MA 226: LAB 4

Competing Species Lab

This lab is due Tuesday, December 6 in class. Late labs will not be accepted.

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. 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 that this equation is the default equation in the tool called Competing Species.

1. First compute all equilibrium points for this system that lie along either the x or y axis. 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 a and b drawn from the interval [0, 2]. You answer here and elsewhere may depend upon the parameters a and b.

2. Using linearization, determine the types of the equilibrium points that you found lying along the x and y axes.

3. Now find all equilbrium points that satisfy x > 0 and y > 0. Again your answer should depend on both a and b. For which values of these parameters do such equilibria exist?

4. Now determine the regions in the a,b-plane where this system has qualitatively different phase portraits. Draw a diagram indicating what the phase plane looks like in each of these regions.

5. Next determine what happens to the x and y populations in each of these regions, i.e., determine regions of coexistence, extinction, and so on. Include a brief essay describing what happens in each of these regions as well as a diagram. In this question, you need only be concerned with initial conditions for which both x and y are positive. In particular, we are not concerned about solutions along the x or y axes.

6. Discuss all of the bifurcations that occur in this family, i.e., describe what happens as you cross the various boundaries of the regions above.

7. What can you say about the special parameter a = b=1?