FISHERY BULLETIN, VOL. 72. NO. 1 



not change yield per reci*uit. If age at reciniit- 

 ment is held constant and fishing mortality is 

 doubled, when M = 0.6 yield per reci-uit de- 

 creases by some 20% ; when M = 0.8 yield per 

 recruit increases on the order of 5% ; and when 

 M = 1.0 yield per recruit increases about 30%. 

 If effort is doubled and age at recruitment is 

 raised to 1.83 yr, when M = 0.6 or M = 0.8 

 yield per recruit increases on the order of 20% ; 

 and when M = 1.0 yield per recruit increases by 

 about 40% . 



Estimation of t 



r 



In employing a knife-edged approximation to 

 size-specific recruitment protracted over some 

 time period, the first problem is to determine 

 the proper age at recruitment {t^') such that 

 the integration reflects the same yield per re- 

 cruit as the size-specific recruitment case. There 

 are two problems in doing so. First, there are 

 two values for t^.' that will give the same yield 

 per recruit as the size-specific recruitment case, 

 unless eumetric fishing obtains. Often, however, 

 this may be of little consequence, since one of 

 the two values for t ' could be obviously infea- 

 sible. Second, t ' will depend on the fishing 

 mortality. 



Two estimators of t ' are provided, at least 

 implicitly, by Beverton and Holt (1957): (1) the 

 age corresponding to the mean selection length, 

 and (2) the resultant of a formula depending on 

 Z and the average age, T (or average length. 

 /). in the catch. The mean selection length is 

 the 50% selection length if the selection curve is 

 symmetrical, and it is not dependent on the 

 magnitude of the fishing mortality coefficient, 

 F. The second estimator of t ' is 



r 



t; - 1 -HZ 



or, in terms of length 



i; = J-K{L^-J)jZ. 



(la) 



(lb) 



These two equations were obtained from manip- 

 ulations of the Beverton and Holt yield equation. 

 Several computations of yield per recruit 

 with the program GXPOPS were made utilizing 

 F = 0.1 and F - 2.0. M = 0.8, the von Bert- 

 alanffy equation for Atlantic yellowfin tuna, 

 and an arbitrary age-specific selection curve 

 (Figure 4) in order to demonstrate the two 



1.0,- 



0.8 



06 



04 



0.2 



50% SELECTION AT 21 mo. 



|<— F = 20 tr' = 24 mo 



F = I tr = 19 mo. 



10 



20 



30 40 



AGE (mo) 



50 



60 



70 



Figure 4. — Arbitrary age-specific recruitment curve. 



problems and to evaluate the two estimators of 

 t ', . At F — 0.1, the values of t ! giving the 

 same yields per recruit as the selection curve 

 are <8 mo (^q of the von Bertalanffy growth 

 curve is 7.48 mo) or 24 mo, and 19 or 45 mo for 

 F = 2.0. Since the state of the simulated fishery 

 is not eumetric for either value of F, there are 

 two knife-edged approximation locations. The 

 effect of the magnitude of F on the true t ' 

 is obvious, with the lower value increasing from 

 <8 to 19 mo and the upper value increasing 

 from 24 to 45 mo as F is changed from 0.1 to 

 2.0. The reasonable values for t ' to approx- 

 imate the selection curve, however, are 24 mo 

 for F = 0.1 and 19 mo for F = 2.0, a change of 

 5 mo. 



Estimator 1, the mean selection age, is 21 

 mo and is shown along with the reasonable 

 values in Figure 4. Using 21 mo for t^' would 

 result in yields per recruit that are 4% and 15% 

 too high for F = 0. 1 and F = 2.0 respectively. 

 Estimator 1 does not change with F, of course, 

 but in this case it lies intermediate between the 

 true t^' values. Estimator 2 gives 19 mo for F 

 = 0.1 and 18 mo for F = 2.0. We emphasize 

 that this estimator does depend on the magni- 

 tude of F. 



Neither estimator is exact in this examj^le 

 where the catches, their ages, and the selection 

 curve are known without error. This places 

 doubt on their estimates from the usual catch 

 at age data where considerable random error 

 would be involved. Encouraging, though, is 

 that both estimators indicate the proper direction 

 that the fishery's selectivity should proceed to 

 approach the optimal yield per recruit — about 



44 



