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



521 



When fishing and the intrinsic rate of increase exactly bal- 



ance fa^= r^-F^.=0): 



^v+l = 



'"' l + b.B, 



(2) 



Note that biomass decHnes lB^,^j<B 



when r^-F^ = be- 

 cause r^ is a maximum value defined in the Hmit as bio- 

 mass approaches zero (Eq. 1). 



Following Beddington and May (1977) and May et 

 al. (1978), natural variation in population growth rates 

 was included in our analysis by adding process errors to 

 simulated r^, values. We hypothesized that the intrinsic 

 rate of population increase was more important than car- 

 rying capacity in simulating population growth rates at 

 low biomass levels, and in rebuilding overfished stocks. 

 We focused on stochastic variation in r^ because it likely 

 varies annually (e.g. due to variation in recruitment and 

 growth). We could hypothesize reasonable lower bounds 

 which included negative values; and variability in r^, could 

 be reasonably described in statistical terms (e.g. mean, 

 variance, and autocorrelation) based on available data. 

 Carrying capacity (K) was assumed constant over time 

 because no information about potential covariance in r^, 

 and K was available. 



In the deterministic logistic population model, F^gY=rl2 

 and Bf^gY=KJ2 (Schaefer, 1954). For the sake of simplicity, 

 we assumed K=l so that Bj^^f=Bi^gy~0.5 and biomass B^, 

 was measured in relation to K (e.g. B^-K=l at carrying 

 capacity). It was useful to express biomass in relation to K 

 because estimates of ratios like B,IBf^gY[=2BilK) are often 

 more precisely estimated than either biomass B^, or carry- 

 ing capacity K (Prager, 1994) and because the approach 

 makes results easier to apply to other stocks. 



We used six types of logistic population growth models 

 (Table 1) based on a wide range of initial biomass (Bg), 

 two levels of uncertainty about F^jgy, variance in process 

 errors (stock dependent), and autocorrelation in process 

 errors (also stock dependent). The number of years (an 

 integer) required for the stock to rebuild to 0.955^,gy was 

 recorded in each simulation run. Recovery in the simula- 

 tion model was at 0.95B^^c;y, rather than B^^y, because 

 biomass in the deterministic logistic production model at 

 ^~^MSY approaches asymptotically (but never reaches) 



B 



MSY' 



(this convention had negligible effect on results). 

 Stochastic simulation model results were derived from 

 2000 individual model runs (the maximum length of each 

 run was 2000 years) starting from each point in a grid of 

 31 values of F (i.e. 0, F,^sy/30, 2F,^gy/30, ..., 29F^,sy/30, 

 Fj^jgy) and 35 values of initial biomass (i.e. Bq = 0, 5 x 10"-', 

 5 X 10-2, 5 X 10-1, 5_ 25, ..., 305, where 5 =0.9999 x 0.95 x 

 B,,g^JK/30). 



We calculated distributional statistics including the 

 mean, median (^50.7, ), and various quantiles (e.g. QgQ,y^ 

 for the ninety-percent quantile) for recovery times from 

 all runs at each point in the grid of F and initial biomass 

 levels. We then plotted isopleths (contours) for the dis- 

 tributional statistics. For example, to produce 10-year 

 median rebuilding time isopleths, we calculated median 

 recovery times for each point in the grid of F and initial 

 biomass, and then drew contours (isopleths) by connecting 

 points with 10 year median rebuilding times to identify 

 fishing mortality rates that, if held constant, would give a 

 509r probability of rebuilding from the initial biomass to 

 the target in ten years. We smoothed the isopleths in plots 

 by using LOESS (locally weighted regression smoothing) 

 regression (Cleveland and Devlin, 1988; Cleveland et al., 

 1988) to remove variation caused by the contouring algo- 

 rithm and coarse grid of fishing mortality and biomass 

 starting points. 



