Systems of Differential Equations: Models of Species Interaction
Part 2.2: Graphical Representation of Solutions
Now let's look at a particular example of the predator-prey equations. We'll set a =1, b = 0.03, c = 0.4, and p = 0.01 and use x(0) = 15 and y(0) = 15 as the initial conditions. So we want to describe functions x(t) and y(t) satisfying
dx/dt = x - 0.03xy
dy/dt = -0.4y + 0.01xy
x(0) = 15, y(0) = 15
- Use your computer algebra system to calculate numerical approximations to x(t) and y(t). On your worksheet graph both x(t) and y(t) for t between 0 and 100. In order to compare x and y at the same times, display both of the graphs on the same plot.
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(a) What is the period of x(t)?
(b) What is the period of y(t)?
(c) How much does y seem to lag behind x?
- Another way to see the relationship between x and y is to plot y(t) versus x(t). On a single copy of the xy-plane, plot the points (x(t), y(t)) for t = 0, 2, 3, 4, 5, 6, 8, and 10. Identify each point in the plot with its corresponding t-value. Are the points being traced out clockwise or counterclockwise?
- If we let t vary continuously, we trace out a curve in the xy-plane. This is an example of a curve given parametrically. (The variable t is the parameter.) This particular curve is called a trajectory of the system of differential equations. Use your computer algebra system to trace out the trajectory for this system with the given initial conditions.
Different initial conditions may generate different trajectories. The applet below will trace out trajectories for a typical predator-prey model. Just click on an initial condition; the corresponding trajectory will be traced out. The display below the trajectory gives a representation of the relative numbers of each species. As the number of prey increases and the number of predators remains roughly constant or decreases, squares will switch from yellow (the predator color) to blue (the prey color). On the other hand, when the number of predators is increasing and the number of prey is constant or decreasing, the colors will switch in the other direction.
Predator-Prey Trajectories
- What is the most interesting common feature of the trajectories? What is the corresponding property of the functions x(t) and y(t)?
- Return to the model described at the beginning of this part. Use your computer algebra system to plot the parametric curve in space with coordinates (t, x(t), y(t)). You can "grab hold" of this picture and rotate it. Do that so you are looking in along the t-axis. You should see the original two-dimensional trajectory plot.
- Rotate the image created in Step 7 so that you see a plot of x and t. Now rotate the image so that you see a plot of y and t.
