RAFAIL: STUDY OF FISH POPULATION BY CAPTURE DATA 



and 



Qk-i = Qk = Qk>\ = Qk 



(1.16) 



where R k , M k , and q k are constant parameters 

 per unit time during the (£-l)th, /? th, and 

 (ft + l)th sampling surveys which should belong 

 to the same season. 



R k - M k = B k (a constant). (1.17) 



10. If T k = Tk-i = Tk+ i and similar to Equations 

 (1.8), (1.10), and according to (1.17), we get 



M k T k ---- M' k , R k T k ---- R k \ and B k T k -= B' k (1.18) 



where M' k ,R' k , andB*. represent the instantaneous 

 rates of natural mortality, recruitment, and the 

 difference between them during single surveys 

 (not per unit time) belonging to the same season 

 when the durations of the surveys are made equal. 



11. The number of fish captured by the sam- 

 pling, commercial, and the total fleet during the 

 /?th survey are denoted by C ks , C kc , and C k , 

 respectively. 



12. The catch per unit efforts during the Mh 

 survey obtained from sampling, commercial, and 

 total fleet are respectively 



(C/f') ks , (C/f)kc, and(C/f>* 



where fis primed (f ) according to previous nota- 

 tions to designate exerted effort during a whole 

 sampling survey and not per unit time. 



13. The following expressions are used to obtain 

 simpler mathematical equations: 



(explA*) - l)lA' k = a k (1.19) 



a* 2 /a*-i ' a k + 1 = a' k (1.20) 



(C/f) k 2 KC/f )*_! • {Clf) k+l = (Clf)' k . (1.21) 



A MODIFICATION FOR 



THE EXPRESSION ESTIMATING 



CATCHABILITY 



Rafail (1974) developed an estimate for q k ac- 

 cording to his equation (4.16) briefly as follows 

 when the whole fleet is engaged for sampling: 



C k = N  exp 



,k \ . 



C?,4 



F k  a k (2.1) 



and 



C k+1 ■■-- N  exp(^V A/)- F{ +i  ak +l (2.2) 



C 



k i i 



cr =ex p ,A ^ m -z 



a * + i F'k + \ 



F' k 



(2.3) 



and 



C k a k F' k 



= exp(A^_!) •  -^— (2.4) 



C k -\ 



a*-i F k _i 



and 



C t 2 



C k -\ ' C k +\ 



.2 F '2 



af 



exp(Aj;_i) _ 



exp(A^) a*_i • a k + 1 F' k ^  F k + 1 



(2.5) 



According to Equations (1.7) and (1.16) we get 



Qk 2  f'k 2 



F'k-i  F'k+i Qk 2 ' f'k-i " fk+i 

 f'k 2 



As we have 



fk-i ' fk+i 

 exp(A' k -i) 



(2.6) 



exp(A^_! - A' k ) and 



exp(A^) 

 according to Equation (1.12), we get 



exp(A;_! - A' k ) = exp((Rk-i - M*_i - *V-i)7*_i 

 - (R k - M k -F k )T k ). 



Again according to Equations (1.14) and (1.15), 

 as well as (1.7) and (1.16), we get 



exp(A*_! - A k ) =exp(R k - M k )(T k ^ - T k ) 



- q k if'k-i ~ /*)> '2-7) 



From Equations ( 1 .20), (2.5), (2.6), and (2.7) we get 



r Ck2 r - exp{iR k - M k )(T k - X - T k ) 



f'2 



~ Qk^f'k-l ~ f'kU ' a 'k ' ~FT~ ~£t • 

 ' fk-1 Ik+1 



Rearranging and according to assumption 12 we 

 get 



(Clf')k 2 



tc/Dk-i  (C/f)k +1 



exp^R,- M^Tk-i- TS 



- Wk-i ~ /*))' «*• 



563 



