Maximum Sustainable Yield. The number y of mushrooms in a wooded area is described by the logistic equation dy/dt = r(1-

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Maximum Sustainable Yield The Number Y Of Mushrooms In A Wooded Area Is Described By The Logistic Equation Dy Dt R 1 1 (57.44 KiB) Viewed 34 times

Maximum Sustainable Yield. The number y of mushrooms in a wooded area is described by the logistic equation dy/dt = r(1-y/K)y where t is time in years. You wish to forage for mushrooms in a sustainable fashion. Consider two different foraging mecha- nisms. Fixed effort: mushrooms are picked at a fixed proportion relative to the number mushrooms. Fixed yield: a constant number of mushrooms are picked each year. (a) Write a differential equation for the number of mushrooms y with respect to time t in years taking into account the foraging H(y,t). (b) Assume a fixed effort H(y, t) = Ey model where E <r. Find the non-zero equilibrium quantity of mushrooms y(t) = yo. Explain the behavior of mushroom availability as t → ∞o when the initial quantity of mushrooms is near the equilibrium quantity. (c) The sustainable yield Y= Eyo is a rate at which mushrooms can be picked indefinitely. Determine E so as to maximize the sustainable yield, and find the maximum sustainable yield. (d) Now assume you forage for mushrooms ensuring you have a fixed yield H(y, t) = h. Show if h> regardless of the initial number of mushrooms, there will be none left in a finite amount of time. (e) If one assumes initially there are a lot of mushrooms and what is the long term number of mushrooms? What is the maximum sustainable yield in a fixed yield scenario? Compare with part (b).