MATH 19A Midterm: MATH19 Harvard Midterm1F0419 Fall 04
Math 19. Name
Mathematical Modeling
Exam I—Fall 2004
T. Judson
Do not write in this space.
Problem Possible Score
Number Points
1 25
2 6
3 8
4 10
5 10
6 10
7 12
8 11
9 8
Total 100
Directions—Please Read Carefully! You have two hours to take this midterm. Make sure
to use correct mathematical notation. Any answer in decimal form must be accurate to three
decimal places, unless otherwise specified. Pace yourself by keeping track of how many problems
you have left to go and how much time remains. You do not have to answer the problems in any
particular order, so move to another problem if you find you are stuck or spending too much time
on a single problem. To receive full credit on a problem, you will need to justify your answers
carefully—unsubstantiated answers will receive little or no credit (except if the directions for that
question specifically say no justification is necessary, such as a True/False section). Please be sure
to write neatly—illegible answers will receive little or no credit. If more space is needed, use the
back of the previous page to continue your work. Be sure to make a note of this on the problem
page so that the grader knows where to find your answers. You may use a calculator on this exam,
but no other aids are allowed. Good Luck!!!
1
1. (25 points) In an isolated region of the Canadian Northwest Territories, a population
of arctic wolves, x(t), and a population of silver foxes, y(t), compete for survival. (For
each population, one unit represents 100 individuals). The two species have a common,
limited food supply, which consists mainly of mice. The interaction of the two species
can be modeled by the following system of differential equations,
dx
dt =x−x2−xy
dy
dt =3
4y−y2−1
2xy,
where the proportionality constants were obtained from observation.
(a) Find the nullclines of the system for x≥0 and y≥0.
(b) Find all of the equilibrium solutions for x≥0 and y≥0.
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(c) Using linearization, determine the nature of the equilibrium solution lies strictly
in the first quadrant. That is, determine the stability of the equilibrium solution
for x > 0 and y > 0. What is the long term situation for the foxes and the wolves?
Can the two species survive together?
3
Document Summary
You have two hours to take this midterm. Any answer in decimal form must be accurate to three decimal places, unless otherwise speci ed. Pace yourself by keeping track of how many problems you have left to go and how much time remains. You do not have to answer the problems in any particular order, so move to another problem if you nd you are stuck or spending too much time on a single problem. Please be sure to write neatly illegible answers will receive little or no credit. If more space is needed, use the back of the previous page to continue your work. Be sure to make a note of this on the problem page so that the grader knows where to nd your answers. You may use a calculator on this exam, but no other aids are allowed.
