The maximum sustainable yield management of an age-structured salmon population: Fishing vs. conservation

Transcription

1 The maximum sustainable yield management o an age-structured salmon population: Fishing vs. conservation Anders Skonhot Department o Economics Norwegian University o Science and Technology (Anders.skonhot@svt.ntnu.no) Abstract This paper develops a sustainable yield harvesting model or the wild Atlantic salmon (Salmo salar), where the population comprises dierent age classes. It is shown that the weight ecundity relationship o the spawning population, comprising young and old ish, is crucial or the maximum sustainable yield ishing composition. First, the optimal harvest is ound when ecundity is approximated by weight. Second, we analyze what happens when weight is an inaccurate approximation or ertility. Finally, the conservation perspective is taken into account when the maximum sustainable yield is maximized subject to a minimum viable size o the harvestable, or spawning, population. It is shown that the conservation perspective does not aect the qualitative structure o the optimal harvest policy in which the old mature ish are more aggressively targeted than the young mature population. Key words: Salmon ishery, age classes, maximum sustainable yield JEL: Q22; Q7; C6... Thanks to Yajie Liu or assistance with preparation o this paper.

2 . Introduction For many years, the North Atlantic salmon (Salmo salar) has been one o the most important ish species in Norway because o its social, cultural, and economic importance. It was traditionally harvested or ood, but today is most important to recreational anglers. Norwegian rivers are the most important spawning rivers or the East Atlantic stock, and about 0% o the remaining stock spawns there. The wild salmon are harvested by commercial and recreational isheries. The marine harvest is commercial and semi commercial, whereas the harvest in the spawning rivers is recreational (NOU 999). The amount harvested in marine and river isheries has been more or less similar over the last ew years. However, the value o river ishery is much higher, because o the higher willingness to pay or sport ishing (NOU 999, Olaussen and Skonhot 2008). However, the abundance o wild salmon stocks has been declining during the last ew decades. Stock development has been especially disappointing since the 990s because o a combination o various actors, such as sea temperature, diseases, and human activity, both in the spawning streams and through the strong growth o salmon sea arming (NASCO 200). As the wild stock began to decrease during the 980s, the Norwegian government imposed gear restrictions to limit the marine harvest. Drit net ishing was banned in 989, and the ishing season o bend net ishing, taking place in the jords and close to the spawning rivers, has been restricted several times. At the same time, the sport ishing season in the spawning rivers has been subject to various restrictions (NOU 999). However, despite all these measures taken to secure and rebuild the stock, the abundance o wild salmon seems to be at only hal the level experienced in the 960s and 970s. It is believed that the rapid expansion o the armed salmon industry has played the most important role in this decline and today, armed salmon is regarded as the main threat to the viability o the wild salmon population because o the spread o diseases, escapees, and environmental pollution (Hindar et al. 2006). Wild salmon ishing has been studied in many papers rom an economic perspective. Routledge (200) studied a mixed stock versus single stock ishery related to Paciic salmon while Laukkanen (200) analyzed the northern Baltic salmon ishery in a sequential ishing biomass model. Olaussen and Skonhot (2008) also analyzed a sequential harvesting biomass model, but with recreational ishery in the rivers as its ocus. The economics o the Baltic salmon ishery is studied in an age-structured model in Kulmala et al. (2008). This is a dynamic model, comprising migration and seasonal harvest and competing harvesting by 2

3 commercial and recreational ishermen. Uncertainty is also included, and the model is parameterized and solved numerically or a Finnish river stock. In the ollowing ew pages, an age-structured wild salmon bioeconomic salmon model is analyzed as well, but within a much simpler ramework than that o Kulmala et al. The goal is to study, rom a theoretical point o view, how the harvesting o dierent age classes inluences recruitment and stock abundance, and the main ocus is to ind the harvest composition that maximizes the yield in biological equilibrium.. Neither possible interactions rom the armed salmon sector nor the problem o nonselectivity in the harvest is considered. However, the conservation perspective is taken into account as the maximum sustainable yield policy given a minimum viable spawning population is studied. Age-structured models are ar more complex than biomass models. On the one hand, it is relatively straightorward to ormulate a reasonably good age-structured model and numerically simulate the eects o variations in ishing mortality between age classes and over time (e.g., Kulmala et al. 2008). On the other hand, it is notoriously diicult to understand the various biological and economic orces at work in such models. Tahvonen (2008, 2009) has recently published papers dealing with some o these issues, in which he inds some results in a dynamic setting, but under quite restrictive assumptions. Early contributions analyzing age-structured models include Reed (980), who studied the maximum sustainable yield problem. He ound that optimal harvesting comprises, at most, two age classes. Further, i two age classes are harvested, the elder is harvested completely. Getz and Haight (988) reviewed various age-structured models, and ormulated the solution or the maximum sustainable yield problem as well as the maximum yield problem over a inite planning horizon. The ollowing analysis has similarities with Reed (980) and Getz and Haight (988), but we study a dierent biological system in which all the spawning ish, i.e., salmon, die ater spawning. This contrasts with Reed s model, where the spawning ish (e.g., cod) survive and enter an older year class ater spawning. As will be seen, this dierence has important implications or the optimal harvesting policy. While our analysis is directly related to Atlantic salmon, we will ind that it its various Paciic salmon species, such as pink and chum salmon, which also die ater spawning (see, e.g., Groot and Margolis 99). The paper is organized as ollows. In the next section, the population model is ormulated. The model is somewhat stylized ( generic ) as we consider only two harvestable, and hence

4 two spawning, age classes. In section three, we ind the maximum sustainable yield ishing policy under dierent assumptions, whereas in section our, we take the ishing value into account. The question o conservation is analyzed in section ive and the theoretical reasoning is numerically illustrated in section six. Section seven summarizes and concludes the paper. 2. Population model Atlantic salmon is an anadromous species that has a complex lie cycle with several distinct phases. Freshwater habitat is essential in the early development stages, as this is where it spends the irst one to our years rom spawning to juvenile rearing beore undergoing smoltiication and seaward migration. Then, it stays or one to three years in the ocean or eeding and growing and, when mature, returns to the natal or parent rivers to spawn. Ater spawning, most salmon die, as less than 0% o the emale salmon spawn twice (Mills 989). The Atlantic salmon is subject to ishing when it migrates back to its parent river. In Norway, most sea ishing takes place in jords and inlets with wedge-shaped seine and bend nets. This ishing is commercial or semi commercial. In the rivers, salmon are caught by recreational anglers with rods and hand lines. As indicated, the recreational ishery is by ar the most important rom an economic point o view. In what ollows, a speciic salmon population (with its native river) is considered in terms o a number o individuals at time t structured into recruits N 0,t ( yr < ), three young age classes, N,t ( yr < 2), N 2,t ( 2 yr < ) and N,t ( yr < ), and two adult, spawning classes, N ( yr < ) and N,t ( yr 6). Recruitment is endogenous and density dependent, and,t the old spawning salmon has higher ertility than does the young spawning salmon (more details are provided below). Natural mortality is ixed and density independent and, as an approximation, it is assumed that the whole spawning population dies ater spawning. It is urther assumed that the proportion between the two mature age classes is ixed. This proportion may be inluenced by a number o actors, such as the type o river ( small salmon river vs. large salmon river) and environmental actors (NOU 999). As ishing takes place when the ish returns back to its native river (see also above), only the mature salmon N,t and N,t are subject to ishing. Figure presents the model lie cycle o a single cohort o the salmon population considered. A ar more detailed description o the lie cycle o the Atlantic salmon is ound in, e.g., Verspoor et al. (200).

5 Figure about here With B t as the size o the spawning population, adjusted or dierent ertility among the two spawning classes (see below), the stock recruitment relationship is deined by: () N0, = RB ( ). t t RB ( t ) may be a one-peaked value unction (i.e., o the Ricker type) or it may be increasing and concave (i.e., o the Beverton Holt type). In both cases, zero stock means zero recruitment, R (0) = 0. Next, the number o young, depending on natural mortality, is as ollows: (2) Na, t sn a at, = ; a = 0,, 2, + + where s a is the age-speciic natural survival rate, assumed to be density independent and ixed over time. Finally, we have the mature age classes that are subject to ishing mortality (marine as well as river ishing), in addition to natural mortality. With 0< σ < as the proportion o the mature stock that returns to spawn in the irst year, the number o spawning ish o this part o the adult population (young) is: () N, t sn, tσ, t = + ( ), where,t yields the ishing mortality. Accordingly, H, t = sn, tσ, t is the number o harvested young mature ish in year t. As indicated, the parameter σ may depend on various actors, but is considered as ixed and exogenous. The rest o this cohort sn ( ), σ stays one year more in the ocean. When subject to natural t mortality, as well as subsequent ishing mortality, on migration back to the home river, the size o the next year s (old) spawning population becomes: () N, t+ 2 sn, t σ s, t+ = ( ) ( ). Hence, H, t sn, t( σ ), t+ = is the number o harvested old mature salmon in the cohort year t +. With γ and γ as the ecundity parameters o the young and old mature population, respectively, and where the old mature class is more productive than the young mature class, γ > γ, the spawning population in year t may be written as B = γ N + γ N, or t, t, t

6 () Bt γsn, t σ, t γsn, t 2 σ s, t = ( ) + ( ) ( ). The ecundity parameters will be considered as dimensionless parameters; that is, when scaling the ertility o the young mature population to one, γ =, the ertility o the old mature population, γ >, simply indicates that B t is measured as the ertility-weighted number o spawning salmon. Equation (2) implies that N =, t sssn, or , t (6) N =, t+ sr( Bt), when also using equation (2) and where s= sss 0 2comprises the previous years survival rates. Subsequently, N,t will be reerred to as the potentially harvestable population, or simply the harvestable population. For given ishing mortalities, equations (6) and () yield a system o two dierence equations o degree ive or the two variables N,t and B t. As already indicated, we are concerned only with equilibrium ishing, or sustainable harvesting, in this paper. The population equilibrium or ixed ishing mortalities is deined or N, t N = and Bt = B or all t such that: ( ) B [ γsσ( ) γs( σ) s( )] N and = +, (6 ) N = sr( B). In what ollows, ( ) is reerred to as the spawning constraint, whereas (6 ) represents the recruitment constraint (see also Skonhot et al. 200). An internal equilibrium ( N > 0 and B > 0 ) holds only i either or, or both, are below one; that is, to exclude depletion, both mature classes cannot be totally ished down. Notice that this is a necessary but not suicient condition. Figure 2 illustrates the internal, unique equilibrium when the recruitment unction is o the Beverton Holt type, i.e., R (0) = 0, R/ B = R' > 0 and R '' < 0 (see also section six). As the spawning constraint ( ) is a linear unction, whereas the recruitment constraint (6 ) is a strictly concave unction with Beverton Holt recruitment, an internal solution requires that the slope o the spawning constraint is less steep than that o the recruitment constraint or a zero stock level. t Figure 2 about here 6

7 In line with intuition, we ind that higher ishing mortalities shit the spawning constraint up ( ) and yield smaller equilibrium stocks. On the other hand, higher survival rates yield more ish as the spawning constraint ( ) shits down (both through s and s ), and the recruitment constraint (6 ) shits up (through s ). For a larger raction o the young mature stock through a higher value o σ, we ind more spawning ish as well as more harvestable ish i the mortality-corrected ertility parameter is higher or the young mature stock than or the old stock, i.e., γ( ) > γs( ). For equal targeted stocks, =, this simpliies to γ / s > γ, indicating that the biologically discounted ertility o the young mature stock dominates the old mature stock ertility.. The maximum sustainable yield harvesting program Initially, we analyze the optimal sustainable harvesting program without any conservation concerns. With w and w as the ixed weights (kg per ish) o the young and old mature population, respectively, where w > w, the equilibrium biomass harvested (in kg) is deined by Y wh wh [ ws σ ws ( σ) s] N = + = +. Then, the maximum sustainable yield problem is described by inding the ishing mortalities that maximize Y subject to the spawning constraint ( ) and the recruitment constraint (6 ). The Lagrangian o this problem may be written as L = [ wsσ + ws( σ) s] N λ[ N sr( B)] µ { B [ γ sσ( ) + γ s ( σ) s ( )] N }, where λ > 0 and µ > 0 (both in kg per ish) are the shadow prices o the recruitment and spawning constraints, respectively. Following the Kuhn Tucker theorem, the irst-order necessary conditions (assuming that N > 0 and B > 0 ) are: / ( ) 0 < ; 0, / ( ) 0 < ; 0, (7) L = N w µγ (8) L = N w µγ (9) L/ N = and w sσ + w s ( σ) s λ + µγ [ sσ( ) + γs ( σ) s ( )] = 0, (0) L / B = λ sr '( B) µ = 0. 7

8 Control condition (7) indicates that the ishing mortality o the young mature population should take place at the point where the marginal biomass gain is equal, below or above its marginal biomass harvest loss, determined by the ecundity parameter and evaluated by the spawning constraint shadow price. Condition (8) is analogous or the old mature population. The stock condition (9) says that the harvestable population should be managed so that the recruitment constraint shadow price λ is equal to the total marginal harvest gain plus the total marginal spawning biomass gain, evaluated at its shadow price. Finally, stock condition (0) indicates that the recruitment growth, evaluated at its shadow price, should be equal to the spawning constraint shadow price µ. From the control conditions (7) and (8), it is observed that only the weight ecundity ratio w / γ ( i =, ) determines the ishing mortality and the ishing composition and, hence, no i i other actors play a direct role. This outcome diers rom the seminal paper by Reed (980), who ound that weight together with natural mortality directly determined the ishing composition. As already indicated, the reason or this discrepancy is the dierent biological characteristics o the ish stocks, and the act that the mature ish die ater spawning in our salmon model, whereas the spawning ish survive and enter older age classes in the Reed model. The above solution is analyzed in two steps. First, we look at the situation where weight is considered as an approximation or ertility. Second, we analyze the optimal harvest option when the weight ecundity ratio is highest or the old mature stock.. Proportional weight ecundity relationship Weight and ertility are related, and larger, heavier, and older ish, in most instances, i not always, indicate higher ertility (e.g., Getz and Haight 989). When irst assuming that weight may serve as an approximation or ertility, indicating a proportional relationship between weight and ertility, w / w = γ / γ or w/ γ = w/ γ, both control conditions (7) and (8) must hold as equations. That is, w µγ = 0 with 0< < ( i =, ). This is the only i i possibility, because w / γ = w / γ > µ and, thereore, i = indicates extinction, whereas w / γ = w / γ < µ and = 0 ( i =, ) imply a zero biomass yield. Given this weight i ecundity assumption, the spawning constraint shadow price µ is determined by condition (7) i (or 8) as µ w/ γ w/ γ = = (the superscript indicates optimal values). Next, rom 8

9 equation (9), ater some small rearrangements, we ind the recruitment constraint shadow price rom = [ + ( ) ]. Thereore, when inserting µ and λ into condition (0), λ s wσ w σ s the size o the optimal spawning biomass is described by w / γ R'( B ) = µ / sλ =, or R'( B ) =. When urther ss [ w σ + w ( σ) s ] ss [ γσ+ γ( σ) s ] plugging in the recruitment constraint (6 ), the sustainable yield maximizing harvestable population is determined by N = sr( B ). Thereore, somewhat surprisingly, we ind that the optimal spawning population as well as the size o the harvestable population are determined independently o the ishing mortalities. Further, it is seen that higher values o both the ecundity parameters reduce R'( B ) and, hence, yield a higher spawning as well as a harvestable population. In line with this intuition, higher survival rates yield similar eects, whereas generally the eects o a higher σ are ambiguous. The task o inding the optimal ishing mortalities remains. Because, under the proportional weight ecundity assumption, conditions (7) and (8) give the same inormation, there is one degree o reedom in the system o equations deining the maximum sustainable yield policy. Thereore, the remaining equation describing the optimal policy, the spawning constraint ( ), must be satisied or all combinations o ishing mortalities that keep this equation in balance. When rewriting ( ), we then ind that all < ( i =, ) (but not 0 i = = ) satisying () ( B / N ) γσ ( ) = + γs( σ) s γ( σ) s are in accordance with the maximum sustainable yield ishing policy. This is stated as the ollowing proposition. Proposition. When ecundity is approximated by weight, ishing both adult subpopulations will represent the maximum sustainable yield harvesting policy. This optimal policy can be reached by an ininite number o combinations o ishing mortalities. The above optimal ishing mortality rontier () describes as a decreasing unction o. Both 0< < and the opposite case can be in accordance with this maximum sustainable harvesting policy. Dierentiation o () yields d = γσ d γ( σ) s, 9

10 indicating that a one percentage point increase o the young adult ishing mortality must be γσ accompanied by a reduction o γ ( σ)s wσ, or w ( σ ) s, o the old adult stock to sustain the optimal harvesting policy. This reduction may be above or below the one percentage point increase, but it will certainly be below it i the proportion σ o the stock that return to spawn in the irst year is small. With similar ishing mortalities, = =, we ind that B / N =. Thereore, not surprisingly, a higher optimal s [ γ( σ) s + γσ] spawning harvestable population ratio must be accompanied by a less aggressive similar ishing mortality among the two stocks..2 Weight as an inaccurate approximation or ecundity According to McGinnity et al. (200), weight is a questionable ertility approximation or the wild Atlantic salmon. Instead, they assume ertility to be described by a strictly concave unction o weight (and age), indicating that the weight ertility ratio increases with weight. With w/ γ w/ γ > and, hence, a higher marginal gain loss ratio or ishing the old spawning ish, the maximum sustainable yield harvesting policy, given by conditions (7) and (8), indicates a higher ishing mortality or the old than the young mature subpopulation. This is stated as the ollowing proposition. Proposition 2. With a higher weight ecundity ratio or the old adult subpopulation, the maximum yield harvesting policy is governed by a higher ishing mortality o the old subpopulation. There are three possible cases, all corner solutions, that may represent this optimal policy: i) = and 0< <, ii) = and = 0, and iii) 0< < and = 0. The spawning constraint ( ) will be steeper in case i) than in case ii), which again will be steeper than that in case iii). Thereore, when taking the recruitment constraint (6 ) into account (again, see Figure 2), we ind that the size o the spawning population as well as the harvestable stock will be highest with harvest option iii) and lowest i case i) represents the optimal policy. 0

11 In case i), the spawning constraint shadow price is determined through condition (7) as =. In combination with equation (9), this yields µ w / γ λ s wσ w σ s = [ + ( ) ]. Thereore, as both shadow price values are similar to the above proportional weight ecundity relationship in section., equation (0) λ sr '( B) = µ together with the recruitment constraint (6 ) indicates similar sizes o the optimal spawning biomass harvestable population N as well. I case ii) where = and B, and the = 0 represents the maximum sustainable yield policy, the spawning constraint ( ) simply reads as B= γsσn. In this case, we ind that this equation together with the recruitment constraint (6 ) N = sr( B) alone determines N and B. Hence, the optimal stock sizes are not directly inluenced now either by the ertility parameter o the old mature group or the ish weights. In case iii), where 0< < and = 0, the spawning constraint shadow price is ixed as µ = w / γ, whereas the recruitment constraint shadow price becomes λ = sw[ γσ/ γ + ( σ) s]. Thereore, just as in case i), the size o the harvestable population and the spawning biomass are determined through condition (0) and the recruitment constraint (6 ). The yield maximizing ishing mortality is again determined through the spawning constraint ( ), in this case as B = s [ γσ+ γ( σ) s ( )] N. I w/ γ is substantially higher than w / γ, intuitively, we may suspect that it is beneicial or the manager to invest in the salmon population by leaving the young mature population unexploited and harvesting the whole old adult population. Hence, case ii), where = and = 0, should maximize the sustainable yield. On the other hand, with a small weight ertility dierence, either case i), with harvesting o both mature populations, or case iii), with harvesting o the old mature population only, may represent the optimal solution. Thereore, values o γ and γ that make the gap w / γ > w / γ larger may indicate that case ii) is more likely to take place. Moreover, or given ecundity parameters, we can ind the same outcome when the gap widens through a higher weight discrepancy. However, as will be demonstrated in the numerical section, other parameter values, such as natural survival rates, will play a role as well.. The maximum economic yield policy

12 As indicated, the Atlantic salmon is targeted both by commercial ishermen and sport ishers, with the sport ishing taking place in the rivers with rods, whereas the commercial or semicommercial ishery takes place in the jords and inlets, with wedge-shaped seine and bend nets. When p ( i =, ) are the ishing values (NOK per kg) and when the value is i higher in the sport ishery than in the commercial ishery, but p p in both isheries (Olaussen and Liu 200), then π = [ pws σ + pws ( σ) s] Ndescribes the yearly gross money value in our salmon ishery. Because, typically, stock dependent variable costs in both isheries are small and negligible (or a description o these isheries, see, e.g., NOU 999), the gross money value may approximate gross proit when considering the ish prices as net prices. See Olaussen and Skonhot (2008) or a urther discussion o the cost and beneit structure o recreational salmon ishery. The control conditions or maximizing the sustainable economic yield value π subject to the spawning and recruitment constraints are as ollows: / ( ) 0 < ; 0, (2) L = N pw µγ and / ( ) 0 < ; 0, () L = N pw µγ where the spawning constraint shadow price µ is now measured in NOK per ish. The stock conditions o this new problem are basically the same as in the above maximum sustainable yield problem. Equation (0) is similar, whereas equation (9) changes slightly as the marginal beneit is now measured in NOK per ish. With w / γ > w / γand p p, we have pw / γ > pw / γ and corner solutions with a higher ishing mortality o the old than o the young adult population. Thereore, the structure o the solution o this problem is the same as the previous solution or the situation where weight is considered to be an inaccurate approximation or ecundity. The only thing to note is that case ii) is more likely to represent the optimal policy, as the marginal gain loss ratio o the old mature population may become relatively higher.. The conservation perspective So ar, the maximum sustainable ishing policy has been analyzed when only harvest values have been taken into account. However, wild Atlantic salmon typically also carry 2

13 nonconsumptive beneits such as existence and biodiversity values (see Verspooor et al. 200, and also the classical Krutilla 967). These stock values may have increased during the last ew decades not only because o lower ish abundance and the extinction o some stocks (each river has its own stock), but also because o interbreeding and genetic pollution rom escaped armed salmon (c. introductory section; see also Verspoor et al. 200). There are at least two dierent ways to take the stock value and conservation perspective into account within our social planner ramework. One possibility involves attaching a nonconsumptive value to the population size and inding ishing mortalities that maximize the composite utility, comprising the harvesting (low) value and population (stock) value. Another possibility is to impose a minimum population size constraint, and ind ishing mortalities that maximize the sustainable economic yield subject to this constraint, in addition to the biological constraints. Here, we considering the second option, in line with the upcoming management practice in Norway whereby the Environmental Department (through the Directorate or Natural Resource Management) aims to regulate salmon ishery in rivers with weak and threatened populations by imposing a minimum viable size o the spawning population (Hindar et al. 2007). In our analysis, this type o policy is ormulated by demanding that the ollowing holds: () N N. Notice that this restriction is similar to imposing a minimum size o the spawning population B (see also below). Now, the planning problem is to maximize π = [ pws σ + pws ( σ) s] Nsubject to the spawning and recruitment constraints ( ) and (6 ), in addition to the minimum harvestable population size constraint (). The Lagrangian o this problem is L = [ p w s σ + p w s ( σ) s ] N λ[ N sr( B)] µ { B [ γ sσ( ) + γ s ( σ) s ( )] N } θ( N N ] with θ 0 as the stock (conservation) shadow price (NOK per ish). Now, we ind that (2) and () also describe the irst-order control conditions, whereas the stock conditions (with N > 0 and B > 0 ) are given by (0) together with: () L/ N = p w sσ + p w s ( σ) s λ + µγ [ sσ( ) + γs ( σ) s ( )] + θ= 0.

14 Thereore, still under the assumption that pw / γ > pw / γ, the optimal harvesting policy should again be governed by a higher ishing mortality o the old than o the young adult stock either through case i) = and 0< <, case ii) = and =, or case iii) 0 0< < and = (but see below). This is stated as the ollowing proposition. 0 Proposition. Imposing a minimum viable population size constraint relecting the conservation perspective leaves the structure o the optimal harvest composition unaected. The interesting situation, o course, is when this new constraint binds. When it binds, we have N = N together with θ > 0. Then, the spawning population is determined directly through the recruitment constraint (6 ) as N sr( B ) =. Furthermore, the ishing mortalities must satisy the spawning constraint ( ) written as B [ γsσ( ) γs( σ) s( )] N = + (see also Figure 2). Thereore, case ii) with = and = 0 cannot represent the optimal solution (except by accident). Moreover, because case i) with = and 0< < indicates a more aggressive ishing policy than case iii) with 0< < and = 0 (see also section three above), we ind that this last case yields the only possible optimal solution when the stock regulation policy is tight and, hence, the stock constraint N is high. With this case as the optimal solution, the old adult ishing mortality is determined through B = [ γ sσ + γ s ( σ) s ( )] N. On the other hand, with case i) as the optimal option, the young adult ishing mortality is determined through B = γ sσ( ) N. In the various cases, we have π = θ (envelope theorem). / N 6. Numerical illustration 6. Data The above theoretical reasoning will now be illustrated numerically. Hansen et al. (996) estimated a salmon recruitment unction or a small river in Norway (the Imsa River) based on the Shepherd recruitment unction, which includes three parameters. In our generic model, we choose a simpler approach and use the Beverton Holt unction (c. Figure 2). This B unction may be speciied as RB ( ) = r + B/ K, with r > 0 as the intrinsic growth rate, or maximum number o recruits per (ertility adjusted) spawning salmon, and K > 0 as the stock

15 level or which density-dependent mortality equals density-independent mortality. The size o rk yields the maximum number o recruits and scales the system ( size o the river ), which is assumed to be 0,000 (number o recruits). The value o r indicates the quality o the river, and we choose r = 00 (number o recruits per spawning salmon). Then, we ind K = 00. Table shows these values, as well as the other baseline parameter values used in the numerical analysis. When normalizing the ertility parameter or the young to one, γ =, and using the ertility weight unction in McGinnity et al. (200), we ind that γ = 2. under the assumption o (average) ishing weights o w = 2.0 and w =.(kg/salmon). These weights it a typical medium-sized Norwegian salmon river (NOU 999). Thereore, or the given weigh and ecundity values, the weight ertility ratio is higher or the adult stage than or the young stage, w / γ > w / γ. The survival parameters are based on NOU (999), whereas the ishing prices are related to recreational ishery, which, as indicated, is ar more important economically than the marine ishery. The assumption here is that the ishing permit price in a reasonably good river is about 200 NOK per day (see also Olaussen and Liu 200). Based on average catch success, this permit price may translate into ishing prices in the range o (NOK/kg), or even higher. We assume the same price or old and young and use p = p = 0 (NOK/kg). Table about here 6.2 Results Because the weight ertility ratio is highest or the old adult population and the ish price is similar in the baseline scenario, and hence pw / γ =.8 > pw / γ = 00 (NOK/ish), the economic yield maximizing ishing mortality will be highest or the old adult population (Proposition 2). Table 2 (irst row) demonstrates where case i) with = and = 0. yields the optimal ishing mortality. Increasing γ, while keeping all the other parameters ixed, eventually leads to similar marginal gain loss ratio with γ = 2.7, pw / γ = pw / γ = 00 (row two). Then, the maximum sustainable economic yield policy can be reached by an ininite number o combinations o ishing mortalities obeying equation () (Proposition ), ranging rom 0. to.00 or the young and rom.00 to 0.0 or the old

16 mature population. As analyzed above (section three), the harvestable population size the size o the spawning population N and B are similar to those o case i) (irst row). This will be the case or the yield as well as or the proit. On the other hand, reducing the gain loss ratio o the young mature subpopulation by lowering p while keeping all other parameters at their baseline values, so that the discrepancy between pw / γ and pw / γ increases, leads to the optimal ishing policy described by case ii), with no harvesting o young ish (row three). Table 2 about here The last row in Table 2 indicates what happens when the natural survival rate o the young s is reduced while all other parameters are kept at their baseline values. Such a reduction may be the result o inection through transmission o lice rom armed salmon. Indeed, as already indicated, this is considered to be one o the most important threats to the wild Atlantic salmon (see, e.g., Verspoor et al. 200). A 0% reduction yields quite dramatic eects. The spawning biomass declines signiicantly and the proit is reduced by more than 0%. Again, case i) with harvesting o the entire old adult population represents the optimal ishing policy. Table demonstrates what happens when conservation through the harvestable population constraint () (section ive) is taken into account. The qualitative structure o the harvesting composition with more aggressive ishing o the old mature stock is unchanged (Proposition ). The optimal harvesting policy is determined by case i) with = and 0< < when the conservation constraint starts to bind. Increasing N urther eventually leads to case iii) with 0< < and = 0 because case ii), as analyzed above (section ive), cannot represent an optimal ishing option when conservation is taken into account. (However, this case becomes the optimal harvest option by accident when N =, 600, although this is not shown in the table). The cost o conservation eventually becomes quite signiicant (column seven), as also indicated by the shadow price value θ (last column). Thereore, increasing the minimum viable size o the harvestable population by one single ish rom, say, N =,700, reduces the proit by NOK 2. Table about here 6

17 7. Concluding remarks In this paper, we have studied, rom a theoretical point o view, the maximum sustainable yield management o an age-structured wild Atlantic salmon (Salmo salar) population with two spawning and harvestable classes. The basic inding is that the weight ecundity ratio discrepancy between the harvestable classes determines the optimal ishing mortality and the ishing composition, and no other actors play a direct role. This outcome diers rom the seminal paper by Reed (980), who ound that weight together with natural mortality directly determined the maximum sustainable yield ishing composition. The reason or this discrepancy is the dierent biological characteristics o the ish stocks, as the mature salmon die ater spawning in our model, whereas in the Reed model, a ixed raction o the spawning ish survive and enter older age classes. Our analysis and indings are based on the Atlantic salmon, but the results will also apply to, e.g., the various Paciic salmon stocks, which also die ater spawning. Our model is analyzed in three steps. First, the situation with a proportional weight ecundity relationship is considered. Second, the optimal harvest options are analyzed when the weight ecundity ratio is highest or the old mature stock. Finally, conservation through a minimum viable size o the harvestable population is also taken into account. The analysis provides three propositions about the maximum sustainable ishing yield or the Atlantic salmon. Our model may also be extended to the possibly more realistic situation with three harvestable age classes. Given that the weight ecundity ratio increases with weight (and age) (McGinnity et al. 200), we ind that the maximum sustainable yield policy includes ishing o the two oldest year classes at most. This will be the straightorward outcome when analyzing the Kuhn Tucker control conditions o this new, extended problem. One important management implication o our analysis is that a balanced ishing policy that implies equal ishing mortalities among the ishable stocks will generally lead to economic losses, and that such losses will typically increase with the weight ecundity discrepancy among the spawning ish stocks. Hence, any quota policy should be based on speciied ishing mortalities or the harvestable classes, which will generally be dierent, and highest or the old mature salmon. This will also be the case i the management policy includes imposing a minimum viable size o the spawning population. Selecting between harvesting old and young mature ish is diicult, but can be inluenced by actors such as ishing gear and mesh size, as well as seasonal regulation. Seasonal regulation will aect ishing selectivity, because young 7

18 mature and old mature salmon to some extent migrate back to the parent river sequentially, with the broad pattern being that the young mature return beore the old (see, e.g., NOU 999). Hence, seasonal regulation combined with mesh-size regulation in the marine ishery are possible ways to more eiciently target the Atlantic salmon ish stock. Reerences Getz, W. and R. Haight 988. Population Harvesting. Princeton University Press, Princeton Groot, G. and L. Margolis (eds.) 99. Paciic salmon lie histories. UBC press, Vancouver Hansen, L. P., B. Jonsson and N. Jonsson 996. Overvåking av laks ra Imsa og Drammenselva (in Norwegian). NINA Oppdragsmelding 0, Trondheim Hindar, K., I. Flemming, P. McGinnity and O. Diserud Genetic and ecological eects o armed salmon on native salmon: modeling rom experimental results. ICES Journal o Marine Science 6: 2-27 Hindar, K. et al Gytebestandsmål or laksebestander i Norge (in Norwegian). NINA Report 226, Trondheim Krutilla, J Conservation reconsidered. American Economic Review 7: Kulmala, S., M. Laukkanen and C. Michielsens Reconciling economic and biological modelling o a migratory ish stock: Optimal management o the Atlantic salmon shery in the Baltic Sea. Ecological Economics 6: Laukkanen, M A bioeconomic analysis o the Northern Baltic salmon ishery. Environmental and Resource Economics 88: 29- McGinnity, P. et al Fitness reduction and potential extinction o wild populations o Atlantic salmon, Salma salar, as a result o interactions with escaped arm salmon. Proceedings o the Royal Society B 270: 2-20 Mills, D Ecology and management o Atlantic salmon. Chapman and Hall, New York NASCO 200. Report on the activities o the North Atlantic salmon conservation organization ( NOU 999. Til laks åt alle kan ingen gjera? (in Norwegian) Norges Oentlige utredninger999:9, Oslo. Olaussen, J. O. and Y. Liu 200. Does the recreational angler care? Escaped armed versus wild Atlantic salmon. Working paper Department o Economics NTNU, Trondheim Olaussen, J.O. and A. Skonhot A bioeconomic analysis o a wild Atlantic salmon recreational ishery. Marine Resource Economics 2: 9-9 8

19 Reed, W. J Optimum age-speciic harvesting in a nonlinear population model. Biometrics 6: Routledge, R Mixed-stock vs. terminal isheries: A bioeconomic model. Natural Resource Modeling : 2-9 Skonhot, A., N. Vestergaard and M. Quaas 200. Optimal harvest in an age structured model with dierent ishing selectivity. Submitted Tahvonen, O., Harvesting an age-structured population as biomass: does it work? Natural Resource Modeling 2: 2-0 Tahvonen, O Economics o harvesting age structured ish populations. Journal o Environmental Economics and Management 8: Verspoor, E., L. Stradmeyer and J. Nielsen (eds.) 200. The Atlantic Salmon. Genetics, Conservation and Management. Blackwell Publishing, New York 9

20 Table. Biological and economic baseline parameter values Parameter Description Value s Natural survival rate 0.0 young s s r K Natural survival rate young adult Natural survival rate old adult Intrinsic growth rate recruitment unction Scaling parameter recruitment unction (# o recruits/ ertility adjusted spawner) 00 (# o spawners) σ Migration parameter 0. w Weight young adult 2.0 (kg/ish) w Weight old adult. (kg/ish) γ γ Fecundity parameter young adult Fecundity parameter old adult.0 2. p Fish price young adult 0 (NOK/ kg) p Fish price old adult 0 (NOK/kg) 20

21 Table 2: The maximum sustainable economic yield problem N (#) B (#) H (#) H (#) Baseline values 0..00, % increase [0. - [ ],20 2 [ ] [82 79 ertility coeicient.00] 89] old ( γ = 2.7 ) π (000 NOK) 200% reduction price young adult ( p = 0 ) 0% reduction natural survival rate young ( s = 0.0) , Table : The maximum sustainable economic yield and conservation N B (#) H (#) H (#) π (000 NOK), , , , , ,6 θ (NOK/ish) 2

22 Figure. Schematic representation o the lie cycle o a wild Atlantic salmon or a single cohort (the time index is omitted). See main text or deinition o symbols. Age 0 N 0 s 0 Harvest H = H + H 2 Recruitment, R Age N Spawning biomass B s Age 2 N 2 s 2 (- ) Age N (- ) σ s Age N (-σ) s Age N 22

23 Figure 2. Internal equilibrium or ixed ishing mortalities 0, 0 (but not = = ). Beverton-Holt type recruitment unction. N Eq.( ) (spawning constraint) Eq. (6 ) (recruitmet constraint) B 2