AMATH_383_Solution_5 - AMATH 383 Solution#5 Winter 2010...

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AMATH 383 Solution #5 Winter 2010 Problem 9.6 6. For an open access fishery, the profit a fisherman can earn is governed by the following equation: () Pt pH cE Where H is the harvest rate governed by: H tq E N The change in Effort expended by fisherman is directly proportional to the amount of profit the will earn. Subbing in the above equations, you get the following differential equation based on Effort and the amount of fish: (, ) ( ) ( ) . dE f E N a pqEN cE aE pqN c dt  (0.1) The fish population is governed by the following equation: (1 ) dN N rN H dt K  By subbing in H(t) above, we get the following differential equation governing the fish population: ( , ) ( 1) [ ( dN N N gEN rN qEN Nr qE dt K K  ] . (0.2) To determine the equilibrium (E*,N*) of the coupled equations above, we set f(E,N) = g(E,N)=0 and solve for each variable: ** * * * (, )0 ( ) f E N a pqE N cE * *0 , o r c EN pq (0.3) * * * (, )0 ( 1 ) N gE N qEN K * * , o r * 0 and * , or ) and * .
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AMATH_383_Solution_5 - AMATH 383 Solution#5 Winter 2010...

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