9.1 THE SCHAEFER AND FOX MODELS
9.2 GULLAND'S FORMULA
9.3 CADIMA'S FORMULA
9.4 MSY ESTIMATORS BASED ON THE SURPLUS PRODUCTION MODEL
9.5 MUNRO AND THOMPSON PLOT
9.6 STANDARDIZATION OF EFFORT
9.7 THE DERISO/SCHNUTE DELAY DIFFERENCE MODEL
In contrast to Chapters 3 to 8, Chapters 9 and 13 do not deal with "analytical models", but with "holistic models" (cf. Fig. 1.8.1), wherein the stock is considered as one big unit of biomass and wherein no attempt is made to model on an age or length base. The "surplus production models" which are discussed in this chapter deal with the entire stock, the entire fishing effort and the total yield obtained from the stock, without entering into any details such as the growth and mortality parameters or the effect of the mesh size on the age of fish capture etc. Surplus production models were introduced by Graham (1935), but they are often referred to as "Schaefermodels" (see below).
The objective of the application of "surplus production models" is to determine the optimum level of effort, that is the effort that produces the maximum yield that can be sustained without affecting me longterm productivity of the stock, the socalled maximum sustainable yield (MSY). The theory behind the surplus production models has been reviewed by many authors, for example, Ricker (1975), Caddy (1980), Gulland (1983) and Pauly (1984).
Because holistic models are much simpler than analytical models, the data requirements are also less demanding. There is, for example, no need to determine cohorts and therefore no need for age determination. This is one of the main reasons for the relative popularity of surplus production models in tropical fish stock assessment. Surplus production models can be applied when data are available on the yield (by species) and of the effort expended over a certain number of years. The fishing effort must have undergone substantial changes over the period covered.
The maximum sustainable yield (MSY) can be estimated from the following input data:
f(i) = effort in year i, i = 1,2,...,nY/f = yield (catch in weight) per unit of effort in year i.
Y/f may be derived from the yield, Y(i), of year i for the entire fishery and the corresponding effort, f(i), by
Y/f = Y(i)/f(i), i = 1,2,...,n..........(9.1.1)
The simplest way of expressing yield per unit of effort, Y/f, as a function of the effort, f, is the linear model suggested by Schaefer (1954):
Y(i)/f(i) = a + b*f(i) if f(i) £ a/b..........(9.1.2) 
Eq. 9.1.2 is called the "Schaefer model".
The slope, b, must be negative if the catch per unit of effort, Y/f, decreases for increasing effort, f, (see Fig. 9.1.1). The intercept, a, is the Y/f value obtained just after the first boat fishes on the stock for the first time. The intercept therefore must be positive. Thus, a/b is positive and Y/f is zero for f = a/b. Since a negative value of catch per unit of effort Y/f is absurd, the model only applies to fvalues lower than a/b.
An alternative model was introduced by Fox (1970). It gives a curved line when Y/f is plotted directly on effort, f, (see Fig. 9.1.1), but a straight line when the logarithms of Y/f are plotted on effort:
ln (Y(i)/f(i)) = c + d*f(i)..........(9.1.3) 
Eq. 9.1.3 is called the "Fox model", which can also be written:
Y(i)/f(i) = exp(c + d*f(i))..........(9.1.4)
Both models conform to the assumption that Y/f declines as effort increases, but they differ in the sense that the Schaefer model implies one effort level for which Y/f equals zero, namely when f = a/b whereas in the Fox model, Y/f is greater than zero for all values of f.
This can easily be seen in Fig. 9.1.1 where the plot of Y/f on f gives a straight line in case of the Schaefer model and a curved line, which approaches zero only at very high levels of effort, without ever reaching it (asymptotic) in the case of the Fox model.
In Section 8.3 it was demonstrated that CPUEw(t) = q*B(t) (Eq. 8.3.1). Since Y/f is also the catch per unit of effort in weight, we can write
Y(i)/f(i) = q*B = a + b*f(i) for the Schaefer model and
Y(i)/f(i) = q*B = exp(c + d*f(i)) for the Fox model
where B is the biomass and q the catchability coefficient (a constant).
Fig. 9.1.1 illustrates another basic feature of the two models. For f close to zero Y/f takes the maximum value and so does the biomass because Y/f = q*B, and q is a constant. The biomass corresponding to f = 0 is called the "virgin stock biomass" or the "unexploited biomass", denoted by "Bv". Thus, replacing Y/f by q*Bv in Eqs. 9.1.2 and 9.1.4 gives:
q*Bv = a or Bv = a/q (Schaefer)
q*Bv = exp(c) or Bv = exp(c)/q (Fox)
Fig. 9.1.1 Illustration of the different assumptions behind the Schaefer model and the Fox model
The Bv for the two models must be the same. When increasing f from zero to level A (see Fig. 9.1.1) the two curves are approximately equal, but to the right of A the differences become larger. Thus, the choice between the two models becomes important only when relatively large values of f are reached. It cannot be proved that one of the two models is superior to the other. You may choose the one you believe is the most reasonable in each particular case or the one which gives the best fit to the data. However, the Beverton and Holt model is more in agreement with the Fox model, because they have a similar curvilinear relationship between catch per unit effort and effort (cf. Figs. 8.3.1 and 9.1.1) and between the mean biomass per recruit B/R and F (cf. Figs. 8.3.2 and 9.1.1).
Example 31: Schaefer and Fox models, demersal fish, Java Sea
Fig. 9.1.2 shows an example of plots of CPUE, Y(i)/f(i), against effort, f(i), and of the yield, Y(i), against effort. Data are from the demersal fishery off the North coast of Java during the years 19691977 (from Dwiponggo, 1979). In this case yield, Y, refers to the annual catch measured in units of 1000 tonnes and Y/f refers to yield per thousand standard vessels each year. Effort is given in units of standard vessels per year.
Fig. 9.1.2 Trends of yield, Y(i), and catch per unit of effort, Y(i)/f(i), off the North coast of Java (based on data from Table 9.1.1)
Because the fleet consisted of a number of different boat types, the effort of each boat category has been converted to a standard unit before summing to obtain total effort. Not surprisingly, the trend of Y/f shows a decline for increasing effort. Considering the stock biomass as a limited resource shared by the boats participating in the fishery, we expect a smaller share per boat the more boats enter the fishery.
There is no clear trend in the relationship between yield, Y, and effort, f, in this case. We will now show how the two models can be applied to this type of data.
The estimation procedures for the parameters (Schaefer: a and b, Fox: c and d) will be explained on the basis of the data given in Table 9.1.1 and Fig. 9.1.2. Since we are dealing with a straight line in the case of the Schaefer model and a curve which has been linearized by taking the logarithm in case of the Fox model, the determination of a,b and c,d requires two linear regressions, of f(i) on Y(i)/f(i) and f(i) on ln(Y(i)/f(i)) respectively. The results of the two regressions are presented in Table 9.1.1, including the standard deviations of the slopes and the intercepts. The lines are shown in Fig. 9.1.3. We have thus determined the relationships between catch per unit of effort and effort for both models.
Table 9.1.1 The calculation procedure for estimating the MSY and f_{MSY} by the Schaefer model and by the Fox model using catch and effort data from the trawl fishery off the North coast of Java (Dwiponggo, 1979) 1)
year 
yield 
effort no. of standard vessels 
SCHAEFER 
FOX 
i 
Y(i) 
f(i) 
Y(i)/f(i) 
ln(Y(i)/f(i)) 
(x) 
(y) 
(y) 

1969 
50 
623 
0.080 
2.523 
1970 
49 
628 
0.078 
2.551 
1971 
47.5 
520 
0.091 
2.393 
1972 
45 
513 
0.088 
2.434 
1973 
51 
661 
0.077 
2.562 
1974 
56 
919 
0.061 
2.798 
1975 
66 
1158 
0.057 
2.865 
1976 
58 
1970 
0.029 
3.525 
1977 
52 
1317 
0.039 
3.232 
mean value 
923.22 
0.0667 
2.7648 

standard deviation 
485.14 
0.02171 
0.3873 

intercept, a or c 
0.1065 
2.0403 

slope, b or d 
0.00004312 
0.0007848 

variance of slope 



sb^{2} = [(sy/sx)^{2}  b^{2}]/(92) 
2.041*10^{11} 
3.087*10^{9} 

standard deviation of slope, sb 
0.00000452 
0.0000556 

confidence limits of slope 



b + t_{92}*sb, t_{7} = 2.37 
0.000032 
0.00065 

b  t_{92}*sb 
0.000053 
0.00092 

variance of intercept 



sa^{2} = sb^{2}*[sx^{2}*(n1)/n + _{}^{2}] 
0.00002167 
0.003277 

standard deviation of intercept, sa 
0.0049 
0.0572 

confidence limits of intercept 



a + t_{92}*sa 
0.118 
1.90 

a  t_{92}*sa 
0.095 
2.18 

MSY 
Schaefer: 0.25*a^{2}/b 
65.8 


Fox: (1/d)*exp(c1) 

60.9 

f_{MSY} 
Schaefer: 0.5*a/b 
1235 


Fox: 1/d 

1274 

1) In the formulas given in this table a and b should be replaced by c and d for the Foxmodel 
Fig. 9.1.3 The Schaefer and the Fox models illustrated by the demersal fisheries off the North coast of Java (based on data from Table 9.1.1)
The objective is, however, to obtain an estimate of the maximum sustainable yield (MSY) and to determine at which level of effort MSY has been or will be reached. To that purpose we have to rewrite Eqs. 9.1.2 and 9.1.4 expressing the yield as a function of effort, by multiplying both sides of the equation by f(i):
Schaefer: Y(i) = a*f(i) + b*f(i)^{2} if f(i) < a/b..........(9.1.5)
or
Y(i) = 0 if f(i) = a/bFox: Y(i) = f(i)*exp[c + d*f(i)]..........(9.1.6)
Eq. 9.1.5, the Schaefer model, is a parabola (see Fig. 9.1.3), which has its maximum value of Y(i), the MSY level, at an effort level
f_{MSY} = 0.5*a/b..........(9.1.7)
and the corresponding yield:
MSY = 0.25*a^{2}/b..........(9.1.8)
Eq. 9.1.6, the Fox model, is an asymmetric curve with a maximum (the MSY level), with a fairly steep slope on the left side and a much more gradual decline on the right of the maximum (see Fig. 9.1.3).
The MSY and f_{MSY} for the Fox model can be calculated by formulas which are derived from Eq. 9.1.6 by differentiating Y with respect to f and solve dY/df = 0 for f,
f_{MSY} = 1/d..........(9.1.9)
MSY = (1/d)*exp(c1)..........(9.1.10)
The results of the calculations of the example are presented at the bottom of Table 9.1.1.
From Table 9.1.1 and Fig. 9.1.3 it can be seen that the two models give slightly different results. According to the Fox model the MSY level is 60900 tons, at an effort level f_{MSY} of 1274 standard vessels, while according to the Schaefer model the MSY level is much higher (65800 tons) and at a lower f_{MSY} of 1235 standard vessels. According to both models the effort level surpassed f_{MSY} in 1976 and the yield was below MSY (see Fig. 9.1.2).
So far we have dealt mainly with the computational procedure, which was straight forward. One could now ask why we should be concerned with complicated models like the Beverton and Holt yield per recruit model when estimates of MSY can be obtained so easily from the surplus production models. One answer is that what we gain in simplicity with the surplus production models has the cost of having to make a number of assumptions about the dynamics of fish stocks, which may be (and nearly always are) impossible to justify. Some of these assumptions are discussed below. The reasoning given below is based on the Schaefer model, but it also applies to the Fox model.
(See Exercise(s) in Part 2.)
The assumption of an equilibrium situation
To explain the concept of an equilibrium situation we consider a situation where a virgin stock starts to be exploited in, say, 1971 (see Fig. 9.1.4), by, say, 1000 boats. Suppose the "Schaefer line" in Fig. 9.1.4 applies to this stock. According to the Schaefer model the yield in 1971 corresponding to 1000 boats should be x.
However, it turned out to be y, i.e. a larger value than predicted by the model. This is because when fishing started in 1971 the biomass was still the virgin stock biomass, Bv, and only after a certain period of exploitation the biomass declines.
When fishing continued in 1972 the biomass was reduced due to the removal by fishing in 1971 and the 1972 catch therefore became smaller than that of 1971. Each year the resource is reduced, the reduction being smaller the longer time has elapsed since the introduction of the 1000 boats. Eventually, the system will stabilize at the Y/flevel x. We say that the system has reached an "equilibrium situation" after a "transition period".
Fig. 9.1.4 Illustration of the concepts of an equilibrium situation and the transition period (for further explanations, see text)
For the equilibrium situation the production of biomass per time unit, equals the removal by fishing, the yield per time unit, plus the amount of fish dying of natural causes. This has also been illustrated in Fig. 9.2.1.
The "equilibrium situation" in the surplus production models is comparable to the "stabilized constant parameter system" in the Beverton and Holt models (cf. Section 8.1).
The biological assumptions
The biological reasoning behind the model was adequately formulated by Ricker (1975) as follows:
"1. Near maximum stock density, efficiency of reproduction is reduced, and often the actual number of recruits is less than at smaller densities. In the latter event, reducing the stock will increase recruitment.2. When food supply is limited, food is less efficiently converted into fish flesh by a large stock than by a smaller one. Each fish of the larger stock gets less food individually; hence a larger fraction is used merely to maintain life, and a smaller fraction for growth.
3. An unfished stock tends to contain more older individuals, relatively, than a fished stock. This makes for decreased production, in at least two ways:
(a) Larger fish tend to eat larger foods, so an extra step may be inserted in the food pyramid, with consequent loss of efficiency of utilization of the basic food production.(b) Older fish convert a smaller fraction of the food they eat into new flesh  partly, at least because mature fish annually divert much substance to maturing eggs and milt."
However, it is also possible to consider these models as purely empirical. For instance, if observations of Y/f plotted on f give a curve complying with the Fox model, this model may be applied without any concern for a possible biological explanation.
Assumptions on the catchability coefficient
We assume that fishing mortality is proportional to effort (Eq. 4.6.1):
F = q*f
This assumption in itself is not controversial (if f is a reasonable measure of effort). The problems come when f is measured in, for example, the number of boat days per year over a series of years. In most cases the efficiency of the boats has changed over a long period; often the boats have become larger and better equipped. Thus, 100 boat days in, say, 1978 may create a larger fishing mortality than 100 boat days in 1968. This means that q becomes a function of time or rather, a function of the technical development which is usually a function of time. It has proved very difficult to account for changes in q, caused by increased fishing efficiency and usually it is assumed that q remains constant. Therefore, one should be cautious not to include too long a time series of data in the surplus production analysis. Alternatively, changes of q must be taken into account. In some fisheries on small pelagic species in upwelling areas, for example, the anchoveta in Peru and Chile it occurs that the fish concentrate in small areas because of changes in the environmental conditions. In such cases there is no direct relationship between q, f and F, and surplus production models cannot be applied.
In this section we consider the case of poorly investigated stocks. Time series of catch and effort data are not available, but some estimates of overall biomass and the natural mortality have been obtained.
Several empirical formulas have been developed with the objective of providing a first rough estimate of the MSY based on such scanty data. These formulas have found wide application after first estimates were obtained of the standing biomass after one or a series of exploratory bottom trawl surveys and/or acoustic surveys. The first formula was developed by Gulland (1971), a modification was proposed by Cadima (in Troadec (1977) and finally a set of formulas strictly based on the Schaefer and Fox surplus production models was developed by Garcia, Sparre and Csirke (1989).
Gulland (1971) suggested the following way of estimating maximum sustainable yield:
MSY = 0.5*M*Bv..........(9.2.1)
where Bv is the virgin stock biomass and M the natural mortality.
This formula has been used especially on sparsely investigated and lightly exploited stocks. Bv is often estimated by the "swept area method" (Chapter 13) and M is often a value estimated for similar species in a sea area which is believed to be similar to the one under investigation. As Gulland's formula requires an estimate of the virgin stock biomass, Bv, it is in practice applicable only to unexploited stocks. There is no proper scientific justification for Eq. 9.2.1 (Gulland, pers. comm.). However, the following statements which were made already by Tiurin (1962) and Alverson and Pereyra (1969) appear reasonable:
1. MSY must depend on the virgin stock biomass, Bv
2. A high M corresponds to a high production (this is further discussed below)
3. If the biomass = 0.5*Bv and F = M under optimum exploitation Eq. 9.2.1 is fulfilled.
Fig. 9.2.1 Illustration of Gulland's formula MSY = 0.5*M*Bv (for further explanation, see text)  EQUILIBRIUM CONDITIONS
Item 2 above is illustrated in Fig. 9.2.1. We consider the annual biomass budget for two virgin stocks, A1 with a high production rate and B1 with a low production rate. For stock A1 the loss caused by natural mortality is large and to maintain the stock the net production must be of equal size. For stock B1 the loss due to natural deaths is small and only a small net production is required to counterbalance this loss. The cases A2 and B2 deal with the same two stocks but after they have been for a while under optimum exploitation and produce the maximum sustainable yield. In A2 a large part of the natural deaths has been replaced by deaths due to fishing. This is also the case for B2, but the potential for replacing natural deaths by deaths due to fishing is smaller in case B2 than in case A2.
Although widely used, Eq. 9.2.1 has been criticized by a number of workers. Caddy and Csirke (1983) showed that the third assumption, that F equals M under optimum exploitation, does not apply in many cases, especially stocks of prey species (e.g. shrimps). Based on simulation studies, Beddington and Cooke (1983) concluded, that Eq. 9.2.1 generally overestimates MSY by a factor of 2 to 3. Thus, replacing "0.5" by "0.2" in Eq. 9.2.1 might perhaps give a better (and consequently much lower) estimate of MSY.
A generalized version of Gulland's estimator was proposed by Cadima (in Troadec, 1977) for exploited fish stocks for which only limited stock assessment data are available.
Cadima's estimator has the form:
MSY = 0.5*Z*_{} ..........(9.3.1)
where _{} is the average (annual) biomass and Z the total mortality. Since Z = F + M and Y = F*_{}, Cadima suggested that in the absence of data on Z, Eq. 9.3.1 could be rewritten:
MSY = 0.5*(Y + M*_{})..........(9.3.2)
where Y is the total catch in a year and _{} is the average biomass in the same year.
As most stocks in the world are now already being exploited this equation is quite frequently used in developing and some developed fisheries, where catch and effort time series are not yet available, but where biomass estimates are occasionally obtained from, for instance, trawl or acoustic surveys.
9.4.1 Validation of estimates of MSY based on empirical formulas
Based on the considerations in the foregoing section, Garcia, Sparre and Csirke (1989) suggested two alternative ways to estimate the potential yield of exploited fish stocks which have basically the same foundation and applications as the Gulland and Cadima estimators, but which are consistent with the underlying models. The two estimators have been derived from the Schaefer model and the Fox model.
Both methods assume that the observations:
_{}(average biomass) and Y (current yield)
are available for one year only. They also assume that natural mortality, M, is known and that there is a relationship between M and f_{MSY} of the form:
f_{MSY} = k*M..........(9.4.1)
where k is a constant.
As f = Y/_{} and Y/f = _{}, we can write the surplus production models Eqs. 9.1.2 and 9.1.3 in the form:
Schaefer: _{} = a + b*(Y/_{})..........(9.4.2)
Fox: ln_{} = c + d*(Y/_{})..........(9.4.3)
Suppose observations B1 and Y1 are available and we have a "guesstimate" of M, then combined with the assumption of Eq. 9.4.1 we get:
Schaefer: B1 = a + b*(Y1/B1) and f_{MSY} = k*M = a/2b..........(9.4.4)
Fox: ln B1 = c + d*(Y1/B1) and f_{MSY} = k*M = 1/d..........(9.4.5)
These equations can be solved for a and b, c and d in the Schaefer and the Fox model, respectively:
Schaefer: _{} ; _{} ..........(9.4.6)Fox: c = ln (B1) + Y1/(B1*f_{MSY}); d = 1/f_{MSY}..........(9.4.7)
Once we have (a,b) or (c,d) we can estimate the MSY by Eqs. 9.1.8 and 9.1.10:
Schaefer: MSY = 0.25*a^{2}/b
Fox: MSY = (1/d)*exp(c1)
and draw the yield curves (cf. Eqs. 9.1.5 and 9.1.6).
The MSY expression corresponding to the Schaefer model is found by inserting Eq. 9.4.6 into Eq. 9.1.8:
_{}
The yield curve is determined by a and b (Eq. 9.4.6). When f_{MSY} is not known (as is most often the case) it may be replaced by k*M. In the special case where k = 1 and f_{MSY} = M we get:
_{}
If the stock is unfished (i.e. when f = 0, Y = 0 and B = Bv) Eq. 9.4.9 becomes Gulland's original formula (Eq. 9.2.1).
If the stock in question responds better to the Fox production model (Eq. 9.1.10), we get the expression:
MSY = f_{MSY}*_{}*exp[Y/(f_{MSY}*_{})  1]..........(9.4.10)
The yield curve is determined by c and d (Eq. 9.4.7). In the special case where k = 1 and f_{MSY} = M, Eq. 9.4.10 becomes:
MSY = M*_{}*exp[Y/(M*_{})  1]..........(9.4.11)
When Y = 0, the estimate of MSY comparable to Gulland's estimator becomes
MSY = M*_{}*exp(1) = 0.37*M*_{}..........(9.4.12)
Thus, one pair of observations (B1,Y1) and assumptions on M and the relationship between M and f_{MSY} (f_{MSY} = k*M) are sufficient information to get a first rough estimate of the yield curve (Schaefer: a,b or Fox: c,d) from which a first rough estimate of MSY can be derived.
The interested reader is referred to the original paper (Garcia, Sparre and Csirke, 1989) for further details.
When working with mathematical models the fishery scientist should check whether the basic assumptions of the models are fulfilled. This applies in particular to the abovementioned empirical formulas (Garcia, Sparre and Csirke, 1989). For example, the biomass (_{}) is meant to be the exploited average biomass, and both the catch and the biomass referred to should be comparable and have the same age (or size) structure. For instance, the biomass figure should not include small sizes which are not available to the fishery. This biomass is the annual biomass value and seasonal oscillations caused by change in growth, mortality or recruitment, which are likely to be more important in shortlived species such as shrimps, squids and anchovies, should be taken into account and as far as possible be levelled off to obtain an appropriate annual average of the total biomass.
If seasonal oscillations of biomass are caused by migrations, then the peak biomass representing the real size of the stock should be used. If another country is exploiting the same stock at another season of the year, in another area, the catch of that country should be included in the calculations and the MSY estimate should refer to the whole unit stock. The estimated potential yield could also be validated by comparison with other similar stocks for which better information might be available.
Some of the questions to be asked are: How does an estimate of the density, expressed as MSY/km^{2} stand with respect to similar estimates for other stocks of the same species in ecologically similar areas exploited under similar fishing regimes? Does the size structure of the catch provide support for assessment implying that the stock is heavily overfished (e.g. predominance of juveniles) or underfished (e.g., predominance of large, old fish)?
It should be noted that a closer look at the lengthfrequency composition of the catch would give some guidance on relative levels of exploitation and this should be among the first data to be collected in any developing fishery in order to allow some estimation of the total mortality rate for crosschecking with other methods (cf. Chapters 3 and 4).
If the stock in question has been exploited for some time it is likely that a time series of catch data is available, which should also be examined. Even if no detailed effort data are available, the indication that after a period of sustained increase, the total catch has been stable for some time may mean that the MSY has been reached at least for the present regime of exploitation, while if the catch has dropped from a previous high level it may mean that the stock is overfished and an average of the highest catches experienced in the past may provide an independent approximation to the MSY. In interpreting catch time series as suggested above, one assumes that such variations in catches are caused by changes in fishing effort and not by environmental or socioeconomic changes.
The surplus production models are usually applied to time series of CPUE and effort. Munro and Thompson (1983 and 1983a), however, applied the surplus production model to a set of data from the Jamaican coral reef fishery all collected in the same year but representing different fishing grounds fished at different levels of effort. Fig. 9.5.1 shows a schematic map of Jamaica divided into 9 areas, which except for area B coincide with the parishes of Jamaica. The fishery studied by Munro and Thompson (1983) is a local trap fishery operated from canoes. Coral reef fish are not considered to be very mobile and it was assumed that each area (Fig. 9.5.1) has its own stocks which are independent of the neighbouring stocks (little mixing). The basic assumption is that the "ecological regimes" in the sea areas opposite the various parishes do not differ substantially around the island. Based on that assumption it makes sense to further assume that the relation between the yield and fishing effort in the different areas will follow the same model.
Table 9.5.1 shows the CPUE and effort data collected in the various parishes (Fig. 9.5.1) for the Jamaican shelf fishery on shelfdwelling species in 1968.
Effort is expressed in units of canoes per km^{2} per year to accommodate the assumption that each area has the same relative potentials, i.e. can support the same production per area unit. Thus, if exploited at the same rate (same effort per unit area per year) all areas should have the same yield per unit area per year (kg/km²/year) (cf. column D of Table 9.5.1).
The yield per unit area was based on the area of the shelf and the proximal banks of each parish. For further details the reader is referred to the original papers (Munro and Thompson, 1983 and 1983a).
The relationship between CPUE and effort is here assumed to follow the Fox model (Eq. 9.1.3). Fig. 9.5.2 shows the plot of ln (CPUE) on effort as well as the yield per no. of canoes per km^{2}. Munro and Thompson had reasons to exclude area F from the regression analysis.
When applying a Munro and Thompson plot care should be taken to exclude all those fish which can move freely between areas, such as large pelagics.
The Munro and Thompson plot may be useful in situations where only limited data are available from certain parts of a region which have similar fisheries on coral reef stocks or other resources with a similar low mobility.
Table 9.5.1 Input data for Munro and Thompson Plot (for further details, see text). From Munro and Thompson (1983a)

A 
B 
C = ln B 
D = A*B 
parish (stock) 
effort canoes per km^{2} 
CPUE kg per canoe per year 
ln (CPUE) Fox model 
yield kg per km^{2} per year 
A 
1.63 
2367 
7.769 
3858 
B 
0.38 
3279 
8.095 
1246 
C 
3.09 
1407 
7.249 
4348 
D 
5.63 
556 
6.321 
3130 
E 
4.43 
974 
6.881 
4315 
(F) 
5.51 
1306 
7.175 
7196 
G 
4.58 
564 
6.335 
2583 
H 
4.20 
767 
6.642 
3221 
I 
1.49 
1875 
7.536 
2794 
Fig. 9.5.1 Schematic map of Jamaica showing the parishes used for the "Munro and Thompson plot" (Fig. 9.5.2)
Fig. 9.5.2 Munro and Thompson plot. Based on data in Table 9.5.1 (from Munro and Thompson, 1983)
In Section 4.3 it was suggested that effort is proportional to fishing mortality. This is true, of course, if we define effort as something proportional to fishing mortality, but such a definition has no practical application. In practice we will have to choose a measure for effort which we believe is related to fishing mortality or rather "fishing power". There are many possible choices. For a trawl fishery we may consider:
number of trawlers
number of trawlerdays
number of standard trawlers (taking into account the boat type)
number of standard trawlerdays
... etc.
For a handline fishery it may be more appropriate to consider the number of fishermandays or the number of hooks used times the number of days. In this case it may be necessary to take into account that the fishermen on the same boat compete, so that effort is not a linear function of the number of fishermen.
In general, a measure which can be shown to be linearly related to the catch rate is a suitable measure. That is, if it can be shown that two units of effort catch twice as much as one unit of effort when operating under equal conditions the effort measure is a suitable one. For example, the number of fishing hours times engine horsepower may be a suitable measure of effort in some bottom trawl fisheries, whereas in a gill net fishery the boat type and the number of hours are likely to be less important than the number of gill nets set per day. In both of the above mentioned cases the number of fishermen may not be linearly related to the fishing power.
There are already considerable difficulties in defining suitable measures of effort for a single gear as discussed above, but when trying to define the effort for combinations of gears exploiting the same resources some rather really intricate problems are encountered. In tropical fisheries many different gears are used to capture the same resources, therefore several methods for standardization of effort units will be discussed here.
Relative effort
Before we start the discussion on standardization of effort units we notice that:
_{ }as CPUE = yield/effort
In the following we shall be using this relation, not for yield and effort, but for quantities proportional to yield, effort and CPUE, so that the final result is a measure proportional to effort. Therefore we call it "relative effort". We also assume that all the effort units defined are suitable ones.
Table 9.6.1 Example (hypothetical) to illustrate summation of effort for different effort units
Example 32: Summation of effort for different effort units
In the example given in Table 9.6.1 we consider the effort of four gears measured in numbers of units per year. As can be seen, the original effort units are not compatible. The yields corresponding to the various efforts are also available from a sampling scheme. The four sampled gears given in the Table are assumed to constitute only one tenth of the total catch of the stock in question. To make the different gear types (effort units) compatible each unit must be converted into CPUE, which then in turn is converted into "relative CPUE" as shown in the Table. The relative catch per unit of effort of gear i in year y is defined as follows:
_{}where
_{}
when a time period over the years y1, y1+1, ..., y2 is considered. (In Table 9.6.1, y1 = 1971 and y2 = 1975).
As the relative CPUE of a gear has no dimension you may say that we have obtained compatible units of CPUE by the conversion into the relative CPUE. The relative CPUEs can be summed.
In the hypothetical case that all CPUE observations were proportional to the population size (cf. Section 4.3) the relative CPUEs would become identical for all gears.
However, in reality some gears are less important than others. The purse seine in Table 9.6.1 is the dominating one as far as development in yield is concerned, whereas the pole and line fishery is rather unimportant in terms of yield. This is accounted for by calculating the sum of the relative CPUEs weighted by the corresponding yields. For example for year 1971 (see Table 9.6.1):
_{}
Dividing the total yield of the species under consideration, YT(y), including the yield not covered by the catch/effort sampling scheme, by the weighted sum of relative CPUEs gives a figure proportional to the total effort, R(y), shown in the column "YT(y)/R(y)" in Table 9.6.1. The last column contains the normalized relative effort, E(y). (This concept is introduced in order not to confound relative effort with absolute effort.)
The first gear in Table 9.6.1, the purse seine, is the important one, in the sense that the trend in the yield of this gear is the same as the trend for the normalized relative effort. The beach seine shows the opposite trend, but because it is a relatively unimportant gear it has less influence on the combined effort. If the purse seine catches had been only 10% of what they were the trend would be changed as the beach seine would then become a relatively more important gear. The E(y) values would then have the same trend as the yields of the beach seines (see Table 9.6.2).
Table 9.6.2 Exploring the concept of normalized relative effort for several gears combined (cf. Table 9.6.1)
year 
from Table 9.6.1 
if purse seine catches were only 10% of those given in Table 9.6.1 
E(y) 
E(y) 

1971 
0.653 
1.014 
1972 
0.734 
1.018 
1973 
0.903 
1.035 
1974 
1.146 
0.984 
1975 
1.564 
0.949 
The method described above has been used by the North Sea Round Fish Working Group of ICES (ICES, 1980). This method does not require a direct comparison of the different boat types. It requires only a type of data which is often available. The method can be questioned, and in fact, the results achieved by the ICES working group were not "overwhelmingly convincing" when the ICES working group correlated the normalized relative effort figures with the fishing mortalities obtained from VPA.
Relative fishing power
A more direct (and probably more dependable) method to standardize effort is the one suggested by Robson (1966) (discussed in Gulland, 1983). It does, however, require additional data. The method works with the concept of "relative fishing power". With the fishing power of vessel B relative to vessel A we mean:
_{}
when the two boats are fishing under the same conditions (at the same time and in the same area). Vessel A is often called the "standard vessel".
Suppose the boats participating in a certain fishery can be divided into 5 homogeneous groups, so that each group consists of boats with similar fishing powers. Suppose also that the CPUE is in units of catch per time unit (e.g. catch per trawling hour), and further that the following data have been collected:
Boat type 
A (standard) 
B 
C 
D 
E 
Fishing power (PA) 
1.0 
PA(B) 
PA(C) 
PA(D) 
PA(E) 
Number of boats (N) 
NA 
NB 
NC 
ND 
NE 
Average number of fishing days per boat (d) 
dA 
dB 
dC 
dD 
dE 
The total effort would then be estimated by:
total effort = 1.0*NA*dA + PA(B)*NB*dB + PA(C)*NC*dC + PA(D)*ND*dD + PA(E)*NE*dE..........(9.6.3)
In certain cases it can be assumed that the fishing power is proportional to some characteristics of the boat or gears which are relatively easy to obtain, such as GRT (tonnage) or HP (horsepower) or their product for trawlers and, for example, the number or length of nets for gill netters. As we are usually only interested in the relative effort, the PA'S (fishing power) in Eq. 9.6.3 can simply be replaced by the boat/gear characteristics.
A family of models which attempts to be a compromise between the surplus production models and the agestructured models has been presented by, for example, Deriso (1980), Ludwig and Walters (1985), Ludwig (1987) and Schnute (1985, 1987). These models, however, have limited use in tropical areas as they were developed for longlived (slow growing) species, which are not exploited in the first part of their life. They are based on a series of rather strong assumptions which make them inapplicable to certain species. Schnute (1985) states about the models: "Among other things, they reflect the undeniable reality that the population consists of cohorts that get one year older each year". This clearly shows that these models are not intended for stocks where this "reality" can be denied, e.g. shrimps. Actually, the methods give nonsense results for such species (e.g. negative biomass).
The theoretical biological basis for these models does not go beyond the models already introduced (the basic reasoning is similar to that of Pope's cohort analysis), but the mathematical theory applied for the estimation procedure is somewhat more sophisticated than for most of the models presented in this manual. The main difference from the agestructured models given in this manual is that the FordWalford equation replaces the von Bertalanffy model for growth in weight (Eq. 3.1.2.1).
The models are highly sophisticated and not quite simple to use. Their description in nonmathematical language would require a very long chapter. The interested reader is therefore referred to the abovementioned papers by Deriso, Schnute and others.