Restrepo and Legault: Approximations for solving the catch equation 



309 



cumstances. It seems that the decision of whether or 

 not a plus group should be used in an assessment de- 

 pends on the ageing errors and characteristics of the 

 population being assessed (Restrepo and Powers, 1991 ). 

 The basic plus group dynamics can be presented 

 as follows. Assume that M is the same for the plus 

 group and the previous age (we will drop age and 

 year subscripts for M) and letp denote the plus group 

 age category. The number of fish alive at the begin- 

 ning of time period t+1 is given by 



N p . t+ i = N„ t e 



-z„ 



+ N p-U e ~ 



(2) 



and the corresponding catch equation is 



AT 



p.t+i 



F„Ae z >-<-l) 



F p - U (e Z ^'-D 



(3) 



Given known values of C p t , C p _ lt t , iV p t+1 , and M, 

 no unique joint solution exists for the two unknowns, 

 F pt and F p _ lt in Equation 3. In this study, we use the 

 simplifying assumption that the value of a in 



a 



F P,t 



P-U 



(4) 



purposes, we also use the same approach for catch 

 equations not involving a plus group. 



Range of values examined 



In order to evaluate the accuracy of the approxima- 

 tions and to develop empirical corrections, we gener- 

 ated 1,000 uniform random values of each param- 

 eter in the following ranges: N , [500, 1,500], N x t 

 [500, 1,500], M [0.05, 1.0], F p _[ , [0.05, 3.0], a [0.5, 

 2.0]. These ranges are arbitrary but we felt that they 

 represent realistic extremes: the stock size of the plus 

 group and the preceding age can differ by a factor of 

 3; the range in M is representative of a wide range 

 in lifespans; the fishing mortality range is extremely 

 wide because our purpose was to find reasonable 

 approximations for a wide range of Fs; and the range 

 in a allows the fishing mortalities of the plus group 

 and the previous age to differ by a factor of 2. 



Initial approximations 



Pope's (1972) approximation to F a t in Equation 1 

 can be explained as follows. Consider the equations 

 (from Appendices A and B in Pope, 1972) 



N. 



a,t 



N a + U+l e 



Z, 



(5) 



is known so that F p t in Equation 3 can be replaced 

 by ocFp.j ( . Now the solution consists of a single fish- 

 ing mortality rate, F p _ h t . Theoretically, one or more 

 a values can be estimated as parameters in an age- 

 structured model. However, estimation of several a 

 values is difficult, particularly for the last years for 

 which catch data are available (Powers and Restrepo, 

 1992). Thus, in many assessment applications, a 

 values are assumed from a knowledge of the popula- 

 tion and the fishery being examined (Powers and 

 Restrepo, 1992). For example, selectivity studies of 

 the fishing gear may indicate that fish of ages p-1 

 and older are equally vulnerable, giving a = 1. 



and 



N. 



M 



a + l,t + l' 



N„ 



ZaA 1 ' 



FaA 1 - 



,Z„ 



) 



(6) 



Pope (1972) demonstrated that, over a range of 

 fishing and natural mortality values, the function 

 multiplying C a , in Equation 6 can be reasonably 

 approximated by e ,M/21 . Making use of this approxi- 

 mation and substituting Equation 5 into Equation 6 

 gives 



N. 



M 



a+l,t+l' 



N 



a+U + l' 



,z a . 



C a , t e 



Mil 



Approach 



The approach we follow is similar to that used by 

 Sims ( 1982 ). We first provide simple approximations 

 to F p _ h t , similar to those developed by Pope (1972). 

 On the basis of simulated parameter values, we then 

 empirically estimate correction factors that can be 

 used to improve upon the initial approximations. 

 Finally, we use the corrected approximations as start- 

 ing guesses for Newton's Method (see Sims, 1982), 

 which can be used to obtain a more accurate numeri- 

 cal solution to the catch equation. For comparative 



which can be solved for F a t as 



^=ln 



A' 



C aJ_ e -MI2 + 1 



a+M+1 



(7) 



We followed a similar approach for the purpose of 

 obtaining an initial analytical approximation to F 1 t 

 in Equation 3. Consider the special case when the 

 fishing mortalities of the plus group and the preceding 

 age are the same (i.e. a=l in Eq. 4, giving F t = F^ ,). 

 Then, the equation analogous to Equation 5 is ob- 

 tained from Equation 2: 



