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Fishery Bulletin 98(1) 



tion, however, only one level of fishing will satisfy 

 this condition. Note that the converse is not true; for 

 a given level of fully recruited fishing mortality, nu- 

 merous partial recruitment functions can satisfy the 

 above condition. Because of the nature of these rela- 

 tionships, I will focus on those situations where the 

 partial recruitment function is specified and solve for 

 the level of fishing that is sustainable. Once the selec- 

 tion pattern and fully recruited fishing mortality are 

 set, Ap can be found by the power method as described 

 by Johnson and Riess ( 1981 ). 



Example of maintenance fishing mortality: 

 deterministic case 



Data from Georges Bank haddock (Melanogrammus 

 aeglefinus) are used to illustrate the computation and 

 application of this reference point. For ease of discus- 

 sion, I first present general results assuming knife- 

 edge recruitment to the fishery at age t^, with full vul- 

 nerability thereafter. 



Age-specific fecundity equivalents (X(i); Table 2) 

 were computed as 



Xa) = W(i)xPM{i), 



where W(/) = mean weight (kg) at age /; and 



PMii) = proportion of females mature at age i. 



and spawning stock biomass was computed as the 

 product of fecundity equivalents and number of fish 

 at age. Mean weight at age iW{i))and proportion of fe- 

 males mature ^PM^ i ) ) reported by O'Brien and Brown^ 

 were used in this analysis. The instantaneous natural 

 mortality rate (Mii)) of haddock age 1 and older is 0.2 

 (Clark et al., 1982), and a maximum age of 15 was 

 used following Gabriel et al. ( 1989). 



I computed annual R/SSB (Table 3) from the ratio 

 of number of female fish at age 1 to their parental 

 female spawning stock biomass (Clark et al., 1982; 

 O'Brien and Brown^; Hayes and Buxton^) for the pe- 

 riod 1931-94. For the entire data series, R/SSB aver- 

 aged 0.5902. As noted by Gabriel et al. (1989), how- 

 ever, the R/SSB ratio (reflecting age-0 survival) ap- 

 pears to have declined following the collapse of the 

 Georges Bank haddock stock during the early 1960s. 



2 O'Brien, L., and R. W. Brown. 1996. Assessment of the 

 Georges Banlt haddock stock for 1994. Northeast Fisheries 

 Sdence Center Reference Document 95-13. National Marine 

 Fisheries Service, Northeast Fisheries Science Center, Woods 

 Hole, MA. 107 p. 



■■' Hayes, D. B., and N. G. Buxton. 1991. Assessment of the 

 Georges Bank haddock stock. Papers of the Northeast Regional 

 Stock Assessment Workshop, Research Document SAW 13/1, 

 126 p. 



At present the causes of the decline in the R/SSB ra- 

 tio are not known. Several hypotheses explaining the 

 obsei"ved decline in R/SSB have been put forth, in- 

 cluding depensatory mortality on age-0 haddock (Col- 

 lie and Spencer, 1993), changes in oceanographic con- 

 ditions (Myers and Pepin, 1994), and increased pre- 

 dation or competition with elasmobranchs (Collie and 

 Spencer, 1993 1. Although the cause is not known, a cru- 

 cial consideration is the choice of an appropriate time 

 period where R/SSB is representative of current pop- 

 ulation abundance and biomass and current environ- 

 mental conditions. 



One strategy to obtain a mean R/SSB value repre- 

 sentative of "current" conditions is to average R/SSB 

 from the most recent data point back several years. 

 The philosophy behind this strategy is to smooth an- 

 nual variations in R/SSB by averaging over a suffi- 

 ciently long time period. The problem, however, is to 

 define a time period sufficiently long to achieve ad-, 

 equate precision without introducing excessive bias. 

 Averages over short time periods suffer from low pre- 

 cision and can vary considerably because of annual 

 variation in R/SSB. If averages are taken over a time 

 period spanning a wide range of population levels or 



