84 



Fishery Bulletin 98(1) 



cause it can fully represent the information contained 

 in the distribution of observed R/SSB values. Also, a 

 stochastic simulation can be used to provide a mea- 

 sure of the uncertainty associated with estimates of 

 the biological reference point. The use of the mean 

 population growth rates instead of the mean of the 

 population sizes is justified on theoretical grounds 

 (Tuljapurkar, 1989). As shown in Table 6, use of the 

 mean R/SSB in a deterministic analysis or the rate 

 of growth of the mean population size in a stochastic 

 simulation tends to result in higher estimates of sus- 

 tainable fishing mortality (or, alternatively, a higher 

 estimate of population growth rate for a given fishing 

 mortality rate) than does the stochastic simulation 

 where the mean of population growth rates are used 

 as the output. 1 feel that the reason for this difference 

 hinges on the distinction between projections and fore- 

 casts (Caswell, 1989) . If we view the long-term simu- 

 lation results as forecasts, this implies that we could 

 use the distribution of population size ( and the mean 

 of the distribution) as the best estimate of the future 

 state of the population. In these simulations, however, 

 the mean population size is strongly influenced by the 

 very high values that occur in the right-hand tail of 

 the lognormal distribution. These estimates are far 

 above what has ever been observed for this stock, and 

 are probably not biologically realistic. As such, they 

 should not be treated as true forecasts of the future 

 population. In contrast, if we view the analysis as a 

 projection, the goal is not to forecast future popula- 

 tion size but rather to use the results to detennine the 

 population growth rate that is represented by the cur- 

 rent Leslie matrix. By knowing the curi'ent population 

 growth rate, it is possible to find the fishing mortality 

 rate that maintains an expected value for population 

 growth of zero, which would result in a statistically 

 stationary population. 



In a stochastic setting, the entire distribution of 

 FJ/SSB ratios is used to portray the reproductive suc- 

 cess for the stock. When a deterministic analysis is de- 

 sired, however, it is necessary to choose among several 

 possible measures of central tendency for the R/SSB 

 ratio. Sissenwine and Shepherd (1987) advocated the 

 use of the median R/SSB ratio as a way of robustly 

 portraying the reproductive success of a stock. Their 

 rationale was that the frequency of relatively poorer 

 recruitment is balanced by years of better recruitment 

 when the median R/SSB is used. Table 6 illustrates, 

 however, that using the median of the observed R/SSB 

 ratios can result in estimates of sustainable fishing 

 mortality that are substantially different from the sto- 

 chastic simulations, which are arguably the best to 

 represent the population's dynamics. The use of the 

 mean of the observed R/SSB ratio as the measure 

 of central tendency can likewise result in estimates 



of sustainable fishing mortality that differ from the 

 standard set by stochastic simulations. The primary 

 reason for this difference is that use of a mean of 

 the observed ratios is biased high in relation to the 

 preferred estimator of the ratio ( in this case the sum 

 of recruitm'ent over the sum of spawning stock bio- 

 mass; Cochran, 1977). As such, the use of the mean 

 of the observed R/SSB values can also be inaccurate. 

 Among the measures presented here, the estimator 

 Ireciaiitment/Sspawning stock biomass should be used 

 as the measure of central tendency for the R/SSB ratio 

 in deterministic analyses. The use of this measure re- 

 sults in point estimates of sustainable fishing mortal- 

 ity that are essentially the same as a full stochastic 

 analysis. 



Although the Leslie matrix is a useful tool to por- 

 tray the dynamics of harvested populations and to de- 

 termine appropriate reference points, several issues 

 arise that are of considerable practical importance. 

 As alluded to earlier, a significant challenge is how 

 to detennine what is an appropriate distribution for 

 the R/SSB ratio. Because of the variability in R/SSB 

 over time and the occurrence of occasional large year 

 classes, it is very difficult to determine what time 

 frame is representative of the present. Although the 

 answer to this question is beyond the scope of this pa- 

 per, 1 feel that the best approach is to plot the mean 

 R/SSB ratio over progressively longer time periods 

 back from the present to determine if there are any 

 temporal trends or epochs in the data set. The analyst 

 should then use his or her judgment based on other 

 biological information over time (such as stock size, 

 mean weight per individual at age, and maturation 

 schedule) to determine an appropriate period to use as 

 the basis for stochastic simulations. It is important to 

 emphasize that the dilemma of choosing a representa- 

 tive time period is not unique to analyses in which the 

 Leslie matrix is used, and the same problem arises for 

 computing any biological reference point. 



In addition to the difficulty of determining what is a 

 representative time period for the present population, 

 a fundamental question is how to represent the dy- 

 namics of populations with a density-dependent stock- 

 recruitment relationship. In principle, this can be ap- 

 proached by altering the R/SSB ratio as a function of 

 stock size (e.g. Quinn and Szarzi, 1993). Although I 

 agree with Quinn and Szarzi's (1993) approach, the 

 challenge of accurately specifying the distribution of 

 R/SSB ratios at different stock sizes is even greater 

 than specifying the current distribution. 



As a final comment, biological reference points for 

 fish populations are not necessarily targets for fish- 

 ery management (Mace, 1994), nor are they inviolate 

 boundaries that may not be crossed. Rather, they are 

 most useful as a means of comparing the consequences 



