Jacobson and Cadrin: Stock-rebuilding time isopleths and constant-f stock-rebuilding plans for overfished stocks 



527 



60 120 180 240 300 



1.0 

 0.5 

 0.0 



1.0 

 0.5 

 0.0 



1.0 



0.5 



0.0 



60 120 180 240 300 



Rebuilding time (yr) 



Figure 5 



Distribution of simulated recovery times for coweod rockfish in the Southern CaUfornia Bight from 

 six types of logistic population growth models ( type- 1 deterministic, others stochastic, see Table 1 ) 

 with F^js,.)~0. 0.018 /yr, CV for uncertainty in F^^g^— zero or 20%, variance for production process 

 errors zero or 0.037, and autocorrelation in process errors p=zero or 0.9. Results are for 2000 model 

 runs starting from an initial biomass ot B/ B^f^y =0.19 and constant F=0.011/yr. 



ing and evaluating rebuilding plans. This recommenda- 

 tion is based on a narrow technical consideration, i.e. 

 that median recovery time calculations are less sensitive 

 to model assumptions. A median rebuilding time plan is 

 risk neutral in the sense that the probability of rebuilding 

 times less than intended is the same as the probability of 

 rebuilding times longer than intended (i.e. both 50%). Of 

 course, rebuilding plans based on median rebuilding time 

 goals will be more liberal in terms of short-term catch 

 (i.e. have higher F) and have longer rebuilding times 

 on average than plans based on mean or, for example, 

 Qggcr, rebuilding time goals. In summary, a manager who 

 is willing to accept a 50% chance of rebuilding times 

 greater that desired and who is concerned about model 

 uncertainties, might choose a median rebuilding time 

 approach. 



A potential advantage in using percentiles other than 

 the median is that managers can specify risk levels in try- 

 ing to rebuild stocks. For example, managers could choose 

 and evaluate rebuilding plans based on a Qg^cy, isopleth to 

 insure at least a 90% chance of rebuilding in specified time 

 period. However, Qg^,- isopleths may be sensitive to model 

 assumptions. 



Approaches to using rebuilding time isopleths without 

 relying on uncertain estimates of rebuilding time distribu- 

 tions (e.g. mean or Qgg,- ), are an important area for future 

 research. Cadrin (1999) used deterministic rebuilding 

 time isopleths with the tenth percentile of the estimate 

 for r. Isopleths for median rebuilding time might be used 

 with lower bounds on confidence intervals for Bg and Bg/K 

 or upper bounds on current F, F^gy, or F/F^jgy Estimates 

 of uncertainty in these parameters are often available 

 (Prager, 1994) and can be incorporated in an ad hoc fash- 

 ion. For example Butler et al.^ suggest that Bj^^glK for 

 coweod rockfish is 7% (CV about 30%) and that the F in 

 1998 was 0.085/yr (CV 34% ). A risk-averse manager might 

 implement a rebuilding plan that reduces that Fjf^^^^f^^,^ to 

 a point on the 75-year rebuilding time isopleth that lies 

 above the lower boundary of a 95% confidence interval for 

 biomass. Similarly, a risk-averse manager might select a 

 rebuilding plan that reduces ^T-^r.s/io/rf '-° account for the 

 upper bound on uncertainty in estimating F. 



Uncertainty in Fj^jgy is an important factor to consider 

 in rebuilding plans as F increases from low levels towards 

 Fj^^gy Expected rebuilding times increase at high F be- 

 cause F may exceed true F^fgy (assumed known but with 



