) 
(slope) 

11.1818.55 
2.81 





18.5522.15 
1.30 





22.1525.27 
2.96 





25.2727.58 
3.18 





27.5829.06 
2.00 





29.0630.87 
1.89 





30.8733.16 
1.78 





33.1636.19 
0.98 





36.1940.50 
0.63 





40.5047.50 
0.63 





Details of the regression analyses:
lower length 
slope 
number of obs. 
Student's distrib. 
variance of slope 
stand. dev. of slope 
confidence limits of slope 
L1 
Z/K 
n 
t_{n2} 
sb^{2} 
sb 
Z/K ± t_{n2} * sb 




























Exercise 4.5.1 Beverton and Holt's Zequation based on length data (applied to shrimp)
The same data as for Exercise 4.4.6a (from Jones and van Zalinge, 1981) on Penaeus semisulcatus are given in the worksheet below. L_{¥ } = 47.5 mm (carapace length).
Tasks:
Estimate Z/K using Beverton and Holt's Zequation (Eq. 4.5.1.1) and the worksheet (start cumulations at largest length group).
Worksheet 4.5.1
A 
B 
C 
D 
E 
F 
G 
H 
carapace length group 
numbers landed/year 
cumulated catch 
midlength 
*) 
*) 
*) 
*) 
L' (L1)  L2 
C(L1, L2) 
S C(L1, L_{¥ }) 
_{} 
_{} 
_{} 
_{} 
Z/K 
11.1818.55 
2.81 






18.5522.15 
1.30 






22.1525.27 
2.96 






25.2727.58 
3.18 






27.5829.06 
2.00 






29.0630.87 
1.89 






30.8733.16 
1.78 






33.1636.19 
0.98 






36.1940.50 
0.63 






40.5047.50 
0.63 






*) Column E: catch per length group * mid length 
Exercise 4.5.4 The PowellWetherall method
Forklength distribution (in %) of the bluestriped grunt (Haemulon sciurus) caught in traps at the Port Royal reefs off Jamaica during surveys in 19691973, are given in the worksheet below (from Munro, 1983, Table 10.35 p. 137).
Tasks:
1) Complete the worksheet, from the bottom.2) Make the PowellWetherall plot and decide on the points to be included in the regression analysis.
3) Estimate Z/K and L (in forklength).
4) What are the basic assumptions underlying the method?
Worksheet 4.5.4
A 
B 
C 
D *) 
E *) 
F *) 
G *) 
H *) 
L1  L2 
C(L1, L2) (% catch) 
_{} 
S C(L',¥) 
_{} 
_{} 
_{} 
_{} 
(x) 






(y) 
1415 
1.8 
14.5 





1516 
3.4 
15.5 





1617 
5.8 
16.5 





1718 
8.4 
17.5 





1819 
9.1 
18.5 





1920 
10.2 
19.5 





2021 
14.3 
20.5 





2122 
13.7 
21.5 





2223 
10.0 
22.5 





2324 
6.3 
23.5 





2425 
6.4 
24.5 





2526 
5.3 
25.5 





2627 
3.3 
26.5 





2728 
1.8 
27.5 





2829 
0.3 
28.5 





*) Column D: sum column B (from the bottom) 
Exercise 4.6 Plot of Z on effort (estimation of M and q)
For the trawl fishery in the Gulf of Thailand the effort (in millions of trawling hours) and the mean lengths of bulls eye (Priacanthus tayenus) over the years 19661974 were taken from Boonyubol and Hongskul (1978) and South China Sea Fisheries Development Programme (1978) and presented in the worksheet below (L_{¥ } = 29.0 cm, K = 1.2 per year, Lc = 7.6 cm).
Tasks:
1) Calculate Z, using the worksheet.
2) Plot Z against effort and determine M (intercept) and q (slope).
3) Calculate the 95% confidence limits for the estimates of M and q.
Use the following two sets of input data (years):
a) The years 19661970
b) The years 19661974 and comment on the results.
Worksheet 4.6
year 
effort a) 
mean length 
_{} 
1966 
2.08 
15.7 
1.97 
1967 
2.80 
15.5 

1968 
3.50 
16.1 

1969 
3.60 
14.9 

1970 
3.80 
14.4 

1071 
no data  
1972 
no data  
1973 
9.94 
12.8 

1974 
6.06 
12.8 

a) in millions of trawling hours 
Exercise 5.2 Agebased cohort analysis (Pope's cohort analysis)
Catch data by age group of the North Sea whiting (from ICES, 1981a) are presented in Tables 5.1.1 and 4.4.3.1.
Tasks:
1) Calculate fishing mortalities for the 1974 cohort (catch numbers given in Table 5.1.1 and M = 0.2 per year) by Pope's cohort analysis under the two different assumptions on the F for the oldest age group:F6 = 1.0 per year
F6 = 2.0 per year2) Plot F against age for the two cases above as well as for the case of Table 5.1.1, where
F6 = 0.5 per year3) Discuss the significance of the choice of the terminal F (F6). Which of the three alternatives do you prefer? (Base your decision on the solution to Exercise 4.4.3, which deals with the same data set).
Exercise 5.3 Jones' lengthbased cohort analysis
As in Exercises 4.4.6a and 4.5.1 we use the landings of female Penaeus semisulcatus of the 74/75cohort from Kuwait waters (from Jones and van Zalinge, 1981). These data were derived from the total number of processed prawns in each of ten market categories (cf. Worksheet 5.3).
Tasks:
1) Using Worksheet 5.3 and the formulas given below, estimate fishing mortalities and stock numbers by means of Jones' lengthbased cohort analysis, using the parameters:K = 2.6 per year
M = 3.9 per year
L_{¥ } = 47.5 mm (carapace length)2) Give your opinion on our choice of terminal F/Z (= 0.1).
3) Is the cohort analysis a dependable method in this case? (The value of M is a "guesstimate").
Worksheet 5.3
length group 
nat. mort. factor 
number caught 
number of survivors 
exploitation rate 
fishing mort. 
total mort.  
g) 
a) 

b) 
c) 
d) 
e)  
L1  L2 
H(L1, L2) 
C(L1, L2) 
N(L1) 
F/Z 
F 
Z  
11.1818.55 

2.81 



 
18.5522.15 

1.30 



 
22.1.525.27 

2.96 



 
25.2727.58 

3.18 



 
27.5829.06 

2.00 



 
29.0630.87 

1.89 



 
30.8733.16 

1.78 



 
33.1636.19 

0.98 



 
36.1940.50 

0.63 



 
40.5047.50 

0.63 f) 



 
a) 
_{}  
b) 
N(L1) = [N(L2) * H(L1, L2) + C(L1, L2)] * H(L1, L2)  
c) 
F/Z = C(L1, L2)/[N(L1)  N(L2)]  
d) 
F = M * (F/Z)/(1  F/Z)  
e) 
Z = F + M  
f) 
N(last L1) = C(last L1, L_{¥ })/(F/Z)  
g) 
carapace lengths in mm corresponding to the market categories (in units of number of tails per pound): 
no/lb: 
400 
110 
70 
50 
40 
35 
30 
25 
20 
<15 
L1: 
11.18 
18.55 
22.15 
25.27 
27.58 
29.06 
30.87 
33.16 
36.19 
40.5 
L2: 
18.55 
22.15 
25.27 
27.58 
29.06 
30.87 
33.16 
36.19 
40.5 
47.5 
Exercise 6.1 A mathematical model for the selection ogive
Tasks:
Draw a selection curve using the parameters:L50% = 13.6 cm and L75% = 14.6 cmUse the logistic curve S_{L} = 1/[1 + exp(S1  S2 * L)]
Exercise 6.5 Estimation of the selection ogive from a catch curve
Data on catch by length group of Upeneus vittatus were taken from Table 4.4.5.1. K = 0.59 per year, L_{¥ } = 23.1 cm, t_{0} = 0.08 year
Tasks:
1) Estimate the logistic curve S_{t} = 1/[1 + exp(T1  T2 * t)]
2) Estimate L50% = L_{¥ } * [1  exp(K * (t_{0}  t50%))] and L75%
3) Evaluate the choice of first length interval given in Table 4.4.5.1.
Worksheet 6.5
A 
B 
C 
D 
E 
F 
G 
H 
I 
length group 
t 
D t 
C(L1, L2) 
ln (C/D t) 
S_{t }obs. 
ln (1/S  1) 
est. 
remarks 

(x) 




(y) 


67 
0.56 
0.102 
3 
3.38 



(not used) 
78 
0.67 
0.109 
143 
7.18 




89 
0.78 
0.116 
271 
7.76 




910 
0.90 
0.125 
318 
7.86 




1011 
1.03 
0.134 
416 
8.04 




1112 
1.17 
0.146 
488 
8.11 




1213 
1.32 
0.160 
614 
8.25 




1314 
1.49 
0.177 
613f) 
8.15 



used for the analysis to estimate Z (see Table 4.4.5.1) 
1415 
1.67 
0.197 
493 f) 
7.83 




1516 
1.88 
0.223 
278 f) 
7.13 




1617 
2.12 
0.257 
93 f) 
5.89 




1718 
2.40 
0.303 
73 f) 
5.48 




1819 
2.74 
0.370 
7 f) 
2.94 




1920 
3.15 
0.473 
2 f) 
1.44 




2021 
3.70 
0.659 
2 
1.11 




2122 
4.53 
1.094 
0 
 




2223 
6.19 
4.094 
1 
1.40 




2324 
 
 
1 
 




a) t [(L1 + L2)/2], age corresponding to interval midlength b) ln(C/D t), dependent variable in catch curve regression analysis c) S(t) obs. = C/[D t * exp(a  Z * t)], observed selection ogive Z = 4.19 and a = 14.8 (from Table 4.4.5.1) d) ln(1/S  1), dependent variable in regression e) S(t) est. = 1/[1 + exp(T1  T2 * t)], theoretical (estimated) selection ogive f) points used in the catch curve analysis (cf. Table 4.4.5.1) 
Exercise 6.7 Using a selection curve to adjust catch samples
Tasks:
1) Adjust the lengthfrequencies for Upeneus vittatus (from the data given in Table 4.4.5.1) using the results of Exercise 6.5:
L50% = 13.6 cm and L75% = 14.6 cm
S1 =
S2 =
S_{L }=
2) Draw a histogram of the original and the adjusted frequencies excluding the raised (estimated unbiased) frequencies which you think are not safely estimated.
Worksheet 6.7
length group 
midpoint 
observed biased sample 
selection ogive 
estimated unbiased sample 
67 

3 


78 

143 


89 

271 


910 

318 


1011 

416 


1112 

488 


1213 

614 


1314 

613 


1415 

493 


1516 

278 


1617 

93 


1718 

73 


1819 

7 


1920 

2 


2021 

2 


2122 

0 


2223 

1 


2324 

1 


Exercise 7.2 Stratified random sampling versus simple random sampling and proportional sampling
This exercise illustrates the gain in precision obtained from stratification. Use Table 7.2.2.
Tasks:
1) Estimate the variance of the mean landing Y from three different sampling methods, when the total sample size is n = 20, using the worksheets.a) Simple random sampling
b) Proportional sampling: a sample of 20% from each stratum
Worksheet 7.2 for a) and b)
stratum 
s(j) 
s(j)^{2} 
N(j) 
_{} 
_{} 
1 large 





2 medium 





3 small 





total 





_{} 

a) Simple random sampling _{} b) Proportional sampling _{} 
Worksheet 7.2 for c)
stratum 
s(j) * N(j) 
_{} 
_{} 
1 large 



2 medium 



3 small 



total 

1.00 
n = 20 
c) Optimum stratified sampling _{} 
2) Calculate the standard deviations and compare the allocations per stratum.

random 
proportional 
optimum 
_{} 



allocation per stratum 



1 large 



2 medium 



3 small 



Exercise 8.3 The yield per recruit model of Beverton and Holt (yield per recruit, biomass per recruit as a function of F)
Pauly (1980) determined the following parameters for Leiognathus splendens (cf. Exercise 3.1.2). W_{¥ } = 64 g, K = 1.0 per year, t_{0} = 0.2 year, Tr = 0.2 year, M = 1.8 per year.
Tasks:
1) Draw the Y/R and the B/R curves, for three different values of Tc: Tc = Tr = 0.2 year, Tc = 0.3 year and Tc = 1.0 year.
Worksheet 8.3

Tc = Tr = 0.2 
Tc = 0.3 
Tc = 1.0  
F 
Y/R 
B/R 
Y/R 
B/R 
Y/R 
B/R 
0.0 






0.2 






0.4 






0.6 






0.8 






1.0 






1.2 






1.4 






1.6 






1.8 






2.0 






2.2 






2.4 






2.6 






2.8 






3.0 






3.5 






4.0 






4.5 






5.0 






100.0 






2) Try to explain why MSY increases when Tc increases (without the use of mathematics). Is the above statement a general rule, i.e. does it hold for any increase of Tc?
3) Read the (approximate) values of F_{MSY} and MSY/R from the worksheet. Comment on your findings under the assumption that the present level of F is 1.0.
Exercise 8.4 Beverton and Holt's relative yield per recruit concept
For the swordfish (Xiphias gladius) off Florida, Berkeley and Houde (1980) determined the parameters:
L_{¥ } = 309 cm, K = 0.0949 per year and M = 0.18 per year
Tasks:
Draw the relative yield per recruit curve, (Y/R') as a function of E, for two different values of the 50% retention length:Lc = 118 cm and Lc = 150 cm.
Worksheet 8.4

Lc = 118 cm 
Lc = 150 cm 

E 
(Y/R)' 
(Y/R)' 
(F) 
0 


0 
0.1 


0.020 
0.2 


0.045 
0.3 


0.077 
0.4 


0.120 
0.5 


M = 0.180 
0.6 


0.270 
0.7 


0.42 
0.8 


0.72 
0.9 


1.62 
1.0 


¥ 
Exercise 8.6 A predictive agebased model (Thompson and Bell analysis)
In the (hypothetical example) given in the table below a fish stock is exploited by two different gears, viz. beach seines and gill nets. These gears account for the total catch from the stock. A sampling programme for estimation of total numbers caught by age group and by gear has been running for the years 19751985.
Based on the total numbers caught a VPA has been made and the estimated F values for the last data year (1985) have been separated into a beach seine component, FB and a gill net component FG (cf. Eq. 8.6.1). The average recruitment (number of 0group fish) for the years 1975 to 1985 has been estimated from VPA to be 1000000 fish. The natural mortalities are assumed to take the agespecific values. These data are presented in part a of the worksheet.
Tasks:
Use Worksheet 8.6a to solve the following problems:1) Under the assumption that fishing mortality remains the same as in 1985 and that the recruitment is of average size, predict (based on the assumption of equilibrium):
1.1) The number of survivors (stock numbers) by age group.
1.2) Numbers caught by age group for each gear.
1.3) Yield of each gear.Use Worksheet 8.6b to solve the following problems:
2) Under the assumption that the gill net effort remains the same as in 1985 but that the beach seine fishery is closed (and that the recruitment is of average size) predict as 1.1, 1.2 and 1.3 above.
3) Would you, based on the results of 1) and 2) recommend a closure of the beach seine fishery?
Worksheet 8.6
a. No change in fishing effort:
age group 
mean weight (g) 
beach seine mortality 
gill net mortality 
natural mortality 
total mortality 
stock number 
beach seine catch 
gill net catch 
beach seine yield 
gill net yield 
total yield 
t 
w 
FB 
FG 
M 
Z 
'000 
CB 
CG 
YB 
YG 
YB + YG 
0 
8 
0.05 
0.00 
2.00 

1000 





1 
283 
0.40 
0.00 
0.80 







2 
1155 
0.10 
0.19 
0.30 







3 
2406 
0.01 
0.59 
0.20 







4 
3764 
0.00 
0.33 
0.20 







5 
5046 
0.00 
0.09 
0.20 







6 
6164 
0.00 
0.02 
0.20 







7 
7090 
0.00 
0.00 
0.20 







total  
Z = FB + FG + M 
N(t + 1) = N(t) * exp(Z)  
CB = FB * N * (1  exp(Z))/Z 
CG = FG * N * (1  exp(Z))/Z  
_{} 
_{} 
b. Closure of the beach seine fishery:
age group 
mean weight (g) 
beach seine mortality 
gill net mortality 
natural mortality 
total mortality 
stock number 
beach seine catch 
gill net catch 
beach seine yield 
gill net yield 
total yield 
t 
w 
FB 
FG 
M 
Z 
'000 
CB 
CG 
YB 
YG 
YB + YG 
0 
8 










1 
283 










2 
1155 










3 
2406 










4 
3764 










5 
5046 










6 
6164 










7 
7090 










total 
Exercise 8.7 A predictive lengthbased model (Thompson and Bell analysis)
For this exercise a hypothetical example is used:
M = 0.3 per year, K = 0.3 per year, L_{¥ } = 60.0 cm
_{}
Recruitment, N(10, 15) = 1000
length class 
fishing mortality 
mean body weight g 
price per kg 
natural mortality factor 
L1  L2 
F (L1, L2) 
_{} 
(L1, L2) 
H (L2, L2) a) 
1015 
0.03 
19.5 
1.0 
1.05409 
1520 
0.20 
53.6 
1.0 
1.06066 
2025 
0.40 
113.9 
1.5 
1.06904 
2530 
0.70 
207.9 
1.5 
1.08012 
3035 
0.70 
343.3 
2.0 
1.09544 
3540 
0.70 
527.3 
2.0 
1.11803 
40L_{¥ } 
0.70 
767.7 
2.0 
 
a) H(L1, L2) = ((L_{¥ }  L1)/(L_{¥ }  L2))^{M/2K} 
Tasks:
Do the lengthconverted Thompson and Bell analysis on the example.
Worksheet 8.7
length class 
P(L1, L2) 
N(L1) 
N(L2) 
mean biomass 
catch 
yield 
value 
1015 
0.03 
1000 





1520 
0.20 






2025 
0.40 






2530 
0.70 






3035 
0.70 






3540 
0.70 






40L_{¥ } 
0.70 


f) 




Total 
_____ 





a) N(L1) of a length group is equivalent to the N(L2) of the previous length group c) C(L1, L2) = F(L1, L2) * Nmean(L1, L2) * D t _{} e) value(L1, L2) = yield(L1, L2) * price(L1, L2) _{} 
Exercise 8.7a A predictive lengthbased model (yield curve, Thompson and Bell analysis)
Tasks:
1) Do the same exercise as in Exercise 8.7 but under the assumption of a 100% increase in fishing effort (Worksheet 8.7a).
Worksheet 8.7a
length class 
F(L1, L2) 
N(L1) 
N(L2) 
mean biomass 
catch 
yield 
value 
1015 

1000 





1520 







2025 







2530 







3035 







3540 







40L_{¥ } 



f) 




Total 
_____ 





a) N(L1) of a length group is equivalent to the N(L2) of the previous length group _{} where Nmean(L1, L2) * Dt = [N(L1)  N(L2)]/Z(L1, L2) c) C(L1, L2) = F(L1, L2) * Nmean(L1, L2) * D t _{} e) value(L1, L2) = yield(L1, L2) * price(L1, L2) _{} 
2) Use the result of 1) combined with the solution to Exercise 8.7 and the results given in the table below to draw the yield, the mean biomass and the value curves.
Ffactor 
yield 
mean biomass 
value 
x 

* D t 

0.0 
0.00 
1445.41 
0.00 
0.2 
116.38 
865.89 
226.11 
0.4 
154.48 
585.63 
296.49 
0.6 
165.12 
426.42 
312.70 
0.8 
164.75 
326.87 
307.56 
1.0 



1.2 
153.25 
213.94 
277.35 
1.4 
146.23 
180.15 
260.38 
1.6 
139.37 
154.84 
244.14 
1.8 
132.95 
135.40 
229.10 
2.0 



MSY = 165.8 at X = 0.69 biomass at MSY = 378.8  
MSE = 312.9 at X = 0.61 biomass at MSE = 405.7 
Exercise 9.1 The Schaefer model and the Fox model
In Worksheet 9.1 are given total catch and total effort in standard boat days for the years 1969 through 1978 for the shrimp fishery in the Arafura Sea. Catches are mainly composed of the five species Penaeus merguiensis, Penaeus semisulcatus, Penaeus monodon, Metapenaeus ensis and Parapenaeopsis sculptilis (from Naamin and Noer, 1980).
Tasks:
1) Calculate Y/f (kg per boat day) and ln (Y/f) and plot them against effort.
2) Estimate MSY and f_{MSY} by the Schaefer model.
3) Estimate MSY and f_{MSY} by the Fox model.
4) Plot yield against effort and draw the yield curves estimated by the two methods.
Worksheet 9.1
year 
yield (tonnes) 
effort 
Schaefer 
Fox 
i 
Y(i) 
(x) 
(y) 
(y) 
1969 
546.7 
1224 


1970 
812.4 
2202 


1971 
2493.3 
6684 


1972 
4358.6 
12418 


1973 
6891.5 
16019 


1974 
6532.0 
21552 


1975 
4737.1 
24570 


1976 
5567.4 
29441 


1977 
5687.7 
28575 


1978 
5984.0 
30172 


mean values 




standard deviations 




intercept (Schaefer: a, Fox: c) *) 



slope (Schaefer: b. Fox: d) *) 



*) a, b replaced by c, d for the Fox model 
continuation of Worksheet 9.1

Schaefer 
Fox 
variance of slope 


standard deviation of slope, sb 


variance of intercept 


standard deviation of intercept 


MSY 
 a^{2}/(4b) = 
(1/d) * exp(c  1) = 
fMSY 
 a/(2b) = 
 1/d = 
Worksheet 9.1a (for drawing the yield curves)
f 
Schaefer 
Fox 
5000 


10000 


15000 


20000 


25000 


f_{MSY} 


30000 


35000 


f_{MSY} 


40000 


45000 


Exercise 13.8 The swept area method, precision of the estimate of biomass, estimation of MSY and optimal allocation of hauls
The data for this exercise were taken from report no. 8 of Project KEN/74/023: "Offshore trawling survey", which deals with the stock assessment of Kenyan demersal resources from surveys in the period 197981. The data used here are a modified set on the catch of the smallspotted grunt, Pomadasys opercularis. The data are given as catch in weight per unit time (Cw/t) in kg per hour trawling for 23 hauls covering two strata (in Worksheet 13.8). The vessel speed, current speed, both in knots (nautical mile per hour) and trawl wing spread (hr * X2) are also given.
Tasks:
1) Apply Eq. 13.5.3 to calculate the distance, D, covered per hour and Eq. 13.5.1 to calculate the area swept per hour, a, for each haul. Calculate the yield, Cw, per unit of area for each haul using Eq. 13.6.2 (data in the worksheet, 1 nautical mile (nm) = 1852 m).2) Calculate for each stratum the estimate of mean catch per unit area Ca and the confidence limits of the estimates (using Eq. 2.3.1). Calculate using Eqs. 13.7.5 and 13.6.3 an estimate of the mean biomass for the total area, when A1 = 24 square nautical miles (sq.nm), A2 = 53 sq.nm and X1 (catchability) is assigned the value 0.5.
3) Estimate MSY using Gulland's formula, with M = Z = 0.6 per year (i.e. we assume a virgin stock).
4) Construct a graph showing the maximum relative error for the mean catch per area against the number of hauls for each of the two strata. We define (cf. Section 7.1, Fig. 7.1.1)
_{}where s is the standard deviation of the estimate of the catch in weight per unit area:
_{}5) Assume that you have financial resources to make 200 hauls. Allocate these 200 hauls between the two strata for optimum stratified sampling (cf. Section 7.2).
Worksheet 13.8
STRATUM 1:
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
HAUL 
CPUE 
VESSEL 
CURRENT 
TRAWL 
AREA 
CPUA  
no. 
Cw/h 
speed 
course 
speed 
direction 
spread 
distance 
swept 
Cw/a = Ca 
1 
7.0 
2.8 
220 
0.5 
90 
18 



2 
7.0 
3.0 
210 
0.5 
180 
16 



3 
5.0 
3.0 
200 
0.3 
135 
17 



4 
4.0 
3.0 
180 
0.4 
230 
18 



5 
1.0 
3.0 
90 
0.5 
270 
17 



6 
4.0 
3.0 
45 
0.4 
160 
18 



7 
9.0 
3.5 
25 
0.4 
200 
18 



8 
0.0 
3.0 
210 
0.3 
300 
18 



9 
0.0 
3.5 
0 
0.4 
0 
18 



10 
14.0 
2.8 
45 
0.6 
0 
18 



11 
8.0 
3.0 
120 
0.3 
300 
18 











_{}  
STRATUM 2:  
12 
42.0 
4.0 
30 
0.5 
160 
17 



13 
98.0 
3.3 
215 
0.4 
90 
17 



14 
223.0 
3.9 
30 
0.0 
0 
17 



15 
59.0 
3.8 
35 
0.3 
180 
17 



16 
32.0 
3.5 
210 
0.5 
270 
17 



17 
6.0 
2.8 
210 
0.5 
330 
17 



18 
66.0 
3.8 
45 
0.5 
30 
17 



19 
60.0 
4.0 
30 
0.5 
180 
18 



20 
48.0 
4.0 
210 
0.5 
180 
18 



21 
52.0 
3.8 
20 
0.4 
180 
18 



22 
48.0 
4.0 
30 
0.5 
190 
18 



23 
18.0 
3.0 
210 
0.3 
190 
18 











_{} 
Confidence limits of _{}  
stratum 
number of hauls 
_{} 
standard deviation 

Student's distr. 
confidence limits for _{} 

n 

s 
s/Ö n 
t_{n1} 
_{} 
1 






2 






Worksheet 13.8a (for plotting graph maximum relative error)
number of hauls 
Student's distribution 
stratum 1 
stratum 2 
n 
t_{n1} 
e a) 
e a) 
5 
2.78 


10 
2.26 


20 
2.09 


50 
2.01 


100 
1.98 


200 
1.97 


_{} 
Worksheet 13.8b (optimum allocation)
stratum 
standard deviation of Ca 
area of stratum 




s 
A 
A * s 
A * s/S A * s 
200 * A * s/S A * s 
1 





2 





Total 




