520 



Fishery Bulletin 100(3) 



10-year Isopleth 

 ■• • 5-year Isopleth 

 ^ 4-year Isopleth 

 ^— 3-year Isopleth 

 ^^ 2-year Isopleth 

 i^» Control Rule 

 ^ VPA 



4 6 



Relative biomass (S/S^^jy.) 



Figure 1 



Isopleths for median rebuilding times based on a deterministic logistic 

 population growth model (model type 11 for Georges Bank yellowtail 

 flounder with Ff^^^^^O.'i. Also shown are a common harvest control rule, 

 and the biomass-F trajectory during 1996-99 for Georges Bank yellow- 

 tail flounder. The harvest control rule specifies a maximum (threshold) 

 F as a function of stock biomass level. The biomass-F trajectory shows a 

 time series of F and biomass estimates from virtual population analysis 

 (VPA, Cadrin'"'!. 



Stock-rebuilding time isopleths 



Catirin ( 1999 ) calculated theoretical recovery times for 

 Georges Bank yellowtail flounder (Limanda ferruginea ) 

 and used rebuilding time isopleths to depict trends in stock 

 biomass in relation to fishing mortality. Calculations were 

 based on a deterministic logistic population growth model 

 with a range of constant annual fishing mortality rates 

 (F=zero to Fy^y) and a range of initial biomass levels less 

 than the target level (fi,|=zero to Bj^jgy). Recovery time was 

 the number of years required for stock biomass to increase 

 from an initial overfished biomass level 'fio<fi7/,ri.s/,ou* to 

 the biomass target ^Tare^-t-^MSY' assuming a constant 

 annual fishing mortality rate. Rebuilding time isopleths 

 were formed by connecting points of initial biomass and 

 constant fishing mortality (5,,, Fg ) with the same recovery 

 time (Fig. 1). For example, beginning at the initial biomass 

 level Bq<Bj^ f, any constant fishing mortality rate Fg on 

 the 10-year isopleth would theoretically rebuild the stock 



to B 



.^Targcl 



in ten years. In contrast, any constant-F value 

 <Fg (below or to the right of the isopleth) would rebuild 

 the stock sooner and any constant F values >Fg (above 

 or to the left of the isopleth) would rebuild the stock later. 

 Rebuilding time isopleths were used to develop overfish- 

 ing definition options for nine overfished New England 

 groundfish stocks (Applegate et al.-*). 



In this article, we calculate stock-rebuilding time iso- 

 pleths based on stochastic population dynamics models 

 and characterize statistical distributions (mean, median, 

 and percentiles) of stock-rebuilding times under different 

 assumptions about uncertainty and process error (pro- 



cess errors are uncertainty in population dynamics due 

 to natural variability in growth, recruitment, and other 

 biological factors, Hilborn and Walters, 1992). Like Cadrin 

 (1999), we use logistic population growth models, but our 

 analysis includes uncertainty about F\,,,.y and autocorre- 

 lated process errors in production. We analyze rebuilding 

 times for two stocks (cowcod rockfish, Sebastes levis, and 

 Georges Bank yellowtail flounder) with different life histo- 

 ries, levels of Fycjy, and autocorrelation in production pro- 

 cess errors (the calculations are examples only and not for 

 use by managers). We also describe how stock-rebuilding 

 time isopleths from deterministic and stochastic models 

 can be used to develop and evaluate rebuilding plans and 

 to monitor their progi-ess. 



Materials and methods 



Following Prager ( 1994), we used the continuous time ver- 

 sion of the logistic population dynamic model in simula- 

 tion calculations.^ In particular, for the logistic population 

 growth parameter r^, (subscripted to represent the value in 

 yeary) carrying capacity K, b = r IK, and a = r -F^O: 



B„ 



So. 



a+bRJe"' -1) 



(1) 



SAS simulation program code available from the senior author 



