Part 1. Using the textbook (Sections 6.1 and 6.2)
• define the critical points of an autonomous system of linear equations;
• classify critical points;
• give sufficient conditions for stability of linear and almost linear systems.
Part 2. Use Part 1 to study the behavior of certain concrete systems of differential equations.
1. Find and classify all critical points (equilibrium solutions) of the following resource
competition model
x
0 = x(3 − x − 2y),
y
0 = y(2 − x − y).
The system models the populations of rabbits x and sheep y in a hypothetical nature preserve,
where time t is measured in months.
2. Find and classify all critical points (equilibrium solutions) of the following system
x
0 = y + y
2
e
x
,
y
0 = x.
Instructions.
• The report must be written in complete sentences and all the arguments must be justified. The project will be graded both for mathematical quality and for expository
quality. There should be an introduction, the detailed plan, and some closing comments. The plan should have enough details so that a reader could actually carry out
the steps. You can work in groups of up to three
Requirements: 1 day
