FISHERY BULLETIN: VOL. 76, NO. 4 



(S,) for a fixed number of years / into the past is 

 linear with respect to A? or/? (Figure D.ThisHnear- 

 ity can be seen in Equation (6) since n and k enter 

 only as linear terms in the numerator. Positive 



30i 



Figure l.— Sensitivity of the model Sffn,;fe,H in 1959 for a range 

 of deviations in the initial number (n ), for a range of deviations in 

 the kills (k), or for a range of deviations in the net reproductive 

 rate (r), for Stenella attenuata in the eastern tropical Pacific. 



values of either n ov k yield positive deviations in 

 the back estimates. However, the farther back the 

 population is projected in time, the smaller the 

 contribution of N,, to the back estimate becomes 

 relative to the contribution of the kills. Thus the 

 effect of bias in the estimate of the initial numbers 

 (n) becomes progressively smaller the farther 

 back in time the population is projected, while the 

 consequence of a consistent bias in the kill esti- 

 mates (k) becomes larger (Figure 2). Since the 

 annual kills have no simple relationship to time, 

 the effect of a particular value of « or ^ over time 

 (Figure 2) cannot be described by any simple func- 

 tion. This trade off in the sensitivity of the back 

 projected estimates between n and k is exact in the 

 sense that for any decrease over time in the slope 

 of S with respect to n there is an equivalent in- 

 crease in the slope of S with respect to k. This can 

 be seen by evaluating the partial derivates of S 

 with respect to n and with respect to k and noting 

 that they sum to 1. 



The effects of bias in the estimates of the net 

 reproductive rate vector are more complicated 

 than for the other two factors. Positive deviations 

 in the net reproductive rates (r) yield negative 

 deviations in the back projected estimate (Figure 

 2). The effect of r tends to increase over time (Fig- 

 ure 2). S approaches being linear with respect to r 

 for any particular year, but unlike the relation- 

 ship for k and n, this result is not exact ( Figure 1 ). 

 The approximate linearity of the sensitivity ofN^ 



£. 10- 



.^ 5 



-5- 



,o- 



.- 0-' 



.-O' 



,o' 



.-cr 



,0' 



FIGURE 2.— Sensitivity of the model St 

 (n,k,r} over time to a 30% deviation in 

 the initial number (n = 0.3), in the kill 

 vector (k = 0.3), and in the net reproduc- 

 tive rate vector (r = 0.3) when all factors 

 are held constant (or Stenella attenuata 

 in the eastern tropical Pacific. 



774 



