FISHERY BULLETIN: VOL. 81, NO. 4 



If the values of No r , PS, q, a„ and Ws. (i = r, . . ., n) 

 are known, the system of n — r + 2 algebraic 

 Equations (9) and (4) can be solved with respect to E 

 and No, (i = r+l,...,n + l). Note that this equation 

 system can be reduced to an n — r + 1 th order 

 polynomial equation and solved for E using a stan- 

 dard computer routine which finds the zeroes of a 

 function. The values of No, can then be found by sub- 

 stitution. Meaningful solutions for E are constrained 

 by the conditions 



E>0 



and 



No,> i = r+ 1, ...,n + 1. (10) 



Knowing E and No,'s, the C, values can be deter- 

 mined on the basis of Equation (8). 



Method III (Limited Control Over 

 Age Composition) 



If a fishery can be divided into components, each of 

 which is characterized by a unique set of q, values, 

 some control over the age composition of the total 

 catch can be exercised by varying the relative amount 

 of fishing effort expended in these components. In 

 such a case, it is appropriate to replace q,E in E qua- 

 ff 



tion (9) with X q 7 E ; wherey denotes one of k com- 



ponents of the fishery. The system of Equations (4) 

 and (9) (modified) may then have a number of solu- 

 tions (i.e., sets of E ; 's and No,'s), but only the manage- 

 ment strategies defined by nonnegative solutions will 

 be possible for implementation. Determination of 

 the meaningful solutions and the associated catches 

 will be helpful for fisheries managers in selecting a 

 fishing strategy that is most suitable for achieving 

 their objectives (e.g., the attainable maximum yearly 

 catch weight). 



VALIDITY OF ASSUMPTIONS AND 

 MANAGEMENT IMPLICATIONS 



Assumption (a) limits the number of species for 

 which the methods can be applied. It is not satisfied 

 for most tropical species (see review in Saila and 

 Roedel 1980), but does hold well for many temperate 

 species (see reviews in Gulland 1969, 1977; Ricker 

 1975). 



The magnitude of recruitment to most fish stocks is 

 affected to some degree by environmental variation; 



therefore, assumption (b) will rarely be strictly satis- 

 fied. However, as long as the environmentally in- 

 duced variation in recruitment is random and not 

 large, results derived on the basis of the methods 

 should provide a good indication of the stabilizing 

 catch level. 



Assumptions (c) to (e) are standard for most fish- 

 eries analyses (see reviews in Gulland 1969; Ricker 

 1975) although their validity is not always obvious. If 

 both the age structure of the population and the 

 environment are stable, assumptions (c) to (e) will 

 likely be satisfied. The assessment of compliance 

 with assumption (c) is extremely difficult. Simple 

 methods used for estimating M, ( Gulland 1 969, 1 97 7 ; 

 Ricker 1975) are usually unsuitable for testing this 

 assumption. More complex methods are available 

 (e.g., Majkowski 1981), but these have considerable 

 data requirements and are frequently impractical. 

 Assumptions (d) and (e) can usually be tested, es- 

 pecially if a technique of direct age determination is 

 available for the species under consideration. 



A management policy defined by the values of C and 

 f/s (satisfying Equations (3) and (4)) or E and q/s 

 or E/s and q,/s (satisfying Equations (4) and (9)) 

 can be effective immediately if the age structure 

 of the population at the beginning of the first year of 

 policy implementation is identical to that defined by 

 the calculated values of No,'s (corresponding to the 

 values of C and f,'s, E and q/s or E/s and q,/s). This 

 will rarely be the case because of historical variation 

 in catches. As a consequence, the parental biomass 

 during an initial period of harvesting CB may fall 

 below or increase above the specified level. As long as 

 this has no effect upon recruitment, the population 

 age structure will approach that defined by the calcu- 

 lated No,'s over the life span of the species. 



The accuracy of the input parameters is implicitly 

 assumed. Uncertainty in the management recom- 

 mendations (i.e., in the value of CB) due to inac- 

 curacies (caused by estimation errors and/or natural 

 variability) in estimates of the input parameters for 

 the procedures is generally difficult to predict, but 

 can be examined for each specific application of the 

 procedures using a sensitivity analysis technique 

 (see reviews in Majkowski et al. 1981a; Majkowski 

 1982, in press; Majkowski and Hampton 1983). 



EXAMPLE 



Application of the methods described is demon- 

 strated by using southern bluefin tuna population 

 and fishery data collected prior to 1981. 2 This 



! This analysis has since been updated (Hampton et al. in press) 



726 



