130 



Fishery Bulletin 89(1), 1991 



will be estimated for values of release age t, M, M/K, 

 and F/M, which are representative estimates for rock- 

 fish populations. 



To evaluate the short-term releases of hatchery- 

 reared juveniles to restore a depleted rockfish popula- 

 tion, a Deriso-Schnute delay-difference age-structure 

 model is used (Zheng and Walters 1988). Deriso (1980) 

 derived a population model that combined simple sur- 

 plus production models with more detailed age-struc- 

 ture models. Schnute (1985) modified this model with 

 a three-parameter Brody growth model. This Deriso- 

 Schnute model depends on a Brody growth parameter 

 (p), the age of recruitment to a fishery (k), body weights 

 at ages k and k - 1 (W k and W k _ ] , respectively), and 

 total annual survival in year t (s t ). Thus, the biomass 

 in year t + 1 (B t + 1) is described as 



B t+ i = (l + p)s t B t - pstSt.jBt.j + R t + 1 



PSt— — Rt, 

 Wk 



where R t is the recruitment to the fishery in year t 

 (Schnute 1985). A Ricker stock-recruitment relation- 

 ship is used to model the recruitment (in weight) from 

 the natural rockfish population and from hatchery- 

 released 1 -year-old juveniles to the fishery to obtain the 

 total recruitment (in weight) function (R t ): 



R t = AS t _ k exp(-BS t _ k )exp(z) 



+ H t - k + 1 W k exp(-M(k-l)); 



where S t is the spawning biomass, H t is the number 

 of hatchery-released 1-year-old juveniles, M is natural 



0.05 0.1 10.18 0.25 0.33 0-43 0.54 0.67 0.62 1.00 1.22 1. 50 1 86 2.33 3 00 4.00 5.67 9 0019-00 



F • M" 1 



Figure 1 



Beverton and Holt yield-per-recruit as a fraction of asymp- 

 totic weight at optimum size at entry to the fishery as a func- 

 tion of F/M for three levels of M/K. 



mortality, and A and B are constants. This model 

 assumes hatchery juveniles are added to the natural 

 population without any density-dependence. The term 

 exp(z) is used to add a stochastic element to the natural 

 recruitment function when the variable z represents 

 a random variable, which has a normal distribution, a 

 mean of 0, and a variance of a 2 . When o 2 is set to 0, 

 the stochastic term is eliminated, and deterministic 

 recruitment is assumed. The Ricker stock-recruitment 

 relationship appears to represent an appropriate model 

 of recruitment in a natural rockfish population (Archi- 

 bald et al. 1983). 



The Deriso-Schnute model is fit to catch and fishing 

 mortality data for Pacific ocean perch Sebastes alutus 

 from Queen Charlotte Sound, Canada, during 1963-77 

 when the population was fished from an estimated 

 biomass of 82,000 metric tons (t) to 13,000 t (Archibald 

 et al. 1983). The model with its estimated parameters 

 is then used to estimate the equilibrium yield curve to 

 determine the biomass and fishing mortality that 

 achieve maximum sustainable yield (MSY). Then the 

 model with stochastic recruitment is used to simulate 

 the recovery of this depleted stock to the biomass level 

 that supports MSY under several management strate- 

 gies, including the release of hatchery-reared juveniles. 



Results 



Values of M for rockfishes typically range from 0.01 

 to 0.25 per year, K from 0.09 to 0.21 per year, and M/K 

 ratios from 0.44 to 2.27 (Table 1). Values of yield- 



