730 



Fishery Bulletin 91|4). 1993 



The solution to Equation A7 can be written as a poly- 

 nomial of degree n+\ as follows: 



n i n 



(A8) 



X(l[t%k(::\)](n) k K">- 



i=0 \ 4=0 



[ ltt% ) k K"^ q y MS) : - qF MS / ♦* = 0. 



In the special case where <7=0, Equation A8 gives 

 the following polynomial solution for F' max : 



X f I [ (":%k(::\)(n) k K" k ])F; n J=0. (A9) 



i=0 \ *=0 / 



Equation A7 can be solved explicitly for q. The locus 

 at which F' MSY -\ is given by 



[2 \ JMf)'] 



(A10) 



Note that Equations 6-8 constitute the special cases 

 of Equations A8-A10 where n=l. 



K>0 When growth follows the form of Equation 22, 

 stock biomass per recruit can be written 



, ~ /-l_p-Kla-a l-KIK"\n 



BPR(F')=l w r [ 1 _ e - KI K J e- Ma + FHa -"Jda 



(All) 



/ w r \ /y (-lHl)e- l " UK \ 

 \M( l-*- K < K T)\t'o kK + 1 + F ) 



Substituting Equation All into Equation A6, multi- 

 plying through by MF\ and differentiating gives the 

 following expression: 



dY(F') 

 dF' 



MB(F') /l 



-\ 



(1-9)1 



kK'+l + F / 



(A12) 



Letting 

 z(F) 



*=" (kK' + 1 + F') 2 

 MB(F') 



ll-q)( 



"(-lt(l)e'' K K \ " 



(A13) 



Equation A12 can be rewritten as 



dY(F') 



dF 



n\tf 



= z(F')\ L ((-l)*(" t )e-* K ' ,A '"[(l-<7)x 



(kK'+l)-qF'](Y\ imK'+ 1+F') 2 ))]. 



m*i J 



(A14) 



The term enclosed in large square brackets in Equa- 

 tion A14 can be expanded to polynomial form. Pro- 

 ceeding in steps, first note that 



ll (miT + l)= X (-D m s(7i, n-*n)K' m . (A15) 



in=0 m=0 



Letting 



a*„,= X (-lY"+*(n-k),s(n-A,n-m) (A16) 



A=0 



(except for a „= n\), Equation A15 can be extended to 

 £( II imK' + 1)\ =X ( X a*, m if'"Y (A17) 



*=0\n.=* / 4=0 \ "1=0 / 



Next, let 



/?,,,,„ = «*.,„( '7") ' ,A18) 



in which case Equation A17 can be extended to 

 Xf ll (wA"+l+F'))=Xfl(l/J,,,„F''"V'Y (A19) 



*=0\m*k / *=0\/=0\m=0 / / 



Then, let 



y„*=l( X A,,!,,.-,! 



/=0 \ m=minl0,7-n) / 



(A20) 



in which case Equation A19 can be extended to 



if n ( W F-+i+F)A=if ifi yilJ K\F\ <a2d 



*=0\m«k / M>\>0\*=0 / / 



Finally, the solution to Equation A14 can be written 

 as a polynomial of degree 2n + l as follows: 



i [ X f X (-1 WjX*y 4 w. * + r, , *» e ^ /A ' ) *' J - 



1=0 L 7=0 V=0 / 



fXfi<-l>*(;>(/^,,,-,, + ?,.,*+ (A22) 



\ J ,=0 \*=0 



