FISHERY BULLETIN: VOL. 70, NO. 2 



The Lagrangian for the maximizing problem 

 now becomes: 



L = 



r-6 





t=0 



(1 + r)' 



i^t [Px/(fcO — Pe /bt) 



+ At [Rt — Rtfih) — -St] 



+ 2 TT^TF^^^'^' 



[12] 



fc=r-5 



given 72(0) = R^, R{\) = /2i, -R(2) = R2, 

 R{Z) =R3,Ri4) = R4,aindR{5) = R5. Note 

 that in this model Ro, R\, R2, Rs, Ra, and R5 are 

 the given initial run sizes and that the terminal 

 value condition must be modified to account for 

 the more complex spawner-return relationship 

 now being used. Also, it is implicitly assumed 

 that run sizes in the last five periods have no 

 direct effect on one another, but are determined 

 by spawner stocks in previous periods. Once 

 T -» 00 is allowed this assumption becomes un- 

 important. 



The necessary conditions for a maximum of L 

 with fct > 0, iSt > 0, Xt > are satisfied if: 



(Px - Xt) f'ikt) =Pe, t = 0,..., r-6; [13] 

 Xe-4 = (1 I ^y ([Px/(A;t) -PEkt] 



+ \t [1 - fikt)] 



dRt 



+ (^l I ry (t^-^<^'+0 -PEkt+^1 



+ At-i [1 - /(/ce + i)]) 1^ 

 + -(rTlV (t^-/(^'^2) -P£/c;t+2] 



+ X,-2 [1 — f(kt^2)] 



dR 



t + 2 



3ot-4 



t = 4,..., T—S; [14] 



^^-" = (1 I ^)4 ([^x/(A:t-7) - PEkT-7} 

 + Xr-r [1 -/(fcr-7)]) |f^ 

 + (I I r)' ([^x/(A:t-6) - PEkr-s] 

 + Xt-6 [1 — fikr-e)] 



dR 



T-ll 



^ (1 + ry dRr-, dSr-n ' ^^^^ 



^^-1" — (I ■!- y)4 ( [PxfikT-e) — PEkr-e'] 



+ Xt-s [1 - fikr-sn) 11^ 

 , 1 dG dRT-5 



(1 + r)5 dRT-5 dSr-io 



1 dGr '^Rt — 4 



(1 + r)6 dRT-i dSr-io ' 



[16] 



Xt-9 = 



dG 



dR 



T-5 



(1 + r)4 dRT-5 dST-9 



1 dG dRT-4 



(1 + r)5 di2T-4 dST-9 

 1 dG dRT-3 



Xt-8 = 



(1 + r)6 C^/?T-3 dST-9 ' 

 1 ttG 3/VT-4 



(1 + r)4 d/?T-4 3^717 



1 dG dRr-z 



(1 + r)5 d/2r_3 9St-8 



1 dG 9i?T-2 



[17] 



Xt-7 — 



+ 



(1 + r)6 dRT-2 dSr-s ' 



1 dG dRT-3 



(1 + r)" dRT-3 dSr-i 



1 dG Si?T-2 



[18] 



(1 + r)5 d/2T-2 dSr-i 



502 



