FISHERY BULLETIN: VOL. 76. NO. 4 



earlier andi?. is the net reproductive rate ib-d). 

 The above model was modified in the procedure 

 used by NMFS to account for situations when the 

 kills occur throughout the time interval instead of 

 instantaneously at the end of the interval, as: 



r+i "• T 



(3) 



This equation can be repeatedly applied to give 

 estimates any number of years it) into the past. 

 When rearranged to explicitly display the popula- 

 tion size t years earlier, and relabeling so that the 

 initial abundance is N„, one obtains 



1974 and the annual incidental kills and repro- 

 ductive rates from 1959 to 1974. Several se- 

 quences of estimated annual kills and reproduc- 

 tive rates were considered, incorporating the 

 uncertainty in the data. 



In the present paper the sequences of annual 

 kills and net reproductive rates given in Table 1 

 are used to illustrate several general aspects of the 

 behavior of Equation (4). These correspond to the 

 "high kill" and "central reproductive rate" se- 

 quences for the bridled dolphin, Stenella at- 

 tenuata, in the Workshop report. The estimate of 

 1974 abundance used by us and the Workshop was 

 3.5 million. 



A^„ 



A^. = 



t 



n 



(1+^p 



t 



X.(l+i?./2) 



(4) 



Note in this form that the time-index t runs back- 

 wards from zero. As is apparent in this form, the 

 estimation of abundance t years earlier involves 

 2t+\ parameters. The sequences of annual kills 

 and net reproductive rates can be termed the kill 

 and the net reproductive rate vectors, each com- 

 posed of t elements. 



The data used here to explore this estimation 

 procedure is from the report of NMFS Workshop 

 discussed above (see footnote 3).^ From existing 

 unpublished data and reports the Workshop par- 

 ticipants used estimates of the population size in 



''It should be noted that the estimates used here are based on a 

 number of assumptions currently under investigation and that 

 these estimates are subject to significant change in the near 

 future (I. Barret, Director, Southwest Fisheries Center, La Jolla, 

 CA 92038, pers. commun. April 1978). 



Estimation of Bias 



A sensitivity analysis was done to examine the 

 effects of biased parameter estimates on the 

 backcalculated abundance. A new population size 

 1 yr earlier, from Equation ( 3 ), when each parame- 

 ter is changed by a specified amount is 



N^{n, k, r) = 



N^{l+n)+0.bK^{l+k) 



and in general for t years earlier. 



(5) 



iV;(n, k, r) = 



N^il+n) 



U^ (l+/2.(l+r)) 



, KiUk) (l-H(/2 (l+r)/2)) 



-H_2— ^-^ (6) 



S.(l+/?.(l+r)) 



Table l. — Estimates used for kill and reproductive rate vectors 

 of Stenella attenuata in the eastern Pacific. 



where Nq, R^, and K^ are defined as above, and 

 n = the proportion that A^q deviates from its 



estimate 

 r = the proportion that all elements of the net 

 reproductive vector deviate from their es- 

 timates 

 k = the proportion that all elements of the kill 

 vector deviate from their estimates. 



N'j in,k,r) was then compared with A^, from Equa- 

 tion (4) or equivalently N ', (0,0,0). As a measure of 

 the sensitivity of the basic model, S, {n,k,r) is 

 defined to equal the percent that A^', (n,k,r) de- 

 viates from A^^ 



772 



