FISHERY BULLETIN: VOL. 87, NO. 3, 1989 



populations were constructed with data from a 

 single region-month stratum using five combina- 

 tions of growth coefficients for yolk-sac (a?-, a 

 temperature-specific coefficient) and feeding lar- 

 vae (a„,, a season-specific coefficient) (see Table 

 8). Each population was sampled repeatedly and 

 an average mortality coefficient calculated as- 

 suming standard growth coefficients. These 

 mortality coefficients were then compared with 

 the fixed mortality coefficient used to construct 

 the populations. 



RESULTS 



The simulation model was used to estimate the 

 following: 1) the mortality coefficients and their 

 standard errors for various sample sizes when 

 the true mortality coefficient was fixed, 2) the 

 difference between two mortality coefficients 

 and its standard error for various sample sizes, 

 and 3) the mortality coefficients, assuming var- 

 ious growth rates. 



Estimates of (3 with Various Sample Sizes 



The mortality coefficient (p) was fixed at 1.5 

 for the inshore area (regions 4, 7, 8, 11, and 13; 

 Fig. 2) and at 0.05 for the offshore area (regions 

 5, 9, and 14). The lower coefficient was required 

 to generate simulated catch curves similar to 

 those observed in offshore areas. The low mor- 

 tahty coefficient observed in offshore areas was 

 likely the result of transport of older larvae from 

 inshore to offshore regions (Power 1986). The 

 average mortality coefficient ((3), weighted by 

 area of each region, was 1.41. 



For each sample size (50, 100, 200, 300, and 

 400 plankton tows) 100 computer runs were 

 made, and an estimate of the mortality coeffi- 

 cient (b) was calculated. The mean mortality 

 coefficient, its standard error, and the coefficient 

 of variation (cv) are listed in Table 4 for each 

 sample size. The mean mortality coefficient for 

 all sample sizes, except 50, slightly overesti- 

 mated the true value of p = 1.41. The cv de- 

 creased with increasing sample size. 



The relationship between cv and the number 

 of positive tows (n) was quantified by assuming 

 that half of the tows contained anchovy larvae 

 (the actual portion of positive tows in 1984 was 

 0.5) (Table 2).The curve (Fig. 8) may be de- 

 scribed by the power function: 



From Figure 8 and the above expression, cv may 

 be expected to be 0. 10, 0.06, or 0.05 for 20, 60, or 

 100 positive tows. For 7i > 100, cv may be ex- 

 pected to decrease at a slow rate. Thus a survey 

 of 120 tows, yielding 60 positive tows, is suffi- 

 cient to estimate the mortahty coefficient with 

 an expected cv = 0.06. Data from annual surveys 

 conducted between 1980 and 1987, where the 

 portion of positive tows ranged from 0.47 to 0.98, 

 are also shown on Figure 8. The variation of 5, as 

 related to sample size during 1980-87, follows 

 the relationship estimated from a single year's 

 data and implies that the relationship can be 

 used as a guide for sample size determination. 



Table 4. — Mean, standard error (SE), and coef- 

 ficient of variation (cv) of estimates of tfie mor- 

 tality coefficient (b) for various sample sizes (N), 

 viHU 50% positive for anchovy larvae (n = 0.5 

 N), from 100 computer runs of eachi simulated 

 survey. 



cv{b) = 0.418 n 



-0.47 



Estimates of D with Various Sample Sizes 



The mortality coefficient ((3) was fixed at 1.0, 

 1.5, 2.0, 2.5, and 3.0 for the inshore area (regions 

 4, 7, 8, 11, and 13). The inshore area was rela- 

 tively well sampled and contained relatively high 

 abundances of larvae; the proportion of positive 

 stations in these regions was approximately 0.6 

 (Tables 1, 2). Estimated mortality coefficients 

 (b) were determined for five simulated popula- 

 tions (corresponding to each of the five mortality 

 coefficients ((3)) using sample sizes of 50, 100, 

 and 200 plankton tows with 60% of them positive 

 for anchovy larvae. 



The average estimated mortahty coefficient 

 and its standard error were determined after 100 

 computer runs and listed in Table 5. As ex- 

 pected, standard errors decreased with in- 

 creased sample size. The estimated mortality 

 coefficient was biased slightly low for (3 < 2 and 

 biased slightly high for p > 2. The biases are 

 negligible although they appeared to increase in 

 magnitude as p departed from 2. The estimates 

 of mortality rates and their standard errors were 

 used to determine minimum sample size by two 

 methods. 



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