82 



Fishery Bulletin 98(1) 



Population growth rate 



Figure 4 



Distribution of annual population growth rates (lambda! to 

 year 10 for a stochastic projection of the Georges Bank Had- 

 dock stock. The distribution of growth rates for each time 

 interval is based on 5000 simulations. 



mean value (Table 5) with a decreasing variance (Fig. 

 5). This result has both theoretical and pragmatic im- 

 plications. Of theoretical importance is the concept 

 that although the variance of projected population 

 abundance increases over time, the variance of the 

 growth rates declines over time. Thus mean popu- 

 lation growth rate can be used in defining a main- 

 tenance fishing mortality rate. The practical conse- 

 quence of the above result is that at least two different 

 strategies can be used to compute A for a stochastic 

 population. One strategy is to project the population 

 for a long period of time (e.g. hundreds of years) to 

 make a precise estimate of the long-term population 

 growth rate. This strategy makes use of the fact that 

 the variance of the long-term population growth de- 

 clines as the period of projection is lengthened. A prob- 

 lem with this approach, however, is that for projec- 

 tions over a long period of time, population abundance 

 can become so large or small that it cannot be directly 

 represented on a digital computer, causing a numeric 

 overflow or underflow. A preferable strategy is to com- 

 pute A for a large number of simulations over a shorter 



time period (e.g. 150 years). This method avoids the 

 problem of numeric overflow and achieves precision in 

 the estimate of mean A by having a large number of 

 simulations. 



Based on the current partial recruitment vector to 

 the fishery, a fishing mortality of 0.450 (F,,) would 

 result in an average population growth rate of zero 

 (Table 6). The fishing mortality that results in a zero 

 growth rate for the mean population size was higher, 

 at 0.517 (Table 6). Interestingly, this is nearly the 

 same as F^^ computed for the deterministic case by us- 

 ing the mean R/SSB. The estimate of F^^^^y with these 

 same data is much lower than F,, — only 0.069 (Table 

 6). When a deterministic Leslie matrix analysis was 

 made with the corresponding median R/SSB, it re- 

 sulted in an estimate of F,, that was nearly the same 

 as F^^^ (Table 6). This finding illustrates the basic 

 connection between these methods when they are op- 

 erating on the same inputs. As an additional compari- 

 son, I computed SSB/R as the ratio ISSB/SR instead 

 of the mean (or median) of the individual year ratios. 

 This method is based on sampling theory that sug- 

 gests that the ratio of the sums is a less biased estima- 



