BOOTH: MODEL FOR OPTIMAL SALMON MANAGEMENT 



dG 



^.Rt-i 



Xt-6 — 



(1 + r)6 (IRt-i dST-7 ' 



dG 



[19] 



+ 



+ 



(1 + r)4 dRT-2 dST-G 

 1 dG dRr-i 



(1 + r)' dRr-i dSr-e 

 1 dG -dRr . 



(1 + ry dRr dSr-e ' 



[20] 



St = Rt[l—f{kt)'\,t = 0,...,T-6. [21] 



For a steady state (/c*, S*, X*) these necessary 

 conditions reduce to the following: 



(Px-X*) fik*) = Pe, 



dR^ 



[22] 





^ l-D[l-/(n]^ 



S*==i2*[l-/(fc*)], 



[23] 



[24] 



where D = 



a4 



as 



Oe 



(1 + r)4^ (1 + ry^ (1 + r)« 



dR* 

 In order for X* to be positive, [1 — /(fc*)] -r^ 



must be less than 1/D. The economic interpre- 

 tations of [22] , [23] , and [24] are the same as 

 the corresponding interpretations of [5'], [9"], 

 and [8'] in the second section. 



Some biologists believe that for certain spe- 

 cies and spawning grounds spawned salmon in 

 time period t will deplete the spawning grounds 

 of food to such a degree that food sources will 

 not be replenished sufficiently in time period t + 1 

 to support an equally large number of spawned 

 fish. The nature of this phenomenon has not 

 yet been very well specified, but one possible 

 expression of it is the following spawner-return 



relationship where feeding interaction between 

 years is accounted for by modification of the 

 power term for e: 



Rt = St-4 aiae^-^'^t-4- "=5^-5) 



+ St-5 a5ae^~''i^t-5- b^st-e' 

 + St-6 aeae^-^^^t-G- ^^t-7^ 



[25] 



The necessary conditions in a steady state for 

 the model using this spawner-return relationship 

 are the same as [22], [23], and [24], except 



dR* 

 that -7^ must be replaced by 



rJR* h 



One thing should be noted about the two 

 models presented in this section. In order to 

 attain the steady state solution, it is necessary 

 to set the run size equal to its equilibrium so- 

 lution level for the first six periods in the first 

 model and the first seven periods in the second 

 model. Hence, the problem of attaining the 

 equilibrium solution is of greater magnitude here 

 than in the simple model, where it was necessary 

 to set the run size equal to its equilibrium so- 

 lution level only in the initial period. This 

 problem can, perhaps, be more adequately dealt 

 with in the framework of a specific model and, 

 in any case, requires further research. 



NEEDED REFINEMENTS 



Thus far, the analysis considers only necessary 

 conditions for the existence of a maximizing so- 

 lution. Sufficient conditions for existence are 

 satisfied if the Lagrangian is concave in all var- 

 iables. Unfortunately, concavity neither can be 

 proved or disproved for the models examined in 

 this paper. Again, the proof may be possible 

 in the context of a more specific model, where all 

 parameters and functions are assigned. 



A practical application of the model presented 

 here would involve significantly difficult estima- 

 tion problems. Some work has already been 

 done on estimating spawner-return functions, 

 but the results in most cases have not been too 

 promising (Mathews, 1967). The spawner-re- 



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