We now extend the model from step 2 to the general case, where the fish are...

 

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We now extend the model from step 2 to the general case, where the fish are also being caught at a constant rate H fish per year. Then the model from step 2 is modified as dP = 2P Н. 200 (1) Sketch graphs of the dP/dt against P for each of the values H = 50, 75, 90. Mark the corresponding equilibrium solutions on each plot. (2) For each of the H values above, what is the range of the initial condition such that the fish population does not extinct eventually? How does this range change when H is increasing from 50 to 90? (3) (Bonus 0,5pts) What happens to the equilibrium solutions as the value H + 0 (H decreases to 0)? What is the significance of this in terms of the fish pond population dynamies? (4) (Bonus 0.5pts) What happens to the equilibrium solutions as the value H 100 (H increase to 100)? What is the significance of this in terms of the fish pond population dynamics? (5) (Bonus 0.5pts) What happens to the equilibrium solutions when H > 100 ? What is the significance of this in terms of the fish pond population dynamics? (6) (Bonus 0.5pts) Based on your analysis above, recommend a harvest strategy to the fish farmer to ensure long-term survival of the fish population while maximize the harvest. Figure 1: Left: Slope fiekl for Step 1 model. Right: slope field for Step 2 model. We now extend the model from step 2 to the general case, where the fish are also being caught at a constant rate H fish per year. Then the model from step 2 is modified as dP = 2P Н. 200 (1) Sketch graphs of the dP/dt against P for each of the values H = 50, 75, 90. Mark the corresponding equilibrium solutions on each plot. (2) For each of the H values above, what is the range of the initial condition such that the fish population does not extinct eventually? How does this range change when H is increasing from 50 to 90? (3) (Bonus 0,5pts) What happens to the equilibrium solutions as the value H + 0 (H decreases to 0)? What is the significance of this in terms of the fish pond population dynamies? (4) (Bonus 0.5pts) What happens to the equilibrium solutions as the value H 100 (H increase to 100)? What is the significance of this in terms of the fish pond population dynamics? (5) (Bonus 0.5pts) What happens to the equilibrium solutions when H > 100 ? What is the significance of this in terms of the fish pond population dynamics? (6) (Bonus 0.5pts) Based on your analysis above, recommend a harvest strategy to the fish farmer to ensure long-term survival of the fish population while maximize the harvest. Figure 1: Left: Slope fiekl for Step 1 model. Right: slope field for Step 2 model.

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