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Fishery Bulletin 88(4). 1990 



species models in order to evaluate changes in policy 

 performance as additional species were included. The 

 selected species (bocaccio, chilipepper, widow rockfish, 

 splitnose rockfish S. diploproa, and shortbelly rockfish 

 S. jordani) all occur off California, support or could sup- 

 port significant commercial fisheries, and have a wide 

 range of life-history types. I used bocaccio and chili- 

 pepper for the two-species model because they were 

 estimated to have similar biomass levels. I included 

 widow rockfish for the three-species model because it 

 was estimated to have about twice the maximum sus- 

 tainable yield level of biomass of bocaccio and chili- 

 pepper. For the five-species model, shortbelly and split- 

 nose rockfish were added as examples of relatively 

 short- and long-lived rockfish species, respectively. 

 Shortbelly rockfish biomass also is substantially higher 

 than that of the other four species. 



Model structure 



I used Schnute's (1985) generalization (eq. 2.7) of 

 Deriso's (1980) delay-difference model to represent the 

 dynamics of each stock: 



used in this study. To introduce a degree of density- 

 dependence into the model, I assumed arbitrarily that 

 recruitment decreased by 10% when spawning stock 

 decreased by 50%. That assumption has been used in 

 earlier stock-assessment studies (Lenarz 1984, Henry 

 1986, Methot and Hightower 1988, Tagart 1988) to ob- 

 tain harvest recommendations assumed to be conser- 

 vative when information on the stock-recruitment rela- 

 tionship was unavailable. The normal random variate 

 Zt was assumed to have mean and variance o~. I 

 assumed that R; . . R|^ were produced by a stock equal 

 in size to Bj because the actual B values producing 

 those year-classes were not known. 



Harvesting policies 



In the harvesting policies used in this study, the annual 

 fishing mortality rate for species j in year t (F|[t]) 

 was either a constant 



Fj[t] = bjo (constant F), (4) 



a function of stock biomass 



B, = (1+P)S,_,B 



t-i'Jt-i 



PS,_,S,.2Bt. 



Fj[t] = bjo + bj,i Bj[t] / Bcc.j (variable F), (5) 



-H Rt - P(v/V)S,_iRt. 



(1) 



where Bj was the fishable population biomass at the 

 beginning of year t; St was the survival rate in year 

 t; P, v, and V were parameters for the growth curve; 

 and R, was the recruitment biomass at the beginning 

 of year t. Growth parameters p, v, and V were esti- 

 mated by fitting the following curve relating mean 

 weight at age a (w^) to age: 



w,, = V -h (V-v)(l-pi 



^)/(l 



(2) 



where V and v were parameters representing mean 

 weight at the age of recruitment (age k) and at age 

 k- 1, respectively, and p was a parameter describing 

 the growth rate (Schnute 1985, eq. 1.14). Following 

 Kimura et al. (1984), I assumed that B,, (needed to 

 calculate B2) equaled the starting biomass level B]. 

 Recruitment was calculated from a Beverton-Holt 

 stock-recruitment curve (Kimura 1988): 



R..k = R^(Bt/B„)/(l-A(l-B,/B^))exp(z,) 



(3) 



where R^ was the virgin recruitment level associated 

 with virgin stock biomass B^, A was the level of 

 density-dependence assumed in the stock-recruitment 

 relationship, and exp(Zt) was the lognormal error term 

 used to introduce random variability in recruitment. 

 Estimates of A were not available for any of the stocks 



or a function of the combined biomass of all other 

 stocks 



Fj[t] = b,u + b,,,XB,[tJ/lB„„ 



(multispp. var. F) (6) 



Policies (5) and (6) were similar in form to a linear 

 model Hilborn (1985) used to calculate catch as a func- 

 tion of stock biomass. Biomass estimates were scaled 

 by virgin biomass so that the policy parameters were 

 relatively independent of absolute biomass levels. 

 Estimates of the policy parameters (b|(i, bj , ; j = 1, . . ,s 

 species in model) were obtained using stochastic ap- 

 proximation (Ruppert et al. 1984). 



Objective functions 



As in earlier optimization studies (Ruppert et al. 1984, 

 Hilborn 1985), I obtained policies for maximizing total 

 catch 



max 5!('[tJ, where C[t] = XCj[t] 



(maxh objective function) 



and the natural logarithm of total catch 



