FISHERY BULLETIN: VOL 76. NO. 4 



be small or nonexistent. This will be globally true 

 for the relationship between k and n, but will be 

 true for the relationship between /?, r, and /; only 

 when the net reproductive rate is small. The rela- 

 tive importance of bias in A'/s, /?/s, or 7V„ on A^, 

 depends upon the actual values of the parameter. 

 In the bridled dolphin example, after 15 yr, the 

 back estimates were most sensitive to bias in the 

 kill estimate, slightly less sensitive to bias in N„, 

 and considerably less sensitive to bias in the net 

 reproductive rate. However, the importance of 

 bias in A/^o will diminish with the number of years 

 in the back estimate with a proportionate increase 

 in the importance of bias in the kills. 



The sensitivity analysis developed in this paper 

 will include the extremes of a complete sensitivity 

 analysis of the model. The values forS/ (0,/?,0) are 

 limiting values to a complete sensitivity analysis 

 of the individual elements of the kill vector on N,. 

 Similarly S, ( 0,0,/') is a limit to complete sensitiv- 

 ity analysis of the individual elements of the net 

 reproductive rate. Given the additivity of S, with 

 respectto«,r, and^, the surface S^ («,/?,r) contains 

 the extremes of a sensitivity analysis in all 2t+\ 

 dimensions. If in fact the elements within the kill 

 vector and within the reproductive vector are 

 highly interdependent (as is the case for the data 

 used here), then the sensitvity analysis used to 

 look at the effects of bias in this paper approaches 

 a total sensitivity analysis of the back projected 

 estimate given these constraints. 



The variance approximations also indicate that 

 variability in the parameter estimates does not 

 result in compounding uncertainty in the back 

 projected estimates. When estimates of the 

 parameters are independent and the net reproduc- 

 tive rate is low, the CV of the back estimate will be 

 smaller than the CV of the input parameters. In 

 our example if all the CV's were equal, the vari- 

 ance of A^o would make the largest contribution to 

 the estimated variance oiN,. In general this will 

 be true as long as the kills in any one year do not 

 approach the initial abundance. This is a direct 

 consequence of the basic additivity of the model 

 when the net reproductive rate is small. 



In Smith and Polacheck (see footnote 4), an al- 

 ternative probability structure was considered in 

 which the elements within the kill vector and 

 within the net reproductive rate vector were 

 highly interdependent. In this situation, the vari- 



ance of Nt is not completely dominated by the 

 variance of N^^. The variances of N, calculated 

 using this interdependent probability structure 

 are larger than the variances presented here in 

 which all the parameters are assumed indepen- 

 dent. However, the CV of TV, for the dolphin data 

 within this interdependent probability structure 

 is still less than the CV of the parameters if all 

 parameters have equal C V. It appears that even in 

 the situation in which a high degree of inter- 

 dependence exists within the kill estimate or the 

 net reproductive estimates, the variability in the 

 parameter estimates does not induce compound- 

 ing uncertainty in the back projected estimate. 



The comparison of the results from the basic 

 model [Equation (3)] with the simpler model 

 [Equation (2)] indicate that there are no sig- 

 nificant differences between the two models as 

 long as the net reproductive rate is small. Thus it 

 appears that there is no reason to favor the more 

 complex model over the simpler. 



In conclusion, it appears that this back projec- 

 tion procedure (either model) has reasonable 

 statistical properties, at least when the net repro- 

 ductive rates are small. However, Equation ( 1 ) is a 

 simplified description of how the abundance of a 

 population changes through time, especially in 

 not accounting for changes in age structure. The 

 authors feel that caution should be used in apply- 

 ing estimates from this procedure to the manage- 

 ment of long-lived species since changes in the age 

 structure for long-lived species are likely to be 

 important. 



ACKNOWLEDGMENTS 



Financial support for this study was supplied by 

 the U.S. Marine Mammal Commission (Contract 

 MM74C006). We wish to acknowledge the journal 

 editor and an anonymous reviewer for their help- 

 ful comments. 



LITERATURE CITED 



Perrin, W. F. 



1969. Using porpoise to catch tuna. World Fish. 

 18(6):42-45. 



Seber, G. a. F. 



1973. The estimation of animal abundance and related 

 parameters. Hafner Press, N.Y., 506 p. 



778 



