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Fishery Bulletin 90|4). 1992 



y(i) is the yield associated with the ith interval of the 

 histogram. The expected potential yield foregone can 

 be plotted against the corresponding TAG (Fig. 2). 

 Here, y(i)-x is a possible value of the yield foregone 

 provided it is non-negative; negative values are elim- 

 inated by the indicator function 6{\); p(i) is the prob- 

 ability that the yield foregone is equal to ci(i) (y(i)-x). 

 In practice, the expected yield foregone would be com- 

 puted by setting all simulated catches which are less 

 than the TAG equal to zero and then computing the 

 mean of the 1000 values minus the TAG. The results 

 can then be plotted versus the TAG for various choices 

 of TAG (Fig. 2). 



It should be noted that this cost relates to the up- 

 coming year only. One can also calculate the fate of the 

 biomass left in the water at the end of the upcoming 

 year. That is, one can ask whether this biomass left in 

 the water will increase or decrease over the year. In 

 general, for a quantity of biomass left in the water, the 

 relative change in its biomass over the year is given by 



relative change in unfished biomass = 



IPaWa 



Here, Pa is the proportion of the stock that is age a; 

 Wa, the average weight of animals at age a; M, the 

 (constant) instantaneous natural mortality rate; and the 

 summations are over all age groups of interest. 



Trade-offs in decision malting 



The manager can now choose how to trade off poten- 

 tial yield and risk. For example, consider the option 

 of a TAG of 210,000 mt as a means of maintaining the 

 fishing mortality at a constant level. From Figure 2a, 

 the perceived risk of the fishing mortality exceeding 

 the target mortality is ~14%. The expected value of 

 the potential yield foregone for this TAG is ~14,000 

 mt. If, instead, a TAG of 215,000mt is selected, the 

 risk of exceeding the target fishing mortality becomes 

 26% and the expected value of the potential yield 

 foregone becomes 10,000 mt. Thus, an increase in the 

 TAG of 5,000mt would almost double the risk of ex- 

 ceeding Fstatus quo while reducing the expected poten- 

 tial yield foregone by 30%. 



Another way to present the results of the SPA sim- 

 ulations is to plot percentiles of output distributions 

 versus the TAG selected. For example, for each SPA 

 run on simulated data, one can take the estimated 

 population size and iteratively seek the fishing mor- 

 talities that will result in each of several TAGs. Then, 



120 140 160 180 200 220 240 260 

 total catch (1000 mt) 



Figure 3 



Percentiles of the distribution of relative change (%) in biomass 

 of cod Gadus mo7-hua age 3 and above, as a function of the 

 total catch selected for 1991. Dotted lines: (top) 97.5th percen- 

 tile; (middle) 50th percentile; (bottom) 2.5th percentile. 



for any value of TAG one can compute the median and 

 2.5th and 97.5th percentiles of the distribution of 

 fishing mortalities. Since instantaneous fishing mortal- 

 ity may not be meaningful to some interested parties 

 (such as fishing industry groups), one may wish to look 

 at the distribution of changes in population biomass 

 associated with particular choices of the TAG (Fig. 3). 



Thus, we have two approaches which we can sum- 

 marize as follows. The first approach is to select a goal 

 or objective (such as Fgtatusquo or Fo i) and then quan- 

 tify the chances of achieving that goal as a function of 

 the TAG or effort restriction selected. The second is 

 to quantify the consequences of choosing different 

 quotas or effort restrictions. Both approaches are 

 useful to managers. A manager might first ask how a 

 specific management objective like Fo i can be met. A 

 graph similar to Figure 2a makes it clear that the trade- 

 off between risks and costs must be balanced. The 

 manager might also want to know the consequences 

 of picking particular quotas or effort restrictions. For 

 example, for economic or political reasons, it may be 

 difficult to stick with a management policy if a large 

 quota reduction is called for. In this case, the conse- 

 quences to the stock of maintaining the status quo or 

 reducing the quota by various intermediate amounts 

 may be of interest. A graph similar to Figure 3 may 

 be helpful for this. 



Managers and industry have a strong interest in 

 maintaining stability in a fishery. Conflicts can easily 

 arise when annual assessments provide only point 

 estimates of the quota required to achieve a specified 

 goal. This is because random error in the estimates 



