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Fisheries Research 37 (1998) 179±191 On the bioeconomics of predator and prey ®shing Ola Flaaten* The Norwegian College of Fishery Science, University of Tromsù, Breivika, N-9037, Tromsù, Norway Abstract A brief review of biological, bioeconomic and non-use objectives in ®sheries management is given. It is shown that management strategies, with respect to optimal long run effort levels and stock sizes, might differ considerably with the choice of objective, and of a single-species or multispecies framework. Within a deterministic two-species predator±prey biomass model, equilibrium catch and resource rent of the predator are positively affected by an increased prey stock level. The effects on the equilibrium harvest and resource rent of the prey of a change in the predator stock level are ambiguous, although a negative relationship is most likely. In a simpli®ed model with a linear equilibrium relationship between the two stocks this negative effect is always present. Such a simpli®ed approach is applied to the case of northeast Arctic cod's consumption of its commercially most important prey species: capelin, herring, shrimp and small cod, i.e. cannibalism. It is shown that on average for the years 1984±92 capelin constituted about 75% of the feed, on the basis of weight, of age-speci®c cod of age 2 and older. Shrimp is the second most important prey, constituting about 10±20% of the feed; the higher ®gure occurs for 2-year old cod and the lower for the 7 age group. Using Norwegian data on prices of ®sh and cost of effort for the years 1991±92, and data on stomach content 1984±92, it is shown that shrimp constitute 40±80% of the feed costs for all age groups of 2 years and older, whereas capelin constitutes only 15±20%. For the 7 age group cannibalism is just above half the feed costs. This, despite the fact that capelin is by far the major feed in weight. The assumption of the speci®c functional forms in biological multispecies models are of importance for the management implications derived. It might be that the inclusion of economic factors will weaken the relative importance of some biological factors for management strategies. The partial bioeconomic analyses in this paper strongly indicate that biological and economic factors should be considered simultaneously in management analyses. # 1998 Elsevier Science B.V. All rights reserved. Keywords: Predator±prey; Management objectives; Functional forms; Predation costs; Northeast Arctic cod 1. Introduction This paper analyses and discusses some underlying assumptions and objectives of multi-species management of ®sheries within the framework of simple predator±prey models. Within ®sheries systems there *Corresponding author. Tel.: +47 77 64 40 00; fax: +47 77 64 60 20; e-mail: [email protected] 0165-7836/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved. PII S0165-7836(98)00135-0 are several types of interactions. For the purposes of bioeconomic modelling the most important are: (1) biological interactions, (2) harvest technical interactions, and (3) market interactions. The biological interactions of most importance in boreal ecosystems are predator±prey relationships and competition between species. The interactions between the ®sh community and the oceanic environment may also be relevant. Harvest technical interactions, such as when ®shing gear targeting one particular species also gives 180 O. Flaaten / Fisheries Research 37 (1998) 179±191 a bycatch of other species, are prevalent in many ®sheries in the north Atlantic. Market interactions are present when the quantity of one species supplied affects the market price of another species. This paper focuses on biological interactions. In the case of predator±prey interactions it is well known from the ecological literature that the reduction of the predator stock level may increase the surplus production of the prey. When it comes to predators like whales and seals, however, harvesting is often controversial, as the following quotation demonstrates: ``An early exploration (of multispecies fisheries), May et al. (1979), has proved very influential, and now forms the basis for a very controversial piece of work, a bioeconomical analysis of the Barents Sea fishery by Flaaten (1988). Flaaten's work is controversial because of his conclusion that sea mammals should be heavily depleted to increase the surplus production of fish resources for man. This is in harmony with Norwegian government policy, which, ever since the European Community ban on the importation of the skins of whitecoats (nursing harp seals) and bluebacks (hooded seals<1-year old) rendered sealing commercially unviable, has subsidized sealing in the Barents Sea.'' (Yodzis, 1994, p. 51.) Harvesting is, however, not the only utility generated from sea mammals. It has long been acknowledged that non-use values included in the objective function may have implications for stock management. The following quotation demonstrates this: ``It should, however, be stressed that this result may be somewhat modified if the resource is assigned an optional value from people's willingness to pay for keeping the stock at higher level. A biological argument that also may weaken our result is the eventual existence of critical depensation for lower stock levels.'' (Flaaten, 1988, p. 114.) The aim of this paper is twofold. First, to review and discuss in a simple way bioeconomic objectives for ®sheries management within single-species and multispecies framework. Second, to analyze and discuss, within the framework of predator±prey biomass models, the importance of the assumptions regarding the speci®cation of the model for the conclusions of multispecies management studies like, e.g. May et al. (1979); Flaaten (1988); Yodzis (1994). In addition to the theoretical analysis an empirical example of costs in¯icted upon the prey ®shing industries by the predator is presented. This is done for the northeast Arctic cod (Gadus morhua) and its commercially most important prey species: capelin (Mallotus villosus), herring (Clupea harengus), shrimp (Pandalus borealis) and small cod, i.e. cannibalism. Different kinds of uncertainty are related to the three types of interactions above. Following Charles (1992) the three key forms of uncertainty in ®sheries modelling are: ``(1) random fluctuations (in fish stocks, fish prices, etc.) within an otherwise fully determined model, (2) parameter uncertainty due to imprecise parameter estimates in an established fishery model, and (3) basic ignorance about the appropriate model to describe the fishery. Indeed, in this sequence of increasing uncertainty, the initial step can be considered as that involving deterministic models, which ignore the uncertainties in fishery systems but are useful particularly for the qualitative insights they provide. In contrast, the third degree of uncertainty, basic ignorance, often reflects the most realistic situation, but such circumstances are notoriously difficult to model...'' (Charles, 1992, p. 192.) In recent years new modelling approaches have been developed by, e.g. the International Council for the Exploration of the Sea (ICES, 1994, 1997) for the evaluation of management strategies under uncertainty. This paper, however, ignores type 1 and 2 uncertainty. For management purposes the extension of a singlespecies biological model into a bioeconomic model, by including ®sh prices, operating costs, capital costs, etc. is a step towards reducing type 3 uncertainty. The development of a multispecies biological model for an ecosystem, in addition to single-species models, may also be a step towards the reduction of type 3 uncertainty, even if the multispecies model is deterministic. However, in order to keep models simple and operational, additional information, e.g. about biological interdependence, should perhaps not be included if the O. Flaaten / Fisheries Research 37 (1998) 179±191 recommendations, the best performing management strategy or control laws, remain unchanged. Management objectives and the long run harvest strategies implied by single-species and multispecies analyses, and the differences between them, are brie¯y reviewed in the next section. In Section 3 a general predator±prey biomass model is designed to analyze the effects on each species' equilibrium harvest and resource rent of changes in the other species' stock level. Section 4 designs a predator±prey model with linear equilibrium relationship (isoclines) between the two stocks, and analyses how changes in the slope of the isoclines affect the harvest and resource rent of each ®shery. The discussion of the effects on predator equilibrium harvest and resource rent of changes in the prey stock level is related to the sea-mammal±®sh case. Section 5 presents the empirical ®ndings of feed costs of northeast Arctic cod, using stomach content data and Norwegian price of ®sh and cost of effort data. The concluding section discusses the main ®ndings of the paper. 2. Management objectives In the biology literature objectives for managing ®sh stocks are usually related to the maximum sustainable yield (MSY), yield per recruit (Y/R) or some related concepts. In cases of two or more biologically interdependent species, maximum sustainable yield frontiers (MSF) might replace the single-species MSY concept (see, e.g. Beddington and May, 1980; Flaaten, 1988). However, the fallacy of biological management objectives is that they do not consider the economic bene®ts and costs of ®sheries. Many ®sh stocks are deliberately not ®shed due to low market price and/or high catch cost. In the Barents Sea, for example, there are more than 100 ®sh species, but only about 10 are commercially targeted. Some international organizations and agreements have established their own objectives for ®sheries management. The Food and Agriculture Organization of the United Nations (FAO) formulated the following objective: ``Recognizing that long-term sustainable use of fisheries resources is the overriding objective of conservation and management, states and sub- 181 regional or regional fisheries management organizations and arrangements should, inter alia, adopt appropriate measures, based on the best scientific evidence available, which are designed to maintain or restore stocks at levels capable of producing maximum sustainable yield, as qualified by relevant environmental and economic factors, including the special requirements of developing countries.'' (FAO, 1995.) Thus, even though the FAO's Code of Conduct establishes the single-species concept maximum sustainable yield as the main management objective, it is quali®ed by relevant environmental and economic factors. Contrary to the management objectives above, economic objectives are strongly related to social welfare theory that emphasizes the net economic results to society of utilizing natural resources. `Society' in this context usually means a country, but it could also mean a group of indigenous people, a region within a country, or a group of countries. The resource rent is the gross catch value minus the harvest costs. Maximizing the long run resource rent, or the present value of discounted future resource rent, is the main economic management objective, when harvested ®sh is the only bene®t to society. If the stock possesses public non-use values, the related bene®t ¯ow should be included in the management objective. The existence of organizations working to abolish, e.g. whale, seal and dolphin harvesting re¯ects the importance of non-use values. There exists a large literature on methods that can be used to evaluate non-use bene®ts from natural resources (see, e.g. Hanley and Spash, 1993). The main results of single-species bioeconomic analyses are shown in Fig. 1. Panels (a) and (b) show how the sustainable revenue and the total cost of harvesting vary with ®shing effort and stock level, respectively. Generally speaking, the optimal level of , is less than the open access level, ®shing effort, Ess oa , and the optimal stock level, Wss , is higher than the Ess open access level, Wssoa . These general results are valid whether the optimum is derived by maximizing annual economic rent or the present value of rent. However, the economic optimal effort and stock levels approach their open access levels if the interest rate goes to in®nity. 182 O. Flaaten / Fisheries Research 37 (1998) 179±191 Fig. 1. Open access (OA) and optimal (*) effort (E) and stock level (W) in a single species (SS) model. The arrows in panel (a) indicate that the optimal long run fishing effort is higher for the predator and lower for the prey when biological interactions are included, than in the single species case. Correspondingly, the arrows in panel (b) indicate that the optimal long run stock level is lower for the predator and higher for the prey. In single-species models, the biological constraint of the optimization problem is the yield-effort or yield-stock curve on which the revenue curves shown in Fig. 1 are based. Moving from single-species to two-species models changes the biological constraint to, for example, the MSF shown in Fig. 2. Maximizing the yield from each of the two species as if it is independent of the other, gives the combined yields at point S in Fig. 2. However, this is not a sustainable combination of yields since it is outside the MSF. Any Fig. 2. The maximum sustainable yield frontier (MSF) gives the maximum possible yield of one species for a given yield of the other, based on a simple analytical model. Each cross might represent the possible result of a given management strategy for a complex, detailed simulation model. The *S represents the combination of the maximum single species yield of the predator and the maximum single species yield of the prey, but this point is not sustainable. point on or inside the MSF would be sustainable (see e.g. Flaaten, 1988, 1991). The reasoning so far is based on simple deterministic models. For complex, detailed simulation models each set of management strategies gives a combined average annual yield of the two species that might be represented by the crosses in Fig. 2. In this case the MSF is based on the most ef®cient management strategies. Any management strategy below the MSF is inef®cient, biologically speaking. In the case of harvest technical interactions, say with two selectivity patterns, the results may be illustrated by drawing two intersecting MSFs instead of the one in Fig. 2. Which combination of yield should be chosen depends in general on the management objective and on the price of ®sh ± cost of effort ratios for the two stocks. If the stocks are jointly managed, the objective might be to maximize the combined average annual resource rent or the present value of the resource rent obtained from them. Using the present value of the combined resource rent from the two stocks as a management objective, or performance criterion, could give an optimal solution inside the MSF, e.g. at D in Fig. 2. This is due to the positive effect on the unit harvesting cost of an increased stock level, and the discounting of future bene®ts and costs. To illustrate the span of possibilities in predator and prey ®sheries, let me use two simpli®ed examples. In both cases I assume that the stocks can be harvested independent of each other. O. Flaaten / Fisheries Research 37 (1998) 179±191 2.1. Example 1: valuable predator and cheap prey Let species 2 be a predator of high net value per unit harvest and species 1 a prey species with a low net value. In this case the optimal long run combined equilibrium harvest is in the vicinity of B in Fig. 2, where the prey is mainly kept in the sea as feed for the predator. In this case the effort of the predator ®shery does not have to be increased (much) compared to its single species effort shown in Fig. 1(a), whereas the effort of the prey ®shery should be decreased. The effects on the stock levels are the opposite. 2.2. Example 2: predator of low net value and prey of high net value If the predator is of low market value and/or expensive to harvest, its net value per unit harvest is low. Likewise, if the prey is of high market value and/or cheap to harvest its net value per unit harvest is high. In this case the optimal long run combined equilibrium harvest is in the vicinity of A in Fig. 2, where the predator stock during a transitional period was ®shed down to a low stock level to leave more prey to be harvested by the ®shermen (see, e.g. Beddington and May, 1980; Flaaten, 1988). In some cases it even pays to subsidize ®shermen to harvest more predators than they otherwise would have done. In this case the optimal effort of the predator ®shery should be increased and the stock level of the predator reduced compared to the single species case, as indicated by the arrows in Fig. 1. If a species has a public non-use value this might be represented by adding a concave increasing utility function of the stock level to the management objective, e.g. U(W) with U>0, U0 >0 and U00 <0. The question of how to estimate people's willingness to pay for public goods has been extensively dealt with in the literature (see, e.g. Hanley and Spash, 1993 and references therein), but will not be pursued in this paper. The implications for optimal long run effort and stock levels are, however, similar to that of the prey in example 1 above. The positive non-use externality af®liated to the stock implies lower equilibrium effort and higher stock level compared to the case without such non-use value. In actual cases non-use values are mostly claimed for species at or near the top of the 183 food chain (see Kuronuma and Tisdell (1993) for the case of minke whale). 3. The general predator±prey model and the bioeconomic effects of harvesting In the population ecology literature there are numerous deterministic models represented by non-linear functions. My starting point is the following model, with continuous time dynamics acting on the total biomass of the prey, W1, and the predator, W2. Let the biomass be measured in weight, e.g. tonnes. There are three basic model functions in the predator±prey biomass model: namely f(W1)0, the intrinsic growth rate of the prey population; k(W1, W2)0, the amount of prey consumed per unit time per unit predator, which is called the predator's functional response; g(W1, W2)0, the per unit biomass growth rate of the predator population, which is called the predator's numerical response. For a review and discussion of the three functions, see, e.g. Yodzis (1994). Suppressing the arguments in the various functions, and with hi being the harvest of species i, the general form of such biomass models can be represented as follows dW1 f ÿ W2 k ÿ h1 ; W1 ; W2 > 0; k 0 dt 2 df W msy ; f 00 d f < 0 0 if W f0 1 1 < dW1 > dW12 @k 0; @W1 @2k 0; @W12 @k 0; @W2 @2k 0 @W22 (1) and dW2 W2 g ÿ h2 ; dt @g @2g @g > 0; 0; 0 2 @W1 @W2 > @W1 msy if W2 < W2 jW1constant ; @2g 0 @W22 (2) By adding the price of harvest, cost of harvesting, discount rate, non-consumptive bene®ts and costs, etc., the model in Eqs. (1) and (2) is extended to a bioeconomic model. In accordance with the aim of this paper, it is suf®cient to include the bene®ts and costs of the harvesting. To simplify the analysis I assume that for each species the net value per unit of catch depends on the respective stock level only. The net 184 O. Flaaten / Fisheries Research 37 (1998) 179±191 value (bene®t) per unit harvest is bi bi Wi ; b00i i 1; 2 b0i d2 bi 0 dWi2 dbi < 0; dWi (3) An in®nitely elastic demand, implying a constant price of harvest, and a unit cost of harvest which increases with the stock level, is the rationale for Eq. (3). Assuming harvest technology that allows independent harvesting of the two species, the resource rents are i Wi bi Wi hi ; i 1; 2 (4) for a given catch, hi. A common approach in bioeconomic analysis of optimal harvesting is to assume a sole owner maximizing the present value of the total resource rent from the two species. This allows the optimal long run steady state to be derived (see, e.g. Clark, 1990, ch. 9; Hannesson, 1983; Flaaten, 1991). For short run analysis, and actual management of ®sh stocks, agestructured biological models would be of great advantage compared to simpler biomass models. However, for the purpose of illustrating the economics and management of biologically interdependent species, biomass-based bioeconomic models are useful. A ®rst step is to study the marginal effects of changes in the other stock level on the harvest rate and the resource rent of each species in our model Eqs. (1)±(4). 4. The effect on prey harvest and resource rent of changes in the predator stock For dW1/dt0 Eq. (1) gives the equilibrium catch of the prey, and it will depend on the prey stock level as well as the predator stock level. The effect on the equilibrium catch of the prey of a marginal change in the predator stock level is @h1 @k @k W2 ÿ 1 (5) ÿ k W2 0 if @W2 @W2 > @W2 k < Thus, the effect on the equilibrium harvest rate of the prey of an increase in the predator stock level is ambiguous. To get the positive sign in Eq. (5) it is necessary that the relative change of per unit consumption, ÿ@k/k, of a marginal increase in the pre- dator stock, is greater than the relative change of the predator stock level, @W2/W2. The equilibrium harvest of the prey is negatively affected by an increased predator stock level if, and only if, the predator's total consumption moves together with its stock level. This is valid for partial changes in the predator stock with the prey stock unchanged. To summarize, the equilibrium catch of the prey is negatively affected by an increased predator stock level when (@/@W2)(W2k)>0, which usually is biologically plausible. However, as seen in Eq. (5), there might be cases where an increase in the predator stock level also may increase the equilibrium harvest of the prey. The effect on the resource rent is derived by substituting the equilibrium harvest rate from Eq. (1) for h1 in Eq. (4): @1 @k @k W2 ÿ1 ÿb1 k W2 0 if @W2 @W2 > @W2 k < (6) The effect on the equilibrium resource rent of the prey ®shery of an increase in the predator stock level is ambiguous. The sign is the same as for the effect on the harvest rate in Eq. (5), and the conditions are the same. As is seen from Eqs. (5) and (6), the effect on the harvest and the resource rent in this case depends on the predator stock level, W2, and the predator's functional response, k, but not on the predator's numerical response, g. In order to estimate the economic impact, data on the net value per unit harvest for the prey ®shery, in addition to estimation of W2 and k, are required. Uncertainty about the predator's numerical response, g, does not matter for the assessment of the predator's marginal impact on the prey ®shery, since the latter is independent of g. 5. The effects on predator harvest and resource rent of changes in the prey stock For dW2/dt0, Eq. (2) gives the equilibrium catch of the predator. From this, the following is obtained @h2 @g W2 >0 @W1 @W1 (7) The effect on the harvest rate of the predator of an increase in the prey stock level is unambiguously O. Flaaten / Fisheries Research 37 (1998) 179±191 positive, and consequently, the effect on the resource rent of the predator ®shery of an increase in the prey stock level is unambiguously positive: @2 @g b2 W 2 >0 @W1 @W1 (8) As is seen in Eqs. (7) and (8), the effect on the harvest and the resource rent depends on the predator's numerical response, g and on the predator stock level. The predator's functional response does not matter in this case. According to Eqs. (7) and (8), resource managers concerned with the negative effect on the predator ®shery of a declining prey stock have little cause for concern if they do not know how the predator affects the prey. However, they should worry about how the prey stock affects the predator, i.e. the predator's numerical response. Note that the results in Eqs. (7) and (8) are valid for any stock level of W1 and W2, given the assumptions in Eq. (2). Therefore, within the model described in Eqs. (1)±(4), the positive marginal effects on the predator harvest and resource rent of an increase in the prey stock level is a robust result. For actual ®sheries the magnitude of the impact will obviously depend on the empirical formulation of the model and the stock levels. Also, more complex ecological models with three or more interdependent ®sh stocks could give ambiguous results compared to Eqs. (7) and (8) (see, e.g., Flaaten, 1988; Bogstad et al., 1998). 6. Results in four typical models A broad class of models can be represented by Eqs. (1) and (2). One speci®c class of models comprises those with functional and numerical responses that give linear isoclines, i.e. with a linear equilibrium relationship between the stocks. Such models are simple to handle and easy to depict graphically. They have been used to analyse the management of marine mammals±krill ®sheries (see e.g. May et al., 1979; Nicole and de la Mare, 1993) and marine mammals± ®sh ®sheries (see e.g. Flaaten, 1988; Yodzis, 1994). The previous section revealed that the predator's numerical response is of importance for the harvest rate and the pro®tability of the predator ®shery. The management of whales, seals and other sea mammals 185 in relation to their prey is often controversial, as demonstrated by the Yodzis (1994) quotation in Section 1. A central point of focus in such controversies is the competition between sea mammals and ®shermen for commercially valuable ®sh stocks. Given a predator±prey model with linear isoclines, this section analyses how the slope of the predator isocline, which is determined by the numerical response, affects management conclusions. However, the analysis starts with the effects on prey equilibrium harvest and resource rent of marginal changes in the predator stock. The population dynamics of the prey in the linear isocline predator±prey model is described by dW1 r1 W1 1 ÿ W1 =K ÿ aW1 W2 ÿ h1 ; dt r1 ; a > 0 (9) The ®rst term of Eq. (9) is the growth rate of the prey population in the absence of the predator. The predator's functional response, aW1, is proportional to the prey stock. Note, however, that in this case the functional response, i.e. the amount of prey consumed per unit time per unit predator, is independent of the predator stock level. If the growth Eq. (9) is substituted for the general prey growth Eq. (1), the results in Eqs. (5) and (6) are reduced to and @h1 ÿk ÿaW1 @W2 50 @1 ÿb1 k ÿb1 aW1 @W2 60 Thus, if we can neglect the fact that the predator's per capita consumption of the prey decreases when the predator stock level increases, it becomes easy to calculate the effects on prey harvest and resource rent of changes in the predator stock level. The results in Eqs. (50 ) and (60 ) will be applied to the case of northeast Arctic cod in the next section. The population dynamics of the predator is described by dW2 r2 W2 1 ÿ W2 = W1 L ÿ h2 ; dt r2 ; > 0 (10) The predator's numerical response, r2 1 ÿ W2 = W1 L, decreases with the predator stock and 186 O. Flaaten / Fisheries Research 37 (1998) 179±191 increases with the prey stock. It also increases with the unspeci®ed resources given by the independent part of the carrying capacity, L. The case of a negative sign in front of L will also be considered. Assuming catch per unit of effort (CPUE) proportional to the stock level, the isoclines are easily derived. The proportional CPUE implies the Schaefer harvest function hi ri Fi Wi ; i 1; 2 (11) which may be substituted for the hi's (i1, 2) in Eqs. (9) and (10). Fishing effort, Fi, has, for simplicity, been normalized such that the catchability coef®cients equal the intrinsic growth rates, ri, in Eq. (11). The isoclines are now derived for dWi/dt0 in Eqs. (9) and (10). The prey isocline is given by r1 W2 1 ÿ F1 ÿ W1 =K; for dW1 =dt 0 (12) a Fig. 4. The linear isoclines of four submodels, of which the predator isoclines intersect at an assumed known equilibrium point W1F ; W2F . and the predator isocline by W2 1 ÿ F2 W1 L; for dW2 =dt 0 (13) The isoclines are shown in Fig. 3 for Fi0 and Fi>0 (i1, 2). Increasing the ®shing effort of the prey ®shery, F1, moves the isocline to the left, and the equilibrium stock levels, W1F and W2F , are both reduced. Increasing the ®shing effort of the predator ®shery, F2, moves the predator isocline downward. As is clear from Fig. 3, the prey stock level increases, whereas the predator stock level decreases for a partial increase in F2. This is because the equilibrium point in Fig. 3. The linear isoclines without and with harvesting. this case moves along the downward sloping prey isocline. The slope of the prey isocline in Eq. (12) depends on only one parameter, a, in addition to r1 and K, whereas the predator isocline in Eq. (13) depends on two parameters, L and . Different combinations of L and will describe different numerical responses of the predator. The following four combinations are considered of greatest interest with respect to the different biological information they contain: L; ; 0; ÿ; 0 The four types of predator isoclines in Fig. 4 are drawn through the same equilibrium point W1F ; W2F by using different combinations of L and . In all four cases the predator is extremely opportunistic; it takes a constant portion of the prey, through the functional response term aW1 in Eq. (9). Assuming we know the stock level of the predator, Fig. 4 illustrates how uncertainty about the predator's numerical response, given partly by the isocline, may affect management decisions. One extreme is submodel 0, where the predator stock level is not affected by changes in the harvesting of the prey. Increased ®shing effort of the prey moves the prey isocline downwards to the left towards the origin, reducing W1F . However, W2F remains constant because of the horizontal predator isocline. O. Flaaten / Fisheries Research 37 (1998) 179±191 At the other extreme is submodel ÿ. If this is the correct description of the predator's numerical response, the predator is very vulnerable to changes in prey harvesting and stock level. If the prey stock is F1 , the predator becomes reduced below the level W F extinct. Thus, W 1 is a critical prey level for the predator in model ÿ. The predator is then highly specialized and goes extinct when not enough food is available. When L>0 the predator can persist without the prey with a minimum carrying capacity L. This means that even in the extreme case where the prey becomes extinct, the predator may survive. In the limiting model 0, the predator develops independently of the prey. It has its own independent carrying capacity, L, yielding the horizontal isocline at W2F in Fig. 4. The results of this analysis can be summarized by using the expressions Eqs. (7) and (8) to study the effects on the equilibrium harvest and resource rent of partial changes in the prey stock. From Eqs. (2) and (10) the following is derived for given levels of W1 and W2 @g r2 W2 @W1 W1 L2 (14) Substituting for @g/@W1 from Eq. (14) into Eqs. (7) and (8) gives the effect of a change in the prey stock level on the equilibrium harvest and resource rent of the predator ®shery. Note that the unambiguously positive results in Eqs. (7) and (8) were based on the assumption that @g/@W1 is strictly positive in Eq. (2). When 0, however, @g/@W10. This implies that @h2/@W1 and @2/@W1 both are equal to zero in this case. The numerator of Eq. (14) equals the square of the carrying capacity of the predator, given in Eq. (10). The effect on the equilibrium harvest of the predator of an increase in the prey biomass, @h2/@W1, is, of course, nil when 0, and positive for >0. From Eqs. (14) and (7) it increases with L. This, simply because the increase in reproductive capacity of the predator by an increase in the prey stock, increases in L. Sea mammals that are opportunistic feeders, e.g. minke whale (Balaenoptera acutorostrata) and harp seal (Pagophilus groenlandicus) (see Haug et al., 1995; Nilssen, 1995) most likely have >0 and L>0 relative to a given prey species. Only very specialized predators would have L<0. 187 7. An example of predation effects ± the case of northeast Arctic cod The analysis in the previous sections includes one predator and one prey. However, the analysis will also be valid in the case of several prey, if the biological interaction between them is negligible or their stock levels are kept constant by adjusting ®shing effort (Flaaten and Stollery, 1996). The basic ideas presented above may also hold in the case of a predator consisting of several cohorts, as long as there exists a long run joint stationary state for the cohorts. In such cases the expressions derived in Eqs. (5)±(8) and (60 ) are applicable. To illustrate the theoretical analysis above, data from the Barents Sea ®sheries will be used. For many years, Russian and Norwegian researchers have conducted studies on `who eats whom' in the Barents Sea area and have modelled these and other multispecies interactions. The model MULTSPEC from the Institute of Marine Research (IMR), Bergen is a biological multispecies model for the Barents Sea ®sh/sea-mammal system (see, e.g. Ulltang, 1995; Bogstad et al., 1998 and references therein). The MULTSPEC model now includes cod, capelin, herring, minke whale, harp seal and species of zooplankton, but not shrimp. The most important ®sh preying predator is cod which seems to eat any available prey of the right size. Fig. 5 shows the northeast Arctic cod's age-dependent average annual consumption of some commercially important prey species. Species included are shrimp, capelin, herring and cod (cannibalism) above 5, 10, 10 and 20 cm, respectively. The ®gures are in grammes of prey per kg cod, for each age class of cod from 1 to 7 years. Fig. 5 shows, for example, that 1 kg of 2-year old cod annually consumed 2000 g of prey from these four species above the given size, and that about 75% of this was capelin. For all age classes, capelin is the main prey among the species and size groups included in Fig. 5. Taking the net value per unit prey species into consideration (see Flaaten and Stollery (1996) for details) gives the results shown in Fig. 6. The net value per unit prey, which equal b1 in Eq. (60 ), is the net contribution which the ®sh in the sea could have given for the prey harvesters if they had less competition from the predator, the cod. The net value per unit 188 O. Flaaten / Fisheries Research 37 (1998) 179±191 8. Discussion Fig. 5. Northeast Arctic cod's age-dependent average annual consumption of some commercially important prey species. Species included are shrimp (Pandalus borealis), capelin (Mallotus villosus), herring (Clupea harengus) and cod (Gadus morhua) above 5, 10, 10 and 20 cm, respectively. In gram prey per kg cod, 1984±92. Sources: Eide and Flaaten, 1998. prey catch was found to be 30% of the landing price in these ®sheries. Fig. 6 shows, for example, that 2-year old cod had an annual feed cost of NOK 1.50 (ECU 0.19) per kg of biomass, and that about 75% of this was in¯icted on the shrimp ®sheries. Apart from age class 7, the feed costs of shrimp dominates the economic ®gures, whereas capelin dominated the biological results in Fig. 5. Fig. 6. Age-dependent average annual feed cost of northeast Arctic cod's consumption of some commercially important prey species. Species included are shrimp (Pandalus borealis), capelin (Mallotus villosus), herring (Clupea harengus) and cod (Gadus morhua) above 5, 10, 10 and 20 cm, respectively. In NOK per kg cod, at 1991±92 prices. Consumption data from 1984±92. Sources: Eide and Flaaten, 1998. The uncertainty about the effects on predator and prey stocks and yield when ®shed, correspond to uncertainty about the functional forms of model equations, in addition to random ¯uctuations and parameter uncertainties. There are several ways to analyze the effects of such uncertainty on derived management strategies and the performance of ®sheries, e.g. by use of scenario modelling, decision analysis, baysian expert modelling, risk and uncertainty analysis and stochastic programming. However, it is also of interest to study the importance of the choice of functional forms for management strategies and performance in simple deterministic multispecies models. This paper has brie¯y reviewed some connections between bioeconomic objectives, management strategies with respect to effort and stock size, and performance within single-species and multispecies frameworks. When the objective is to maximize resource rent from harvesting a prey the optimal long run effort level is lower and the stock level higher within a predator±prey framework than in a singlespecies one. For a predator ®sh the opposite result appears ± the optimal long run effort level is higher and the stock level lower within a predator±prey context compared to the single-species case. When the prey is valuable and inexpensive to catch and the predator has a low market value, it may even bene®t the total resource rent to subsidize predator harvesting. When managing an abundant species at, or near, the top of the marine food chain, we should include an evaluation of the predation effects on commercially important prey species. The use of simple deterministic models could well be a ®rst step in such an analysis. Experimentation with functional forms may reveal whether they are important or not for management strategies. The MULTSPEC model, which was designed and implemented in Norway, (see Ulltang, 1995; Bogstad et al., 1998) has minke whale and harp seal as the top predators. In relation to the predator±prey models above they have carrying capacities independent of their prey stocks, i.e. (L, )(0). The commercially most important ®sh species of MULTSPEC are cod, capelin and herring. Models like MULTSPEC are useful for studying the implications of changes in O. Flaaten / Fisheries Research 37 (1998) 179±191 the stock levels of the top predators for the prey harvest. However, they are of no use in studying how changes in the prey stock affect the predator harvest and ®shing industry. Thus, sea mammals in MULTSPEC are one type of extreme predators. At the other extreme is the highly specialized predator, with (L, )(ÿ), which does not have other sources of food than the modelled prey. The more extreme it is the steeper is the isocline, and the prey harvesting may have a strong impact on the predator stock and harvest. The minimum stock level of the prey that can sustain the ÿ type of predator is a critical boundary (see Fig. 4) for the predator's survival. However, with respect to opportunistic feeders like minke whale and harp seal in the northeast Atlantic (see Haug et al., 1995; Nilssen, 1995) it is highly unlikely that their survival depends critically on any single prey. The 0 type predator, with a carrying capacity proportionate to the prey stock level, is a compromise between the type predator, with its partly prey independent carrying capacity, and the ÿ type predator. The analyses of sea-mammals ± ®sh interactions and harvesting in May et al. (1979); Flaaten (1988) are based on sea-mammals as a 0 type predator. As shown above the effect on the equilibrium harvest of the 0 type predator of a marginal change in the prey stock level falls in between the corresponding effects of the type and the ÿ type predators. Contrast this with Yodzis (1994), who ``... draw the conclusion that the form of model inherited by Flaaten (1988) from May et al. (1979) biases the case against marine mammals, without justification in terms of the underlying biology.'' (Yodzis, 1994, p. 52.) Note, that uncertainty about the predator's numerical response to changes in the prey stock level does not matter for the effects on the prey equilibrium harvest and resource rent of marginal changes in the predator stock level, as shown in Eqs. (5),(6),(50 ) and (60 ). Also note that these effects on the prey harvest and resource rent can be decomposed into two parts: one containing the predator's average per unit time consumption of the prey, and one containing the density dependent change in this of a marginal change in the prey stock level. The latter effect vanishes in the model with linear isoclines, as seen 189 in Eqs. (50 ) and (60 ), which makes it simple to apply to actual cases (see Flaaten and Stollery (1996) for such an application). Within the framework of a simple predator±prey biomass model it has been shown, in Eqs. (7) and (8), that the effects on the predator harvest and resource rent of a marginal increase in the prey equilibrium stock level is unambiguously positive for all prey stock levels. However, the magnitude of these effects might be less for high prey stock levels than for low levels. This happens if there is a decreasing marginal biological productivity of the prey, i.e. @ 2 g=@W12 < 0. Some stocks possess public non-use values that makes them a mixed good. They are private goods in that it is possible to harvest them, and public goods due to their amenity value. Including the positive nonuse value of the stock level in the management objective implies a lower long run effort level and higher stock level compared to the private good only case. Since non-use values of environmental goods, and evils, are subjective by de®nition, the methods used to estimate bene®ts and costs are not trivial. However, there are numerous examples where such methods have been used (for a review see Hanley and Spash, 1993). In the case of marine predators like whales and seals there are few examples of such studies (Kuronuma and Tisdell, 1993). From a scienti®c point of view it is important to distinguish between management strategies derived from purely biological objectives and strategies derived from economic objectives that include resource economic aspects of harvesting and/or non-use values. The former need only input from biologists, in a broad sense, whereas the latter also require input from economists. To illustrate the use of the theoretical analyses of this paper, data on the northeast Arctic cod's consumption of some commercially important prey species have been presented graphically. It is shown, in Fig. 5, that of the four species capelin, herring, shrimp and cod, above 5, 10, 10 and 20 cm, respectively, capelin is the most important prey for cod. On average for the years 1984±92 capelin constituted about 75% of the feed of cod of age 2 and older. Shrimp is the second most important prey, constituting about 10± 20% of the feed, highest for 2-year old cod and lowest for the 7 age group. These ®gures do not fully acknowledge the importance of capelin as prey since 190 O. Flaaten / Fisheries Research 37 (1998) 179±191 they are based on weight. The caloric content of ®sh species varies with the season, but the annual averages of capelin, herring, shrimp and cod are 1.58, 1.74, 1.39 and 1.12, respectively (adapted from Nordùy et al., 1995). The prey species have alternative value as catch for ®shermen. Knowing landing prices and harvest costs, the feed costs, i.e. the net value per unit harvest, of the prey species were calculated. It is shown in Fig. 6 that shrimp constitute between 40% and 80% of the feed costs for all age groups 2 years and older, with the highest percentage for the age 2 group. Capelin is by far the major feed in weight. For the oldest age group, 7, cannibalism is just above half the feed costs. From a feeding ecology point of view capelin is the most important prey of cod in the Barents Sea. However, the economic ®gures shown in this paper indicate that the costs per kg of cod biomass in¯icted on the prey ®shing industries is much higher for shrimp than for capelin and herring. Even though this is a partial bioeconomic analysis it strongly indicates that biological and economic factors should be considered simultaneously in management analysis. The assumption of functional forms of biological multispecies models are of importance for the management implications derived. However, it might be that the inclusion of economic factors will weaken the relative importance of some biological factors for management strategies. Acknowledgements I gratefully acknowledge the generous help from S. Mehl for providing the cod consumption data and E. Kolsvik for research assistance. I am indebted to two anonymous referees and co-editor T. Schweder for their valuable comments. 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