Somerton and Kikkawa: Population dynamics of Pseudopentaceros wheelen 



769 



Appendix 



Variances of several of the estimators described in Materials and Methods were approximated by using the Delta 

 method (Seber 1973) and assuming all covariance terms were negligible. Variance of (Rj - R-,) was estimated as 



Var(Ri-Ro) = (P,,t,i-1)2 Var(Bo,t.i) + (B 



o.t.i)2 Var(Pr,t,i) + e-2M [Var(Bo,t) + Var(Ct)+(Bo,t- Ct)^ Var(M)], 



(15) 



where all variables are defined in text Equations 5 and 6. Variances of Bq t and Bo,t+i were computed by using 

 the method in Polovina (1986). Variance of Ct was assumed to be negligible because catch was measured by U.S. 

 observers. Variance of M was estimated as the variance of the slope of the regression of log-relative abundance 

 on postrecruitment age (in years). Variance of Pr,t+i was estimated with a bootstrap method (Efron and Gong 

 1983). Bootstrap estimates were obtained from trawl samples of armorhead biological data by iteratively repeating 

 the following steps: (1) A subsample of n fish from each sample was randomly chosen with replacement, where 

 n is equal to the size of the original sample; (2) an FI frequency distribution was constructed from the subsample; 

 (3) Pr,t+i was estimated by fitting the distribution mixture model to the FI frequency distributions. In all cases, 

 variance of Pr,t+i was calculated as the variance among 100 bootstrap estimates. 

 Variance of B* during Period 2 was as 



Var(B*t) = 



qtPf 



Var(Ut) + 



UtPf 

 (qtPf)' 



Var(qt) + 



UtQt 

 (qtPf)' 



Var(Pf) 



(16) 



where all variables are defined in Equation (9). Variance of Ut was estimated as the variance among the daily 

 U within each year. Variance of qt was estimated as the variance of the slope of the Leslie model. When more 

 than one vessel fished in each year, however, variance of qt was the average of the individual variance estimates 

 weighted by catch. 



Variance of Pf was estimated by using a Monte Carlo model. Each iteration of the model consisted of generating 

 a random value of Bot for each year and a value of M, assuming all were normally distributed with means and 

 variances equal to the original estimated values. With these generated values, B* was estimated for each year 

 with Equation (9) and Pf t was estimated as Boj/B* . Mean estimates of Pf and B*go,85 (B*o in Eq. 7), were ob- 

 tained by using the iterative procedure to minimize the weighted sum of squares of the Pf t estimates. In all cases 

 variances of Pf and B*go,85 were estimated from 100 iterations of the Monte Carlo model. 



Variance of B* during Period 3 was estimated using Equation (16) but with qt replaced by qj. Variance of 

 Uf was estimated as the variance among the monthly means. Variance of qj was estimated as the variance among 

 the q estimates in Table 1. Variance of Pf is the same as for Period 2. 



Variance of B* during Period 1 was estimated as 



Var(B*t) = 



Wt 



qi 



Var(Ut) + 



qi 



Var(Wt) + 



UtWt 



q^ 



Var(q,), 



(17) 



where all variables are defined in Equation (11). Variance of Wt was estimated from the biological samples from 

 each research cruise. Variance of Ut was estimated as the variance of U among the four depth strata. 

 Variance of qi was estimated as 



Var(q,) = 



UssWss 



B 



80.85 



Var(P8o) + 



U85P80 



B*80,85 



Var(Wg5) + 



P80W85 



B* 

 80,85 



VaKUgg) + 



U85P80W85 



B*, 



80.85 



Var(B 



80,85.1 



(18) 



where all variables are defined in Equation (12). Variances of Wgj and Ugs were estimated as described above 

 for Wt and Uf. Variance of Pgo was estimated as described for Pr,t+i- Variance of B*8o,85 was estimated with the 

 previously described Monte (]arlo model. 



