RAFAIL: STUDY OF FISH POPULATION BY CAPTURE DATA 



or 



2A' k -Ak-i- AJ+i = (R k - M k )(2T k -T k ^- T k + l ) 



- QkWk " f'k . - /Z + i>.<4.3) 





0.5(f* 



*-l 



f'k + ] 



(5.3) 



Denoting 



a 2 (2A* 2 -A* 2 , -A' k l x ) 



(4.4) 



of Equation (4.1) by (f>A' . 



Equations (4.3) and (4.4) can be used to evaluate 

 log t , a' k given by Equation (4.1) as 



log e a^ = a x (R k - M k )(2T k - T k . x - T k + l ) 



- a.q^n - fk-l - fk + l) + M'. (4.5) 



Equation (4.5) can be inserted in Equation (2.8) 

 to have another expression for q~k as follows: 



Equation (3.3) shows that o^ is estimated at 0.04 

 so that <J)A ' becomes according to Equation (4.4) as 



c/>A' = 0.04(2A^ 2 - A'^ - A'klx). (5.4) 



The correction term 4>A ' given in Equation (5.4) 

 can be put in another form by the inspection of 

 the term A' shown by Equation (4.2) 



A' k = (R k - M k )T k - F k . 



The parameters R k and M k are supposed to be 

 constant during any group of sampling surveys 

 according to assumption 9, and Equation (1.17) 

 we have 



Qk^f'k-i ~ fk) 



4>A' + «i(«* " M k )(2T k - T k - X - T k+1 ) 



- a A q k (2£ k - f' k -i -fk+i) 



+ (R k - M k )(T k - X -T k )- log e (C/f)'k 



Rk 



A' k 



M k = B k a constant 

 B-kTk - F'k 



(5.5) 



and 



or 



A' k 2 = B k 2 T k 2 - 2B k T k F' k + F^ 2 (5.6) 

 and (£>A' of Equation (5.4) .becomes 



<7*<A-i ~ f'k + 2aifk ~ «i/*-i ~ «i/**i> 

 = 4>A' + [R k - M k ][T k (2 ai - 1) 



+ TViU " «i» " ^T k + l ] - \og e (C/f)' k <t>A' = 0.04(R k - M k ) 2 (2T k 2 - T k * x - T k * x ) 



- 0.08(R k - M k )(2F' k Tk - F' k . x T k . x - F' k+1 T k+1 ) 

 or + 0.04(2F' k 2 - F' k 2 - X - F^ ). (5.7) 



Qk 



cbA' + [R k - MJ\\T k (2ax - 1) 4- Tk-tl - a,) - a.T k ^}~ \og e (C/f)' k 

 A(2a x - 1) + Aid - ^) - a,n + 1 



(5.1) 



According to Equation (3.3) we find that 0.5 is 

 a very good estimate for a x which can be inserted 

 in Equation (5.1) to obtain 



Qk = 



<bA' + 0.5(R k - M k )(T k -x ~ ?W - \og e (Clf)' k 



0.5( A 



£-1 



/*+l) 



(5.2) 



If sampling surveys are carried out during 

 equal time intervals, i.e., T k -\ = T k = T k + \, Equa- 

 tion (5.2) becomes 



If T k = T k -\ = T k + x and according to Equation 

 1.18) we have 



M k T k = Mk and R k T k = R' k 



.'.</>A' = -0.08iR' k - M'k)(2F' k - F' k + 1 - F' k+l ) 

 + 0.04(2F* 2 - F'fcx - Fkli). (5.8) 



If Equation (3.2) is used to evaluate a k . 



a'k- 



(expto^A* + a 2 A* 2 + o^A* 3 )) 2 



ex\o{a x A' k ~i + o^Ak 2 ! + a^Ak 3 ^  expi^A'k + x + a 2 A' k 2 +l + <VU 3 -i) 



565 



