**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.

- (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.

- There is a stable node at
- (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. - (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.

- (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?
- (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.

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

Thu Oct 24 15:18:54 EDT 1996