FISHERY BULLETIN: VOL. 75, NO. 3 



Using Equation (1.21), the above equation is 

 transformed to 



log e (a k ) + (Rk ~ MkMT^ - T k ) - \og e (C/f)' k 



Qk ~ f'k-i - f'k 



(2.8) 



If sampling surveys are arranged to have equal 

 durations (or T k - X = T k = T k + l ), then Equation 

 (2.8) reduces to 



Qk = 



\o g e (a' k ) - \og e (C/f)'k 

 f'k-i ~ f'k 



(2.9) 



Equations (2.8) and (2.9) will be modified if a 

 part of the commercial fleet is engaged with the 

 sampling surveys so that (Clf)' k will be replaced 

 by (C/fi'ks, so that the last expression will be evalu- 

 ated from the catch per unit effort of the sampling 

 vessels '\Clf') ks of assumption 12," while all other 

 items will remain the same. 



Again it is important to note that the data of 

 three successive surveys should be used to obtain 

 a single q-estimate because in case of unsucces- 

 sive data the fraction exp(A' k _i)/exp(A' k ) of Equa- 

 tion (2.5) will be biased and Equations (2.8) and 

 (2.9) will not hold good. 



Equations (2.8) and (2.9) can be used to estimate 

 qi, by a number of iterations which is large when 

 fish abundance is increasing and much fewer- with 

 decreasing abundance (Rafail 1974). 



The modification of Equations (2.8) and (2.9) is 

 based on the fact that a k shown by Equation (1.19) 

 can be evaluated as a function of A' k . Paloheimo 

 (1961) gave the following approximation: 



a k 



= (l - exp(-A'))/A' - exp(-0.5A'). (3.1) 



Rafail (1974) has shown that when the instan- 

 taneous rate of change offish abundance is nega- 

 tive, then a k of Equation (1.19) can be represented 

 as in Equation (3.1). In fact a k is more precisely 

 expressed as 



a*=exp(a,A; +a 2 A' k 2 + a 3 A' k 3 ) (3.2) 



where a 1; a^, and a 3 denote certain constants. A 

 simpler and sufficient precise expression for a k is 

 fitted here as 



a k « exp(±0.5A* + 0.04A* 2 ). 



(3.3) 



Table 1 shows a comparison between the values 

 564 



TABLE 1. — A comparison between a^-values calculated accord- 

 ing to the exact Equations (1.19) and (3.3). 



of a k calculated by the exact Equation (1.19) 

 and those calculated by Equation (3.3). 



Table 1 shows that Equation (3.3) can be used 

 to calculate a k with a maximum error less than 

 1% when A' lies between ±3.00, i.e., an error 

 which is practically negligible. Again, the smaller 

 the value of A' the smaller is the error so that 

 when A' lies between ±2.5, the error is less than 

 0.29c, and Equation (3.3) can be considered as a 

 highly precise expression in that range which is 

 always encountered in fisheries studies. Equation 

 (3.3) can be used to evaluate a' k given by Equation 

 (1.20) as 



ak 



(explc^A^ + tyU 2 ))' 



expioCiA'k-i + a 2 A' k 2 -i)  expta^Afc + i + «2-A* + i) 



and 



log, a^ = a l {2A' k - A^_j -- A' k + 1 ) 



+ a 2 (2A' k 2 - Ai 2 -! - AklO- (4.1) 



According to Equations (1.12), (1.14), (1.15), and 

 (1.16) we get 



A' k ---- [R k - M k )T k - F' k 



(4.2) 



2AL - Ai 



*-l 



= 2T k (R k - M k )_- 2F k ' 



- 7Vj (R k - M k ) + F' k -i 

 _T k + l _(R k - M k ) + F' k + 1 



= {R k -M k )(2T k -T k . 1 -T k+1 ) 



- 2F' k + Fk-i + F k \, 



