HORWOOD: BIAS IN ALLEN'S RECRUITMENT RATE METHOD 

 A--1 k-l 



^ Pr,2 + Pk,2 H'n - Qk.1.2 ( 2: Pu) H'f'l exp(Z« - Z{)/{Q,^, ^^^ 



r.> 



fe-i 



I P,,2 + (f,/<|>^) 1 P,,2 



i = A; 



(2) 



Compare this result with that given by Allen 

 (1966, 1968). He gives 



A--1 



r, = Pi,2 - 2: (1.0 - T,/B,^,yP,,,^„ 



1=1 



where 5,_i = P,+i,2-Qam /(Pu"Qa-+i,2) 



T, = exp(Z;' - Zi). 

 Consequently, 



*: A--1 



r, = Ip,.2 - Ti(Q,,i,2/Q^.i) 2: P,i. (3) 



l=X l=X 



To satisfy the above Equations (2) and (3) it is found 



that <H/<I'2 — -'■•O- '^^^^ ^S' ^^^ proportion fished in 

 the two recruited age groups in the second year 

 must be the same. If it is then assumed that the 

 natural mortality rate, M, is the same in each group, 

 then it is necessary that F|, = Fg. From Equation 

 (2) we then have 



k k-l 



l=X l=X 



Consequently to satisfy Allen's model. Equation (3), 

 it is also necessary that ^"^/^^= 1.0, and as above 

 this implies F{ = F«. Hence Z'l = Z\ and T^ = 

 exp(Zj - Z\) is necessarily unity, and there is no 

 flexibility in the choice of T as Allen (1966, 1968) 

 and Ricker (1985) suggest. 



If one then assumes that, within each year, the 

 mortality rate of all recruits is similar, then Equa- 

 tion (2) reduces to a very simple form 



r,,i = a - Pil - a)/(l - P), 



where a = z. Pij+i 



i=l 



k-l 

 P = .? P^,f 



DEVELOPMENT OF 

 A VALIDATION PROCEDURE 



In order to test the robustness of the estimator 

 given by Equation (4) we need a population model. 

 If in year 1 the population is assumed to have a 

 stable age structure and has been increasing at a 

 rate of A per year, such that the population vector, 



Nt.r = A AT,, 



and where the mortality rate, Z, has been constant 

 over time, then with an arbitrary number of 1-yr 

 olds the numbers at age in year 1 can be calculated 

 from the recurrence relationship: 



ATj 1 = 5,000. 



A^^.1,1 = A^ut^M exp(-2i) 



+ (1.0 - f/,i) exp(-M)]/A, 

 A^2i.i = A^2o,rexp(-Zi)/{A(1.0 - exp(-Zi))}. 



It is assumed that [/go.i = 1.0, i.e., k < 20 and N21 

 is a "plus-group" of ages > 21. The age subscripts 

 have now been dropped from Z and F. Consequent- 

 ly the numbers in the second year are given by 



A^i,2 = A -5,000. 



A^,>i,2 = N,AU,,expi-Z,) 



+ (1.0 - [/,i)exp(-M)] 



^21,2 = (A^2o.i + A^2i,i) exp(-Zi) 



Prj = {N,,t f/^)/? i^^,t Uu)- 



21 

 I 



(5) 



1=1 



(4) 



If we wish to consider the effects of stochastic 

 catches at age or problems in aging, then Pj , be- 

 comes a variable, Pj,, and can be expressed in 

 terms of the catch, so that 



119 



