BOOTH; MODEL FOR OPTIMAL SALMON MANAGEMENT 



[A-5']. Utilizing the following derivatives, it 

 is possible to consider a phase diagram analysis 

 of this equation system. 



dXt/dSt 



( 



+ { IP^fikt) -PEkt'] + Xt[l— /(A;t)] 

 p + l — ll —fikt)] dRt/dSt 



d^Rt 



d^ 



[A-6] 



dXt/dSt 



(Px - Xt)/' 



plT^r 



f(kt)] 



dRt 



dE7 



'W 



dk 



A. constant 



■( 



- IP.f(kt) - PEkt] + Xtll-f{kt)] 



d^Rt 

 d^ ' 



[A-8] 



dS 



WTt 



Sf constant 



Rtf'jkt') 

 {P. — Xt)f"{kt) 



[A-9] 



Before proceeding with the analysis, note that: 



(i) /' > 0, /" < 0, P. - Xt > by [A-4'] , and 

 [(Px — Xt)f{kt) — PEkt + Xt] > since 

 P, — \, = Pe//' and f > fkt. 



(ii) If a steady state solution exists, such 



that Xt = and St — 0, which satisfies the ne- 

 cessary conditions for a maximum, the equilib- 

 rium value for St, S*, will be such that S* ^ St 

 where St maximizes Rt as a function of St. The 

 reason for this was discussed in the section on 

 Notation and Assumptions. It then follows that 

 dRt/dSt > 0, and d"Rt/dSt^ < 0. 



(iii) < [1 — fikt)] ^ 1 since St = 

 Rt [1 — fikt] ^ and St ^ Rt. 



(iv) dRt/dSt = 1 evaluated at St\ where 

 St" is the spawner stock required for maximum 

 sustained yield; if St < St^ dRt/dSt > 1, and 

 if St > St', dRt/dSt < 1. 



It is now possible to attach signs to [A-6] 

 through [A-8] as follows: 



dkt/dSt 



X = 



< for St" < St ^ St, 



indeterminant for St < St'; 



[A-10] 



dXt/dSt 



S = 



> for St' ^ St ^ St, 



[A-7] interminant for St < St' ; 



dX 



dSt X constant 



> for St ^ St 



dS 



SxT 



St constant 



> 0, V Su 



[A-11] 



[A-12] 



[A-13] 



dR 



If [1 — fikt)] j^ ^ 1, then there are no sign 



ambiguities in the relevant range, and the equil- 

 ibrium solution exists. From the phase diagram 

 in Appendix Figure 1, it is clear that the equil- 

 ibrium_ is a saddle point; i.e., given an initial 

 So ^ St, there exists a time path for Xt, kt, and 

 St converging to the steady state equilibrium, 

 along which the necessary conditions for a max- 

 imum are satisfied. Hence, the optimal policy 

 for reaching a steady state solution can be spe- 

 cified even if So ^ S*. Again there is no reason 

 for S* = St', where St' corresponds to maximum 

 sustained yield. 



It is now possible to calculate the Jacobian 

 determinant / of the system of equations [A-2] 

 and [A-5'] evaluated at the steady state equil- 

 ibrium, using [A-4'] to eliminate one of the var- 

 iables. In order to calculate the determinant, 

 the following derivatives are required: 



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