SMITH AND POLACHECK; ANALYSIS OF SIMPLE MODEL 



to r appears to be a general feature of this proce- 

 dure when r is small. This can be seen by examin- 

 ing S, expressed as a function of r, which can be 

 obtained explicitly by substituting the definitions 

 of N, [Equation (4)] and iV',[Equation (6)] into 

 Equation (7) and simplifying. 



The consequences of having two factors varying 

 simultaneously are shown in the series of contours 

 of equal values of S from Equation (7) (Figures 

 3-5). These contour plots present a visual picture 

 of the sensitivity of the back projection to the dif- 

 ferent factors. From this set of contour maps, it 

 can be seen that the surface generated by S [Equa- 

 tion (7)] tends to be nearly linear. Since S has no 

 nonlinear terms with respect to n and k, the sur- 

 face described by S in these two dimensions is 

 simply a plane (Figure 4). There are nonlinear 

 effects between the net reproductive rate and both 

 initial abundance and the sequence of kills. For 

 the example examined here, the nonlinearity be- 

 tween k and r is insignificant. For instance, if r and 

 k both equal 0.50, S deviates from a linear model 

 by <Wc . In general the nonlinearity between k 

 and r will be insignificant as long as the kills in 

 any one year do not represent a large proportion of 

 the population and as long as r is relatively small. 

 Also, for the data considered here, the nonlinear- 

 ity between the net reproductive rates and initial 

 abundance is small but not insignificant. For 

 example, if both n and r equal 0.50, S deviates 



from a linear model by as much as 5%. This in- 

 teraction effect is negative, resulting in a surface's 

 bending downward from a strictly linear model 

 when n and r have the same sign. 

 If all three factors vary together, the surface 



3 



2 



,1 



k 



-.1 



-.2 

 - 3 



- 5 



Figure 4. — Contours of equal sensitivity of the back estimated 

 abundance in 1959 for a range of deviations in the initial number 

 (n ) and in the kill vector (^ ) when the net reproductive rate vector 

 is held constant for Stenella attenuata in the eastern tropical 

 Pacific. 



Figure 3. — Contours of equal sensitivity of the back estimated 

 abundance in 1959 for a range of deviations in the initial number 

 (n ) and in the net reproductive rate (r) when the kill vector is held 

 constjmt for Stenella attenuata in the eastern tropical Pacific. 



Figure 5. — Contours of equal sensitivity of the back estimated 

 abundance in 1959 for a range of deviations in the kill vector (k) 

 and the net reproductive rate vector (r) when the initial number 

 is held constant for Stenella attenuata in the eastern tropical 

 Pacific. 



775 



