310 



Fishery Bulletin 93(2), 1995 



L1 * p,t T ly p-i,t J Iv p,<+i K > 

 and the equation analogous to Equation 6 is 



z p _ u a-e- F »->-) 



(8) 



M 



N pM1 e M =N Pit -C Ptt 



F p _ u (l-e- z ^-) pU 



Z p _ u (\-e- F >-") 

 * F p _ u a-e- z >->>) 



(9) 



~~ Cp-l,t 



which, using the same approximation given by Pope 

 (1972), becomes 



N p>t+1 e M = [N p>t + N p _ u ] - [C pit + C p _ u ]e M ' 2 . 



Substituting Equation 8 into the last equation re- 

 sults in 



N pJ+l e M =N PiM e z -u -[C pJ +C p _ u \e' MI \ 

 which can be solved for the fishing mortality rate as 



of parameter values; 2) to plot values of the ratio 

 A F I T F against a and M ( parameters usually assumed 

 to be known a priori) and to identify types of func- 

 tions that can adequately describe the observed re- 

 lationship, if any; and 3) to estimate the coefficients 

 of such functions and use them to correct A F so it 

 becomes closer to T F. This empirical approach is simi- 

 lar to the multiple regression approximations sug- 

 gested by Allen and Hearn (1989). For comparative 

 purposes, we also followed the same approach using 

 the approximation given by Equation 7 for a case 

 without a plus group. 



Newton's Method 



Sims (1982) suggested the use of Newton's Method 

 for solving F in the catch equation with a desired 

 degree of precision. For Equation 1, the functional 

 equation of interest is 



f(F a , t y- 



F a j(e z °<-1) 



N, 



a+l,t+l , 



F P-U = ln 



(Cp.t + ^p~l,t ) -MI2 i 



N Ptt+1 



(10) 



We found that Equation 10 is often a poor approxi- 

 mation for values of a / 1 (see Results section). A 

 much better approximation can be obtained by in- 

 troducing a into the equation as 



Fp-u=\n 



(C pA la + C p _ u ) uii + ^ 



N P,M 



(11) 



Empirical correction factors 



In many cases, given the widespread availability of 

 computers, the approximation given by Equation 11 

 can be used as an adequate starting guess for an it- 

 erative procedure to get a more accurate solution (e.g. 

 see the next section). In some cases, however, it is 

 desirable to improve upon this approximation in or- 

 der to obtain either a better starting guess, or to ob- 

 tain as close as possible to an accurate solution be- 

 cause iterative computations are expensive. The lat- 

 ter is the case of solving multiple catch equations 

 while conducting a cohort analysis on a computer 

 spreadsheet and is the motivation for this study. 



The empirical approach we used was simple. De- 

 note the approximation in Equation 11 hy A F and the 

 true fishing mortality by T F. Our approach was 1) to 

 generate a large number of plausible combinations 



and a solution to F a t is obtained when f(F a t ) = 0. 

 Sims (1982) showed that all requirements for con- 

 vergence were met in order for Newton's Method to 

 converge to that solution. One iteration of Newton's 

 Method (denoted by i) changes the estimate of F a t 

 as follows: 



F a Ai + l) = F a Ai)- 



f((Fg, t W) 



r{F a , t d)) 



(12) 



with 



f'(F„ t ) = - 



C a ,(e z °'(F a 2 l+ F aJ M + M)-M) 



Fl t (e z °< - 1) 2 



For the application of Newton's Method to the so- 

 lution to the catch equation involving a plus group, the 

 functional equation of interest is (from Equation 3) 



f(F p -Xt)- 



C pJ (aF p _ 1J + M) 

 aF p _ u (e aF ^- +M -l) 



, C p _ u (F p _ u+ M) 

 F p _ Xt (e F ^ +M -l) P ' M 



and its derivative with respect to F t t xs 



