Math 306 - Lab 2
A Predator-Prey System with a Parameter

This lab is due Thursday, February 26th in class.

IMPORTANT: The work you submit should be your own and nobody else's. Any exceptions to this will be dealt with harshly.

The goal of this laboratory exercise is to understand the phase plane for a nonlinear system of differential equations, namely the predator-prey equations. The particular system you will deal with depends upon two parameters. In this lab, you will need to use the number B, which is the last nonzero number in your UB person number.

The predator-prey system that you will use is: given by

You should think of x as being the "population" (in some units) of prey (rabbits) and y as the "population" of predators (foxes). The goal is to understand what happens to these populations for various k-values as time increases. Remember that the numbers x and y represent scaled populations --- the units are unspecified, but they may represent hundreds or thousands of predators or prey. We are only interested in the cases where both x and y are non-negative.

One can define "bifurcation" for systems of differential equations in a way analogous to the definition for a scalar equation. If the phase plane is "qualitatively different" for values of k less than k₀ compared to values k greater than k₀, then k₀ is called a bifurcation point. Two phase portraits are qualitatively different, e.g., if they have different numbers of equilibria, or if one has periodic solutions (i.e, solutions which return to their starting point) and the other one doesn't. Your goal is to investigate phase portraits for the predator-prey equations and report what happens as k changes.

• 1. First list your person number and the value B.

• 2. Next find all equilibrium points for this system. Your answer should, of course, depend upon k.

• 3. Now let k = 0. Use Maple to sketch the phase plane for this system. The phaseportrait command is helpful here. There is an example of the use of the command here
  Turn in the phase portrait, and in a brief paragraph, discuss the nature of the phase portrait and what happens to the populations of predators and prey.

• 4. Discuss the bifurcation that occurs when k becomes (slightly) positive. Explain what happens now to the populations in a brief essay complete with appropriate graphs.

• 5. As k continues to increase, a second bifurcation occurs (remeber, we are only intersted in solutions with x and y greater than or equal to 0). First find the exact k value at which this bifurcation occurs. Next describe what happens to the populations before and after this bifurcation. Does disaster befall one of the populations? That question was actually a hint! Provide appropriate graphs.