596 



Fishery Bulletin 89(4). 1991 



Nii ( t + 4„ = {a[bN 11(t , + N 12(t) ] e-« 4t + [N„ (t) - aN 12(t) ] e- 4 '}. 



1 +ab 



N 12(t+ 



12(t + At) 



1 + ab 



{[bN nm + N 12(t) ] e-«" - b[N 11(t) - aN 12(t) ]e-v 



N 



= N 13(t) e-As" + -^- N 11(t) [e- A 3 At - e-*i«] + -^- N 12(t) [e^t _ e -M t]i 

 A1-A3 A 2 -A 3 



13(t + At) 



N 2 i ( t + At) = {a[N 22(t) + bN 21(t) ] e- u4t + [N 22(t) - aN 22(t) ] e -*"] 



1 + ab 



N 22(t+ At) = ; {[N 22(t , + bN 21(t) ] e-' 4t - b[N 21(t) - aN 22(t) ] e^ 1 } 



1+ab 



and 



N 23( t + At) = N 23(t) e-A 3 At + -J^_ N 22(t) [e-A 3 At _ e -A 2 At] + _^A3_ 2i(t) [e _ Ll4t]j 



A 2 -A 3 



A, -A, 



where a and b, the coefficients of the decoupling transformation, are roots of two quadratic equations, 



T 12 a 2 - (Aa-Ajja - T 21 = 0, 



T 21 b 2 - (A 2 -Ai)b - T 12 = 0, 



and 



A! + bT 21 + abA 2 + aT 12 

 1 + ab 



Ao-aT,., + abA,-bT., 



u = 



12 + dUrt.j-u i 2 i 



1 + ab 



Substituting Equations (6) into Equations (5) and integrating, the estimated numbers of tagged fish of each 

 category recaptured between times t and t + At are obtained: 



r ii(t) = - , <a 

 1 + ab 



bN 11(t) + N 12(t 



(l_ e -«At) + 



N, 1(t) -aN 



12(t) 



(l-e~ v4t ) 



r i-(t) = 



R 2 F 2 

 1 + ab 



bN nm + N 12(t) 



(l_ e -uAt) _ b 



Nn,t)-aN 12 



2(t) 



(l_e-vAt) 



l'i:iiU = R3F3 



13 



A, -A 



Nn(t, + N 13 



(t) + ^13(t) 



1-^-3 



(l_e-A 3 At) T (l-e- A i At ) 



~~ A 7 Nll( " ~T~ 



A 3 A, -A 3 A, 



r L'lll| 



R,F, 

 1 + ab 



N 22 (t) + bN 21(t) 



(l-e- uAt ) + 



N 21(t) -aN 22(t 



(l-e^ 1 ) 



