Exploration of the Phase Lines Applet
PROFESSOR: The Phase Lines mathlet helps us to understand the behavior of nonlinear autonomous equations. The large graphing window shows a direction field. The horizontal axis is t, and the vertical axis is y. As you see, the direction field is independent of t. This is the significance of an autonomous equation. The applet opens with the equation 1 minus y times y minus a, but I want to talk to you about a different equation, namely this one, which can be written as the quantity a minus y, times y, minus 1/4. This is a logistic equation with harvesting. Specifically, this is a model of the oryx population in a certain game preserve in Kenya, measured in kilooryx and years. The Kenyan government wants to sell permits to kill 1/4 kilooryx per year, and it wants to know how large a game preserve it should create to guarantee a stable oryx population. The size of the preserve determines what would be the limiting population of oryx which is the number a. We can use this app...