Logistic Population Models with Harvesting In this lab, we consider logistic models of...

Logistic Population Models with Harvesting

In this lab, we consider logistic models of population growth that have been modified to include terms that account for “harvesting.” In particular, you should imagine a fish population subject to various degrees and types of fishing. The differential equation models are given below. (Your instructor will indicate the values of the parameters k, N, a₁, and a₂ you should use. Several possible choices are listed in Table 1.10.) In your report, you should include a discussion of the meaning of each variable and parameter and an explanation of why the equation is written the way it is.

We have discussed three general approaches that can be employed to study a differential equation: Numerical techniques yield graphs of approximate solutions, geometric/ qualitative techniques provide predictions of the long-term behavior of the solution and in special cases analytic techniques provide explicit formulas for the solution. In your report, you should employ as many of these techniques as is appropriate to help understand the models, and you should consider the following equations:

Your report: In your report you should address these three questions, one at a time, in the form of a short essay. Begin Questions 1 and 2 with a description of the meaning of each of the variables and parameters and an explanation of why the differential equation is the way it is. You should include pictures and graphs of data and of solutions of your models as appropriate. (Remember that one carefully chosen picture can be worth a thousand words, but a thousand pictures aren’t worth anything.)

Consider the same equation as in Part 2 above, but let a = a₂. What will happen to the fish population for various initial conditions with this value of a?