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.)
- Use your helper application
to plot the prey population x(t) as a function of time t.
- Similarly, plot the predator
population y(t) as a function of time t, and overlay this
plot on the plot of the prey function.
- What feature of these function
plots corresponds to the "closed cycles" you saw in the plot
of trajectories in Part 2?
- 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.
- 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.
- Make another copy of the
plot in the preceding step. Change the viewpoint so you are looking directly
along the x-axis. Explain what you see.
- Make another copy of the
3-D plot. Change the viewpoint so you are looking directly along the y-axis.
Explain what you see.
- Make another copy of the
3-D plot. Change the viewpoint so you are looking directly along the t-axis.
Explain what you see.