Restrepo and Legault: Approximations for solving the catch equation 



f'(F p . u ) = 



C pJ (e aF ^' +M (a 2 F^ u + aF p _ u M + M)-M) 

 aF 2 _ u (e aF »^ +M -l) 2 



Cp- U (^ u+M ^p-u + F p-u M + M)-M) 

 F*(e F »» +M -l) 2 



One iteration of Newton's Method proceeds in the 

 same manner as explained above in Equation 12, if 

 the empirically corrected estimates of fishing mor- 

 tality (see previous section) as initial values are used. 

 Although we did not carry out a rigorous analysis of 

 conditions for convergence, as Sims (1982) did, we 

 did not encounter any cases where an iteration did 

 not result in an improvement. 



Results 



Table 1 provides some statistics 

 of the ratio A F I T F for the 1,000 

 random combinations of inputs, 

 with and without a plus group. 

 In each case, the first column 

 provides these statistics for the 

 initial approximation i A F from 

 Equation 7 for the case without 

 a plus group and A F from Equa- 

 tions 10 and 11 for the plus 

 group approximation). The next 

 column provides the statistics 

 for the ratios after an empirical 

 correction function is applied (the 

 coefficients of these correction 

 factors are presented in the fol- 

 lowing subsections). The last two 

 columns give the statistics after 

 one and two iterations of New- 

 ton's Method. Implications of 

 these results are explained in 

 more detail below. 



tio indicated an overall 3% bias and the largest er- 

 ror was slightly greater than 8%. 



Visual inspection of a plot of A F/ T F against M in- 

 dicated that a linear relationship would improve the 

 approximation. We fitted the model A F I T F = a + b M 

 by minimizing the sum of absolute deviations be- 

 tween the observed ratios and those predicted by the 

 model. The parameter estimates were 



a = 0.9970, and b = 0.0808. 



Thus the empirical correction to the initial approxi- 

 mation to F was 



emp - A F = imLA F/(a + bM). 



(13) 



Much improvement in the approximation was ob- 

 tained by use of this simple correction function (Table 

 1, second column, and Fig. 1, middle panel). The larg- 

 est observed error was now 3%, which compares fa- 

 vorably with the errors reported by Pope ( 1972 ) over 

 a much narrower range of fishing and natural mor- 

 tality rates. Application of a single iteration of 

 Newton's Method resulted in virtual conver- 



Case I: without a plus 

 group 



The initial approximation pro- 

 vided by Equation 7 (from Pope, 

 1972) was reasonable, as ex- 

 pected (See Table 1 and Fig. 1, 

 top panel). The mean A F I T Fra- 



