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An Icelandic fisheries model
European Journal of Operational Research 33 (1988) 191-199 North-Holland
191
An Icelandic fisheries model Thorkell HELGASON
and Snjrlfur OLAFSSON
Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland Abstract: A model has been developed for the Ministry of Fisheries in Iceland which is being used as a decision support tool in fisheries management. With this model predictions can be made about catches of cod and other demersal species several years ahead. Furthermore earnings and costs in the fisheries are calculated. The starting point is the size of the fish stocks in the beginning of the planning period together with the size and classification of the fishing fleet. Changes in the fleet composition are among the control variables. Emphases are on details such as geographical distribution of the catches. The model is used by the Ministry, both in short term management regarding quota systems and for long term investment planning. Results suggest that in the long run the fishing fleet is now far too big. Parameters for the model are obtained from an intensive data base on catches of every boat for every month in the last decade, disaggregated by species and landing harbour. Parallel to this model a new method of stock size estimation was developed. The current model was preceded by an optimization model which was not politically accepted! The paper touches on the problems of technical consultation on a sensitive political issue. Keywords: Decision support systems, fisheries, fisheries management, government, modelling, planning
1. Introduction This paper deals with a model of the Icelandic fisheries. In Section 2 we survey the background and history of fishing in Iceland. In Section 3 we deal with basic ideas in fisheries modelling and our previous attempts in this direction. Section 4 describes the data used and in section 5 the main model is explained in some detail. Section 6 deals with prediction errors. Then in Section 7 we explain how the model is used and mention some results from the model. Finally in Section 8 we discuss the use of models in political decision making. The sections need not be read in consecutive order.
2. Background Fishing is the backbone of the Icelandic economy. About 70% of Iceland's exports are fish and fish products. The fisheries can be divided into three main categories by fish species. Most important are demersal species. Of those, cod is the Received September 1986; revised January 1987
major one, accounting for 55% of the demersal catch. The second category includes pelagic fish, mainly capelin. The third category covers all kinds of shellfish. There is little overlap between these categories as, for the most part, each fishery is carried out by specialized boats. This paper focuses on the demersal fisheries. Figure 1 shows the development of demersal fisheries in Icelandic waters over the past few decades. In the year 1976 there was a turning point. Until then foreign vessels had been taking about half of the total catch. But then Iceland gained full sovereignty over her fishing grounds. Anticipating this development, Icelandic fishing companies had already started to modernize and increase their trawler fleet. Around 1970, trawlers played an insignificant role in Icelandic fishing, but then, in five years or so, a large and modern fleet of stern trawlers was acquired. Now they number over 100 and account for about half of the demersal catch, the remainder being taken by some 700 boats of different sizes. One of Iceland's main arguments in the dispute over the fishing limits concerned stock protection. Reliable stock size estimation methods emerged rather late. The important VPA-method was intro-
0377-2217/88/$3.50 © 1988, ElsevierScience Publishers B.V. (North-Holland)
Th. Helgason, S. Olafsson / An Icelandic fisheries mode/
192 YIELD
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duced in 1949 but was first applied in the sixties and used for stock estimation in Iceland in the early seventies. Stock size estimates done in 1975 showed that the cod stock was down to less than half of its average size for the post-war period. Early prediction models (e.g. [5]) also show that the applied effort was about twice the maximum yield effort and eventually three times the economically optimal effort. Furthermore the age structure of the catch was unfavourable. Most of this was blamed on a fleet of up to several hundred foreign trawlers fishing in Icelandic waters. Foreign fishing vessels disappeared almost entirely after 1976. With great optimism but little planning they were replaced by a new domestic fleet of trawlers. Gradually it was realized that the total fishing effort had hardly decreased in spite of the disappearance of the foreign trawlers. The size of the stock and consequently of the catch did not improve. This fact was obscured somewhat by one exceptionally good year-class of cod and at the same time an unexpected immigration of cod from Greenland which together boosted the catch in 1981-1983. Nevertheless the cod catch reached an all-time low in 1983. The authorities and the fishing industry gradually realized that the fishing fleet and hence the effort expended was too large. In the late seventies measures were taken to slow down and then gradually halt further expansion of the fleet. Secondly, the effort expended was restrained. This was initially done by restricting the fishing period, both in general (by enforcing an extended Christmas
and Easter vacation for fishermen!) and also by setting a ceiling on the allowed annual operating time of each trawler. Then in 1984 a general quota system was introduced. 3. Fishing models Models of renewable resources are numerous in mathematical economics (see e.g. [4]). Computer models of real fishing situations have also been developed in many places, e.g. [10]. Most such models are focussed around the population dynamics of the species and fishing effort is measured as fishing mortality rate. The link between fishing mortality and physical quantities of effort (such as number of boats) causes problems. Consequently it is also difficult to assign costs to the fishing operation. Several models of this kind had been developed for the Icelandic demersal fisheries prior to 1980 (see e.g. [5]). One of the present authors constructed a dynamic optimization model of this type, [6 and 7]. It has two annual decision controls, one being attributed to total effort in terms of fishing mortalities. The other parameter controls the age pattern of the fishing mortalities and can therefore be interpreted as an indicator of mesh size or other measures to protect the young fish. The objective function of the model sums up the present value of total profits in the fishing industry over an infinite planning horizon. A later version of this model takes account of randomness in fish
Th. Helgason, S. Olafsson / A n Icelandic fisheries model
recruitment and in the actual catch for a given (expected) fishing mortality [6]. This and other similar models showed it to be optimal to cut the total effort down to about half of the actual effort in 1977. These models were criticized by the fishing industry and politicians on several grounds. Their comments can be summarized as follows: (1) Fishing effort is not a global one-dimensional quantity. It is generated by hundreds of different boats. Any management policy will affect the individual boats differently. Therefore total fishing effort is meaningless as an independent control parameter. (2) Although the fleet may be too big in general, a global reduction is economically and politically unfeasible. In some areas employment considerations even call for more boats. The models lack disaggregation and details to take care of this. (3) The linear relationship between stock size and catch is too naive. Doubling of the (catchable) stock may cause a doubling of the catch per trawling hour (say), but then relatively more time is spent on non-fishing operations like sailing and landing. Hence the catch per boat will not double. (4) Ecological feed-back is ignored in the models. The benefits of effort reduction come through stock increase. The calculations are based on the assumption that the growth rate of individual fish is independent of the stock size. This is unrealistic. Limited food supply may hamper the growth rate. Hence the beneficial effects of reduced effort are over-estimated in the long run. (5) The models in question are optimization models, where the objective is the present value of an infinite stream of net profits. A multi-dimensional goal function cannot be reduced down to one-dimensional profit. Also, discounting future earnings in order to lump them together over-simplifies the (political) facts of life. Optimization models are far too monolithic for planning an entire national economy. The last argument certainly applies to a variety of models, not only fisheries models. It is increasingly our belief that optimization is largely out of place in global models. This is not an argument against using optimization in, say, blending or product-mix problems. Optimization has the flavour of commanding a decision and in areas like ours the decision maker does not like to be commanded; at most he seeks guidance or more
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likely decision aid. It is therefore not surprising that OR practitioners have more and more adopted the term 'decision support system' for their models (and even christened old optimization models so!). The criticism under (1) and (2) above are also largely politically motivated. Political decisions are very seldom of a general nature. It lies in the foundation of democratic, pluralistic societies that even minor consequences cannot be ignored or at least not overseen. Hence political decision support models must display details to as large an extent as the model maker can possibly justify on the basis of his data. Critical remark (3) is well founded and, as will be seen later, was extensively dealt with and resolved in the model described in this paper. Remark no. 4 on ecological effects is more difficult to handle. Both in Iceland and abroad much effort has been spent on modelling the relationship between growth rate and stock size. The extensive 'North Sea' model is of this kind. (See [1], and also [8,11,15]). In an another Icelandic fisheries model [3] such relationship is derived and shown to have a noticeable effect on the optimal policy resulting in an increase in the optimal effort. We have so far not incorporated such an ecological relationship permanently in our model. We have however tested some hypothetical versions of a stock dependent growth rate and thereby seen that, whereas the relationship may have some effect, it seems not to overturn any major conclusions drawn from the model. To sum up, earlier optimization models were criticized for being too simple, not showing enough details, linearizing non-linear relationships, and not least for just being optimization models. Then in 1979 the Ministry of Fisheries established a working group consisting of members from the University of Iceland, the (Icelandic) Marine Research Institute and others, to develop a model of the demersal fisheries. It should be a decision support tool for the Ministry for longand short-term fisheries management. Short term management involves closing areas for fishing (depending on boat type and size), temporary bans, mesh size regulations, and quota systems. Through its control of the banking system the government also controls the investment in new vessels. Thus in the long run the fleet size and its composition is manageable.
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4. Data on the fisheries
5. A general description of the model
Extensive data on fishing in Iceland have been collected in the last few decades, more than in most other countries. Thus the data base for various modelling activities in fishing is available. The data are considered to be fairly reliable as government involvement in the transactions between fishermen and the fish processing industry has made it beneficial for both partners to report catches and other data properly. On the other hand, there is considerable randomness in the data, the main cause being the harsh and unstable weather on the fishing grounds. The data used in our model can be divided into four groups, namely: (i) Landings. Monthly records of each boat are available, containing description of the boat, number of trips, days of operation, landing harbour, gear used, and landings by species. (ii) Stock sizes. Current or past stock sizes measured in numbers of fish of the involved species classified by age are estimated. Virtual Population Analysis is the standard method for stock estimation. Members of our group have, however, developed a statistically sounder and, at least for our purpose, a more suitable method which is described in [12 and 13]. (iii) Fishing power and selectivity. The boats use different gear and operate in different areas. Hence the age (or more precisely size) pattern of the catch differs. Traditional fisheries models tend to ignore this effect but we have found it essential to take care of this difference. Therefore we have determined with statistical analysis the age selectivity for the different types, mainly the different gear. Simultanously the fishing power, i.e. the expected catch for a given unit stock size, has been estimated for different boat sizes, types, etc. More information on this estimation process is to be found in [9 and 10]. (iv) Economic data. Financial accounts of a majority of the fishing vessels are available and are used in an aggregated form to predict future losses or profits for the fleet as a whole but also in classifications by boat-size, location, etc. These data make some special studies possible, e.g. the effect of oil price changes on different types of vessels under different management plans.
In this section we review the basic formulas relating the catch to the stock size together with the dynamics of the stocks. Details are left out but we refer in general to [2, 11 and 13]. Assuming we are dealing with one species (cod, haddock, saithe, redfish, Greenland halibut or 'other fish') we define the following indices: a: age in years of a fish; t : time measured in years; g: gear (trawl, longline (and handline) and gillnet). The basic variables concerning a particular stock are: N(a, t) : stock size in numbers of a year old fish in the beginning of year t; B(g, t) : exploitable biomass for gear g in year t; sel(a, g): age-selectivity of gear g; w(a, t) : weight of a year old fish in year t. The selectivity used is actually computed for the two halves of the year separately. This refinement is left out in the current description. Furthermore, weight is usually kept independent of time t in runs of the model (see Section 3). The model on which the calculation of fishing power is based assumes that the expected catch per unit effort is directly proportional to the 'exploitable biomass', B(g, t), which is the sum of the biomasses of the year classes weighted by the exploitation pattern, i.e. selectivity:
B(g, t ) = E s e l ( a , g) * w(a, t) * N(a, t). a
Now, the expected catch of a given boat is proportional both to the exploitable biomass and the effort of the boat: Catch = constant * Effort * B. The next step involves modelling the effort. The effort of a (standard) boat is proportional to the time actually spent fishing. On the other hand, we want the number of boats, or more precisely their total operating time, to become the independent control variable. Thus we have to compute the effort as a function of the operating time. This function must include the stock itself, for evidently with increased stocks the boats become fully loaded sooner, so relatively more time is
Th. Helgason, S. Olafsson / An Icelandic fisheries model
spent on non-fishing operations such as sailing to shore. To cut a long story short, we have derived a function of the following type: Effort = constant * Operating time * B C2-1 where c 2 is a certain power which theoretically should take values between 0 and 1. Combining the last two equations we obtain the following model of the catch: Catch(g, t) = c 1 * Operating time * B ( g, t ) c2. (5.1) This formula refers to a boat of specific type and size. The power c 2 depends on the gear (g) and the season and is different for boats in the north and south (see [9]). The other constant, q , is the so-called fishing power, which we have dealt with in Section 4. In c~ we cover as much detail as is justifiable from the data. Thus, the fishing power is different for different harbour areas, etc. These constants, q and c 2, have been estimated in an elaborate regression analysis (see [9]). Actually the model involves a negative correlation between the catches of different species. The total catch, Totcatch(g, t), of the species caught in gear g is obtained by summing up the catch in a given year t (actually season). Then the fishing mortality rates, F(a, t), can be computed implicitly from:
F(a, t)/[ F(a, t ) + M ( a ) ] • [1 - e x p ( - F ( a , t) - M ( a ) ) ] = ~ s e l ( a , g ) * Totcatch(g, t ) / B ( g , t). g
Here M ( a ) refers to the natural mortality rate, which is supplied by the biologists. It may depend on the stock sizes but we have so far not incorporated any ecological feed-back of this kind. Finally we have all ingredients for updating the stock from year to year: N ( a + 1, t + l )
= N(a, t) * e x p ( - F ( a ,
t) - M(a)).
6. Prediction errors
The precision of predictions made by the model are of course of much concern. The main check on the precision was obtained by three different
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classes of simulations where the code-catches in the period 1977-1983 were predicted. In all cases the fleet is equal to the real one. On the other hand the simulations differed with respect to the uncertainty about the status of the stocks: (1) In the first simulation the 'true' biomasses were given in each year. Thus we are only testing the precision of the basic catch equation, (5.1) and not that of the stock dynamics. Even for a given stock the real catch varies, owing to weather among other things. Expected total catch based on equation (5.1) will therefore be different from the real one. (2) In the second simulation, however, the stock at the beginning of the planning period, i.e. 1977, was given. The stock was then updated from year to year according to the amount caught. Natural mortality, recruitment (including some immigration of mature fish), and the weight of fish at age were all given as in case (1). This simulation tests how the errors in the estimation of fishing power spread through time, owing to the dynamics of the stock. (3) Finally we simulated a realistic prediction where nothing is known about the future state of the system, so it is assumed to remain constant. Thus weight-at-age was fixed at the 1977 values, recruitment was set constant at an average value, and no immigration assumed. This simulation is, however, unrealistic in so far as the initial stock is now better known than it would have been in 1977. The average error in annual total catch is about 2% in the first test and about 5% in the other two. As is to be expected, it gradually increases in the second simulation where the stock is dynamic, reaching about 10% at the end of the period. Figure 2 shows the annual cod catch in the third and most realistic prediction in comparison with the real catch. The model is used for disaggregated predictions. For example, we predict the catch for the main boat and gear types (gillnet, line and trawlboats and stern trawlers) separately. The (weighted) average error in these categories over the 7 years is 8%, 13% and 14% for the three tests respectively. We finally remark that the absolute error is not the concern. The model is used mainly to compare different policies influencing the fleet size and composition. Biases will normally have the same
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Annual
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sign in different runs, thus causing little harm in such comparative studies.
7. Examples of experiments with the model As said before, the model has been used both in long term studies and for predictions one or two years ahead (e.g. when the quota system was being tested). In the last two years, it has mainly been used in long term planning and we will only present results from such experiments here. We begin by describing how the model is operated, i.e. what basic options the user has and what choices he can and must make. Then we report on certain experiments made at the request of the Icelandic authorities.
7.1. Running the model The model is purely deterministic. First, the base year is selected and then the model computes the expected catch, economic outcome and other statistics, year by year, for a given number of years. The main control variables govern the size and composition of the fishing fleet from year to year. Care has been taken to make it easy for the user to play with these controls. The relevant parameters are inputted in a spreadsheet form.
Voluminous output is a problem with large, detailed models. Our model was designed before good graphic facilities became available to us so the output is in the form of tables, but the user can choose from menus how many tables, and how detailed he wishes to see. He can get the catch for each boat size of each of the five species in both seasons. Also, he can get the distribution of the total catch by months and areas. Further, income, expenditure and profits for the various size-classes of boats are supplied. Finally, the stock sizes and fishing mortality can be obtained.
7.2. Sustainable yieM under different policies A government committee on planning the fishing fleet has used the model to compare sustainable yield for different fleet size and types. Sustainable yield refers to the equilibrium catch for a given constant effort and standard environmental factors like average recruitment. Yield is of course never sustainable in practice due to all kinds of environmental variations. Nevertheless, sustainable yield (or more appropriately: sustainable profit) is generally considered in fisheries management to be the proper indicator for comparing the benefits of different policies. The year 1983 is used as the base year in these studies. Thus the fleet, the recruitment, and the weights are kept constant from 1983 onwards.
Th. Helgason, S. Olafsson / A n Icelandic fisheries model
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Figure 3. Sust~nable yield of cod as a function of the size of the fishing fleet relative to the 1983 fleet
With this choice, fishing power and other related parameters refer to the situation prior to the introduction of the quota management system and hence refer to more or less unrestricted fishing. After running the model through approximately 10 years the sustainable situation has been reached. The main conclusion from these studies is that the fleet is too large. Not only could the present catch be taken with a smaller expended effort but also is there some possibility of simultaneously increasing the catch. However, with a smaller fleet,
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the stocks will increase. Hence there is a possibility that lack of food would affect the growth rate. Unfortunately the current biological knowledge does not establish this potential relationship. This was discussed in Section 3. Following is a short summary of a few experiments with different fleet sizes.
7.2.1. General reduction of the fishing fleet In most fisheries models basically only the total effort is controllable, i.e. the total fleet size. This can also be done with our model. Thus
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Th. Helgason, S. ~)lafsson / An Icelandic fisheries model
sustainable yield can be calculated as a function of the fleet size, with the implicit assumption that the ratio between different units of boats is kept constant. Figure 3 shows sustainable yield of cod as a function of the size of the fishing fleet relative to the 1983 fleet. The figure reveals that the maximum sustainable yield can be achieved with a fleet which is 15-20% smaller than the 1983 fleet. Note, however, how flat the curve is. The catch would be practically the same for a fleet in the size range of 60% to 120% of the 1983 fleet. On the other hand, the profit curve (not shown) has a more distinctive maximum achieved with a considerable reduction of the fleet. The sensitivity of this result to different assumptions about external parameters was tested in various ways. Below in Section 7.3 we deal with one such test: Effects of variability in the recruitment. Although changes in these parameters (within reasonable limits) do move the maximum of the profit curve back and forth, is never reversed the main conclusion that the current fishing fleet is too large.
7.2.2. Changes in the composition of the fleet Although a general reduction of the effort will in the long run increase catches and profits in the fishing industry, such uniform change may not give the best global result a n d / o r may have unacceptable side-effects. The latter issue was of more concern to the relevant government committee; mainly because of the fact that the effort reduction will to some extent shift the landings from the north to the south, which, for employment reasons, is considered unfeasible. If on the other hand total catch or profit maximization is the target the fleet composition should be altered differently. Common sense together with some experiments show that it is mainly the trawler fleet that should be reduced, the main reason being that the trawlers catch younger fish than the boats. The reason for this is that the young fish are generally in the north but move to the south when they grow older. So far we have used catches as a measure of the quality of different fishing fleets. But as said earlier the model incorporates calculations of economic factors. It turns out that the boats benefit much more from a reduction in the effort than the trawlers do. On the other hand the most efficient measure to increase the stocks consists of reducing
the trawler fleet. This contradiction is one of the main obstacles to the implementation of the policy suggested by this model. Furthermore there are arguments for maintaining a certain minimum of trawlers since they are more stable suppliers of fish to the processing industry than boats whose catch has more seasonal variation.
7.3. Random recruitment In the experiments reported in Section 7.2 all external factors are assumed to be constant in time. In reality they are highly random, however, mainly the recruitment of the fish. Having selected management policies that look favourable as regards sustainable yield it is at least of secondary interest to see whether these policies behave differently in their ability to reduce the variance in the catches. High variability in the catches causes unwanted pulsations in the whole economy in a nation which depends so heavily on fishing as does Iceland. (See also [2] in this connection). A simple experiment in this direction was made to compare the variation in yield of the current fleet with a fleet uniformly reduced by one fourth. These two fleet sizes lead to roughly the same sustainable yield (see Figure 3). The recruitment was randomized with a uniform distribution based on real figures from the last two decades. Figure 4 shows the cod catch over a period of 13 years with these two fleets and for two different series of random recruitment. It so happens that one of the series (A in Figure 4) corresponds to a favourable period of recruitment, whereas in the other (B) the recruitment is rather low most of the time. As seen from the figure, the variation is considerably lower with the smaller fleet. This is to be expected as with a smaller fleet the fish stocks are larger and one particular yearclass matters less.
8. Concluding remarks This paper outlines a model which has been used as a decision support tool in fisheries management in Iceland. Many of the technical details have been left out together with spin-off products which are of interest per se. Thus the development of the model necessitated a revision of stock size estimation which then was independently devel-
Th. Helgason, S. ¢9lafsson / An Icelandic fisheries model
oped further (see [12 and 13]). On the other hand we have dealt to some extent with the history of this model, its political background and environment. It is our experience that mathematical models are rather hard to sell, in particular models of imperative character like optimization models. It is in the nature of every decision maker to be sceptical about advice coming from something he feels is nothing more than a 'black box'. This applies generally to any kind of optimization. 'Decision support systems' does sound better in the ears of politicians than 'optimization models'. Long term planning involving renewable natural resources is always problematic because the uncertainties are many and badly predictable. Under these circumstances the OR scientist should concentrate on comparisons of different policies rather than seeking predictions of absolute values. In these comparisons sensitivity analysis are of vital importance. If a particular policy turns out to be better than another one in all or most simulated cases, the former must look more promising to implement. The situation for the politician, i.e. the decision maker here, is in a sense a contrast to this. When he has to explain or rationalize his decisions he is generally forced to present some absolute values. For this reason recommendations obtained from such numeric comparisions of relative superiority are often mistaken for predictions about absolute values by the politician. We have overcome this problem on the one hand by working mainly with sustainable yield, i.e. in stationary situations. But we have also found it to be instructive to make predictions retrospectively, i.e. to predict past experience. The user may be uneasy in predictions about the future due to uncontrollable uncertainties like recruitment or growth rate of fish. On the other hand he is more likely to accept actual values for past periods although he is experimenting with changes which do affect stock size and hence indirectly these same parameters. The OR practitioner, eager to have his models implemented, must have patience and take into account the learning process the user must go through. Having gained experience with simple
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and visual tools the manager may, however, become willing to consider optimal policies.
References [1] Andersen, K.P., and Ursin, Erik, "A multispecies extension to the Beverton and Holt theory, of fishing, with
[2]
[3]
[4] [5] [6] [7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
accounts of phosphorus circulation and primary production", Meddelelser fra Danmarks Fiskeri- og Havundersogelser, N.S. 7 (1977) 319-435. Anonymous, "Size of the cod stock in Iceland, effort and fishing power in the period 1972-1979' (in Icelandic), working paper, 1982. Arnason, R., "Efficient harvesting of fish stocks: The case of the Icelandic demersal fisheries", Ph.D. thesis, Univ. British Columbia, Dept. of Economics, 1984. Clark, C.W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, 1986. Hannesson, R., Economics of Fisheries: Some Problems of Efficiency, Ph.D. thesis, Studentlitteratur, Lund, 1974. Helgason, Th., "On optimal fishing effort for the Icelandic cod" (in Icelandic), Aegir 7 (1979) 654-668. Helgason, Th., "Optimal fishing patterns for Icelandic
cod", in: D.C. Chapman and V.G. Gallucci (eds.), Quantitative Population Dynamics, International Co-operative Publishing House, Fairland, MA, 1981 243-265. Helgason, Th., and Gislason, H., "VPA-analysis with species interaction due to predation", ICES Report C.M. 1979/G:52, 1979. Helgason, Th., and Kenward, M.G., "Estimation of fishing power with relation to exploited biomass", ICES-Report C.M. 1985/D:7, 1985. Garrod, D.J., and Shepherd, J.G., "On the relationship between fishing capacity and resource allocation", in: K. Brian Haley (ed.), Applied Operations Research in Fishing, Plenum Press, 1981, 321-333. Gislason, H., and Helgason, Th., "Species interaction in assessment of fish stocks with special application to the North Sea", Dana 5 (1985) 1-44. Gudmundsson, G., "Statistical considerations in the analysis of catch-at-age observations", J. Cons. Int. Explor. Mer. 43 (1986). Gudmundsson, G., Helgason, Th., and Schopka, S., "Statistical estimation of fishing effort and mortality by gear and season for the Icelandic cod fishery in the period 1972-1979", ICES Report C.M. 1982/G:29, 1982. Gudmundsson, G., and Helgason, Th., "The effect of uncertainty on the management of Icelandic cod fisheries", presented at the Symposium on Resource Management under Uncertainty at the Chr. Michelsens Institutet,
Bergen, 1983. [15] Magnus, R.J., and Magnusson, K.G., "Existence and uniqueness of solutions to the multi-species VPA-equations", ICES-Report C.M. 1983/D:2, 1983.
191
An Icelandic fisheries model Thorkell HELGASON
and Snjrlfur OLAFSSON
Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland Abstract: A model has been developed for the Ministry of Fisheries in Iceland which is being used as a decision support tool in fisheries management. With this model predictions can be made about catches of cod and other demersal species several years ahead. Furthermore earnings and costs in the fisheries are calculated. The starting point is the size of the fish stocks in the beginning of the planning period together with the size and classification of the fishing fleet. Changes in the fleet composition are among the control variables. Emphases are on details such as geographical distribution of the catches. The model is used by the Ministry, both in short term management regarding quota systems and for long term investment planning. Results suggest that in the long run the fishing fleet is now far too big. Parameters for the model are obtained from an intensive data base on catches of every boat for every month in the last decade, disaggregated by species and landing harbour. Parallel to this model a new method of stock size estimation was developed. The current model was preceded by an optimization model which was not politically accepted! The paper touches on the problems of technical consultation on a sensitive political issue. Keywords: Decision support systems, fisheries, fisheries management, government, modelling, planning
1. Introduction This paper deals with a model of the Icelandic fisheries. In Section 2 we survey the background and history of fishing in Iceland. In Section 3 we deal with basic ideas in fisheries modelling and our previous attempts in this direction. Section 4 describes the data used and in section 5 the main model is explained in some detail. Section 6 deals with prediction errors. Then in Section 7 we explain how the model is used and mention some results from the model. Finally in Section 8 we discuss the use of models in political decision making. The sections need not be read in consecutive order.
2. Background Fishing is the backbone of the Icelandic economy. About 70% of Iceland's exports are fish and fish products. The fisheries can be divided into three main categories by fish species. Most important are demersal species. Of those, cod is the Received September 1986; revised January 1987
major one, accounting for 55% of the demersal catch. The second category includes pelagic fish, mainly capelin. The third category covers all kinds of shellfish. There is little overlap between these categories as, for the most part, each fishery is carried out by specialized boats. This paper focuses on the demersal fisheries. Figure 1 shows the development of demersal fisheries in Icelandic waters over the past few decades. In the year 1976 there was a turning point. Until then foreign vessels had been taking about half of the total catch. But then Iceland gained full sovereignty over her fishing grounds. Anticipating this development, Icelandic fishing companies had already started to modernize and increase their trawler fleet. Around 1970, trawlers played an insignificant role in Icelandic fishing, but then, in five years or so, a large and modern fleet of stern trawlers was acquired. Now they number over 100 and account for about half of the demersal catch, the remainder being taken by some 700 boats of different sizes. One of Iceland's main arguments in the dispute over the fishing limits concerned stock protection. Reliable stock size estimation methods emerged rather late. The important VPA-method was intro-
0377-2217/88/$3.50 © 1988, ElsevierScience Publishers B.V. (North-Holland)
Th. Helgason, S. Olafsson / An Icelandic fisheries mode/
192 YIELD
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duced in 1949 but was first applied in the sixties and used for stock estimation in Iceland in the early seventies. Stock size estimates done in 1975 showed that the cod stock was down to less than half of its average size for the post-war period. Early prediction models (e.g. [5]) also show that the applied effort was about twice the maximum yield effort and eventually three times the economically optimal effort. Furthermore the age structure of the catch was unfavourable. Most of this was blamed on a fleet of up to several hundred foreign trawlers fishing in Icelandic waters. Foreign fishing vessels disappeared almost entirely after 1976. With great optimism but little planning they were replaced by a new domestic fleet of trawlers. Gradually it was realized that the total fishing effort had hardly decreased in spite of the disappearance of the foreign trawlers. The size of the stock and consequently of the catch did not improve. This fact was obscured somewhat by one exceptionally good year-class of cod and at the same time an unexpected immigration of cod from Greenland which together boosted the catch in 1981-1983. Nevertheless the cod catch reached an all-time low in 1983. The authorities and the fishing industry gradually realized that the fishing fleet and hence the effort expended was too large. In the late seventies measures were taken to slow down and then gradually halt further expansion of the fleet. Secondly, the effort expended was restrained. This was initially done by restricting the fishing period, both in general (by enforcing an extended Christmas
and Easter vacation for fishermen!) and also by setting a ceiling on the allowed annual operating time of each trawler. Then in 1984 a general quota system was introduced. 3. Fishing models Models of renewable resources are numerous in mathematical economics (see e.g. [4]). Computer models of real fishing situations have also been developed in many places, e.g. [10]. Most such models are focussed around the population dynamics of the species and fishing effort is measured as fishing mortality rate. The link between fishing mortality and physical quantities of effort (such as number of boats) causes problems. Consequently it is also difficult to assign costs to the fishing operation. Several models of this kind had been developed for the Icelandic demersal fisheries prior to 1980 (see e.g. [5]). One of the present authors constructed a dynamic optimization model of this type, [6 and 7]. It has two annual decision controls, one being attributed to total effort in terms of fishing mortalities. The other parameter controls the age pattern of the fishing mortalities and can therefore be interpreted as an indicator of mesh size or other measures to protect the young fish. The objective function of the model sums up the present value of total profits in the fishing industry over an infinite planning horizon. A later version of this model takes account of randomness in fish
Th. Helgason, S. Olafsson / A n Icelandic fisheries model
recruitment and in the actual catch for a given (expected) fishing mortality [6]. This and other similar models showed it to be optimal to cut the total effort down to about half of the actual effort in 1977. These models were criticized by the fishing industry and politicians on several grounds. Their comments can be summarized as follows: (1) Fishing effort is not a global one-dimensional quantity. It is generated by hundreds of different boats. Any management policy will affect the individual boats differently. Therefore total fishing effort is meaningless as an independent control parameter. (2) Although the fleet may be too big in general, a global reduction is economically and politically unfeasible. In some areas employment considerations even call for more boats. The models lack disaggregation and details to take care of this. (3) The linear relationship between stock size and catch is too naive. Doubling of the (catchable) stock may cause a doubling of the catch per trawling hour (say), but then relatively more time is spent on non-fishing operations like sailing and landing. Hence the catch per boat will not double. (4) Ecological feed-back is ignored in the models. The benefits of effort reduction come through stock increase. The calculations are based on the assumption that the growth rate of individual fish is independent of the stock size. This is unrealistic. Limited food supply may hamper the growth rate. Hence the beneficial effects of reduced effort are over-estimated in the long run. (5) The models in question are optimization models, where the objective is the present value of an infinite stream of net profits. A multi-dimensional goal function cannot be reduced down to one-dimensional profit. Also, discounting future earnings in order to lump them together over-simplifies the (political) facts of life. Optimization models are far too monolithic for planning an entire national economy. The last argument certainly applies to a variety of models, not only fisheries models. It is increasingly our belief that optimization is largely out of place in global models. This is not an argument against using optimization in, say, blending or product-mix problems. Optimization has the flavour of commanding a decision and in areas like ours the decision maker does not like to be commanded; at most he seeks guidance or more
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likely decision aid. It is therefore not surprising that OR practitioners have more and more adopted the term 'decision support system' for their models (and even christened old optimization models so!). The criticism under (1) and (2) above are also largely politically motivated. Political decisions are very seldom of a general nature. It lies in the foundation of democratic, pluralistic societies that even minor consequences cannot be ignored or at least not overseen. Hence political decision support models must display details to as large an extent as the model maker can possibly justify on the basis of his data. Critical remark (3) is well founded and, as will be seen later, was extensively dealt with and resolved in the model described in this paper. Remark no. 4 on ecological effects is more difficult to handle. Both in Iceland and abroad much effort has been spent on modelling the relationship between growth rate and stock size. The extensive 'North Sea' model is of this kind. (See [1], and also [8,11,15]). In an another Icelandic fisheries model [3] such relationship is derived and shown to have a noticeable effect on the optimal policy resulting in an increase in the optimal effort. We have so far not incorporated such an ecological relationship permanently in our model. We have however tested some hypothetical versions of a stock dependent growth rate and thereby seen that, whereas the relationship may have some effect, it seems not to overturn any major conclusions drawn from the model. To sum up, earlier optimization models were criticized for being too simple, not showing enough details, linearizing non-linear relationships, and not least for just being optimization models. Then in 1979 the Ministry of Fisheries established a working group consisting of members from the University of Iceland, the (Icelandic) Marine Research Institute and others, to develop a model of the demersal fisheries. It should be a decision support tool for the Ministry for longand short-term fisheries management. Short term management involves closing areas for fishing (depending on boat type and size), temporary bans, mesh size regulations, and quota systems. Through its control of the banking system the government also controls the investment in new vessels. Thus in the long run the fleet size and its composition is manageable.
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4. Data on the fisheries
5. A general description of the model
Extensive data on fishing in Iceland have been collected in the last few decades, more than in most other countries. Thus the data base for various modelling activities in fishing is available. The data are considered to be fairly reliable as government involvement in the transactions between fishermen and the fish processing industry has made it beneficial for both partners to report catches and other data properly. On the other hand, there is considerable randomness in the data, the main cause being the harsh and unstable weather on the fishing grounds. The data used in our model can be divided into four groups, namely: (i) Landings. Monthly records of each boat are available, containing description of the boat, number of trips, days of operation, landing harbour, gear used, and landings by species. (ii) Stock sizes. Current or past stock sizes measured in numbers of fish of the involved species classified by age are estimated. Virtual Population Analysis is the standard method for stock estimation. Members of our group have, however, developed a statistically sounder and, at least for our purpose, a more suitable method which is described in [12 and 13]. (iii) Fishing power and selectivity. The boats use different gear and operate in different areas. Hence the age (or more precisely size) pattern of the catch differs. Traditional fisheries models tend to ignore this effect but we have found it essential to take care of this difference. Therefore we have determined with statistical analysis the age selectivity for the different types, mainly the different gear. Simultanously the fishing power, i.e. the expected catch for a given unit stock size, has been estimated for different boat sizes, types, etc. More information on this estimation process is to be found in [9 and 10]. (iv) Economic data. Financial accounts of a majority of the fishing vessels are available and are used in an aggregated form to predict future losses or profits for the fleet as a whole but also in classifications by boat-size, location, etc. These data make some special studies possible, e.g. the effect of oil price changes on different types of vessels under different management plans.
In this section we review the basic formulas relating the catch to the stock size together with the dynamics of the stocks. Details are left out but we refer in general to [2, 11 and 13]. Assuming we are dealing with one species (cod, haddock, saithe, redfish, Greenland halibut or 'other fish') we define the following indices: a: age in years of a fish; t : time measured in years; g: gear (trawl, longline (and handline) and gillnet). The basic variables concerning a particular stock are: N(a, t) : stock size in numbers of a year old fish in the beginning of year t; B(g, t) : exploitable biomass for gear g in year t; sel(a, g): age-selectivity of gear g; w(a, t) : weight of a year old fish in year t. The selectivity used is actually computed for the two halves of the year separately. This refinement is left out in the current description. Furthermore, weight is usually kept independent of time t in runs of the model (see Section 3). The model on which the calculation of fishing power is based assumes that the expected catch per unit effort is directly proportional to the 'exploitable biomass', B(g, t), which is the sum of the biomasses of the year classes weighted by the exploitation pattern, i.e. selectivity:
B(g, t ) = E s e l ( a , g) * w(a, t) * N(a, t). a
Now, the expected catch of a given boat is proportional both to the exploitable biomass and the effort of the boat: Catch = constant * Effort * B. The next step involves modelling the effort. The effort of a (standard) boat is proportional to the time actually spent fishing. On the other hand, we want the number of boats, or more precisely their total operating time, to become the independent control variable. Thus we have to compute the effort as a function of the operating time. This function must include the stock itself, for evidently with increased stocks the boats become fully loaded sooner, so relatively more time is
Th. Helgason, S. Olafsson / An Icelandic fisheries model
spent on non-fishing operations such as sailing to shore. To cut a long story short, we have derived a function of the following type: Effort = constant * Operating time * B C2-1 where c 2 is a certain power which theoretically should take values between 0 and 1. Combining the last two equations we obtain the following model of the catch: Catch(g, t) = c 1 * Operating time * B ( g, t ) c2. (5.1) This formula refers to a boat of specific type and size. The power c 2 depends on the gear (g) and the season and is different for boats in the north and south (see [9]). The other constant, q , is the so-called fishing power, which we have dealt with in Section 4. In c~ we cover as much detail as is justifiable from the data. Thus, the fishing power is different for different harbour areas, etc. These constants, q and c 2, have been estimated in an elaborate regression analysis (see [9]). Actually the model involves a negative correlation between the catches of different species. The total catch, Totcatch(g, t), of the species caught in gear g is obtained by summing up the catch in a given year t (actually season). Then the fishing mortality rates, F(a, t), can be computed implicitly from:
F(a, t)/[ F(a, t ) + M ( a ) ] • [1 - e x p ( - F ( a , t) - M ( a ) ) ] = ~ s e l ( a , g ) * Totcatch(g, t ) / B ( g , t). g
Here M ( a ) refers to the natural mortality rate, which is supplied by the biologists. It may depend on the stock sizes but we have so far not incorporated any ecological feed-back of this kind. Finally we have all ingredients for updating the stock from year to year: N ( a + 1, t + l )
= N(a, t) * e x p ( - F ( a ,
t) - M(a)).
6. Prediction errors
The precision of predictions made by the model are of course of much concern. The main check on the precision was obtained by three different
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classes of simulations where the code-catches in the period 1977-1983 were predicted. In all cases the fleet is equal to the real one. On the other hand the simulations differed with respect to the uncertainty about the status of the stocks: (1) In the first simulation the 'true' biomasses were given in each year. Thus we are only testing the precision of the basic catch equation, (5.1) and not that of the stock dynamics. Even for a given stock the real catch varies, owing to weather among other things. Expected total catch based on equation (5.1) will therefore be different from the real one. (2) In the second simulation, however, the stock at the beginning of the planning period, i.e. 1977, was given. The stock was then updated from year to year according to the amount caught. Natural mortality, recruitment (including some immigration of mature fish), and the weight of fish at age were all given as in case (1). This simulation tests how the errors in the estimation of fishing power spread through time, owing to the dynamics of the stock. (3) Finally we simulated a realistic prediction where nothing is known about the future state of the system, so it is assumed to remain constant. Thus weight-at-age was fixed at the 1977 values, recruitment was set constant at an average value, and no immigration assumed. This simulation is, however, unrealistic in so far as the initial stock is now better known than it would have been in 1977. The average error in annual total catch is about 2% in the first test and about 5% in the other two. As is to be expected, it gradually increases in the second simulation where the stock is dynamic, reaching about 10% at the end of the period. Figure 2 shows the annual cod catch in the third and most realistic prediction in comparison with the real catch. The model is used for disaggregated predictions. For example, we predict the catch for the main boat and gear types (gillnet, line and trawlboats and stern trawlers) separately. The (weighted) average error in these categories over the 7 years is 8%, 13% and 14% for the three tests respectively. We finally remark that the absolute error is not the concern. The model is used mainly to compare different policies influencing the fleet size and composition. Biases will normally have the same
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sign in different runs, thus causing little harm in such comparative studies.
7. Examples of experiments with the model As said before, the model has been used both in long term studies and for predictions one or two years ahead (e.g. when the quota system was being tested). In the last two years, it has mainly been used in long term planning and we will only present results from such experiments here. We begin by describing how the model is operated, i.e. what basic options the user has and what choices he can and must make. Then we report on certain experiments made at the request of the Icelandic authorities.
7.1. Running the model The model is purely deterministic. First, the base year is selected and then the model computes the expected catch, economic outcome and other statistics, year by year, for a given number of years. The main control variables govern the size and composition of the fishing fleet from year to year. Care has been taken to make it easy for the user to play with these controls. The relevant parameters are inputted in a spreadsheet form.
Voluminous output is a problem with large, detailed models. Our model was designed before good graphic facilities became available to us so the output is in the form of tables, but the user can choose from menus how many tables, and how detailed he wishes to see. He can get the catch for each boat size of each of the five species in both seasons. Also, he can get the distribution of the total catch by months and areas. Further, income, expenditure and profits for the various size-classes of boats are supplied. Finally, the stock sizes and fishing mortality can be obtained.
7.2. Sustainable yieM under different policies A government committee on planning the fishing fleet has used the model to compare sustainable yield for different fleet size and types. Sustainable yield refers to the equilibrium catch for a given constant effort and standard environmental factors like average recruitment. Yield is of course never sustainable in practice due to all kinds of environmental variations. Nevertheless, sustainable yield (or more appropriately: sustainable profit) is generally considered in fisheries management to be the proper indicator for comparing the benefits of different policies. The year 1983 is used as the base year in these studies. Thus the fleet, the recruitment, and the weights are kept constant from 1983 onwards.
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Figure 3. Sust~nable yield of cod as a function of the size of the fishing fleet relative to the 1983 fleet
With this choice, fishing power and other related parameters refer to the situation prior to the introduction of the quota management system and hence refer to more or less unrestricted fishing. After running the model through approximately 10 years the sustainable situation has been reached. The main conclusion from these studies is that the fleet is too large. Not only could the present catch be taken with a smaller expended effort but also is there some possibility of simultaneously increasing the catch. However, with a smaller fleet,
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the stocks will increase. Hence there is a possibility that lack of food would affect the growth rate. Unfortunately the current biological knowledge does not establish this potential relationship. This was discussed in Section 3. Following is a short summary of a few experiments with different fleet sizes.
7.2.1. General reduction of the fishing fleet In most fisheries models basically only the total effort is controllable, i.e. the total fleet size. This can also be done with our model. Thus
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sustainable yield can be calculated as a function of the fleet size, with the implicit assumption that the ratio between different units of boats is kept constant. Figure 3 shows sustainable yield of cod as a function of the size of the fishing fleet relative to the 1983 fleet. The figure reveals that the maximum sustainable yield can be achieved with a fleet which is 15-20% smaller than the 1983 fleet. Note, however, how flat the curve is. The catch would be practically the same for a fleet in the size range of 60% to 120% of the 1983 fleet. On the other hand, the profit curve (not shown) has a more distinctive maximum achieved with a considerable reduction of the fleet. The sensitivity of this result to different assumptions about external parameters was tested in various ways. Below in Section 7.3 we deal with one such test: Effects of variability in the recruitment. Although changes in these parameters (within reasonable limits) do move the maximum of the profit curve back and forth, is never reversed the main conclusion that the current fishing fleet is too large.
7.2.2. Changes in the composition of the fleet Although a general reduction of the effort will in the long run increase catches and profits in the fishing industry, such uniform change may not give the best global result a n d / o r may have unacceptable side-effects. The latter issue was of more concern to the relevant government committee; mainly because of the fact that the effort reduction will to some extent shift the landings from the north to the south, which, for employment reasons, is considered unfeasible. If on the other hand total catch or profit maximization is the target the fleet composition should be altered differently. Common sense together with some experiments show that it is mainly the trawler fleet that should be reduced, the main reason being that the trawlers catch younger fish than the boats. The reason for this is that the young fish are generally in the north but move to the south when they grow older. So far we have used catches as a measure of the quality of different fishing fleets. But as said earlier the model incorporates calculations of economic factors. It turns out that the boats benefit much more from a reduction in the effort than the trawlers do. On the other hand the most efficient measure to increase the stocks consists of reducing
the trawler fleet. This contradiction is one of the main obstacles to the implementation of the policy suggested by this model. Furthermore there are arguments for maintaining a certain minimum of trawlers since they are more stable suppliers of fish to the processing industry than boats whose catch has more seasonal variation.
7.3. Random recruitment In the experiments reported in Section 7.2 all external factors are assumed to be constant in time. In reality they are highly random, however, mainly the recruitment of the fish. Having selected management policies that look favourable as regards sustainable yield it is at least of secondary interest to see whether these policies behave differently in their ability to reduce the variance in the catches. High variability in the catches causes unwanted pulsations in the whole economy in a nation which depends so heavily on fishing as does Iceland. (See also [2] in this connection). A simple experiment in this direction was made to compare the variation in yield of the current fleet with a fleet uniformly reduced by one fourth. These two fleet sizes lead to roughly the same sustainable yield (see Figure 3). The recruitment was randomized with a uniform distribution based on real figures from the last two decades. Figure 4 shows the cod catch over a period of 13 years with these two fleets and for two different series of random recruitment. It so happens that one of the series (A in Figure 4) corresponds to a favourable period of recruitment, whereas in the other (B) the recruitment is rather low most of the time. As seen from the figure, the variation is considerably lower with the smaller fleet. This is to be expected as with a smaller fleet the fish stocks are larger and one particular yearclass matters less.
8. Concluding remarks This paper outlines a model which has been used as a decision support tool in fisheries management in Iceland. Many of the technical details have been left out together with spin-off products which are of interest per se. Thus the development of the model necessitated a revision of stock size estimation which then was independently devel-
Th. Helgason, S. ¢9lafsson / An Icelandic fisheries model
oped further (see [12 and 13]). On the other hand we have dealt to some extent with the history of this model, its political background and environment. It is our experience that mathematical models are rather hard to sell, in particular models of imperative character like optimization models. It is in the nature of every decision maker to be sceptical about advice coming from something he feels is nothing more than a 'black box'. This applies generally to any kind of optimization. 'Decision support systems' does sound better in the ears of politicians than 'optimization models'. Long term planning involving renewable natural resources is always problematic because the uncertainties are many and badly predictable. Under these circumstances the OR scientist should concentrate on comparisons of different policies rather than seeking predictions of absolute values. In these comparisons sensitivity analysis are of vital importance. If a particular policy turns out to be better than another one in all or most simulated cases, the former must look more promising to implement. The situation for the politician, i.e. the decision maker here, is in a sense a contrast to this. When he has to explain or rationalize his decisions he is generally forced to present some absolute values. For this reason recommendations obtained from such numeric comparisions of relative superiority are often mistaken for predictions about absolute values by the politician. We have overcome this problem on the one hand by working mainly with sustainable yield, i.e. in stationary situations. But we have also found it to be instructive to make predictions retrospectively, i.e. to predict past experience. The user may be uneasy in predictions about the future due to uncontrollable uncertainties like recruitment or growth rate of fish. On the other hand he is more likely to accept actual values for past periods although he is experimenting with changes which do affect stock size and hence indirectly these same parameters. The OR practitioner, eager to have his models implemented, must have patience and take into account the learning process the user must go through. Having gained experience with simple
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and visual tools the manager may, however, become willing to consider optimal policies.
References [1] Andersen, K.P., and Ursin, Erik, "A multispecies extension to the Beverton and Holt theory, of fishing, with
[2]
[3]
[4] [5] [6] [7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
accounts of phosphorus circulation and primary production", Meddelelser fra Danmarks Fiskeri- og Havundersogelser, N.S. 7 (1977) 319-435. Anonymous, "Size of the cod stock in Iceland, effort and fishing power in the period 1972-1979' (in Icelandic), working paper, 1982. Arnason, R., "Efficient harvesting of fish stocks: The case of the Icelandic demersal fisheries", Ph.D. thesis, Univ. British Columbia, Dept. of Economics, 1984. Clark, C.W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, 1986. Hannesson, R., Economics of Fisheries: Some Problems of Efficiency, Ph.D. thesis, Studentlitteratur, Lund, 1974. Helgason, Th., "On optimal fishing effort for the Icelandic cod" (in Icelandic), Aegir 7 (1979) 654-668. Helgason, Th., "Optimal fishing patterns for Icelandic
cod", in: D.C. Chapman and V.G. Gallucci (eds.), Quantitative Population Dynamics, International Co-operative Publishing House, Fairland, MA, 1981 243-265. Helgason, Th., and Gislason, H., "VPA-analysis with species interaction due to predation", ICES Report C.M. 1979/G:52, 1979. Helgason, Th., and Kenward, M.G., "Estimation of fishing power with relation to exploited biomass", ICES-Report C.M. 1985/D:7, 1985. Garrod, D.J., and Shepherd, J.G., "On the relationship between fishing capacity and resource allocation", in: K. Brian Haley (ed.), Applied Operations Research in Fishing, Plenum Press, 1981, 321-333. Gislason, H., and Helgason, Th., "Species interaction in assessment of fish stocks with special application to the North Sea", Dana 5 (1985) 1-44. Gudmundsson, G., "Statistical considerations in the analysis of catch-at-age observations", J. Cons. Int. Explor. Mer. 43 (1986). Gudmundsson, G., Helgason, Th., and Schopka, S., "Statistical estimation of fishing effort and mortality by gear and season for the Icelandic cod fishery in the period 1972-1979", ICES Report C.M. 1982/G:29, 1982. Gudmundsson, G., and Helgason, Th., "The effect of uncertainty on the management of Icelandic cod fisheries", presented at the Symposium on Resource Management under Uncertainty at the Chr. Michelsens Institutet,
Bergen, 1983. [15] Magnus, R.J., and Magnusson, K.G., "Existence and uniqueness of solutions to the multi-species VPA-equations", ICES-Report C.M. 1983/D:2, 1983.