5. VIRTUAL POPULATION METHODS

5.1 VIRTUAL POPULATION ANALYSIS (VPA)
5.2 AGE-BASED COHORT ANALYSIS (POPE'S COHORT ANALYSIS)
5.3 JONES' LENGTH-BASED COHORT ANALYSIS
5.4 THE SLICING TECHNIQUE

This chapter deals in principle with the same problems as Chapter 4. However, the analysis and the data requirements are more detailed. The methods of Chapter 4 could be applied to data sets originating from small samples of the commercial catch or from research vessel catches, while the methods of Chapter 5 require estimates of the total numbers caught by commercial fishing.

The methods (or models) described in this chapter are closely linked with those described in Chapter 8. Chapter 5 deals with methods that can be used to analyse the effect that a fishery has had on a particular year class of a stock and Chapter 8 deals with methods that can be used to predict the effect of different levels of fishing effort in the future. The methods in the second category are usually based on the findings of those of the first category. The methods that look at the past, using "historic" data are called "virtual population analysis (VPA)" or "cohort analysis", while those methods dealing with the future, are called "predictive methods" or "Thompson and Bell methods".

VPA and cohort analysis were first developed as age-based methods. However, in recent years also length-based methods have become available, which are of particular interest to tropical fisheries. The age-based methods are discussed in Sections 5.1 and 5.2, while the length-based methods are dealt with in Section 5.3.

Multispecies versions of VPA have also been developed, but they fall outside the scope of this manual. An overview of these models is given by Sparre, 1991.

Information is required on how much was fished, in terms of numbers of fish. The total landings must be distributed over age groups (age-based methods) or length groups (length-based methods). The totals are obtained by raising the age or length distributions of random samples of the landings, using information on the total landings in tonnes. Tables of the total landings in numbers, by age or length, per year or month, have to be prepared, before we can begin an analysis.

5.1 VIRTUAL POPULATION ANALYSIS (VPA)

Virtual population analysis or VPA is basically an analysis of the catches of commercial fisheries, obtained through fishery statistics, combined with detailed information on the contribution of each cohort to the catch, which is usually obtained through sampling programmes and age readings. The word "virtual", introduced by Fry (1949) is based on the analogy with the "virtual image", known from physics. The "virtual population" is a population created by the method, based on real total catch data and assumptions of the level of natural mortality and terminal fisheries mortality.

The idea behind the method is to analyse that what can be measured, the catch, in order to calculate the population that must have been in the water to produce this catch (see Fig. 5.1.1).

The total landings from a cohort in its lifetime is the first estimate of the numbers of recruits from that cohort. It is however, an under-estimate because some fish must have died from natural causes. Given an estimate of M we can do a backwards calculation and find out how many fish belonging to the cohort were alive year by year and ultimately, how many recruits there were. At the same time we learn the values of the fishing mortality coefficient F, because we have calculated the numbers alive and know from the beginning how many of them were caught in any particular year.

Fig. 5.1.1 The basic features of VPA. For further explanation, see text

VPA therefore looks at a population in an historic perspective. The advantage of doing a VPA is that once the history is known it becomes easier to predict the future catches, which is usually one of the most important tasks of fishery scientists.

A complete review of the development of VPA methods is given by Megrey (1989). The method originated in the USSR, where Derzhavin (1922) was probably the first to combine age data with catch statistics. The method was rediscovered by Fry (1949) and subsequently modified by many authors, including Gulland (1965) and Pope (1972). The modification made by Pope is usually referred to as "Pope's cohort analysis". It will be discussed separately in Section 5.2.

Practical reviews of VPA methods are, among others, given by Pauly (1984) and Jones (1984).

The easiest way to explain the concepts of VPA is to follow an example based on real data.

Example 18: Virtual population analysis (VPA), North Sea whiting

The data given in Table 4.4.3.1 are of the kind required for virtual population analysis, i.e. the total number caught by age group by the entire commercial fishery. To explain the concepts of VPA, consider the 1974 cohort of whiting, the underlined figures given in Table 4.4.3.1. The annotation used is the same as in Section 4.4.3. C(y,t,t+1) = number caught in year y of age between t and t+1 years (in millions).

The numbers caught (in units of millions) were:

C(1974,0,1) = 599, number caught between age 0 and age 1
C(1975,1,2) = 860, number caught between age 1 and age 2
C(1976,2,3) = 1071, number caught between age 2 and age 3
C(1977,3,4) = 269, number caught between age 3 and age 4
C(1978,4,5) = 69, number caught between age 4 and age 5
C(1979,5,6) = 25, number caught between age 5 and age 6
C(1980,6,7) = 8, number caught between age 6 and age 7

We start our calculations from the bottom, i.e. with the number caught between age 6 and age 7, C(1980,6,7) = 8 million fish. Suppose that we know that the natural mortality was M = 0.2 per year for all age groups. Then if we would know also the fishing mortality for the age group between 6 and 7 years, the so-called 6-group, we could calculate how many fish there must have been in the sea on 1 January 1980, N(1980,6) to account for a catch of 8 million whiting in 1980, by using the catch equation (Eq. 4.2.7):

If we make an initial guess on F(1980, 6, 7) = 0.5 per year, then Z = 0.5 + 0.2 = 0.7. The catch equation then becomes:

thus

Now, knowing that the number of survivors on 1 January, N(1980,6), which is equal to the number at the end of 1979, we can calculate how many whiting there must have been in the sea on 1 January 1979 to account for the catch (in numbers) during 1979: C(1979,5,6) = 25 million.

However, it is not necessary to guess the value of F again, because we can now also calculate which fishing mortality corresponds to this catch. For the calculation of F, Eq. 4.2.7 is used again, but now in conjunction with the exponential decay model (Eq. 4.2.6). (In this example, in both equations the term (t2-t1) = (t+1-t) has been omitted, because it is equal to 1.)

and

N(1980,6) = N(1979,5)*exp(-Z)

which is equivalent to:

N(1979,5) = N(1980,6)*exp(Z)..........(5.1.2)

Inserting the value for N(1980,6) = 22.2 million, calculated above gives:

N(1979,5) = 22.2*exp(Z)

Inserting this result and the number caught, C(1979,5,6) = 25 million into Eq. 5.1.1 gives:

which after multiplication and rearranging is equivalent to:

M is assumed to be 0.2, then inserting Z = F + M = F + 0.2 gives:

We have thus obtained an equation with F as the only unknown variable. Solving it gives an estimate of F. The equation above, however, is not the type of equation which can be solved by algebraic manipulations. It must be solved by some trial and error method. We shall later discuss how this minor technical problem can be circumvented, but for the time being we forget about it and note that F = 0.696 is the solution, i.e.:

From F = 0.696 and M = 0.2 we can derive

Z(1979,5,6) = M + F(1979,5,6) = 0.2 + 0.696 = 0.896

With the estimate of Z(1979,5,6) = 0.896 the number of fish of age group 5 on 1 January 1979 is easily found by means of the decay model (Eq. 5.1.2):

N(1979,5) = N(1980,6)*exp(Z(1979,5,6)) or

N(1979,5) = 22.2*exp(0.896) = 54.4 million

The results of the calculations made so far may be summarized as follows:

age group	year	number caught during year y	fishing mortality during year y	no. surviving on 1 Jan, year y
t	y	C(y,t,t+1)	F(y,t,t+1)	N(y,t)
0	1974	599	­	­
1	1975	860
2	1976	1071	­	­
3	1977	269
4	1978	69	­	­
5	1979	25	0.70	54.4
6	1980	8	0.50 *)	22.2
*) terminal F, assumed value

The next pair N(1978,4) and F(1978,4,5) can be calculated exactly as those for the year 1979. In this way we can work backwards in time estimating numbers of survivors and fishing mortalities for each age group (as indicated by the arrows).

Note that contrary to the catch curve methods, we do not assume F (and Z) to remain constant. Each age group may have a different F-value. This method thus provides a more detailed analysis of the population than any of the other methods presented so far. The two VPA equations derived above read in a general form:

N(y,t) = N(y+1,t+1) * exp[F(y,t,t+1)+M]..........(5.1.4)

For the year 1978 in the example we get:

,

while M=0.2

Inserting these values into Eq. 5.1.3, we get by trial and error:

F(y,t,t+1) = F(1978,4,5) = 0.757

and by inserting this F-value and the number of survivors at 1 January 1979, in Eq. 5.1.4 we get the number of survivors at 1 January 1978:

N(1978,4) = N(1979,5)*exp[F(1979,4,5)+M] = 54.4*exp[0.757+0.2] = 141.9

By repeating this procedure for the years 1977 to 1974 the estimates of fishing mortalities and stock numbers are obtained as shown in Table 5.1.1.

Table 5.1.1 Results of VPA for the 1974 whiting cohort. (Catch data from Table 4.4.3.1, numbers in millions)

age group	year	number caught during year y	fishing mortality during year y	no. surviving on 1 Jan, year y
t	y	C(y,t,t+1)	F(y,t,t+1)	N(y,t)
0	1974	599	0.16	4390
1	1975	860	0.37	3054
2	1976	1071	1.11	1729
3	1977	269	0.99	465
4	1978	69	0.76	142
5	1979	25	0.70	54.4
6	1980	8	0.50 *)	22.2
*) terminal F, assumed value

Fig. 5.1.2 illustrates the cohort dynamics as described by VPA for North Sea whiting (Table 5.1.1). In this case M is relatively small compared to F, which can be seen by comparing the number caught (Eq. 4.2.8 with t2-t1 = 1):

C(y,t,t+1) = F(y,t,t+1) * (y,t,t+1)..........(5.1.5)

to the number of natural deaths:

D(y,t,t+1) = M(y,t,t+1) * (y,t,t+1)..........(5.1.6)

Fig. 5.1.2 presents the results as the number of survivors, N, but we could as well have chosen the fishing mortality, F, as the basic result, because:

F and C determine N and
N and C determine F

so that there is a one-to-one correspondence between N and F when C and M are known.

Fig. 5.1.2 Number of survivors, N (black bars), number caught C = F* and number of natural deaths, D = M*, as derived by VPA in Table 5.1.1 for North Sea whiting

VPA using a plus-group

In Example 18 we started with the calculation of N(1980,6), but we did not say anything about the fish older than 6 years. This approach is correct, because it is not necessary to account for the older age groups. However, there may be many fish older than 6 years and we may want to account for all these older fish, which are difficult to separate into age groups, by combining them in one so-called plus-group. If we account for these older fish, the formulas must be modified accordingly.

Let us return to the example and replace C(1980,6,7) by the plus-group:

C(1980,6+) = C(1980,6,7) + C(1981,7,8) + C(1982,8,9) + ....

where the sum contains all the non-zero numbers of old survivors, or the catch in numbers of all fish of 6 years and older.

We further assume that the fishing mortalities are the same for all components of the plus-group:

F(1980,6+) = F(1980,6,7) = F(1981,7,8) = F(1982,8,9) = ....

The catch equation for the oldest age group (see Example 18):

C(1980,6,7) = N(1980,6)*(F/Z)*(1-exp(-Z(7-6)))

should now be replaced by

C(1980,6+) = N(1980,6)*(F/Z)*(1-exp(-Z(¥ -6)))

and because [1-exp(-Z(¥ -6)) ] = 1 - 0 = 1 the last term disappears, so that

C(1980,6+) = N(1980,6)*(F/Z)

Thus in general, in cases where the first observation is a plus-group, the VPA is started by:

C(y,t+) = N(y,t)*F(y,t+)/Z(y,t+)..........(5.1.7)

Theoretically, the results should be the same whether or not the last age group is a plus-group.

The biomass concept

The biomass concept associated with Table 5.1.1 is rather straightforward when we consider the weight of the cohort at a particular time. For example, the weight of the cohort in year 1979 is N(1979,5)*w(5), where w(5) is the body weight of a whiting of 5 years old. A biomass concept which reflects the cohort during its entire life span is more difficult to grasp.

The average biomass on 1 of January during the first 6 years of life of the cohort is:

The average annual biomass could be defined as:

where is defined by Eq. 4.2.9 with t2-t1 = 1 and is the average annual body weight. The two biomass concepts are different, and it is not obvious how they should be used. The same sort of problem emerges when trying to define the average number of survivors. We shall come back to the biomass concept in the following sections.

Basic features of VPA

From observations on the numbers caught in each age group the VPA estimates how many fish there must have been in the sea to account for that catch, under the assumption that natural mortality is known (see Fig. 5.1.1). If the catch constitutes a small fraction of the stock (i.e. if F is small) the estimation of the stock size becomes more uncertain. Thus, the higher the fishing mortality the more dependable is the VPA.

Natural mortality, M, is assumed to be known from investigations independent of the VPA, but is actually unknown in most cases. The reliability of VPA is also dependent on the size of M relative to F. Often the estimate of M is rather a "guesstimate" (qualified guess), but if M is small compared to F it may not matter so much that M is not well estimated. What a "guesstimate of M" means was discussed in Section 4.7.

A set of equations can only have a unique solution, when the number of equations equals the number of unknown variables. If there are more unknown variables than equations, there will be infinitely many solutions.

The whole set of VPA equations consists of pairs of Eqs. 5.1.3 and 5.1.4 for each age group. There are apparently three unknown variables in each set of two equations, viz. N(y,t), N(y+1,t+1) and F(y,t,t+1). However, in all cases except in the first set with the oldest age group, N(y+1,t+1) is known from the solution of the preceding set of equations and we do end up with two unknown variables in two equations and therefore with a unique solution.

The problem with the first set of equations pertaining to the oldest age group can be solved by making a plausible assumption and formulating it as an additional equation. We can then obtain a solution that is conditioned on this assumption. The solution in the case of a VPA is to assume a value for the P of the oldest age group, the so-called "terminal F".

For example, we might assume that the terminal F is equal to the F of the second oldest age group, so the additional equation would then be:

F7 = F6 (assuming that 7 is the oldest age group)

We then have four equations, two sets of Eqs. 5.1.3 and 5.1.4, with four unknown variables, viz. F7, N7, N6 and N5.

If there are more equations than unknown variables there is (usually) no solution. In that case we use regression analysis to find the best "fit" to the data to find a solution and we may calculate confidence limits. In regression analysis the concept of "unknown variables" is replaced by the concept of "parameters".

To calculate confidence intervals for estimates of parameters the number of observations must be larger than the number of parameters. The number of parameters in VPA (the N's and the F for the oldest age group) equals the number of observations (the C's) plus one. Therefore it is not possible to calculate confidence limits for the estimates of the N's (or the F's of the other age groups).

The data used in Example 18 to illustrate VPA were obtained from direct age reading (otoliths). However, input data could have been derived from time series of length-frequency data which were resolved into cohort components by for example, the Bhattacharya method (Section 3.4). This aspect is further discussed at the end of Section 5.3.

The VPA is a method to analyse historical data for estimation of population parameters. The ultimate use of such parameters is to determine the optimum fishing strategy, i.e. the array of F-at-age, or the so-called "fishing pattern", which in the long term gives the highest yield from the stock in question. To assess alternative (future) fishing strategies we require a counterpart to the VPA, namely a model which can predict the stock and the catch for various assumptions on the future fishing pattern. The "Thompson and Bell model" (Section 8.6) is the predictive version of the VPA.

Computer programs

Mesnil (1988) presents a package of microcomputer programs, ANACO, ("ANAlysis of Cohorts") which can perform the VPA calculations as described above. The ANACO package also offers a number of additional options, for example sensitivity analysis.

5.2 AGE-BASED COHORT ANALYSIS (POPE'S COHORT ANALYSIS)

As derived from the catch equation, the VPA implied the solution of Eq. 5.1.3 by some numerical techniques (some trial and error method). This is a minor technical problem when one has access to a computer. However, the problem can be circumvented in an easy way, so that VPA can also be carried out on a pocket calculator. The version of VPA suitable for pocket calculators is the "cohort analysis" developed by Pope (1972), reviewed in Jones (1984) and Pauly (1984).

Cohort analysis is conceptually identical to VPA, but the calculation technique is simpler. It is based on an approximation, illustrated in Fig. 5.2.1, which shows the number of survivors of a cohort during one year. The catch is taken continuously during the year, but in cohort analysis the assumption is made that all fish are caught on one single day. This day is chosen to be 1 July, i.e. when one half of the year has elapsed.

Consequently in the first half year the fish suffer only natural mortality so the number of survivors on 1 July becomes:

N(y, t+0.5) = N(y,t)*exp(-M/2)

Then, instantaneously, the catch is taken and the number of survivors becomes:

N(y,t)*exp(-M/2) - C(y,t,t+1)

Fig. 5.2.1 Illustration of the approximation behind Pope's cohort analysis (for further explanation, see text)

This number of survivors then suffers further only natural mortality in the second half year and finally the number of survivors at the end of the year is:

N(y+1,t+1) = [N(y,t)*exp(-M/2) - C(y,t,t+1)]*exp(-M/2)

For convenience of calculation this equation is rearranged as:

N(y,t) = [N(y+1,t+1)*exp(M/2) + C(y,t,t+1)]*exp(M/2)..........(5.2.1)

Note that the F that caused computational problems in the VPA equation does not occur here. Again we shall demonstrate the method on the basis of the same data set, the North Sea whiting cohort of 1974.

Example 19: Pope's cohort analysis, North Sea whiting

To apply Eq. 5.2.1 to the whiting example we start the same way as for the VPA by assuming the F for the oldest age group (the so-called "terminal F") to be known, F(1980,6,7) = 0.5, while M = 0.2, and then start by calculating N(1980,6) from the catch equation:

Then Eq. 5.2.1 is applied to calculate N(1979,5):

N(1979,5) = [N(1980,6)*exp(M/2) + C(1979,5,6)]*exp(M/2) = [22.2*1.1052 + 25]*1.1052 = 54.7

and we continue in the same way:

N(1978,4) = [N(1979,5)*exp(M/2) + C(1978,4,5)]*exp(M/2) = [54.7*1.1052 + 69]*1.1052 = 143.1

N(1977,3) = [143.1*1.1052 + 269]*1.1052 = 472.1

..... etc. ....

Thus, as for the VPA we work backwards in time estimating a new stock number at each step. The calculation procedure for the stock numbers, the N's, is given in the first half of Table 5.2.1 (with a less extensive notation).

From the estimates of the N's the estimate of fishing mortalities are obtained from:

Table 5.2.1 The computation procedure of Pope's age-based cohort analysis, illustrated by the 1974 cohort of North Sea whiting (from Table 4.4.3.1)

POPE'S AGE-BASED COHORT ANALYSIS M = 0.2 per year NATURAL MORTALITY FACTOR: G = exp(M/2) = exp(0.2/2) = 1.1052 GUESS ON TERMINAL F: F6 = 0.5 STOCK NUMBERS: N(1979,5) = N5 = (N6*G + C5,6)*G = N5 = (N6*G + C5)*G = (22.2*G + 25)*G = 54.7 N4 = (N5*G + C4)*G = (54.7*G + 69)*G = 143.1 N3 = (N4*G + C3)*G = (143.1*G + 269)*G = 472.1 N2 = (N3*G + C2)*G = (472.1*G + 1071)*G = 1760.3 N1 = (N2*G + C1)*G = (1760.3*G + 860)*G = 3100.4 N0 = (N1*G + C0)*G = (3100.4*G + 599)*G = 4448.9 FISHING MORTALITIES: F6 = (initial guess of terminal F) = 0.50 F5 = ln(N5/N6) - M = ln(54.7/22.2) - 0.2 = 0.70 F4 = ln(N4/N5) - M = ln(143.1/54.7) - 0.2 = 0.76 F3 = ln(N3/N4) - M = ln(472.1/143.1) - 0.2 = 0.99 F2 = ln(N2/N3) - M = ln(1760.3/472.1) - 0.2 = 1.12 F1 = ln(N1/N2) - M = ln(3100.4/1760.3) - 0.2 = 0.37 F0 = ln(N0/N1) - M = ln(4448.9/3100.4) - 0.2 = 0.16

which follows from the exponential decay model solved for F:

N(y+1,t+1) = N(y,t)*exp[-F(y,t,t+1) - M]

The estimated F's are presented in the second half of Table 5.2.1.

Comparing the VPA-results (Table 5.1.1) to the results of cohort analysis (Table 5.2.1) it appears that they differ, but the differences are small. Taking into consideration all the sources of uncertainty involved in this kind of calculation, one can say that the differences between VPA and cohort analysis are insignificant. Pope (1972) has shown that for values of F < 1.2 and M < 0.3 the differences will be small. The only advantage of the cohort analysis compared to VPA is that it can easily be carried out with a pocket calculator. To do the VPA on a pocket calculator requires the tedious trial and error way of calculating F (unless the calculator is fully programmable).

Example 19a: Cohort analysis for successive year classes

The interpretation of fishing mortalities estimated for one year class during its lifetime (Tables 5.1.1 and 5.2.1) is not clear. The changes with time and age may be due to differences of effort between ages or between calendar years. The sampling, however, always provides data for other year classes (see Table 4.4.3.1) which can be treated in the same way. The results for F and N are given in Table 5.2.2. Year classes 1969-80 are represented in the data by one or several age groups. As terminal F was chosen 0.5 for all year classes in which fish of age group 6 were sampled. The values of F for each age of year class 1974 estimated above were inserted as terminal F's for the year classes which did not yet reach age 6 in 1980. The mean values of F for each age group (last column) differ somewhat from the terminal F values used. The analysis might be repeated with the averages as final F values.

Table 5.2.2 North Sea whiting. Estimation of fishing mortalities and population numbers for all ages and year classes in the data material from which the data for the 1974 year class used above was selected. Estimates of F for year class 1974 (underlined) are adopted as terminal F values for all cohorts that have not yet reached age 6 in 1980

Another possibility is to make use of the average catch by age group (Table 4.4.3.1, last column). Such data form a "pseudo-cohort": it represents the average history of a cohort instead of the history of a specific cohort of fishes all spawned at the same time. Table 5.2.3 shows the result. The estimates of fishing mortalities for the older fish are above the averages of the individually estimated values of F in Table 5.2.2. This could be due to a change in effort or in average mesh size over the years investigated. The oldest fish of 7 years and older (the 7+ group) are included in Table 5.2.3 because the erratic catches of these in individual years are now averaged. The number of fish entering the plus group at 7 years of age is estimated from

N(7) = C(7+) * Z(7+)/F(7+)

which is another formulation of the equation

C(y,t+) = N(y,t)*F(y,t+)/Z(y,t+)

of Section 5.1. F(7+) is the guessed final F value.

The use of pseudocohorts is important for the length-based cohort analysis of Section 5.3.

Table 5.2.3 Analysis of a pseudo-cohort: the average catch of each age group in 1974-80. North Sea whiting. Data from Table 4.4.3.1

age	average catch 1974-80	No. in population at beginning of year	F
0	488	3059	0.19
1	612	2063	0.40
2	601	1135	0.88
3	237	385	1.14
4	62.3	101	1.13
5	15.7	26.6	1.05
6	4.7	7.6	1.15
7+	1.4	2.0	0.50

The catch data of Table 4.4.3.1 are not the original observations. These were the total catch by weight, the length frequency distributions of weighed samples and otolith readings of subsamples of these. Age/length keys (Section 3.2.1) were used to raise the otolith readings to the numbers in the length frequency samples. Based on sample weights and the weight of the total catch these were raised to give the total catch in numbers by age group.

There are many sources of error in such a process. It is therefore important to check the realism of the estimated length frequency distribution of the total catch. This is done by multiplying the numbers-at-age by the weight of the individual fish in each age group as calculated from known parameter values of the growth equation or from mean weights of the sampled fish of each age group. The total weights of each age group are summed and the result compared to the known value of the total catch in weight. A major discrepancy should cause a re-examination of data and procedures.

The biomass in each calendar year is calculated by multiplying the numbers in the population (the N values of Table 5.1.1) by the body weight of the fish.

Fig. 5.2.2 Derivation of the formulas for Pope's cohort analysis in the case of variable time intervals (Eqs. 5.2.3 and 5.2.4)

Eq. 5.2.1 was derived for a time period of one year. As was the case with the catch curve (cf. Section 4.4.4) we may consider the catch during any time period, from t to t+D t. In that case Eq. 5.2.1 should be replaced by the more general expression:

N(t) = [N(t+D t)*exp(M*D t/2) + C(t,t+D t)]*exp(M*D t/2)..........(5.2.3)

The derivation of Eq. 5.2.3 is similar to that of Eq. 5.2.1 and is shown in Fig. 5.2.2. The parallel to Eq. 5.2.2 is:

The year index, y, has disappeared in Eqs. 5.2.3 and 5.2.4, the main reason being that they usually are applied under the assumption of a constant parameter system (cf. Section 4.4.4). Further, when D t varies it will not conform to the year intervals and the notation used for age groups is not suitable any more.

Eqs. 5.2.3 and 5.2.4 applied under the assumption of constant parameters are typical applications of cohort analysis for tropical fish stocks. They could be applied to all cohorts during one year or to the average annual catch during a sequence of years. We shall return to this in Section 5.3.

(See Exercise(s) in Part 2.)

5.3 JONES' LENGTH-BASED COHORT ANALYSIS

In this section we deal with the situation when only length composition data for the total fishery are available for one year (or the average length composition for a sequence of years). The approach is basically the same as for the length-converted catch curve (see Section 4.4.5). The name "length-based cohort analysis" is somewhat misleading, as we are not dealing with real cohorts in the present analysis. The real cohort is replaced by a "pseudo-cohort" which is based on the assumption of a constant parameter system (see Section 4.4.1). Thus, it is assumed that the picture presented by all length (or age) classes caught during one year reflects that of a single cohort during its entire life span. We shall come back to this aspect later on. Also this method will be explained based on an example.

Example 20: Jones' length-based cohort analysis, hake, Senegal

Table 5.3.1 shows a data set for the hake fishery off Senegal (CECAF, 1978), which can be used as input for a length-based cohort analysis.

Table 5.3.1 Length composition of the total catch of hake (Merluccius merluccius) off Senegal (from CECAF, 1978), input data for length-based cohort analysis

length group cm	number caught ('000)
L1-L2	C(L1-L2)
6-12	1823
12-18	14463
18-24	25227
24-30	8134
30-36	3889
36-42	2959
42-48	1871
48-54	653
54-60	322
60-66	228
66-72	181
72-78	96
78-84	16
84-¥	46

As was the case for the catch curve analysis (cf. Section 4.4.5) the length groups can be converted into age intervals by the inverse von Bertalanffy equation (Eq. 3.3.3.2 and Eq. 4.4.5.1 respectively):

For the hake off Senegal (Table 5.3.1) the von Bertalanffy growth parameters and the natural mortality factor have been estimated as:

K = 0.1 per year, L_¥ = 130 cm and M = 0.28 per year.

Putting t₀ = 0 (compare Section 4.4.5) and applying Eqs. 5.3.1 and 5.3.2 gives the relative ages, t(L1) and D t as shown in Table 5.3.2, columns B and C respectively.

To convert the cohort analysis equation (Eq. 5.2.3) into a length-based version, only the term exp(M*D t/2) has to be changed. This is easily done by substituting D t with Eq. 5.3.2.

It is convenient to use a symbol instead of this complicated term, therefore we introduce the symbols:

N(L1) = N(t(L1)) = the number of fish that attain length L1 = the number of fish that attain age t(L1) (also called the number of survivors)

N(L2) = N(t(L1)+D t) = the number of fish that attain length L2 = the number of fish that attain age t(L2) (= t(L1)+D t)

C(L1,L2) = C(t,t+D t) = the number of fish caught of lengths between L1 and L2 = the number of fish caught of ages between t(L1) and t(L2)

N(L1) = [N(L2)*H(L1,L2) + C(L1,L2)]*H(L1,L2)..........(5.3.4)

The calculation procedure of Eq. 5.3.4 is similar to that of the age-based cohort analysis (Eq. 5.2.1). We start with the last group and use the length-based form of the catch equation:

In the case of the hake off Senegal the last group is the hake of length 84 cm and longer:

In this case D t refers to all fish longer than 84 cm, so that D t is very large. Theoretically, the age corresponding to L_¥ is ¥, so theoretically D t = ¥, and therefore:

exp(-Z*¥) = 0

The catch in numbers per length group is known (see Table 5.3.1), thus, approximately:

or

Here again it is necessary to make an initial guess, viz. of F/Z in this case (cf. Table 5.2.1). If F/Z for the last length group is assumed to take the value 0.5 then the number of hake attaining length 84 cm becomes:

N(84) = 46/0.5 = 92

Column D of Table 5.3.2 gives the values of H(L1,L2) for hake off Senegal with M = 0.28 per year and K = 0.1 per year, i.e. M/2K = 0.28/(2*0.1) = 1.4.

The number of hake attaining length 78 cm can be obtained by inserting into Eq. 5.3.4 as N(L2) the value of N(L1) obtained for the larger length group and further the corresponding value for H from Table 5.3.2. In this case N(L2) = N(84) = 92 and H(78,84) = 1.1873.

N(78) = [92*1.1873 + 16]*1.1873 = 148.7

Table 5.3.2 Length groups of the hake (Merluccius merluccius) off Senegal, converted into age intervals, and the factors H(L1,L2). K = 0.1 per year, L_¥ = 130 cm, M = 0.28 per year

A	B	C	D
length group cm L1 - L2	relative age t(L1)	D t	natural mortality factor H(L1 - L2)
6-12	0.473	0.496	1.0719
12-18	0.968	0.522	1.0758
18-24	1.490	0.551	1.0801
24-30	2.041	0.583	1.0850
30-36	2.624	0.619	1.0905
36-42	3.242	0.660	1.0967
42-48	3.902	0.706	1.1039
48-54	4.608	0.760	1.1122
54-60	5.368	0.822	1.1220
60-66	6.190	0.890	1.1337
66-72	7.087	0.984	1.1478
72-78	8.071	1.092	1.1652
78-84	9.163	1.226	1.1873
84-¥	10.389	-	-
column	contents
B	(Eq. 5.3.1, with t0 = 0)
C	(Eq. 5.3.2)
D	(Eq. 5.3.3)
Note: In this case H(L1,L2) can be calculated on the basis of either the age-based or the length-based formula. The results are the same, but the length-based approach is much shorter, because it is not necessary to calculate D t.

Continuing to move backwards in length (and thus in time) the subsequent stock numbers are calculated:

N(72) = [148.7*1.1652 + 96]*1.1652 = 313.7
N(66) = [313.7*1.1478 + 181]*1.1478 = 621.0
.... etc. ....

The stock numbers for all length groups obtained in this way are given in column D of Table 5.3.3.

Table 5.3.3 The calculation procedure of Jones' length cohort analysis illustrated by the hake (Merluccius merluccius) off Senegal.

K = 0.1 per year, L_¥ = 130 cm, M = 0.28 per year.
Terminal F/Z assumed to be 0.5000 (indicated by *)

A	B	C	D	E	F	G
length group cm	natural mortality factor	number caught ('000)	number of survivors ('000)	exploitation rate	fishing mortality	total mortality
L1 - L2	H(L1,L2)	C(L1,L2)	N(L1)	F/Z	F	Z
6-12	1.0719	1823	98919.3	0.1255	0.04	0.32
12-18	1.0758	14463	84392.7	0.5805	0.39	0.67
18-24	1.0801	25227	59475.8	0.7920	1.07	1.35
24-30	1.0850	8134	27623.0	0.6979	0.65	0.93
30-36	1.0905	3889	15967.8	0.6369	0.49	0.77
36-42	1.0967	2959	9861.5	0.6785	0.59	0.87
42-48	1.1039	1871	5500.5	0.6977	0.65	0.93
48-54	1.1122	653	2818.8	0.5792	0.39	0.67
54-60	1.1220	322	1691.5	0.5072	0.29	0.57
60-66	1.1337	228	1056.6	0.5234	0.31	0.59
66-72	1.1478	181	621.0	0.5890	0.40	0.68
72-78	1.1652	96	313.7	0.5817	0.39	0.67
78-84	1.1873	16	148.7	0.2823	0.11	0.39
84-¥	-	46	92.0**	0.5000*	0.28	0.56
column	contents
B
D	N(L1) = [N(L2)*H(L1,L2) + C(L1,L2)]*H(L1,L2)
**	N(84) = C(84,¥)/(F/Z) = 46/0.5 = 92
E	F/Z = C(L1,L2)/[N(L1)-N(L2)]
F	F = M*(F/Z)/(1-F/Z)
G	Z = F + M

To estimate F we could use Eq. 5.2.4, but it is more convenient to calculate F by:

where the exploitation rate F/Z is derived from:

For example, writing for simplicity F/Z for F(L1,L2)/Z(L1,L2), for the length group 72-78 cm we get:

and

The complete results of this calculation procedure for length cohort analysis are given in Table 5.3.3.

Mean number and biomass

We now want to calculate the mean number of fish in the sea and their biomass. Summing the column of N(L1) values in Table 5.3.3 would not give the right number because changing the class interval would give a different sum: The N(L1) values are simply the number alive at any length L1. The procedure is to find the mean number in each class interval and weight it by the time, D t, spent in that class interval. The same problem was dealt with by Eqs. 4.2.6 and 4.2.9. Some manipulation of either of these equations leads to the result:

(L1,L2)*D t = [N(L1)-N(L2)]/Z..........(5.3.8)

which is the annual mean number in each length class.

N(L2) = N(L_¥ ) = 0 may be assumed for the last group. The total mean number of fish in the sea of lengths above the first L1 (here 6 cm) becomes in general:

Correspondingly, we find the annual mean biomass in each length group by multiplying by the mean weight, (L1,L2), in the length group:

The body weight is calculated from

(L1,L2) = q*[(L1+L2)/2]^b..........(5.3.11)

where q and b are the constants of the length-weight relationship described in Section 2.6. The body weight of the last group may be calculated as (L1,L_¥ ) or better by Eq. 5.3.16.

The general sum

is an estimate of the average biomass during the life span of a cohort, or of all cohorts during a year, and is independent of the length class interval.

The body weight may also be used to estimate the yield, i.e. the weight of the catch. The weight of the catch belonging to length group i becomes

Table 5.3.4 shows the calculation of the yield and the average biomass during a year.

Table 5.3.4 The calculation procedure for yield and average biomass in Jones' length-based cohort analysis illustrated by the hake (Merluccius merluccius) off Senegal. q = 0.00001 kg/cm³, b = 3, K = 0.1 per year, L_¥ = 130 cm, M = 0.28 per year

A	B	C	D	E	F	G	H
length group cm	number caught ('000)	number of survivors ('000)	total mortality rate	mean body weight kg	mean N*D t ('000)	mean biomass*D t tonnes	yield tonnes
L1 - L2	C	N(L1)	Z	(L1,L2)	(L1,L2)*D t	*D t	Y(L1,L2)
6-12	1823	98919.3	0.32	0.0073	45369	330.7	13.3
12-18	14463	84392.7	0.67	0.0338	37335	1260.1	488.1
18-24	25227	59475.8	1.35	0.0926	23664	2191.5	2336.3
24-30	8134	27623.0	0.93	0.196	12575	2475.1	1601.0
30-36	3889	15967.8	0.77	0.359	7919	2845.9	1397.6
36-42	2959	9861.5	0.87	0.593	5007	2970.1	1755.3
42-48	1871	5500.5	0.93	0.911	2895	2638.1	1704.9
48-54	653	2818.8	0.67	1.33	1694	2247.1	866.2
54-60	322	1691.5	0.57	1.85	1117	2068.6	596.3
60-66	228	1056.6	0.59	2.50	741	1852.8	570.1
66-72	181	621.0	0.68	3.29	451.1	1481.9	594.6
72-78	96	313.7	0.67	4.22	246.5	1039.9	405.0
78-84	16	148.7	0.39	5.31	144.9	770.1	85.0
84-00	46	92.0	0.56	12.25	164.3*	2012.7	563.5
					Total	26184.6	12977.2
column	contents
E	(L1,L2) = q*[(L1+L2)/2]b
F	(L1,L2)*D t = [N(L1)-N(L2)]/Z
G	*D t = (L1,L2)*[N(L1)-N(L2)]/Z
H	Y(L1,L2) = (L1,L2) * C(L1,L2)
*	(L1,¥)*D t = N(84)/Z(84,¥))

Cohort analysis with several fleets

VPA and cohort analysis, age-based or length-based serve to estimate separately the fishing mortalities created by each of several fleets. A fleet in this context is a group of fishing vessels characterized by for instance their gear or nationality etc. Separate catch statistics for each fleet are required as well as separate sampling and raising of length frequency distributions.

We have the catch equation, Eq. 5.1.5:

C = F*

with the annual mean number in the population of the age or size group specified. The indices of age or length e.g. N(t), N(L) are omitted for convenience. Partitioning the catch upon fleets gives

C(1) = F(1)*
C(2) = F(2)*
:
:
:
C(n) = F(n)* ..........(5.3.13a)

C = C (1) +C (2) + ... +C (n) ; F = F(1)+F(2)+....+F(n)

Dividing Eq. 5.1.5 by Eq. 5.3.13a and rearranging gives the fishing mortalities created by fleet i:

F(i) = F*C(i)/C..........(5.3.13b)

Table 5.3.5 illustrates this simple procedure applied to a constructed example of a length-based cohort analysis with a trawl fleet and a gill net fleet each using one mesh size only. The array of total F values for both fleets combined is not easily interpreted. Separation into trawl mortality, F(1), and gill net mortality, F(2), discloses that F(1) increases toward an asymptote indicating a gear selection curve such as in Fig. 4.5.3.1 whereas the gill net array is bell-shaped.

Basic features of length-based cohort analysis

The length cohort analysis defined by Eqs. 5.3.3 to 5.3.7 (see Table 5.3.3) is called "Jones' length-based cohort analysis". (Jones, 1976, and Jones and van Zalinge, 1981, reviewed in Jones, 1984 and Pauly, 1984). As already mentioned, the method is usually applied to "pseudo-cohorts", i.e. we assume a constant parameter (equilibrium) system. In order to simulate an equilibrium condition it is essential that the data pertain to a relatively long time period (say a year or several years), preferably a number of full years.

A length-frequency sample collected during a relatively short time period is not applicable. The September sample of shrimps shown in Fig. 3.4.2.6 consisting of only one cohort is such an example. The method assumes that a larger specimen is also older, but in this case we assumed that all shrimps, irrespective of length, are of the same age. The descending slope of that sample has something to do with the variation in individual growth rates - it is not related to mortality.

It is possible to apply Jones' length-based cohort analysis method to a real cohort, but that implies that we are able to follow a cohort through time, i.e. that we know its age. If that is the case, however, we might as well use Pope's age-based cohort analysis, as this does not present problems in connection with the conversion of length into age.

Table 5.3.5 Length-based cohort analysis with two fleets. M = 1, L_¥ = 25 cm, K = 0.4. Guessed value of terminal F = 1.4

As Jones' length-based cohort analysis is based on Pope's age-based cohort analysis (Section 5.2) it has the same limitations. The approximation to VPA is valid for values of F*D t up to 1.2 and of M*D t up to 0.3 (Pope, 1972).

The use of a length-based VPA without the approximation of the cohort analysis requires no mere than a slight change in the formulation of Eq. 5.1.3 and Eq. 5.1.4. First, the time interval of one year is changed to D t:

N(t) = N(t+D t)*exp[(F(t,t+D t)+M)*D t]

Let age t correspond to L1 and age t+D t to L2:

where, using Eq. 4.4.5.1:

N(L1) = N(L2)*exp[F(L1,L2) + M)*D t(L1,L2)]..........(5.3.15)

With VPA as well as with cohort analysis the best estimate of the body weight in the plus group is derived from Eq. 8.3.6:

in which

Z = F(L1,¥) + M and S = 1 - L1/L_¥

The mean weight depends on Z because of the effect of mortality upon the size frequency distribution of the stock: the higher mortality the smaller fish. This is of little consequence when small length classes are used. Hence the simple approximative formula of Eq. 5.3.11.

Computer programs

The program "LCOHOR" in the LFSA package of microcomputer programs (Sparre, 1987) can execute Jones' length-based cohort analysis as described above. Also the "COMPLEAT ELEFAN" package (Gayanilo, Soriano and Pauly, 1988) and FiSAT (Gayanilo et al., 1995) contain routines for a length-based analysis similar to LCOHOR.

(See Exercise(s) in Part 2.)

5.4 THE SLICING TECHNIQUE

With the age-based methods of Sections 5.1 and 5.2 a cohort is easily followed year after year through its lifetime such as illustrated in Table 4.4.1.1. This is not the case for the length-based methods because the time spent in a length group is not the same for all length groups. The length-based cohort analysis and VPA handle this problem by analysing a pseudocohort: an average length frequency distribution for a period in which neither effort nor mesh size changed appreciably. The period has to be so long that all humps on the length distribution curve are smoothed, those caused by seasonal recruitment as well as those caused by annual variation of the number of recruits.

When length frequency data are sampled every year, it is desirable to be able to treat each year's data separately, identifying recruitment variation and changes in fishing mortality. To achieve this the fish must be referred to age groups of the same time span, usually one year, when there is annual spawning. The length of the fish at one year of age, two years of age, etc., is calculated from the growth parameters. Fish smaller than the length at one year of age are referred to the 0-group, those of lengths between the length at one year and the length at two years to the 1-group, and so on. Some length classes will have to be distributed proportionally upon two age groups. If, for instance, the length class interval is 1 cm and one-year-old fish are 12.6 cm long, then six-tenths of the fish in the 12-13 cm class are referred to age group 0 and four-tenths to age group 1. If the length interval is 6 cm with a length class of 12-18 cm, then only a fraction of 0.6/6 = 0.1 of the fish goes to age group zero, whereas nine-tenths end up in age group 1.

This technique which is known as "slicing" exists in a number of variations, the respective merits of which are not yet clear. There are two major problems to be considered. One is that the true age of the fish is usually not known. Length-based cohort analysis uses an arbitrary value of t₀ (often zero) because ages proper are not used, only the age differences, D t. This eliminates t₀. With the slicing technique, the choice of t₀ affects the distribution by age groups and the recruitment. This is a minor problem when a pseudocohort is analysed. When annual samples over a series of years are being sliced it is important that all fish of one cohort (year class) go into the same age group: age group 0 in one year becomes age group 1 in the next year, etc., and that such cohorts are not distributed over two age groups by the slicing. Therefore, the value of t₀ should be chosen such that a hump on the size distribution curve caused by a strong year class goes as far as possible into just one age group each year.

Even with a careful choice of t₀ some fish end up in a wrong age group because of the size variation within a cohort, such as illustrated for instance in Fig. 1.4.1. The problem may perhaps be partly overcome by making assumptions on the standard deviation of fish length within a cohort, as estimated for the younger fish with the Bhattacharya method, Section 3.4. Because of these problems it is difficult to say to which extent slicing is going to replace the length-based cohort analysis.

Example 20a: Slicing technique applied to the same data on hake from Senegal as used in Example 20

Tables 5.4.1 and 5.4.2 show a simple example of an ordinary, age-based cohort analysis following a slicing process. The pseudocohort of hake already utilized as an example of length-based cohort analysis (see Table 5.3.1) was used again. In the length-based analysis the recruitment was defined as the number of fish entering the first length group at 6 cm long and the choice of t₀ was without consequence. It was put to zero. However, using t₀ = 0 in the slicing technique leads eventually to an estimate of recruitment at a nominal fish length and age of zero which is not very useful. The estimated age at 6 cm (Table 5.3.2) was 0.473 years. Putting t₀ = -0.473 years allots the age zero to fish of 6 cm. Thus, the age-based cohort analysis which follows the slicing also gives the recruitment of fish 6 cm long. The technique of distributing the catch over age groups by means of slicing is illustrated in Table 5.4.1 for the first three age groups. The ensuing age-based cohort analysis for all age groups is given in Table 5.4.2. The recruitment at 6 cm, N(0), is now estimated to be 99.8 million as against 98.9 million with the length-based cohort analysis: a fair agreement. The fishing mortalities estimated by the two methods are compared in Fig. 5.4.1. The results are similar except for the oldest fish which are few in the catch.

Table 5.4.1 Example of "slicing". The catch numbers ('000) in the first rows of Table 5.3.1 are referred to age groups. Cf. the application of the results in Table 5.4.2

age	length at age	length class	fraction in lower group	total catch	catch by age group
t	L(t)1)	L1-L2		C(L1-L2)	0	1	2
0	6.00	6-12	1.0000	1823	1823
1	17.81	12-18	(17.81-12)/6 = 0.9683	14463	14005	458
		18-24	1.0000	25227		25227
2	28.48	24-30	(28.48-24)/6 = 0.7467	8134		6073	2061
		30-36	1.0000	3889			3889
3	38.14	36-42	(38.14-36)/6 = 0.3567	2559			1055
Total catch referred to age groups 0, 1 and 2 respectively					15828	31758	7005
1) L(t) = 130*[1-exp(-0.1*(t+0.473))]

Table 5.4.2 Age-based cohort analysis following a slicing procedure. The pseudocohort of Senegal hake (Table 5.3.1). M = 0.28

nominal age group	catch ('000)	nos. at beginning of year ('000)	fishing mortality
t	C(t,t+1)	N(t)	F(t,t+1)
0	15828	99818	0.20
1	31758	61680	0.91
2	7005	19008	0.55
3	3426	8276	0.65
4	1044	3276	0.46
5	354	1569	0.30
6	227	878	0.35
7	144	466	0.44
8	64	227	0.40
9+	58	1161)	0.28
1) N(9) = C(9+)*Z(9+)/F(9+) (Eq. 5.1.7)

Fig. 5.4.1 Comparison of estimates of F as a function of age from length-based cohort analysis (broken line) and the slicing technique (drawn line). Senegal hake (see Tables 5.3.2, 5.3.3 and 5.4.1)