Examine the effects of harvesting a renewable resource whose growth is logistic

Examine the effects of harvesting a renewable resource whose growth is logistic

Examine the effects of harvesting a renewable resource whose growth is logistic. An important question is what level of harvesting leads to a sustainable yield. In other words, how much can be harvested without the population depleted in the long run?

Under harvesting, the growth of your sustainable resource is governed by the following differential equation:

dy = ry (1- (y/k)) -hy

dt

Where y measures the amount of resource at time t (in whatever units of time are appropriate for the system under study), r is the growth rate when the amount of resource are small, k is the maximal amount of resource supported by the environment, h is the removal (or harvest) rate of the resource.

1. Study Oxygen. Be as creative as you like.
2. Do some research to find the appropriate values for r and k. You need to use Google Scholar only and provide citations.
3. Solve for the equilibrium values. The point of this project is to determine how big hy can be without depleting the resource. So we do not yet have a value for You will end up with an expression depending on h for one of the equilibrium values.
4. Determine values for h so that the nonzero equilibrium value (2) is larger than zero. ( if this nonzero equilibrium value is stable, the resource will go to it and will not be depleted)
5. Now we you will need to calculate the maximum sustainable harvest using optimization. Maximize z=hy*, where y* is the expression for the nonzero equilibrium value. (y* is the amount of resources available in the long-term, and hy* is the amount of harvested in the long run) Be sure to check that the value of h you find is a maximum.
6. Graph z versus h to confirm your result from (4).
7. Interpret the meaning of your answer from question 4. Explain how this relates to the graph in question 5.
8. Create a slope field using the h value from question 4.
9. Using the slope field in question 7 describe what happens to the renewable resource over time depending on its initial value. Explain how the maximum sustainable harvest relates to the equilibrium value in your slope field.
10. Now create 2 more slope fields: one with a value of h less than that found in question 4 and one with a value of h greater than that found in question 4.
11. Using the slope fields in question 9 describe what happens to the renewable resource over time depending on its initial value. Explain how the maximum sustainable harvest relates to the equilibrium value in your slope field.