ICES mar. Sei. Symp., 193: 1. 1991

Multispecies Models Relevant to Management of Living Resources

Preface

Scientific quality is undoubtedly the most important bers of the Steering Committee. In addition, they would aspect of a contribution to a symposium, but the value of like to thank the referees of the papers selected for a contribution is also greatly heightened by a vivid and publication for their invaluable contribution and ex­ stirring presentation, and the discussion it stimulates. In pertise in formulating many suggestions for improve­ order to underline the importance of the latter in com­ ment. It seems only appropriate that the names of these municating scientific results, it was decided to give scientists be listed here: R. S. Bailey, N. J. Bax, W. awards for the “best presentations”. On the basis of a Brugge, S. Clark, E. B. Cohen, W. Dekker, W. Gabriel, plenary vote, the awards were presented to H. Gislason S. Garcia, H. Gislason, J. Gulland, T. Helgason, J. R. (paper) and to M. Tasker, R. Furness, M. Harris, and G. Hislop, M. Holden, E. Houde, G. R. Lilly, J. R. S. Bailey (poster) for their excellent use of audio­ McGlade, G. Magnusson, R. Marasco, B. M. van der visual aids. Meer, S. Mehl, B. Mesnil, S. A. Murawski, W. J. The Co-conveners would like to express their grati­ Overholtz, O. K. Pâlsson, D. Pauly, E. K. Pikitch, J. G. tude to J. Harwood (UK), A. Laurec (France), J. G. Pope, J. E. Powers, J. C. Rice, A. Rosenberg, B. J. Pope (UK), and H. Sparholt (Denmark), who put great Rothschild, K. J. Sainsbury, F. M. Serchuk, T. Smith, effort into the organization of the Symposium as mem­ H. Sparholt, W. Stobo, K. J. Sullivan, and C. Tucker.

1 I. Multispecies virtual population analysis

ICES mar. Sei. Symp., 193: 12-21. 1991

Introduction to multispecies virtual population analysis

Per Sparre

Sparre, P. 1991. Introduction to multispecies virtual population analysis. - ICES mar. Sei. Symp., 193: 12-21.

Starting from single-species virtual population analysis, the basic features of multi­ species virtual population analysis are introduced, with emphasis on the mathematical description and computational procedures.

Per Sparre: Marine Resources Service, Food and Agriculture Organization, Via delle Terme di Caracalla, 00100 Rome, Italy.

Introduction C = F x N

Multispecies virtual population analysis (MSVPA) has where C is the catch in numbers from a cohort, F is the been developed largely as a joint venture within the fishing mortality coefficient, and N is the average num­ ICES Ad hoc Multispecies Working Group (Anon., ber of survivors from the cohort. 1984a, b, 1986, 1987, 1988). The method integrates the The same mathematical expression applies in the case suitability concept of Andersen and Ursin (1977) for of the number of deaths caused by natural causes (D), describing predator-prey relationships with traditional i.e. all causes other than fishing (predation, disease, virtual population analysis (VPA) as developed by Fry starvation, spawning stress, old age, parasites, etc.) (1949) and Gulland (1965). The first steps in this direc­ tion were taken by Helgason and Gislason (1979), Pope D = M x N (1979), and Sparre (1980). For a historic review of the where M is the natural mortality coefficient. Figure 1 development of these methods the reader is referred to illustrates the flow of fish (number of fish) as described Pope (1991). by Baranov’s equation. The N-boxes symbolize the MSVPA takes into account predation among number of survivors on the birthday of the cohort, i.e. exploited fish populations. Essentially, it is a number- the number at the beginning of a period. Boxes C and D crunching exercise to estimate fishing and predation symbolize the losses in numbers caused by fishing and mortalities by year and age group simultaneously for a number of species on the basis of catch-at-age data and stomach-content data by age group of predator and prey. This contribution aims at introducing the basic fea­ tures of this estimation technique, with particular em­ phasis on the mathematical description and computa­ N, tional procedures. For more details on VPA, see Beyer and Sparre (1983), Jones (1984), and Sparre et al. (1989). Beyer and Sparre (1983) introduce MSVPA in connection with VPA. c,

Single-species VPA VIRTUAL POPULATION The single-species VPA is derived from Baranov’s catch Figure I. Illustration of the model underlying single-species equation (1918) VPA.

12 natural mortality, respectively, during each year of life. is believed to be the dominating source of mortality Boxes D and C are the accumulated numbers of deaths compared to all other sources, including fishing. This during the year and are conceptually different from the idea was inspired by quantitative findings on the con­ N-boxes. The “virtual population” is the C-boxes in sumption of some predatory fish, notably, North Sea cod Figure 1, that is, the part of the population one can (Gadus morhua) (Daan, 1973, 1975). It was therefore actually see when visiting the fishing harbours. By esti­ natural to concentrate on the predation mortality when mating the C’s from sampling the commercial landings, searching for a solution to the “M problem”. and assuming that M is known, N, D, and F can be calculated. With this technique, known as VPA (virtual population analysis), it can be calculated how many fish Andersen and Ursin’s model there must have been in the sea to account for the Andersen and Ursin (1977) developed the basic concep­ number caught. tual framework for predation which allowed extension Fry (1949) introduced the concept of "virtual popu­ of VPA to MSVPA. This framework was actually em­ lation” and derived a method for estimating exploitation bedded in a general and comprehensive model of an rate which is virtually the same as that presented by exploited marine ecosystem. Gulland (1965) - the method usually referred to as Although a valuable model, it was not really “oper­ “V PA” . ational” because of the large number of parameters One feature to be noted here is that in single species which were difficult to estimate. But, by removing a VPA each cohort can be treated separately. The results number of components from their model and by simpli­ of one cohort have no influence on the results of the fying the remaining part of it, the bases of MSVPA are others. As will be shown later this is not true for revealed. MSVPA. Four alternative reductions of the Andersen and Single species VPA requires the natural mortality Ursin model (AU model) to an “operational model” coefficient as input. The VPA equations link F and M in have been suggested by Helgason and Gislason (1979), a functional relationship, so if M is overestimated the Gislason and Helgason (1985), Pope (1979), Sparre stock numbers (N) will also be overestimated and the (1980), and Ursin (in Sparre, 1980). The following fishing mortalities (F) underestimated, and vice versa. descriptions of the AU model refer to the predator-prey The natural mortality coefficient is difficult to esti­ relationship, which I consider the cornerstone of the mate for exploited fish stocks because the number of fish Andersen and Ursin (1977) model, and which has served dying from natural causes (D-boxes in Figure 1) is as the basis for the development of the MSVPA version difficult to observe. The traditional approach is to make of this model. a “qualified guess” for M. The flow of biomass between components of the For a stock assumed to be in an equilibrium situation, ecosystem is illustrated in Figure 2. The components of the total mortality

Z = F + M can be estimated from age-frequency samples. If the FISHERY stock is unexploited, then F = 0 and consequently M = Z. Thus, in this case there is a simple way of estimating M. Mackerel Haddock However, important fish stocks are usually exploited and not in equilibrium. Although we may be able to estimate Z for exploited stocks, there is no reliable way Herring of estimating the split into M and F for individual age groups. Several attempts to devise simple methods of estimating M, based on general knowledge about the Whiting physiology and behaviour of a fish species, have been suggested. They are all indirect methods and are based on assumptions which are questionable. For a further Saithe discussion of these topics the reader is referred to Chapters 4 and 5 in Sparre et al. (1989). Natural mortality occurs through a variety of causes, e.g. predation, diseases, parasites. In principle, the Other Food number of fish dying from predation by fish can be sampled by analysing the contents of stomachs of preda­ Figure 2. Flow chart of biomass between components in the AU model for the North Sea considered by the ICES multi­ tory fish. For young (small) fish the predation mortality species working group.

13 body growth body growth body growth and Fishery recruitment Fishery recruitment Fishery recruitment

Other food Other food Other food

Mackerel Mackerel Mackerel

Saithe Saithe Saithe

Cod

Whiting Whiting Whiting

Haddock Haddock H a ddock

Herring Herring Herring

Sprat

Norway Pout < Norway Pout Norway Pout

Sand Eel Sand Eel Sand Eel

1981 1982 1983 Figure 3. Flow of population numbers along the time axis in the AU model. the North Sea considered by the ICES Multispecies and Working Group (Anon., 1984b, 1986, 1987) arc used as an example. Some stocks are predators only, some are D = (M l + M2) X N prey only, and some are both predators and prey. This classification was based on the results from stomach In MSVPA the predation mortality (M2) is estimated content examinations (Anon., 1984a). In addition to within the model whereas Ml still has to be guessed just exploited fish species, a category “other food” is con­ as M is in single species VPA. Using index p for prey sidered, in which all other prey types are lumped species by age group the number of deaths (P) caused by together. The AU model also describes the dynamics of predation thus becomes the system along the time axis, as illustrated in Figure 3. The flow of numbers of fish in time is associated with Pp = M2p x Np body growth and reproduction (or recruitment). Although the numbers in each cohort are subsequently The total number dying from predation is further reduced due to mortality, the corresponding biomass divided into the numbers eaten by each predator species may (or may not) increase due to growth of individuals. age group (i): This process is linked to the age of the cohort. Thus, flows of both numbers and biomasses between com­ ponents and along the time axis are described in the AU P p = model. The AU model takes food availability into account when calculating body growth rates, whereas and the number eaten by predator type i becomes: this aspect has been ignored in MSVPA. Irrespective of the biomass of prey the predators are assumed to have a Ppi = M2pi x Np constant growth rate in MS VP. In order to model the flows of biomass between the components, the AU The AU model is based on the concept of “food model divides natural mortality into predation mortality suitability” (Upi). Upi is a measure of the suitability of (M2) and “other” natural mortality (Ml): prey p as food for predator i. The Upi’s are fractions and add up to one. To define suitability, we introduce the M = Ml + M2 relative stomach content, Spl, which is defined as:

14 Weight of prey p in the stomach of predator i M2pi = Nj x R, x Upi Sn; = (4) Total weight of stomach content of predator i V Nj x Wj x Uj; Spj is a measurable quantity given accurate food compo­ sition data. The AU model assumes that the relative stomach from which the total predation mortality is obtained by content of prey p in the stomach of predator i is given by: summing over the predator index:

Suitable biomass of prey p S™ = : M2n M2„ Total suitable biomass available to predator p ■I

or “Other food” C _ Np x w„ x UDi P' V-, (2) One problem faced in practice is that the data required ) xwjx WjUjj for estimating the parameters of the AU model are available only for a limited number of stocks of major commercial interest. Thus the components of the model where wp is the average body weight of a prey species p. have to be categorized into stocks for which data are Spj represents a fraction, and the Spi’s summed over prey available and a residual group for which limited infor­ (index p) add up to 1.0. Thus, U pi reflects the diet mation is available. In the North Sea, the latter group composition of the predators relative to the available contains mainly prey species, and was therefore called food. “other food”. Manly (1974) introduced a similar concept in a more The four alternative simplifications of the AU model general ecological context. If R: is the total annual food mentioned above differ only in the way they treat the consumption of predator i. then the total number of prey “other food” compartment. Pope (1979) assumed that p eaten by predator i, Ppi, is “other food” constituted a fixed proportion of the total consumption, i.e. a fixed proportion of the stomach _N,x R,x Spi content stems from “other food”. P' (3) Helgason and Gislason (1979) and Gislason and Hel­ gason (1985) assumed that the biomass of “other food” Combining Equations (1), (2). and (3) we get: in the sea remained constant. The proportion of “other

M2 Pope Sparre ( Uotiw > U )

Sparre ( U o » ,., = U n.h ) Ursin

Sparre ( U d i.. Helgason & Gislason

N, prey abundance Figure 4. Predation mortality (M2) as a function of prey stock size for the four alternative assumptions on “other food” (redrawn from Sparre, 1980).

15 food” in the stomach contents was thus allowed to vary known we can calculate the M2’s using Equation (4). in accordance with the availability of the various prey Once the M2’s are known the N’s and the F’s can be species. calculated using single species VPA techniques. How­ Sparre (1980) assumed that the total biomass of all ever, since the N ’s are not known, this problem can only populations in the sea (including other food) remained be solved using iterative techniques. constant. An initial guess on the M2’s allows us to make a first Ursin (quoted in Sparre 1980) suggested that the total estimate of the N’s. With these N’s and the given biomass of populations weighted by their suitability suitability coefficients, new values of M2 can be calcu­ (including other food) remained constant. This assump­ lated using Equation (4), which in turn can be used to tion would make MSVPA consistent with the model of estimate new values of the N’s. The process is repeated food availability dependent growth rate in the AU until two consecutive iterations converge to marginally model. different M2 values according to predefined criteria. Figure 4 illustrates the differences in the four alterna­ The iterative procedure illustrated in Figure 6 deals tive assumptions in the case of only one predator and with one year’s data at a time only. Thus, the process has one prey. It shows how the predation mortality changes to be repeated for each year of the time series con­ as a function of the prey abundance under each of the sidered. four alternative assumptions. It has not so far been The computation procedure involves in fact two possible to demonstrate that any one model is superior iterations, as illustrated in Figure 7. The “inner” itera­ to the others. Three models were tested by the ICES tion deals with the solution of the catch equation for NMVPA Working Group. Because of its computational given M in respect of F, i.e. to solve C = F x N for F. complexity, Ursin’s suggestion was not tested, although As N is a non-linear function of F (i.e. N = from a purely theoretical point of view I feel that it is the N(1 - exp (-F - M))/(F + M)), it can only be derived most satisfactory one, because it is consistent with the through iteration. assumption on constant body growth rate in MSVPA. The second iteration deals with the estimation of the M2’s. The procedure is repeated for each data year starting with the most recent data year and proceeding Multispecies VPA backwards in time, just as in traditional VPA. Thus, if the suitability coefficients were known the MSVPA is a computation technique by which we can MSVPA could now be run. However, since they are not calculate the amount of fish there must have been in the known an extra iterative procedure on top of the sea to account for the observed catches in fisheries and MSVPA iteration is required. To introduce the “suit­ the observed stomach contents of predators. A basic ability iteration”, we first show how the suitability coef­ assumption is that suitabilities are constant from year to ficients can be derived from the observed stomach con­ year. Figure 5 summarizes the MSVPA in terms of input tent data under the assumption that stock numbers are and output as explained in the foregoing sections. known. It can be shown algebraically that Equation (2) One principal difference compared to VPA is, of together with 2 U pi = 2 Spi = 1.0 implies that (Sparre, course, that MSVPA deals with several stocks at a time. 1980): MSVPA executes a series of parallel linked VP As. In single species VPA each cohort can be treated n Spi/Npi x w„) (5) separately, the results being independent of the results UP' ^ w of the other cohorts. The usual procedure is to work ^ Sji/(Nji x W j) backwards in time, starting with the oldest age group and ending with the recruits. For MSVPA this would not work. All cohorts of all species have to be dealt with Thus, if the N ’s are known the U ’s can be calculated, as simultaneously, as the value of the predation mortality the S’s are given by the stomach content data. depends on the abundances of predators and prey. Thus Equation (5) is the key formula in MSVPA, which MSVPA works on a “by-year basis” rather than on a by­ turns the AU model into an estimation technique: It cohort basis. converts stomach contents into suitability coefficients, The technical computation problem in MSVPA is which in turn may be used to calculate mortalities. that: (1) To calculate the numbers in the sea from the The estimation of suitability coefficients is illustrated catches and from the stomach contents of predators, by a small hypothetical example in Tables 1A-D. The fishing mortalities and predation mortalities must be computation presented in this table corresponds to a known. (2) To calculate the predation mortalities the single step in the iterative procedure described below. number in the sea and the suitability coefficients must be The example deals with three species with 3,2, and 3 age known (cf. Equation (4)). To solve this complex of groups respectively, as shown in Table A. We assume problems, we start by assuming that the suitability that the values in column N are derived from the coefficients are known. This implies that if the N ’s are MSVPA iteration procedure described above. Weights

16 numbers assumption relative body food M1 & caught on stomach weight consumption terminal "other food" contents Fs

MSVPA for given suitability coefficients

estimation of suitability coeffecieints

fishing stock suitability predation mortality numbers coefflcents mortality M2

Figure 5. Inputs and outputs to MSVPA. are inputs from other sources. Table B shows relative in Table D. The biomass of all stocks is in this case stomach contents derived from samples. Thus, Tables A assumed to remain constant. and B represent the input data. Table C is calculated The iterative process used to estimate the suitability from Tables A and B and represents the fraction of each coefficients is shown in Figure 8. Note that the prey category divided by biomass. The suitabilities are “MSVPA” box contains the iterative process illustrated immediately obtained from Table C and they are shown in Figures 6 and 7.

Initial guess on M2

year Y year Y + 1 N, Z, F < species no.1 N, Z, F ^ N, Z, F < N, Z, F <

species no.2 N, Z, F N, Z, F < N, Z, F < N, Z, F

species no.3 N, Z, F n , z , f N, Z, F < N, Z, F < : : : f

New values of M2's

Figure 6. Iterative computational procedure for one year of MSVPA.

17 START

MAKE INITIAL GUESS ON M2, e. g., M2 = 0

y := most recent data year

Solve the VPA equations for all stocks :

C = F N (where N = N(1 - exp(-F-M))/(F+M)) for the given M value with respect of F using iterative techniques (inner iteration)

Calculate for all predators : Available food for predator i =

Other Food Other food,I

Calculate M2 :

M 2 p = l(predator) JM2

M2

M2 One of ‘current no three M2 present options(*)

yes

yes y > first data year

no

STOP

Figure 7. The iterative computation procedure for MSVPA when suitability coefficients arc fixed. Options for other food (*) are (1) amount of other food is constant, (2) amount needed to hold total biomass constant, or (3) zero other food.

18 START

INITIAL GUESS ON SUITABILITY COEFFICIENTS

MSVPA for the years : Most recent year to year Y (The Algorithm in Figure 7)

CALCULATE SUITABILITY COEFFICIENTS FOR YEAR Y USING THE STOMACH CONTENT DATA :

v w Upl= — (5) 2, S)/(NJ|*wj)

NO

current previous

YES

MSVPA for years : Year Y to first data year. (The Algorithm in Figure 7)

Figure 8. The iterative procedure for estimation of suitability coefficients. Year Y represents the year for which stomach content data are available.

Table 1A. Estimation of suitability coefficients from MSVPA An initial guess on the suitability coefficients allows output and relative stomach contents illustrated by a hypotheti­ running the MSVPA back to Y, the year for which cal example (from Sparre, 1980). Output from MSVPA (N) and stomach content data are available. With the estimated body weights (w) necessary for the estimation of suitability N's for year Y in combination with the stomach content coefficients. data, new suitability coefficients can be estimated using Equation (5). This procedure is repeated until suitabili­ Species Age N w Nw ties converge according to predefined criteria. 1 1 200 5 1000 The first MSVPA was carried out by the ICES Multi­ 100 50 5 000 species Working Group at its meeting in 1984 (Anon., 80 4000 3 50 1984b). The computations were made with a FO R ­ 2 1 50000 1 50 000 TRAN 77 program prepared for the meeting, and exe­ 2 20000 5 100000 cuted on the VAX/11/750 mini-computer at the Danish Institute for Fisheries and Marine Research (Sparre, 3 1 1000 5 5 000 1984). The FORTRAN source code makes up some 100 2 500 20 10000 A4 pages, a large part of which comprises the compre­ 3 100 30 3000 hensive comments. A first version of the program was Total fish biomass: 178000 presented in Sparre (1984). One complete run with the Total ecosystem biomass: 1000000 MSVPA program took about 5 min of CPA time on the 822000 Other food biomass: VAX/11/750 mini-computer. The program was ex-

19 Table IB. Relative stomach contents (S(a,a,j,b)).

Predator (j.b)

2 3 j 1

s a b 1 2 3 1 2 1 2 3

1 1 0 0 0.05 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 Prey 2 1 0.5 0.5 0.3 0 0.2 0.2 0.2 0.2 2 0 0.3 0.4 0 0 0 0.1 0.3

3 1 0 0.1 0.15 0 0 0 0 0.1 2 0 0 0.05 0 0 0 0 0 (s,a) 3 0 0 0 0 0 0 0 0

Other food 0.5 0.1 0.05 1.0 0.8 0.8 0.7 0.4

Total 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Table 1C. Fraction of relative stomach contents over biomass (x lO f5), S(s,a,j,b)/[N(y,s,a)w(s,a)].

Predator (j.b)

Nw j 1 2 3 (from s a b 1 2 3 1 2 1 2 3 Table A)

1 1 0 0 5.0 0 0 0 0 0 1000 2 0 0 0 0 0 0 0 0 5 000 3 0 0 0 0 0 0 0 0 4 000 Prey ------2 1 1.0 1.0 0.6 0 0.4 0.4 0.4 0.4 50000 2 0 0.3 0.4 0 0 0 0.1 0.3 100 000

3 1 0 2.0 3.0 0 0 0 0 3.0 5000 2 0 0 0.5 0 0 0 0 0 10 000 3 0 0 0 0 0 0 0 0 3000

Other food 0.061 0.012 0.006 0.122 0.097 0.097 0.085 0.049 822000

Total ecosystem 1.061 3.312 9.506 0.112 0.497 0.497 0.589 3.749 1000000

Table ID. Food suitability matrix; U(s,a.j,b).

Predator (j,b)

i 1 2 3

s a b 1 2 3 1 2 1 2 3

1 1 0 0 0.53 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 Prey 2 1 0.94 0.30 0.06 0 0.80 0.80 0.68 0.11 2 0 0.09 0.04 0 0 0 0.18 0.08

3 1 0 0.60 0.32 0 0 0 0 0.80 2 0 0 0.05 0 0 0 0 0 (s,a) 3 0 0 0 0 0 0 0 0

Other food 0.06 0.01 0.05 1.00 0.20 0.20 0.14 0.01

Total 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

20 tended and modified later (Anon., 1986, 1987; Sparre fisheries. Nauchn. Issled. Ikhtiologicheskii Inst. Izv., 1: 81- and Gislason, 1986; Gislason and Sparre, 1987). 128. (In Russian). Beyer, J., and Sparre, P. 1983. Modeling exploited fish stocks. In Application of ecological modeling in environmental man­ agement, Part A, pp. 485—582. Ed. by S. Jørgensen. Elsevier, Catch prediction Amsterdam. Daan, N. 1973. A quantitative analysis of the food intake of VPA, which deals with the analysis of historical catch North Sea Cod, Gadus morhua. Neth. J. Sea Res 6- 497- data, has a counterpart, the Thompson and Bell model 517. (1934), which deals with predictions or simulations. Daan, N. 1975. Consumption and production in North Sea MSVPA also has a catch prediction counterpart. The Cod, Gadus morhua: an assessment of the ecological status first version of the multispecies prediction program was of the stock. Neth. J. Sea. Res., 9: 24-55. Fry, F. E. J. 1949. Statistics of a lake trout fishery. Biometrics presented in Sparre (1980), whereas a later version 5: 27-67. included some economic features (Sparre, 1983). Gislason, H., and Helgason, T. 1985. Species interaction in These predictive programs are more straightforward assessment of fish stocks with special application to the North applications because they use the parameter values Sea. Dana, 5: 1^14. Gislason, H., and Sparre, P. 1987. Some theoretical aspects of obtained from MSVPA in the simulation and require the implementation of multispecies virtual population analy­ only one iterative loop. sis in ICES. ICES CM 1987/G: 51. Gulland, J. A. 1965. Estimation of mortality rates. Annex to Arctic fisheries working group report. ICES CM Doc. 3. Acknowledgments Helgason, T., and Gislason, H. 1979. VPA-analysis with species interaction due to predation. ICES CM 1979/G: 52. This description of the MSVPA has concentrated on the Jones, R. 1984. Assessing the effects of changes in exploitation mathematical description of the computation pro­ pattern using length composition data (with notes on VPA cedures, the reason being that I worked mainly in that and cohort analysis). FAO Fish. Tech. Paper No. 256. 118 pp. field. However, I am well aware that without the biologi­ Manly, B. F. J. 1974. A model for certain types of selection cal fieldwork carried out by my colleagues and without experiments. Biometrics, 30: 281-294. the input from these field workers there would have Pope, J. G. 1979. A modified cohort analysis in which constant been no MSVPA. My task was to translate their obser­ natural mortality is replaced by estimates of predation levels ICES CM 1979/H: 16. vations and ideas into formulas and computer programs. Pope, J. G. 1991. The ICES Multispecies Assessment Working Amongst them I mention the coordinators of the ICES Group: evolution, insights, and future problems. ICES mar" stomach sampling program (1981) and in particular their Sei. Symp., 193: 22-33. chairman, who suggested that I took up the task to Sparre, P. 1980. A goal function of fisheries (Legion analysis). ICES CM 1980/G: 40. 81 pp. program the MSVPA, and who had a major influence on Sparre, P. 1983. Legion Analysis Game and Simulation Pro­ the design of the MSVPA program. gram (LAGS). NAFO SCR Doc. 83/IX/71. Serial no. N736. 49 pp. Sparre, P. 1984. A computer program for estimation of food References suitability coefficients from stomach content data and multi­ species VPA. ICES CM 1984/G: 25. 60 pp. Andersen, K. P., and Ursin, E. 1977. A multispecies extension Sparre, P. 1986. FORTRAN 77 program for multispecies to the Be verton and Holt theory, with accounts of phos­ forecast. Working document for the 1986 meeting of the Ad phorus circulation and primary production. Meddr Danm. Hoc Working Group on Multispecies Assessment. Fisk.- og Havunders. N.S., 7: 319-435. Sparre, P., and Gislason, H. 1986. FORTRAN 77 program for Anon. 1984a. Report of the coordinators of the stomach multispecies VPA. Working document for the 1986 meeting sampling project 1981. ICES CM 1984/G: 37. of the Ad Hoc Working Group on Multispecies Assessment Anon. 1984b. Report of the Ad Hoc Multispecies Assessment Sparre, P., Ursin, E., and Venema, S. C. 1989. Introduction to Working Group. ICES CM 1984/Assess: 20. tropical fish stock assessment. Part 1. Manual. FAO. Fish. Anon. 1986. Report of the Ad Hoc Multispecies Assessment Tech. Pap. No. 306.1. Rome. FAO. 337 pp. Working Group. ICES CM 1986/Assess: 9. Thompson, W. F., and Bell, F. H. 1934. Biological statistics of Anon. 1987. Report ot the Ad Hoc Multispecies Assessment the Pacific halibut fishery. 2. Effect of changes in intensity Working Group. ICES CM 1987/Assess: 9. upon total yield and yield per unit of gear. Rep. Int. Fish. Baranov, F. I. 1918. On the question of the biological basis of (Pacific Halibut) Comm., 8: 49 pp.

21