## MA 231: LAB 4 Competitive Exclusion Lab

This lab is due Tuessday, November 20 in class. As usual, you may use any technology that you have available: Differential Systems, Mathematica, Matlab, programmable calculator, etc. However, the software tool called Competing SPecies is ideally suited for use with this lab.

You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. It is perfectly acceptable to hand in hand-drawn figures, in case the queues at the printers become too long. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.

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 planes for a nonlinear system of differential equations that model a pair of competitive species whose populations are given by x and y, namely

dx/dt = x(1 - x/400) - (a/400)xy
dy/dt = y(1 - y/400) - (b/400)xy

You should think of x and y as the populations of different species that compete for the same food supply. Here a and b are parameters that we will vary. This means that you will choose different a and b values and get correspondingly different differential equations. Both a and b will be chosen between 0 and 2.

Note: The number 400 is actually given by other parameters in the Competing Species tool, namely m and n. However, 400 is the default value for these parameters so that they need not be changed in the program.

1. First compute all non-negative equilibrium points for this system. Of course, these equilibria will depend on a and b. Remember that, since we are dealing with populations, we are only interested in values of x and y that are non-negative and parameter values in the interval [0, 2].

For the rest of this lab, we will be concerned only with initial conditions for which both x and y are positive.

2. Next determine all of the values of a and b in the interval [0, 2] for which each of the following events occur as time tends to infinity:

• Case 1: Species x dies out and species y survives
• Case 2: Species y dies out and species x survives
• Case 3: Some initial conditions lead to species x surviving and others lead to species y surviving.
• Case 4: All initial conditions lead to coexistence of species x and y
Choose a variety of different initial conditions to see which event occurs. In each case, sketch a typical phase plane, including nullclines, equilibrium points, and typical solutions. You need only draw one such phase plane for each of the four cases above.

3. Now draw a picture of the ab plane, indicating in different colors where the different cases above occur. Be sure to label what these regions correspond to.

4. Describe in a couple of paragraphs the kinds of bifurcations that occur as your parameters move around in the ab plane.

5. Discuss in a short paragraph what happens when both a and b are equal to 1.