Jacobson and Cadrin: Stock-rebuilding time isopleths and constant Fstocl<Tebuilding plans for overfished stocks 



525 



With A:=3400 t (Table 3), r= 0.039 which suggests Fj^g^-^ 

 0.0185/yr. This crude estimate is less than implied by the 

 simple calculation ;■ = '2F^,v;y = 0.11 based on the proxy 

 ^MSY = M = 0.055/yr, possibly because the natural mor- 

 tality rate overestimates F^fi^y for cowcod (Deriso, 1982). 

 ASPIC estimates were similar (F 



MSY 



= 0.018/yr with an 

 80% bootstrap confidence interval 0.00082-6. 039/yr) to 

 estimates based on r,,. In simulations for cowcod, we used 

 ^^gj.=0.018 (from ASPIC), cx-;^ =0.00050 and p=0.9. 



Simulation model runs indicated that cowcod are very 

 unlikely to rebuild to B^gy in ten years. The mean gen- 

 eration time for cowcod is about 35 years (calculated as 

 described by Restrepo et al., 1998). Simulations indicate 

 that the mean time for rebuilding the stock with zero F is 

 approximately 40 years (30-50 years, depending on model 

 type). In accord with National Standard 1 Guidelines, 

 it may be reasonable to develop plans with the goal of 

 rebuilding the cowcod stock in 75 years or less. We there- 

 fore calculated and plotted 75-year, rather than 10-year 

 rebuilding time isopleths, for cowcod rockfish. 



Sensitivity analyses 



We conducted three sensitivity analyses for each stock 

 to determine if the choice of statistical distribution for 

 stochastic r,^ values influenced rebuilding times in simu- 

 lations. Sensitivity analyses used uncorrelated process 

 errors and no uncertainty in Fi,_<,y (model type 3). The 

 first sensitivity analysis run for each stock was a non- 

 parametric bootstrap (Efron, 1982) with r^^ values drawn 

 randomly with replacement from the observed values (Eq. 

 6). The second and third sensitivity analyses run for each 

 stock were parametric bootstraps with r^^ values drawn 

 fi-om a normal or lognormal distribution with the same 

 mean (^) and variance (a^) as the observed values (Tables 

 2-3, Figs. 2-3). To avoid bias in runs with the lognormal 



distribution (Beauchamp and Olson, 1973), log-trans- 

 formed r^ ^, values were drawn from a normal distribution 

 with mean ln(^)-r-/2 and variance r'^=ln(CV^-HlJ, with 

 CV= a//J (Jacobson et al., 1994). For convenience in pro- 

 gramming, r, ^, values in normal and lognormal runs were 

 sampled with replacement from a fixed pool of 200 random 

 numbers drawn from the proper statistical distribution at 

 the outset of the simulation. 



Results 



Mean rebuilding times were longer than median rebuild- 

 ing times in stochastic simulations (run types 2-6) for yel- 

 lowtail flounder and cowcod rockfish because distributions 

 of recovery times were skewed to the right (Figs. 4-5). 

 Skewness (and the extent to which mean recovery times 

 exceeded median recovery times) was more pronounced at 

 lower starting biomass levels, at higher F levels, where 

 there was more uncertainty about F^gy and autocorrela- 

 tion in process errors. 



Skewed distributions for rebuilding times affected the 

 shape of rebuilding time isopleths (Figs. 6-7). For all sto- 

 chastic models of yellowtail flounder and most stochastic 

 models of cowcod rockfish, isopleths for mean and median 

 rebuilding times were widely separated. In addition, iso- 

 pleths were asymmetrical. For example, the distance 

 between median and Qgo^, isopleths was greater than the 

 distance between Qjq,, and median isopleths. Separation 

 of isopleths and asymmetry were greatest for model type 

 6 which had correlated process errors and uncertainty 

 about F^fgY- 



In simulations for both stocks, uncertainty about F^^gy 

 had a greater effect than process error on the shape and 

 separation between isopleths for Qiq%, median, mean, and 

 Qg^,, rebuilding times. For example, in both stocks, there 



