MAJKOWSKI and HAMPTON: CATCH STABILIZING PARENTAL BIOMASS 



Equation (1) is used as the basis of the methods pre- 

 sented in this paper. 



The assumptions associated with Equation (1) can 

 be relaxed by using a time interval smaller than 1 yr. 

 For example, if monthly periods were used, 12 

 equations could be formulated, each expressing the 

 relationship between the monthly catch and the num- 

 bers of fish at the beginning and end of a month. In 

 such a case, M, and the fishing intensity could vary 

 from month to month. 



Method I (Complete Control Over 

 Age Composition) 



Here we consider a fishery which has complete con- 

 trol over the age composition of catches. The 

 dynamics of a single cohort are described in such a 

 case by the system of equations: 



Cf, = No, exp(- 



/ = /'.... '7 



-0.5M,) - No, + 1 exp(0.5M,) 



(3) 



where C is the total yearly catch in number, f, is the 

 fraction of the total catch belonging to age class i (fj = 

 C,/C), andr andrc, respectively, denote the youngest 

 and oldest age classes numerously represented in the 

 fishable portion of the population. The abundances 

 of age classes should satisfy the condition: 



V a.No.Ws = PS 



(4) 



where a, is the fraction of sexually mature fish in the 

 z'th age class and Ws, is the average weight of a sex- 

 ually mature fish belonging to age class i at the time of 

 spawning. 



The system of n — r + 2 algebraic Equations (3) and 

 (4) is solvable for C and No,'s if the values of PS, No r , 

 f„ M„ a„ and Ws, (i = r, . . ., n) are known. Meaningful 

 solutions are restricted by the conditions 



C>0 

 and 



No, > i = r + 1, . . ., n + 1. 



(5) 



Because of these conditions a meaningful solution 

 may not exist for a given set of input values. In such a 

 case, a change in fishing strategy (and also, therefore, 

 in the fj values) may resolve the problem. The stabiliz- 

 ing total catch, CB, can be found from the formula: 



CB= Z 



i = r 



Cf,W, 



(6) 



where W l is the average weight of a fish from the ith 

 age class. 



If the basic management objective is to maintain the 

 parental biomass at its present level, only the coef- 

 ficients fj- may be subject to manipulation. According 

 to their definition, they have to satisfy the 

 following conditions: 



f > i = r, . . ., n 



and 



If,= l 



(7) 



but their individual values can be selected freely to 

 the extent determined by the nature of Equations (3), 

 (4), and (5). If alternative fishing strategies defined 

 by different values off, are feasible, it is of interest to 

 know 1) which of these are possible under the basic 

 management objective of stabilizing the parental 

 biomass and 2) what catches and age structures of 

 the population are associated with these strategies. 

 These questions can be easily addressed by solving 

 the system of Equations (3) and (4). 



It may be desirable to select such a fishing strategy 

 which, in addition to maintaining the parental 

 biomass at its present level, would yield the absolute 

 maximum weight of yearly catch. This strategy could 

 be determined by finding the set of f, coefficients 

 which maximizes CB. The problem is readily solv- 

 able with the aid of linear programming methods 

 using Equation (6) as an objective function (treating 

 Cf, as a single variable C,, thus making the problem 

 linear) constrained by Equations (3), (4), and (5). 



Method II (No Control Over 

 Age Composition) 



Here we consider a fishery which may be age selec- 

 tive, but that selectivity is beyond the fishermen's 

 control. This being the case, changes in the age com- 

 position of catches can only be caused by alterations 

 in the age composition of the population. 



In this case, C, can be expressed as 



C, = qENo, exp(-0.5M,) i = r, 



n 



(8) 



where q, is the catchability coefficient for age class i 

 and E is an index of effective fishing effort. Substitut- 

 ing for C, in Equation (1) we obtain 



qENo, exp(-0.5M,) 



No, exp(-0.5M,) 



- No, +I exp(0.5M,) 

 i = r n. . 



(9) 

 725 



