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Fishery Bulletin 89(1), 1991 



G; = a, - bjSS (5) 



where SS = total adult stock size (numbers), 

 a,b = age specific parameters. 



The increment from ages 2 to 5 was smaller the 

 larger the adult stock. Fish weight at ages 1-5 was the 

 result of growth in the first year and subsequent in- 

 crements at ages 2-5; thus, age-1 growth partially 

 determined the average weight of a fish throughout its 

 lifetime. If the stock was reduced in any given year, 

 the cohort could recover and grow faster. Growth at 

 ages 6 + was assumed to follow trends in the recent 

 data, since by this time cumulative mortality has usual- 

 ly been sufficient to reduce a cohort to lower levels. 



Percent maturity at ages 2 and 3 was assumed to 

 vary based on a relationship between the fraction 

 mature and spawning stock size calculated as 



PM, = a - b(SSB) 



(6) 



where PMj = percent mature at age i, i = 2,3, 

 SSB = spawning-stock biomass, 

 a,b = parameters. 



This submodel was parameterized (Table 4) so that 

 the maturity of age-2 fish can vary from 20 to 50%, 

 while maturities for age-3 fish range from 70 to 100%. 



Natural mortality due to predation (M2) for ages 1 

 and 2 fish was estimated from a relationship between 

 M2 and year-class size-at-age (Fig. 4; Table 4) calcu- 

 lated as 



r 

 12 3 4 5 6 7 8 



Year Class Size (billions) 



Figure 4 



Type 3 functional relationship between predation mortality 

 rate (M2) and year-class size used in the density-dependent 

 simulation model (DDM). 



sponses in regulating this stock. Monte Carlo simula- 

 tions were produced for a variety of different scenarios, 

 and average results from 1000 annual data points were 

 summarized. Results from the model were compared 

 with forecasts from the current standard assessment 

 model (STD). 



M2j = (a*YCi)/[l+(YCi/K) h ] 



(V) 



where M2 



natural mortality due to predation on 

 age i, i = 1,2, 

 YCj = year-class size at age i, i = 1,2, 

 a,b,K = parameters. 



M2 mortalities on age-1 and -2 fish could reach approx- 

 imately 1.0 and 0.6, respectively, with this model. 



This relationship was used since it approximates the 

 findings of our mortality study over an initial range of 

 stock size (Fig. 3A, B), and since it appears to be an 

 appropriate predator prey response model (Holling 

 1965, Murdoch 1973). Although this model does not 

 produce a typical type-3 response (Holling 1965) exact- 

 ly, since there is no inflection point over the initial stock 

 sizes (Fig. 4), it serves as a sufficient functional model 

 to study the natural mortality mechanism. 



The density-dependent simulation model (DDM) was 

 used to study the impact of different levels of fishing 

 mortality, management strategies, and to investigate 

 hypotheses concerning the role of compensatory re- 



Model sensitivity and validation 



The sensitivity of model results to the different density- 

 dependent mechanisms was investigated by compar- 

 ing catch in 1987 and spawning stock in 1988 and 1991 

 for all the different combinations of growth, maturity, 

 and natural mortality at a reference fishing mortality 

 of 0.05. In runs where only a single mechanism was 

 examined, 1987 catch was most affected by changes 

 in the growth pattern (mean weights) of the stock (Fig. 

 5-B). Spawning stock in 1988, on the other hand, was 

 almost equally sensitive to maturity and natural mor- 

 tality. Density-dependent natural mortality influenced 

 1991 spawning stock to the greatest degree (Fig. 5-C). 

 When the results of pairing the mechanisms were ex- 

 amined, weight and natural mortality had the greatest 

 effect on catch, percent maturity, and natural mortality 

 on SSB in 1988 and weight and natural mortality on 

 SSB in 1991 (Fig. 5-D, E, F). When the three mechan- 

 isms were all operating there was no change in the im- 

 pact on 1987 catch, but spawning stock in 1988 and 

 1991 was several percentage points lower (Fig. 5-G). 



