The growth of a population of fish under harvesting is governed by the following
ID: 2851091 • Letter: T
Question
The growth of a population of fish under harvesting is governed by the following differential equation:
dy/dt = 2y(1-y/200) - H
where y is the number of fish (in millions) at time t in years amd fish are removed by fisherman at a rate of H(million) fish per year.
1.) Use the quadratic formula to obtain an expression for the equililbrium values as functions of H
2.) When do the equilibrium values exist? (Hint: For what values of H are they real?)
3.) Use a spreadsheet to predict the annual fish population for 10 years using Eulers method, when H=50 and y(0) = 100. (Hint: Start with 100 million fish and compute the growth rate every 0.1 years). What happens when the intial number of fish y(0)=10?
4.) Graph the fish population over time (for both initial values)
5.) What is happening to the fish population in each case?
6.) Create a slope field for the situation in (3).
7.) Describe what happens to the fish population depending on its intial value
8.) using slope fields or the spreadsheet you created, experiment with different H values to determine what happens to the fish population as H varies from 0 to 100.
9.) Use the results of your analysis in (7) to fill in the table below:
Fishing Rate H Amount of fish needed to Guarantee Survival of Fish Population
0
20
40
50
60
80
100
10.) Recommend a strategy (to NOAA) to maximize fish harvest while laos ensuring long-term survival of the fish population (based on your table in #9). Explain your reasoning. Is your strategy sensitive to the variability inherent in life? (disease outbreaks, bad weather, etc)
Explanation / Answer
The growth of a population of fish under harvesting is governed by the following
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