where C, , is determined as an independent random 

 variable. For the expected catch at age, m > 50 Cjj 

 is distributed as A''[m, „m, , (1 - P,t)] where Pit is 

 calculated from Equation (5), and for m < 50 as 

 Poisson [m^t]. m is obtained as 



and 



m,,, = F,(1.0 - exvi-Zt))N,, U^jlZ^, 



CT,= Iq,. 



If aging of the catch introduces bias or variance this 

 can be investigated using a matrix A where the ele- 

 ment ttj J is the probability that an animal of true 

 age i will be called j. The new catch at allocated age 

 can be given by C where 



C f = A Cf, 



and where C is the column vector of catch at age. 

 From the validation model the true recruitment 

 rate can be calculated as 



FISHERY BULLETIN: VOL. 85, NO. 1 



(i) Fj ^ F, 



From Equation (6) it is evident that the value of 

 Fo {F2 T^ 0) does not affect the recruitment rate and 

 this is reflected in Equation (4) where only propor- 

 tions in the catch each year are needed. The value 

 does however affect the variance of r as shov^ni later. 

 This was also noted by Ricker (1975). 



(ii) A 7^ 1.0 



Equation (7) shows that r is a funtion of A, the rate 

 of increase of the population given F-^. Equation (4) 

 accurately gives the true value of r irrespective of 

 A or F2. 



(iii) k - Incorrectly Chosen 



Let k ' be the selected age at first full recruitment. 

 If A;' > A; then Equation (4) gives the same rate as 

 Equation (6). As a confirmation of Ricker's (1975) 

 findings it is easy to show that in the extreme case 



A--1 



A^i,2f/i.2 + \ (iV,:,i,2f/,.i,2 - ^^^lJ^.^ exp(-Zi)) 



1 = 1 



21 



? iV,,2f/, 



(6) 



! = 1 



1,2 



It can be easily shown that, for C/, j = t/, 2 -^(^ = 

 1) in a stationary age composition, this reduces 

 to 



(A - exp(-Zi))/A. 



(7) 



In the following tests the results using Equations 

 (6) and (4) will be compared when parameters are 

 changed from year to year or when variability is in- 

 troduced. In all tests F^ = 0.05 and M = 0.05. 



RESULTS 



The results of comparing the true recruitment rate 

 from Equation (6) with those obtained from Equa- 

 tion (4) are given below, for the cases when the 

 fishing mortality is different in 2 adjacent years, for 

 A # 1, and for the age at first full recruitment {k) 

 incorrectly chosen when [/,; 2 = ^i, 1 (sections i - iii 

 below). Section iv considers the effect of variability 

 and biases in the age determination of the catch and 

 section v considers stochastic effects, all with [/, 2 

 = t/; 1. Section vi considers the effects oi Ui^ i= 



of knife-edge recruitment k ' can he> k and that if 

 k' is < k then no new recruitment is detected. For 

 U, = OAi, i = 1 to 10 and U, = 1.0, otherwise 

 Table 1 shows the reduction in r using Equation (4) 

 as /c' is reduced from its true value at A; = 10. The 

 reduction is substantial and the effect on the esti- 

 mated net recruitment rate, r', is more so. r' is 

 calculated as (Chapman 1983) 



r' = r - 1 + exp(-M). 



It can be seen that in this example, which is not un- 

 like many examples in whale assessments, an error 

 of 1 year would reduce r' by 20%. (It is worth noting 

 that this equation for the net recruitment rate is ap- 

 proximate and underestimates the true net rate by 

 about the product of F and MorF and Z depending 



Table 1. — Reduction in recruitment rates (r) as k' Is incorrectly 

 chosen <k = 10, and net recruitment rate, r'. 



120 



