Predator-Prey Models

Part 3: Graphical Representations

In Part 2 we saw how to represent a system of two first-order differential equations by a direction field and how to plot trajectories in that field, using the numerical solver in your helper application. In this part of the module we explore graphical representations that display the time variable explicitly. (Note: We begin with the assumption that x(0) = 15 and y(0) = 15.)

1. Use your helper application to plot the prey population x(t) as a function of time t.
2. Similarly, plot the predator population y(t) as a function of time t, and overlay this plot on the plot of the prey function.
3. What feature of these function plots corresponds to the "closed cycles" you saw in the plot of trajectories in Part 2?
4. How are the peaks of the predator curve related in time to the peaks of the prey curve? Relate your answer to your answers for questions 7 and 8 in Part 2.
5. Now use the 3-dimensional plotting capability of your helper application to graph the points (x(t),y(t),t) in 3-space. Vary the viewing point as necessary to get a good picture of the resulting curve in space.
6. Make another copy of the plot in the preceding step. Change the viewpoint so you are looking directly along the negative x-axis. Explain what you see.
7. Make another copy of the 3-D plot. Change the viewpoint so you are looking directly along the y-axis. Explain what you see.
8. Make another copy of the 3-D plot. Change the viewpoint so you are looking directly along the t-axis. Explain what you see.