BOOTH: MODEL FOR OPTIMAL SALMON MANAGEMENT 



X* = 



00 



2 (1 + ry t^^/(^*) -P^k*^ 



t=l 



dRt' 



+ li^ 1 dG S^T s 



r-^ 00 (1 + r)'' 'wr d^- 



[9'] 



Both expressions to the right of the equal sign 

 in equation [9'] must converge for existence. 

 This requires that 



Then equation [9'] can be rewritten as 



[10] 



1 + r 



IP. fik*) 



— Pe k*] 



dR* 

 dS 



: [l-/(r)J dR- [9"] 



^— 1 + r dS~ 



since lim 1 dG dRr* 4. 



r - 00 (1 + r)-^ ' -m d^ converges to 



zero for 



dG 



IR^ 

 be rewritten as 



bounded. Also [5] and [8] can 



profitability of an incremental spawner, to be 

 positive. 



A steady state solution exists if the determi- 

 nant of the Jacobian matrix for [5'], [8'], and 

 [9"], is non-zero. It can be proven that the 

 Jacobian determinant is negative if it is assumed 

 that the expression to the left of the inequality 

 sign in [10] is less than or equal to 1. Unfor- 

 tunately, there is no reason to assume this, even 

 though in the context of a specific model, with 

 functions and parameters assigned, one might 

 expect that it is true." 



Note that the equilibrium solution requires 

 that Ro = 5V(1 — fik*)). Clearly there is 

 no reason to anticipate that the initial run size 

 will be at the desired level for an equilibrium 

 solution, and to reach the desired level may be 

 costly. Hence, the analysis ignores the question 

 of optimal policy for reaching the equilibrium so- 

 lution. Note also that there is no reason to an- 

 ticipate that S*, the equilibrium solution, will 

 correspond to maximum sustained yield. These 

 issues are discussed more extensively in the ap- 

 pendix in the context of a continuous time model. 



MORE COMPLEX MODELS 



(F. -X*) f'ikn = Pe, 

 S* = R* —R*f{k*). 



[5'] 

 [8'] 



Equations [10] and [9"] together suggest that 

 the productivity of the spawner stock in pro- 

 ducing additional spawners must be less than 

 the social rate of discount, r, in order for X*, the 



In a steady state solution, 





t _ dR* ( 



[1 - /(fc-)] g- 



T 



where * denotes evaluation at the equilibrium solution. 

 Note also that 



3Rt _ dR, <-l 



/(Ml 



dRr 

 dSj-i 



dR. 



dS, 



t-i 



t-1 



n 



dSr 



in the general case. 



A simple model of the type presented in the 

 previous section is not completely realistic for 

 certain species of salmon where fish spawned in 

 time period t will return to their spawning 

 grounds in time periods t + i, t + 5, and ^ + 6. 

 Assuming that the percentages of the total re- 

 turn to the river in time periods t + 4, t + 5, and 

 t + 6 from spawning efforts in time period t are 

 constants, 04, as, and ae, the spawner-return re- 

 lationship becomes: 



Rt = St-4 ai a e^'^^t-i> 

 4- Sf-5 as a e^-^^t~5> 

 + St-Q ae a e'^-''^t-6\ 



where 0-4 + as + Oe = 1. 



[11] 



° See the appendix for a proof that the Jacobian de- 

 terminant is negative for the given assumption. 



501 



