(7.9.1)
GROUP I
1. Consider a stock and an interval of time i, (t_{i}, t_{i+1}). Knowing that for this interval of time:
M_{i} = 0.4 year^{1 }T_{i} = 2.3 year
C_{i} = 230 million individualsa) Adopt the value 0.5 year^{1} for the fishing mortality coefficient for the interval and calculate the numbers of survivors at the beginning and end of the interval.
2. Consider the interval of time i, (t_{i}, t_{i+1}). Knowing that in this interval of time:
M_{i} = 0.6 year^{1 }T_{i} = 0.9 year
C_{i} = 98 million individuals
Calculate the value of the fishing mortality coefficient, F_{i}, for the interval, taking the number of survivors, N_{i}, at the beginning of the interval, i, to be 172 million individuals.
3. Consider the interval of time (t_{i}, t _{I+1}). Knowing that in that interval of time:
M_{i} = 0.5 year^{1 }T_{i} = 1 year
C_{i} = 42 million individualsa) Calculate the value of the fishing mortality coefficient for the interval, knowing that the number of survivors at the end of the year was N_{i+1} = 85 million individuals. Calculate the value of F_{i} using the Pope formula.
GROUP II
The data in the following table represent the catches in millions, of a cohort of hake, Merluccius merluccius, in the Iberic Peninsula waters.
Age (years) 
0 
1 
2 
3 
4 
5 
6 
7 
8 
C_{i} (million) 
712 
3941 
8191 
10311 
5515 
4149 
3081 
1185 
549 
Adopt a value of 0.2 year^{1} for the natural mortality coefficient, constant for all the ages.
1. Suppose that the value of the fishing mortality coefficient at the last age (8 years) was 1.0 year^{1}. Calculate, by an iterative method and by the Pope method, for each age of the cohort:
a) The value of the fishing mortality coefficient.
b) The number of survivors at the beginning of the age.
c) Compare the results obtained by the two methods.
d) Represent, on a graph the values of F_{i} estimated against the age, and say what the recruitment of this cohort is at the exploited phase.
GROUP III
1. Aiming to analyse the influence of the chosen F_{terminal,} repeat the calculations of question 1 of Group II, using one of the previous methods, with 0.3 and 1.5 year^{1} for the value of F_{terminal}.
a) Draw a graph with the estimated values of F_{i} and N_{i} against the age.
b) Comment on the differences between the graphs for the different values of F_{terminal}.
2. Aiming to analyse the influence of the choice of M, repeat the calculations of question 1 of Group II, using one of the previous methods, for values of M of 0.1 and 0.4 year^{1}.
a) Represent, on a graph, the estimated values of F_{i} and N_{i} against the age.
b) Comment on the differences between the graphs for the different values of M.
GROUP IV
The annual catches by age class, of a certain resource, for the years of 1985 to 1994, are presented in the following table.
Catches by age class (Million individuals) 

Years 

Age (years) 
1985 
1986 
1987 
1988 
1989 
1990 
1991 
1992 
1993 
1994 
0 
67 
88 
104 
290 
132 
90 
63 
38 
52 
90 
1 
532 
1908 
1841 
1671 
4172 
1915 
1284 
906 
541 
704 
2 
2070 
1756 
4424 
3178 
2534 
6320 
2826 
1911 
1322 
741 
3 
728 
4016 
2256 
4042 
2499 
1972 
4742 
2115 
1382 
890 
4 
353 
945 
3309 
1273 
1926 
1170 
883 
2102 
896 
540 
5 
97 
439 
733 
1730 
558 
827 
479 
356 
807 
316 
6 
16 
107 
300 
333 
656 
207 
291 
166 
117 
243 
7 
25 
8 
73 
136 
126 
243 
73 
101 
54 
35 
8 
5 
7 
5 
33 
52 
47 
85 
25 
33 
16 
The modus operandi of the fishing fleet was constant during the period, but the number of vessels increased significantly. It is considered that, at present, the resource is intensively exploited.
Besides the information on the fishery, the estimates of the growth parameters of this resource and of the natural mortality coefficient are also available:

L_{∞} = 38.5 cm 
a = 0.021 of the relation W(g)L(cm) 

K = 0.25 year^{1} 
b = 2.784 of the relation W(g)L(cm) 

t_{o} =  0.51 year 
M = 0.3 year^{1} 
1. Estimate the fishing mortality coefficient and the number of survivors at the beginning of the year for each age class and each year. Use the Pope Cohort Analyses method.
a) Start by selecting F_{terminal} = 0.5 year^{1} for the last age of every year and for all the ages of the last year.
b) After analysing matrix F obtained in a), select new values for F_{terminal} and repeat the application of Pope’s method.
2. Besides the information given in the previous question, it is also known that the spawning takes place in a restricted period, around the beginning of the year. Research cruises using acoustic methods took place during the spawning period, in order to estimate the spawning biomass (kg/hour of trawl). The results obtained are shown in the following table:
Years 
1985 
1986 
1987 
1988 
1989 
1990 
1991 
1992 
1993 
1994 
Spawning biomass Index 
1270 
1613 
1629 
1424 
1300 
1209 
1000 
718 
476 
326 
The biological information collected during those cruises was also used to estimate the maturity ogive of the stock at the spawning period:
Age (years) 
0 
1 
2 
3 
4 
5 
6 
7 
8 
% Matures 
0 
1 
20 
50 
80 
100 
100 
100 
100 
a) Calculate the spawning biomass in the spawning period of each year from 1985 to 1994 using the results of the Cohort Analyses obtained in question 1.b.
b) Use the information of the acoustic cruises to tune the Cohort Analyses.
c) Comment on the tuning results.