Laksedemografi0510
UNCOMPLETED FIRST ROUGH DRAFT…..
The maximum sustainable yield management of a salmon population. Fishing vs. conservation
Anders Skonhoft
Department of Economics
Norwegian University of Science and Technology
()
Abstract
This paper develops a sustainable yield harvesting model for the wildAtlantic salmon (Salmo salar)and where the population is composed of different age classes. It is shown that the weight – fecundity relationship of the spawning population is crucial for the optimal harvesting composition. When neglecting the conservation perspective, it is first seen how the harvest composition may be when fecundity is approximated by weight. Second, we analyse what happens when weight is an inaccurate approximation for fertility. The conservation perspective is taken into account when the maximum sustainable yield is maximized subject to a minimum viable size of the harvestable population. It is shown that such conservation perspective does not affect the qualitative structure of the optimal harvest composition.
Key words: Atlantic salmon, age classes,maximum sustainable yield
JEL: Q22; Q57; C61
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Thanks to Yajie Liu for assistance with preparation this paper.
- Introduction
The abundance of wild salmon stocks in the North Atlantic (Salma salar) has been declining during the last few decades. Stock development has been especially disappointing since the 1990s due to a combination of various factors, such as the sea temperature, diseases, and human activity, both in the spawning streams and through the strong growth of salmon sea farming (NASCO 2004). Norwegian rivers are the most important spawning rivers for the East Atlantic stock, and about 30% of the remaining stock spawns here. The wild salmon are harvested by commercial and recreational fisheries. The marine harvest is commercial and semi commercial, whereas the harvest in the spawning rivers is recreational (NOU 1999). As the wild stock began to decrease during the 1980s, the Norwegian government imposed gear restrictions to limit the marine harvest. Drift net fishing was banned in 1989, and the fishing season of the bend net fishing taking place in the fjords and close to the spawning rivershave been restricted several times. At the same time, the sport fishing season in the spawning rivers have been subject to various restrictions as well (Olaussen and Skonhoft 2008). However, despite all these measures taken to secure and rebuild the stock, the abundance of wild salmon seems to be only half the level experienced in the 1960s and 1970s. The same sad picture is observed other places (NASCO 2004).
In this paperan age structured wild salmon modelis developed. The goal is toanalyze how harvest of different age classes influence recruitment and stock abundance, and where the main focus is tofind the maximum sustainable yield harvesting policy under various assumptions. Such knowledge may have important management implications. As will be shown, the optimal policy and harvest composition is directly related to the different weight – fecundity ration among the spawning fish classes. The conservation perspective is taken into account when the sustainable yield is maximized subject to a minimum size of the harvestable population. It is shown that this conservation perspective does not affect the qualitative structure of the optimal harvest composition.Salmon fishing has been studied in many papers, also from an economic perspective. Laukkanen (2001) analyzed the northern Baltic salmon fishery in a biomass modelbased on the Charles and Reed (1985) sequential fishing model. Olaussen and Skonhoft (2008) also studied a sequential harvesting biomass model, but where the recreational fishery in the rivers was at the focus. The Baltic salmon fishery is also studied in Laukkanen et al. (2008), but within an age structured dynamic model.
Age structured models are far more complex than biomass models. On the one hand, it is relatively straightforward to formulate a reasonable good age-structured model and numerically simulate the effects of variations in fishing mortality between age classes and over time (e.g., Laukkanen et al 2008). On the other hand, it is notorious difficult to understand the various biological as well as economic forces at work in such models. Olli Tahvonen(2008, 2009) has recently published papers dealing with some of these issues and where he finds some results in a dynamic setting, but under quite restrictive assumptions. Early contributions analyzing age structured modelsinclude Reed (1980)who studied the maximum sustainable yield problem. He found that optimal harvesting comprises at most two age classes. Further, if two age classes are harvested, the elder is harvested completely. Getz and Haight (1988) review various stage structured models, and formulate the solution for the maximum sustainable yield problem as well as the maximum yield problem over a finite planning horizon. The following analysis has similarities with Reed (1980) and Getz and Haight (1988), but we are studying a different biological system where the spawning salmon dies after spawning. This in contrast to Reed’s model where the spawning fish enter an older year class after spawning. As will be seen, this difference has important implications for the optimal harvesting policy. Just as in the Reed paper, only equilibrium fishing is considered.
The paper is organized as follows. In the next section, the population model is formulated. The model is somewhat stylized (‘generic’) as we consider only two harvestable and hence two spawning year classes.In section three, we find themaximum sustainable yield fishing policy under different assumption while we in the next section four also take the fishing value into account. The question of conservation is analyzed in section five, while the theoretical reasoning is numerically illustrated in section six. Section seven finally concludes the paper.
2. Population model
Atlantic salmon is an anadromous specieswith a complex life cyclewith several distinct phases.Freshwater habitat is essential of the early development stages where it spends the first 1-4 years from spawning to juvenile rearingbefore undergoing smoltification and seaward migration. Then it staysfor 1-3 years for feeding and growingin the ocean, and when mature, it returns to their natal, or ‘parent’, rivers to spawn. After spawning, most diesas less than 10% of the female salmonspawn twice (Mills 1989).The Atlantic salmon is subject to fishing when it migrates back to its parent river. In Norway, sea fishing takes place in fjords and inlets with wedge-shaped seine and bend net and is commercial, or semi commercial. In the rivers, salmon are caught by recreational anglers with rod and hand line. The recreational fishery is the far most important from an economic point of view (NOU 1999), but in number of fish and biomass fished these two fisheries are today more or less equivalent (VitenskapeligRaad 2009).
In what follows, a specific salmon population(with its native river)is considered in number of individuals at time structured asrecruits(), three young age classes, (), () and (), and two adult, spawning classes, () and (). Recruitment is endogeneous and density dependent, and the old spawning salmon has higher fertility than the young spawning salmon (McGinnity et al 2003). Natural mortality is fixed and density independent, and as an approximation it is assumed that the whole spawning population dies after spawning.It is further assumed that the proportion between the two mature age classes is fixed. This proportion may be influenced by a number of factors, such astype of river (‘small salmon river’ vs. ‘large salmon river’) and various environmental factors (NOU 1999).As fishing takes place when the fish returns back to its native river (see also above), only the mature salmon andare subject tofishing.Figure 1 presents the model life cycle of a single cohort of the consideredsalmon population.A far more detailed description of the life cycle of the Atlantic salmon is found in e.g., Verspoor et al. (2003).
Figure 1. Schematic representation of the life cycle of a wild Atlantic salmon for a single cohort (the time index is omitted). See main text for definition of symbols.
Withas the size of the spawning population, adjusted for different fertility among the two spawning classes (see below), the stock recruitment relationship is first defined by:
(1).
may be a one-peaked value function (i.e., of the Ricker type) or it may be increasing and concave(i.e., of the Beverton-Holt type). In both cases, zero stock means zero recruitment,. The number of young, depending on natural mortality, follows next as:
(2);
and wheresa is the age-specific natural survival rate, assumed to be density independent and fixed over time. Finally, we have the mature age classes that are subject to fishing mortality (marine as well as river fishing), in addition to natural mortality. Withas theproportion of the mature stock that returns to spawn the first year, the number of spawning fish of this part of the adultpopulation (young) is:
(3)
whenyields thefishing mortality. is accordingly the number of harvested young mature fish year . As indicated, the parametermay depend on various factors, but is considered as fixed and exogeneous.
The rest of this cohort stays one year more in the ocean. When also being subject to natural mortality as well as subsequent fishing mortality when it migrates back to the home river, the size of this next years (old) spawning population becomes:
(4).
is hence the number of harvested old mature of this cohort year . With and as the fecundity parametersof the young and old mature population, respectively, and where the old mature class is more productive than the young mature class,,the spawning population year may be written as , or:
(5).
The fecundity parameters will be considered as dimensionless parameters; that is, when scaling the fertility of the young mature class to one, , the fertility of the old mature class, , simply indicates thatis measured as a fertility weighted number of spawning salmons. Equation (2) implies, or:
(6)
when also using equation (2) and wherecomprisesprevious years survival rates.will subsequently be referred to as the potentially harvestable population, or simply, the harvestable population.For given fishing mortalities, equations (6) and (5)yield a system of two difference equations of degree fivein the two variables and .
As already indicated, we are only concerned with equilibrium fishing, or sustainable harvesting, in this paper. The population equilibrium for fixed fishing mortalities is defined for and for all such that:
(5’)
and
(6’).
In what follows, (5’) is referred to as the spawning constraint while (6’) represents the recruitment constraint. An internal equilibrium (and ) holds only if eitheror, or both, are below one; that is, to omit depletionboth mature classes can not be fished totally down.Notice that this is a necessary, but not sufficient condition. Figure 2 illustratestheinternal, unique equilibrium when the recruitment function is of the Beverton-Holttype, i.e., and and (see also numerical section six).
In line with intuition, we find that higher fishing mortalities shift up the spawning constraint (5’) and yield smaller equilibrium stocks. On the other hand, higher survival rates yield more fish as the spawning constraint (5’) shifts down(both through and ), and the recruitment constraint (6’)shifts up (through ). For a larger fraction of the young mature stock; that is, a higher value of, we find more spawning fish as well asmoreharvestable fishif the mortality corrected fertility parameter is higher for the young mature stock than the old stock, i.e.,. For equal targeted stocks,, this simplifies to which may be interpreted as that the young mature stock ‘biological discounted’ fertility dominates the old mature stock fertility.
Figure 2. Internal equilibrium for fixed fishing mortalities , (but not ). Beverton-Holt type recruitment function.
3. The maximum sustainable yield harvesting program
We start to analyze the optimal sustainable harvesting program without any conservationconcern. With and as the fixedweights(kg per fish) of the young and old mature population, respectively, and where, the equilibrium biomass harvested (in kg) is defined by. The maximum sustainable yield problem is then described by finding fishing mortalities that maximizes subject to the spawning constraint (5’) and the recruitment constraint (6’). The Lagrangian thisproblem may be written as where and (both in kg per fish) are the shadow prices of the recruitment and spawning constraint, respectively. Following the Kuhn-Tucker theorem, the first-order necessary conditions (assumingand )are:
(7) ; ,
(8) ; ,
(9)
,
and
(10) .
Control condition (7) indicates that the fishing mortality of the young mature population should take up at the point where the marginal biomass gain is equal, below or above its marginal biomass harvest loss determined by the fecundity parameter and evaluated by the spawning constraint shadow price. Condition (8) is analogous for the old mature population. The stock condition (9)says that the harvestable population should be managed so that the recruitment constraint shadow price is equal the total marginal harvest gain plus the total marginal spawning biomass gain, evaluated at its shadow price. Finally, equation (10) indicatesthatthe recruitmentgrowth,evaluated at its shadow price,should be equal the spawning constraint shadow price.
From the control conditions (7) and (8), it is observedthat only the weight-fecundity ratio () steer the fishing mortality and the fishing composition. and hence no other factors play adirect role. This outcome differs fromthe seminal Reed(1980) paperwho found that weight together with natural mortality directly determined the fishing composition. As already indicated, the reason for this discrepancy is the different biological characteristics of the fish stocks, and where the mature fish dies after spawning in our salmon model while the fish survives and enter older age classesin the Reed model.The above solution in analyzed in two steps. First, we look at the situation with a proportional weight – fecundity relationship. Second, it is seen what happens when the weight – fecundity ratio is highest for the old mature stock.
Proportional weight – fecundity relationship
Weight and fertility is related, and larger, heavier and older salmon typically indicates higher fertility (e.g., McGinnity et al. 2003). When assuminga proportional relationship between weight and fertility and hence, both control conditions (7) and (8) then hold as equations, with().This is the only possibility because and thereforemean extinction while and () implies zero biomass yield.Under this weight – fecundity assumption, the spawning constraint shadow priceis determined by condition (7) (or 8) as(superscript ‘’ yields optimal values). From equation (9),after some small rearrangements, we next find the recruitment constraint shadow price as.Therefore, wheninserting for andinto condition (10),the size of the optimal spawning biomassis described by, or.When further plugging into the recruitment constraint (6’), the sustainable yield maximizing harvestable populationis determinedby.Therefore, somewhat surprisingly, we find that the optimal spawning population as well as the size of the harvestable population are independent of the fishing mortalities.It is further seen that higher values of the fecundity parameters reduces and hence yield a higher spawning as well a harvestable population. In line with intuition, higher survival rates yield similar effects while the effects of a shiftingis generally ambiguous.
What is left is to find the optimal fishing mortalities. Because conditions (7) and (8) under the proportional weight - fecundity assumption give the same information, it is one degree of freedom in the system of equations defining the maximum sustainable yield policy. Therefore, theremaining equation describing the optimal policy, the spawning constraint (5’), mustbe satisfied for all combinations of fishing mortalities that keepthis equation in balance. When rewriting (5’), we then find that all () satisfying
(11))
are in accordance withthe maximum sustainable yield fishing policy. This is stated as:
Proposition 1. When fecundity is approximated by weight, fishing both adult sub populations will represent themaximum sustainable yield harvesting policy. This optimal policy can be reached by aninfinite number of combinations of fishing mortalities.
The above optimal fishing mortality equation (11) describes as a decreasing function of . Both, as well as the opposite, may be in accordance with this maximum sustainable harvesting policy. The fishing mortality of the old mature population will be above that of the young mature if. With similar fishing mortalities, , we find. Therefore, not surprisingly, a higher spawning – harvestable population ratio must be accompanied by a less aggressive fishing policy. Differentiation of equation(11) yields, indicatingthat 1 %-point increase of the young adult fishing mortality must beaccompanied by the reduction, or ,of the old to sustain the optimal policy. This reduction may be above as well as below 1 %-point.
Weight as an inaccurate approximation for fecundity
FollowingMcGinnity et al. (2003)the wild Atlantic salmon fertility is typically aconcave function of weight. With , or and hence ahigher marginal gain - loss ratio for fishing the old spawning fish,the optimal harvesting policy, described by conditions (7) and (8), indicates a higher fishing mortality for the oldthan the young maturesubpopulation. In this situation there aregenerally three possibilities present; i) and , ii) and, and iii) and. This is stated as: