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Fishery Bulletin 98(1) 



stock were not fished (Gabriel et al., 1989; Clark, 

 1991 ). These reference points are termed percent max- 

 imum spawning potential (%MSP) or spawning per 

 recruit (SPR) reference points. As an example, Gabriel 

 et al. (1989) found that fishing at F,,,^^ for Georges 

 Bank haddock results in a SSB/R ratio of about 30'7( 

 (Fg,),, ) of the SSB/R that would be produced if the stock 

 were not fished. 



There are several limitations to this group of refer- 

 ence points. First, the population-level effect of fishing 

 above or below a given reference point is not immedi- 

 ately obvious. For example, if fishing at F.^,y^ results 

 in a stable population, the rate of population decline 

 when fishing mortality results in a 20% MSP is not 

 clear. Secondly, although the use of the median SSB/R 

 is an attempt to determine the SSB/R ratio with a 

 robust estimator, the effects of variability in recruit- 

 ment have not been closely examined in the estima- 

 tion process. There is also an implication that two fish- 

 ing mortality patterns (i.e. combination of fishing in- 

 tensity and age at entry into the fishery) that produce 

 the same SSB/R will result in equivalent impacts on 

 the population. To my knowledge, this implication has 

 not been investigated. Finally, these reference points 

 treat the R/SSB ratio as being independent of stock 

 size. Although the stock-recruitment relationship for 

 many stocks is so weak and highly variable that this 

 is a reasonable approach (Clark, 1991), this assump- 

 tion should be examined on a case-by-case basis. 



The goal of this paper is to explore a method for 

 computing a reference point based on stock-recruit- 

 ment data that overcomes some of the limitations of 

 ^nud ^^^ related reference points concerned with re- 

 cruitment overfishing, hi particular, this method al- 

 lows for 1 ) a direct determination of the population- 

 level impact of fishing above or below the reference 

 point and, 2) incorporation of information on recruit- 

 ment variability into the estimated reference point. 

 This method does not, however, take into account any 

 curvature in the stock-recruitment relationship (i.e. 

 this method assumes that recruitment is proportional 

 to spawning stock biomass). Although this assump- 

 tion is not always met, the reduction in abundance 

 that occurs for most exploited fish stocks results in 

 a reduced magnitude of density-dependent effects on 

 recruitment. Thus, the use of this reference point is 

 likely to be reasonable for fish stocks that have al- 

 ready been exploited (Francis, 1997). 



Methods 



The proposed method is founded on an eigenvalue 

 analysis of Leslie matrices representing the popula- 

 tion's djmamics under exploitation- As such, the model 



is specifically intended for use for fish with a single 

 breeding season per year. The underlying model is 

 based on one developed by Quinn and Szarzi (1993), 

 which led to determination of the fishing mortality 

 (F^^) that resulted in a stationary population in a Les- 

 lie matrix setting. Their results are extended by exam- 

 ining the effects of recruitment variability on the ref- 

 erence point and the relationship between this method 

 and %MSP methods. This model is conceptually sim- 

 ilar to those used for environmental impact assess- 

 ment of power plants (e.g. DeAngelis et al., 1977; Co- 

 hen et al., 1983: Goodyear and Christensen, 1984) but 

 differs in that it focuses on sustainable harvest rates 

 across several age classes, whereas environmental im- 

 pact assessment models typically focus on mortality 

 of early life stages. Related methods have also been 

 presented by Getz and Haight ( 1989). Their methods, 

 however, are not explicitly framed toward providing a 

 reference point for a fishery. Further, their methods 

 are based on catch (in numbers or weight) whereas 

 my methods are based on fishing mortality rates. 



Developing a Leslie matrix representation of 

 harvesting: deterministic case 



Consider initially a population with no harvest, and 

 where abundance estimates are available on an an- 

 nual basis at the time of breeding. If we assume that 

 the vital rates (i.e. age-specific reproduction and sur- 

 vival) are constant, the dynamics of the population 

 can be represented by a Leslie matrix L,^; (see Table 1 

 for a list of symbols and their definitions): 



where Ed) is age-specific fecundity, Sii) is the annual 

 survival rate from age / to /+1 and s is the maximum 

 age. With this projection matrix, the population at 

 time /+1 can be determine from the population at time 

 / by the equation: 



^,.^=I^.,^r 



Note that estimates of natural mortality rate are also 

 necessary to compute the previously mentioned ref- 

 erence points and are typically available for species 

 where quantitative stock assessments are performed. 

 Also note that although age-specific fecundity is not 

 always measured in fish stock assessments, spawn- 

 ing stock biomass is commonly used as a surrogate 



