Hampton Natural mortality and movement rates of Thunnus maccoyii 



595 



fishery 2 (N 22 ), and fishery 3 (N 23 ). The way in which (assuming, for the moment, that tag losses due to tag 



these quantities change over time can be repre- shedding, tag-induced mortality, and non-reporting are 



sented by six simultaneous differential equations absent): 



dN i: 

 dt 



-(M^Fj + T.^Nh + T 21 N 12 



(att = 0, N„ = N„i) 



dN 



12 



dt 



= -(M 2 + F 2 + T 21 )N 12 + T 12 N n 



(at t = 0, N 12 = 0), 



dN 



13 



dt 



= -(M 3 + F 3 )N 13 + T 13 N„ + T 23 N 12 (at t = 0, N 13 = 0), 



dN 



21 



dt 



(M 1 + F 1 + T 12 )N 21 + T 21 N 22 



(at t = 0, N 21 = 0), 



dN 



22 



dt 



= -(M 2 + F 2 + T 21 )N 22 + T 12 N 21 



(at t = 0, N 22 = N 02 ), 



dN 



23 



dt 



(M 3 + F 3 )N 23 + T 23 N 22 + T 13 N 21 (at t = 0, N 23 = 0), 



(4) 



where N 01 and N 02 are the number of releases into the 

 two fisheries, Mj , M 2 , and M 3 are the rates of natural 

 mortality operating in the three fisheries, Fj , F 2 , and 

 F 3 are the rates of fishing mortality specific to the 

 three fisheries (assumed, for the moment, to be con- 

 stant over time), and T 12 , T 13 , T 21 , and T 23 are move- 

 ment rates (the first subscript denoting the donor 

 fishery and the second subscript denoting the recipient 

 fishery). 



Tag returns from the three fisheries can be classified 

 in a similar fashion to numbers at large, i.e., r n , r 12 , 

 and r 13 are the numbers of returns of fish released in 

 fishery 1 that are recaptured in fisheries 1,2, and 3, 

 respectively, and r 21 , r 22 , and r 23 are the numbers of 

 returns of fish released in fishery 2 that are recaptured 

 in fisheries 1, 2, and 3, respectively. The estimated 

 rates of return in these categories can be written as: 



dfi 



dt 



1 = F,N 



l-^ll: 



df 



12 



dt 



= FoN,, 



I2i 



and 



dfi 3 

 dt 



dr 2 i 

 dt 



dr 22 

 dt 



df 23 

 dt 



F S N 



3^13. 



= F,N 21 , 



= F 2 N 22 , 



= F 3 N 23 . 



(5) 



Equations (4) and (5) are solved by integrating be- 

 tween times t and t + At. For ease of notation, define 

 Ax-Mi + P^Tu+Tm, A 2 = M 2 + F 2 + T 2 i + T 23 , and 

 A 3 = M 3 + F 3 . Integrating Equations (4) is accom- 

 plished by applying a decoupling transformation (see 

 Sibert 1984 for details) and results in the following set 

 of delay-difference equations, 



