HORWOOD: BIAS IN ALLEN'S RECRUITMENT RATE METHOD 



occur at an earlier age in the second year then a 

 large burst of new recruits will appear. If selection 

 occurs much later, then even the recruits of the 

 previous year will not be seen, giving the negative 

 values. Such a feature was noted by Holt and de la 

 Mare (1983). Horwood et al. (1985) fitted a selec- 

 tion pattern with age that was constant over time 

 for minke whales of the Southern Hemisphere and 

 presented the residual differences. A substantial 

 switching of effort on to different age classes was 

 found over a period of years, and it was shown that 

 this was reflected in the calculated recruitment 

 rates. These residuals and recruitment rates are 

 shown in Table 4 and clearly illustrate the character 

 of the estimate. 



The problem is then not of calculation but of inter- 

 pretation, in that we do not know selection has 

 changed, and in using this technique it is assumed 

 that the recruitment pattern is constant. A decreas- 

 ing trend in recruitment rate will be interpreted as 



Table 3. — Recruitment rates calculated from Equation (4) for the 

 model described in section vi. k is the age of first full recruitment 

 used in Equation (4), Icj is the first full recruitment in year 2. r 

 is the average of the two values and nr is the average approximate 

 net recruitment rate. 



a decline in the true rate and not as an increasing 

 age at recruitment and vice versa. As Table 3 shows 

 these rates differ greatly from the 0.095 for con- 

 stant selection, being much higher or lower depend- 

 ing on the trend in recruitment pattern. Consequent- 

 ly a systematic change in recruitment to the fishery 

 will cause substantial problems in interpretation of 

 the recruitment rates. 



Table 3 also indicates what is likely to occur if the 

 age of recruitment systematically fluctuates about 

 a set pattern. It might be hoped that the r values 

 would average to a useful measure of mean recruit- 

 ment rate. For /cg = 6 the recruitment occurs much 

 earlier in year 2, giving a high r-g, and returns to 

 normal in year 3, giving a low r^. However, the 

 average (0.033) is much smaller than the 0.095 and 

 the approximate net recruitment rate is negative. A 

 similar feature is seen for k2 = 14, but as \k2 - 10| 

 tends to zero the discrepancy is less. If the system 

 fluctuated so that we had a series k{t) = 10, 6, 10, 

 14, and 10, an approximate average value of r would 

 be the average of the four values of r on Table 3 

 (0.367, -0.301, -0.340,0.397 = 0.031), and the ap- 

 proximate symmetry gives a similar feature of low 

 average recruitment rates. 



One way of using the recruitment rates would be 

 to multiply the net recruitment rate by an estimated 

 population size obtained over the same period to give 

 a catch quota which should approximately stabilize 

 the population. From the simulation the average of 

 the recruited population in years 1,3, and 4 has been 

 found. This is very near to the average of the 4 years 

 if the basic recruitment pattern is assumed for the 

 second year. A catch was then found which would 

 make the recruited population in year 5 the same 



Table 4.— Direction of residuals after fitting a time constant selection at age to minke 

 whale data showing switching of fishing selection across ages with time. Recruitment 

 rate (r) values reflect this switching. (After Horwood et al. 1985.) 



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