Hayes: A biological reference point based on the Leslie matrix 



83 



tor of the "true" ratio than is the mean of the individ- 

 ual ratios (e.g. Cochran, 1977). The value for SSB/R 

 obtained with this method was 2.685, and when used in 

 place of the median SSB/R in the F„,^,^y calculations, re- 

 sulted in an estimated reference point of 0.449 — essen- 

 tially the same as in the stochastic simulation where 

 the mean population growth rate was zero (Table 6). 



Discussion 



The primary purpose of this paper was to demonstrate 

 how the Leslie matrix can be used to compute a refer- 

 ence point for harvested populations and to contrast 

 this method with the SSB/R method currently in use. 

 One of the major findings was that the SSB/R method 

 and the Leslie matrix approach produce similar es- 

 timates of sustainable fishing mortality when they 

 are based on the same inputs. This is not surprising 

 given the similarity between reproductive value, on 

 which the Leslie matrix is based, and lifetime spawn- 



ing stock biomass per recruit which the SSB/R ap- 

 proach uses. 



Although the two methods produce similar esti- 

 mates of sustainable fishing mortality, the Leslie ma- 

 trix approach is preferable because of the additional 

 direct information it provides regarding the popula- 

 tion response to fishing at levels different from the ref- 

 erence point. Furthermore, when population growth 

 rate is different fi-om zero, equal levels of SSB/R do 

 not result in the same population gi'owth rate for differ- 

 ent partial recruitment vectors. These differences are 

 small, however, in relation to changes in population 

 growth rate owing to changes in fishing mortality. 



Given the various approaches illustrated (e.g. de- 

 terministic vs. stochastic), the basic question is what 

 method to use. On the basis of theoretical advances 

 in the population dynamics literature and results pre- 

 sented here, I recommend the use of a stochastic anal- 

 ysis where the mean of the population growth rates 

 is used as the 'Tjest" measure of growth for harvested 

 populations. A stochastic analysis is preferable be- 



