MA 3431 A-96 Exam 2 Name: .

Show your work in the space provided. Unsupported answers may not receive full credit. If you use Maple to answer a question, make sure you include your Maple output with your exam and that you clearly identify which problem you are working on.

1. (15 pts) Suppose that you are given the following information about a two dimensional model concerning the steady states and their stability.
• There is a stable node at x=0, y=2.
• There is a stable node at x=2, y=0.
• There is a saddle point at x=1, y=1.
Sketch a phase portrait that is consistent with this information.

2. (10 pts) Consider the following model for the interaction of baleen whales and krill, where x represents the population of krill and y the population of baleen whales.

Analyze this model for the following set of parameter values: , r=10, , , and K=8. Your answer should include information on steady states and their stability, and a description of the phase portrait.

3. (15 pts) Consider the linear system , where is the matrix

Show that the origin is a saddle point for this system. Then find an initial condition such that

for this initial condition.

4. (20 pts) Describe our basic existence and uniqueness result for a system of differential equations of the form . Include the conditions that must satisfy to guarantee that the solution exists and is unique. Then consider the krill-baleen whale model above. For which initial conditions does the theorem not guarantee a unique solution to that system?

5. (15 pts) Consider the single equation

where is a parameter. Find all steady states, and determine their stability. Then summarize your results in the form of a bifurcation diagram.

6. (10 pts) Describe what it means for a fixed point of a system of nonlinear first order differential equations to be hyperbolic.