Restrepo and Legault: Approximations for solving the catch equation 



313 



True F 



Figure 2 



Progression in the approximation to the fishing mor- 

 tality rate for a catch equation involving a plus group 

 ( 400 of the 1 ,000 pairs of approximated and true F are 

 shown). (Top) Initial approximation (Eq. 11); (Middle) 

 approximation after application of empirical correction 

 function (Eq. 14); (Bottom) approximation after one 

 iteration of Newton's Method (Eq. 12). 



Summary 



The results of our study indicate that the empirical 

 correction in Equation 13 applied to Pope's (1972) 



a = 2.0 



As V*~°°o 



o 

 o 



~i — ' — i — ' i > i  i 



2 0.4 6 8 10 



U. 



U. 



T3 

 <U 



E 

 x 

 o 



gj Ms 1.0 



M = 05 



Figure 3 



Example of visual analysis to determine the shape of rea- 

 sonable empirical correction functions. (Topi Values of the 

 ratio of initial approximated F to the true F ( A F/ T F) as a 

 function of M for two levels of a, with a linear fit; (Bot- 

 tom) A F/ T F values as a function of a for two levels of M, 

 with a logarithmic fit. See text for definition of parameters. 



approximation (Eq. 7) provides an accurate solution 

 to the catch equation that does not involve a plus 

 group (Table 1, Case I). Over a wide range of plau- 

 sible fishing and natural mortality values, Equation 

 7 gives errors of up to 8% whereas the empirical cor- 

 rection gives errors of up to 3%. These errors are 

 practically eliminated after one iteration of Newton's 

 Method following the empirical correction. 



For the catch equation that involves a plus group, 

 the initial approximations analogous to Pope's ( 1972) 

 approximation may not be very accurate. For a wide 

 range of plausible mortality values, errors of up to 

 68% and 35% are obtained from the use of Equations 

 10 and 11, respectively (Table 1, Case II). The em- 

 pirical correction in Equation 14 applied to the ap- 

 proximation in Equation 11 reduces the errors to 

 within 12% of the exact solution. One iteration of 

 Newton's Method following the empirical correction 



