Population Growth Models
Part 5.3: Solutions of the Harvesting Model
We saw in Part 5.2 that the proportional harvesting model
can be written as a logistic equation
with M = K - QE / r. In Part 4 you found the general solution of this equation, as well as the solution of the initial value problem with P(0) = P0.
- Use your solution from Part 4 to write the solution of the initial value problem
(Alternatively, you can just use your helper application to solve the problem again.) Simplify as much as you can.
- Use your solution formula to plot the population growth with K = 30000, r = 4 x 10 - 5, P0 = 10000, Q = 0.001, and E = 0 over a time period of 8 years. What does zero effort tell you about harvesting? How is this case related to what you did in Part 4?
- Now set E = 500, and plot again, keeping the other parameters the same. How does harvesting at this level affect the eventual state of the population?
- Increase the harvesting effort, and determine how the eventual state of the population changes. Is there an effort level at which the starting population is an equilibrium level? Is there an effort level at which the population declines rather than grows? If so, what is the eventual state of the population? (If necessary, adjust your time scale so you can see what happens in each case.)
- Explain each of your conclusions about the eventual state of the population in terms of the logistic model
with M = K - QE / r. Is it possible that a proportionally harvested population can be driven to extinction? Why or why not?
