One of the important tasks for a biologist working in the area of fish management is to determine the limit of fish that can be taken from a population, such that the population maintains a healthy population. Suppose that a new reservoir is created, then stocked with a popular, fast growing game fish. Below is a census of the population of fish over a period of 25 years, while the reservoir is being studied.

a. Assume that the population of fish satisfies a logistic growth model

Find the general solution to this differential equation. Determine the least squares best fit of this model to the data above. Give the value of the sum of squares error between the model and the data and write the solution using the best parameters r, M, and P₀. (Give 4 significant figures.) What is the carrying capacity of fish in this reservoir?

b. Use Maple to plot the slope field of this model and include the solution graphs for P(0) = 5, 50, 100, 150, 200, 250, 300 and t = 0 to 50.

c. Now suppose that the reservoir is opened for fishing. The logistic growth model with harvesting is given by

where h is the number of fish caught per year. If the level of fishing allows h = 8 (8,000 fish per year), then find the new solution to the differential equation using the parameters that you found in Part a. What is the new carrying capacity of the reservoir?

d. Find all equilibria for the logistic growth model with harvesting. Repeat the process in Part b. for this model. Discuss what behavior you observe for the population of fish based on the slope field plot.

e. What level of harvesting will result in the fish population going extinct?