Abstract. — Virtual population 

 and cohort analyses are sometimes 

 conducted with an age category 

 known as the "plus group." This 

 category is used to keep track of the 

 abundance and catches of older fish 

 that cannot be assigned individual 

 ages accurately. In this study we 

 present a procedure for solving the 

 catch equation backwards in time 

 when it involves a plus group. The 

 procedure consists of an initial ana- 

 lytical approximation, followed by 

 a correction function based on an 

 empirical analysis, followed by the 

 application of Newton's Method. 

 The results indicate that the pro- 

 cedure works well for a wide range 

 of natural and fishing mortality 

 values. For comparison, we also 

 applied the same procedure to the 

 basic catch equation (without a 

 plus group) and found that ad- 

 equate approximation to the true 

 fishing mortality value is achieved 

 even before application of Newton's 

 Method. 



Approximations for solving the catch 

 equation when it involves 

 a "plus group"* 



Victor R. Restrepo 

 Christopher M. Legault 



Division of Marine Biology and Fisheries 



Rosenstiel School of Marine and Atmospheric Science 



University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33 1 49 



A solution to F at in the catch equation 



N. 



CajZa.t 



<z+l,(+l 



KAe 



Za., 



1) 



(1) 



Manuscript accepted 12 September 1994. 

 Fishery Bulletin 93:308-314 (1995). 



where a denotes the age, t denotes 

 the time period, C at is the catch in 

 numbers, N a+1 t+1 is the cohort size 

 at the start of the following time 

 period, F at is the fishing mortality 

 rate and Z at is the total (fishing + 

 natural) mortality rate (Z at = F at + 

 M a t ), is required in many stock as- 

 sessment problems that track the 

 exploitation history of a cohort (co- 

 hort and virtual population analy- 

 ses). Here, Equation 1 is expressed 

 in a form known as "backward in 

 time," meaning that a solution to F 

 for age a during time period t is to 

 be obtained given known values of 

 C a , t ,N a+u+v andM at . 



An analytical solution to F a t is 

 not possible and this prompted sev- 

 eral authors to develop useful ap- 

 proximations (see Pope, 1972; Sims, 

 1982; MacCall, 1986; Allen and 

 Hearn, 1989). Most current computer 

 implementations of age-structured 

 models use numerical algorithms to 

 solve the catch equation; therefore 

 approximations are seldom used in 

 stock assessment applications. How- 

 ever, approximations provide initial 

 estimates that may improve the effi- 

 ciency of the numerical algorithms, 

 an important consideration when the 

 algorithm is used repeatedly. More 

 important, as with analytical solu- 



tions, simple approximations enable 

 scientists to explore relationships 

 between variables easily. 



The objective of this study is to 

 present a simple procedure for solv- 

 ing a catch equation that involves a 

 "plus group." A plus group lumps 

 together a number of the oldest age 

 classes in a population into a single 

 age category. For example, a plus 

 group may be used when large (old) 

 fish in the catches cannot be aged 

 with a desired degree of accuracy or 

 precision as can smaller fish (Res- 

 trepo and Powers, 1991). We are not 

 aware of definitive analyses that 

 have been conducted to evaluate the 

 general merits of data aggregation 

 into a plus group. Using simulated 

 data with ageng errors in an age- 

 structured model, Fournier and 

 Archibald (1982) found that using 

 a plus group gave better results 

 than did ignoring the ageing errors 

 and extending the analyses to a last 

 "true" age. Deriso et al. (1989), also 

 using simulated data in an age- 

 structured model, found that aggre- 

 gation of older fish into a plus group 

 produced comparable estimates to 

 those obtained from disaggregated 

 data. However, Hiramatsu (1992) 

 warned that use of a plus group in 

 virtual population analyses could 

 cause large biases under certain cir- 



* Contribution No. 94/0501 of the Coopera- 

 tive Unit of Fisheries Education and 

 Research, Rosenstiel School of Marine 

 and Atmospheric Science, University of 

 Miami, Miami, FL. 



308 



