SMITH AND POLACHECK: ANALYSIS OF SIMPLE MODEL 



Table 5. — Coefficients of variation (CV) for the back estimates 

 of bridled dolphin in 1959 (N,, ) for a range of CV for the 

 parameters of the model. The ranges of CV's of the kills, net 

 reproductive rate, and initial abundances were selected to illus- 

 trate particular aspects of the behavior of the variances of the 

 back estimates. 



the variance of Nq tends to completely dominate 

 the variance of N, because of the assumed inde- 

 pendence of the kill estimates. Table 5 gives the 

 CV for the back estimated population size of dol- 

 phin in 1959 for a range of CV for the different 

 parameters involved in the estimate ofN^r,. As can 

 be seen in this table, unless the variance of No is 

 near zero or unless the C V of the kill and reproduc- 

 tive vectors are extremely large (>60%), the CV 

 of the back estimate does not exceed the CV of A^q. 



1 + 



and the basic model [ Equation (3)] for the dolphin 

 population examined here is given in Table 6. The 

 simpler model always gives a slightly higher es- 

 timate for the size of the back projected population 

 but the increase in the estimate is always <1%. 

 The sensitivities of the two models are nearly 

 equivalent. When the values for the parameters in 

 these models deviate as much as SO'/t the differ- 

 ence between sensitivities of the two models is 

 <1%. The approximate variances of the back es- 

 timates of the two models are also similar. 



That the difference between the original and the 

 simpler model is small can be shown by analyti- 

 cally comparing the two models. If the projections 

 are made only 1 yr into the past, the ratio of the 

 estimate from Equation (2) to the estimate from 

 Equation (3) is 



1 + 



O.bR^K^ 



N^+K^+O.bR^K^- 



Only if the value ofR^K^ is large relative to Ao +^i 

 can this ratio deviate significantly from 1. This is 

 only possible if /?i is relatively large. The general 

 formula for the ratio of the two models is 



S 0.5KJi.( fl (l+R, J) 



N^+ 1 0.5K. (n (l+R^ J)+ 2 O.bK.in (l+RJ) 



Comparison of Equations (2) and (3). 



A comparison of the estimated back abundance 

 as calculated by the simpler model [Equation (2)] 



As in the case for projecting back only 1 yr, it can 

 be seen that unless the RjKj terms are large rela- 

 tive to Nq and unless the net reproductive rate is 

 also large, the ratio of the two models will be close 

 to 1. 



Table 6. — Comparison of the back estimate of the abundance of 

 bridled dolphin as calculated by the basic model [Equation (3)] 

 and the simpler model [Equation (2)]. 



DISCUSSION AND CONCLUSIONS 



The results of this analysis indicate that errors 

 in the input parameters do not compound in this 

 procedure for estimating historical abundance. In 

 fact, a systematic bias in the procedure for the 

 estimation of a single set of parameters (either A^o 

 or R^'s or K,'s) always induces a bias in the back 

 projected estimate which is less than the bias of 

 the estimated parameters. This conclusion follows 

 directly from the linear or near linear relation 

 between St and n, k, or /• with small rates of 

 change. Moreover, the effects of bias in two or 

 more sets of parameters are nearly additive. The 

 interaction effects of bias in estimates of kills, net 

 reproductive rates, and the initial number tend to 



777 



