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Fishery Bulletin 90(2), 1992 



Figure 2 



Frequency of seasonal, protracted recruitment dates from one 

 trial. Peak recruitment occurs 1 April-1 June, 80% occurs 

 within 3-5 months of the peak, and recruitment levels vary. 



Figure 3 



Frequency of seasonal, contracted recruitment dates from one 

 trial. Peak recruitment occurs 1 April-1 July, 80% occurs 

 within 2-4 months of the peak, and recruitment levels vary 

 20-fold between time periods. 



of recruitment occurred randomly within ± 2-4 months 

 of the peak (Fig. 3). 



The unobserved change rate was simulated both tem- 

 porally invariant and variant. If variant, then Zt~ 

 U(zi,Z2) on each new trial; Zj and Zo were simulation 

 control variables. Catching was simulated either as a 

 single event that occurred once each midperiod or as 

 a continuous event in each period. Fishing mortality 

 was not imposed until period 6 so that the stock would 

 accumulate as soon as possible after simulation in- 

 itialization. In most tests the fishing mortality rate was 

 drawn U(Fi,F2) on each new trial; Fj and F2 were 

 control variables. Period-specific rates were set con- 

 stant over all trials in two tests to guarantee a stock 

 depletion caused by a rapid increase in fishing levels. 



Sampling simulation included the generation of catch 

 estimates, growth parameter estimates, and relative 

 abundance measures. Populations and catches were 

 generated over 20 time-periods. Relative abundance 

 samples and catch estimates were simulated in the last 

 four periods only, but catches were considered to be 

 removed after the date of abundance samples so catch 

 in the last period was irrelevant to estimation (and thus 

 was not computed). 



Size-class and date-specific catch estimates were 

 drawn from a Gaussian distribution with catches from 

 the simulator as the expectations and with variances 

 specified by a cv. The estimator of catches was thus 



unbiased, and estimation errors (estimator variances) 

 were proportional to catches. 



A complete simulation of growth sampling and esti- 

 mation was deemed too costly, so a reasonable proxy 

 of unbiased estimation was used. Let Aj be a growth 

 parameter of fish i such that Ai~N(A, a'[A]. Defining 

 the uniqueness in growth of fish i as tj = Aj- A, o-[A] 

 = ZTi2/N. Let Aj be unbiasedly measured by aj with 

 normal error so that ai = Ai-i-e|, ei~N(0, o-[e]). Since 

 ei = ai-Ai, o2[e] = lei2/N, o-[ai] = E{a,-E[ai]}2 = o2[A] 

 -i-o2[e]-i-2o[T,e] where the last term is null because 

 T and e are independent. Using the sample mean of g 

 fish to estimate A, a-[A] = (a~[A] + a'^[e])lg, so that 

 growth-parameter estimation variance is separated 

 into two parts, that of inherent variability from fish 

 to fish and that of growth measurement error. CV's 

 were used as simulation control input instead of vari- 

 ances, so growth parameter estimates were N(A,A2 

 (cv[A]2-i-cv[e]2)/g) random variables. In reality, all 

 growth parameters are estimated simultaneously. As 

 a rule, growth parameter estimates are highly nega- 

 tively correlated (Gallucci and Quinn 1979, Knight 

 1968, Burr 1988) with correlation coefficients often 

 -0.90 or less. Estimates of k and A of (1) were drawn 

 as normal random correlated variables (Rubinstein 

 1981:86) with a correlation coefficient of -0.95. The 



