FISHERY BULLETIN: VOL. 85, NO. 1 



Q.2 = ti Q_i,i + £,, i = X + 1 to 20, 



and 



^"21,2 = ^21 ^"20,1 + ^22 C^21.1 + ^21 + *^22> 



where Ei^i) = and var(£,) = (^, + 4?) Q-i,i. 

 Consequently, 



A:-l 



r2 = [C,,2 + I (^,,1 - exp(-Zi)) C,i + . I ^ £,]/ 



k 



E 



21 22 



[Cx,2 + 2! C,^i Qi + .2! £,]. 



For any given vector Ni and fixed Zi and Zg the 

 above terms are independent and the approximate 

 variance can be given by 



var(r2) = var(C.2)(9^/9C.2)^ 



21 



+ Z var(C,i)(ar/aQi)^ 



21 



+ Z var(£j)(9r/a£,;)2 



i = x+ 1 



= [^,2 (1 - r)^ 



/t-i 



+ I {^,,i(l - r) - 0exp(-Zi)}2Cu 



k-l 



+ I (1 - r)2(t,,i + eOC.i 



21 



+ Z r^a,;,! + 2^1,) a, ]/{CT,r~.{S) 



i = k 



Comparison of the simulated variances, with the 

 above pattern of recruitment and A; = 10, with those 

 predicted from Equation (8) showed the analytical 

 variance to be a very good approximation, averag- 

 ing about 0.96 times the simulated variances. How- 

 ever, with k = 12 the analytical variance was only 

 0.70 times the simulated variance. It is clear from 

 Equation (8) that the variance will decrease as the 

 square of the second year catch increases but the 

 first year catches play a more linear role, except 

 through the interactions of ^ and 6 vdth the first year 

 catch. 



Equation (8) also shows there is a cost involved 

 with increasing k. This is desirable to avoid any bias, 

 but if too few age classes are considered to be fully 

 recruited, then variance increases. In the example 

 considered, raising k from 10 to 12 yr increased 

 variance by 40% and the simulated variances show 

 the increase may even be greater. 



Additional simulations also revealed that the use 

 of an age-length key might reduce variance by 

 smoothing out real differences in catch at age, but 

 the reduction was nullified by the additional vari- 

 ance due to increasing k. 



(Vi) f/,,2 ^ U., 



Equation (6) still allows a true recruitment rate 

 to be calculated in this case and Allen's (1966) 

 derivation allows U, o ¥" U,^. As k should not be 



1,2 



i,l- 



underestimated when used in Equation (4), then k 

 can be defined as the larger of the two ages of first 

 full recruitment in years 1 and 2. 



In this trial an initial stable age distribution was 

 prescribed with A = 1.0, i^j = 0.05 and 



U^,^ = 



= (i - 5) X 0.2 

 = 1.0 



i < 5 



i = 5 to 10 

 i> 10. 



A deterministic catch C^j was obtained given F^ 

 and the population vector A^2 found. The catch and 

 population vector in year 2 was then calculated with 

 Fg = F^ and a changed Ui2, 



[/,- 2 = i < k2 - 5 



^t,2 = {i + 5 - k2) X 0.2 i = A;2 - 5 to A;2 



C/, 2 =1-0 ^ > h- 



With C/j3 = [/, 1 and F^ = F^ a, third catch was 

 obtained. 



From this simulation two recruitment values can 

 be obtained, r2 and r^, using Equation (4). The 

 results are given in Table 3 and demonstrate 1) the 

 effect of recruitment occurring earlier in the sec- 

 ond year (r2 with k2 < 10, and r^ with k2 > 10); 2) 

 the effect of recruitment occurring later in the sec- 

 ond year (the converse); and 3) the effect of the age 

 of recruitment fluctuating about an average k to 

 ±\k - k2\. 



Under these conditions Equation (4) accurately 

 gives the proportion of new recruits in the popula- 

 tion and, as expected, if selection and recruitment 



122 



