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

** **

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

**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**

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.