75 



Abstract.— For fish populations with an 

 annual breeding cycle, a biological ref- 

 erence point based on the Leslie matrix 

 is presented and compared with percent 

 maximum spawning potential CJMSP) 

 and F^^ reference points. For determin- 

 istic population projections, the reference 

 point is defined as the level of fishing 

 mortality </^,,) that results in a Leslie 

 matrix with a dominant eigenvalue (i.e. 

 finite rate of increase or A) of 1.0. It is 

 shown that for the same input data, F^^ 

 is similar to a reference point based on 

 a '7f MSP approach. For populations that 

 are growing or declining, however, popu- 

 lations with the same A but with different 

 age-specific selectivities have different 

 levels of '5MSP. Previous applications 

 of this reference point are extended to 

 include situations where recruitment is 

 a stochastic process. In stochastic pro- 

 jections, F^i is defined as the level of 

 fishing mortality that results in an aver- 

 age finite rate of increase of 1.0. In an 

 example with Georges Bank haddock, a 

 deterministic analysis with mean birth 

 and death rates resulted in an estimate 

 of F.,, of 0.52. The same estimate of F^, 

 was obtained in a stochastic projection in 

 which the growth rate of the mean popu- 

 lation size was used. Stochastic projec- 

 tions using the mean of the finite rates 

 of increase resulted in a lower estimate 

 of Fj, (0.45). When the value of recruits 

 per unit of spawning stock biomass used 

 in the '7fMSP analysis was calculated as 

 Xrecruits/lspawning stock biomass, the 

 estimated reference point was the same 

 as the stochastic projection. On the basis 

 of these results, I recommend calculating 

 the reference point based on a stochas- 

 tic projection for which the mean of the 

 simulated growth rates is used. A refer- 

 ence point based on a VfMSP approach 

 using the Irecruits/Ispawning stock bio- 

 mass results in an equivalent estimate of 

 the reference point but does not convey 

 important information on the expected 

 population growth rate at higher or lower 

 rates of fishing mortality. 



A biological reference point based 

 on the Leslie matrix 



Daniel B. Hayes 



Department of Fishenes and Wildlife 

 13 Natural Resources Building 

 Michigan State University 

 East Lansing, Michigan 48824 1222 

 E-mail address: tiayesdarna'msu edu 



Manuscript accepted 17 August 1999. 

 Fish. Bull. 98:75-85 (2000). 



Advice to fishery managers on desir- 

 able harvest or exploitation rates is 

 ideally based on full knowledge of 

 fishery dynamics, including informa- 

 tion on the fish population's stock-re- 

 cruitment relationship, growth and 

 maturation schedule, and bioeco- 

 nomic considerations. With a lack 

 of such complete information, guid- 

 ance to fishery managers often takes 

 the form of providing an estimate 

 of fishing mortality and a compari- 

 son of that rate to one or more bio- 

 logical reference points (e.g. Clark, 

 1991; Anonymous^). Numerous bio- 

 logical reference points exist, each 

 concerned with a somewhat differ- 

 ent aspect of population response to 

 harvesting. One class of reference 

 points focuses on yield per recruit as 

 a function of fishing mortality. The 

 general goal of this class of reference 

 points is to optimize harvest rates 

 in relation to natural mortality and 

 growth (i.e. prevent gi'owth overfish- 

 ing; Beverton and Holt, 1957). For 

 example, F,,,^^ is the fishing mortal- 

 ity rate at which yield per recruit 

 is maximized (Beverton and Holt, 

 1957). A related reference point is 

 Fq j which is the fishing mortality 

 rate where the slope of the yield per 

 recruit curve is 10% of the slope at 

 the origin (Gulland and Boerema, 

 1973). Fishing at F„,^^. or F^^ re- 

 sults in maximal or nearly maximal 

 yield from a fishery when recruit- 

 ment is independent of stock size. 

 A limitation of this class of refer- 

 ence points, however, is that reduc- 

 tions in recruitment are often evi- 



dent when stocks are depleted to low 

 levels (e.g. Overholtz et al., 1986). 

 Thus, management advice based on 

 ^max ^^ ^0 1 ^^^ result in declines 

 in abundance through recruitment 

 overfishing (Sissenwine and Shep- 

 herd, 1987), ultimately resulting in 

 reduced total yield from a stock. 



As a counterpart to reference 

 points based on yield per recruit, sev- 

 eral reference points based on stock- 

 recruitment considerations have 

 been developed. The goal of these 

 reference points is to provide a mea- 

 sure of fishing mortality that will 

 likely avoid recruitment overfishing. 

 An example of this type of reference 

 point is F^gj which is based on the 

 median of the observed levels of re- 

 cruits produced per unit of spawn- 

 ing stock biomass (R/SSB) (Sissen- 

 wine and Shepherd, 1987). The ra- 

 tionale behind this reference point is 

 that fish abundance is maintained 

 when the spawning stock biomass 

 produced by a cohort over its lifetime 

 is equal to the spawning stock bio- 

 mass of the parent population when 

 the cohort was spawned. Related to 

 F„j^^ is a set of reference points based 

 on the spawning stock biomass per 

 recruit (SSB/R) in relation to the 

 SSB/R that would be produced if the 



'Anonymous. 1996. Report of the 20th 

 Northeast Regional Stock Assessment Work- 

 shop 1 20th SAW ): SAW Public Review Work- 

 shop. Northeast Fisheries Science Center 

 Reference Document 95-19. National Oce- 

 anic and Atmospheric Administration, Na- 

 tional Marine Fisheries Service, Woods Hole, 

 MA, 52 p. 



