Math 2280-002
Third MAPLE Assignment
Population Models
Due April 12, 2013 (before 11:59 pm)
Begin by downloading this document and opening it in Maple. A shortcut is to use the
"open URL" option in the "File" menu item.
http://www.math.utah.edu/~kocs/maplehw3.mw
Submission Instructions: Create your document so that we can re-generate your
answers by using the Edit/Execute/Worksheet menu option . Before you submit your .
mw solution file on CANVAS, remove all Maple output using the menu option
Edit/Remove Output/From Worksheet. If you're worried that your file may become
corrupted during submission and we won't be able to regenerate your answers, you
may also submit a .pdf or .ps printout which includes all the output, in addition to the
.mw file.
Maple Formatting instructions: Here's some general formatting guidelines if you
choose to use Maple:
When creating your Maple document, choose the "document" format as opposed to
the "worksheet" format.
Remember to only do mathematical computations inside of Math execution groups
Use a header. In Text mode, and using the text formatting tools at the top of the
window, type (4) separate lines with your name; your student number; your Math
2280 section and Professor; and the date. Right justify these lines. Below this
header, and centered, write "Maple Project 3, Math 2280".
For each problem, copy the number and problem from this assignment file into
your new file. You can copy/paste from window to window, or you can retype the
questions.
Remember the key is readability. Every time you initialize a new variable or group of
variables write a short comment explaining what the variables represent.
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At each new step in the process, describe what you are doing in either a comment or a
text paragraph. After completing the computations for each problem, answer the
question (using complete sentences) in "Text" mode.
You own a large safari park that you originally stock with 300 zebras and 150 lions on
January 1, 2013. The following differential equations model the numbers
of
zebras and
of lions months later:
.
a) Find the critical points of this system of differential equations.
b) What is the Jacobian
for this system? You can either use MAPLE or work
out the Jacobian by hand. Either way include you answer with explanation.
c) Compute the eigenvalues of the Jacobian at each critical point, and then use
these eigenvalues to classify the type and stability of each critical point.
d) Plot the direction field together with the solution curve corresponding to our
initial conditions
.
Explain what this phase portrait tells you about the interaction of the populations.
e) We can't solve this system of differential equations explicitly in MAPLE and get
some sort of comprehensible answer. Instead, we'll use the approximation techniques
we learned in Chapter 2. Write an algorithm that uses the Runge-Kutta method with
the given inital values for and . Because appears in the DE for and appears in
the DE for , your for-loop will have to handle and simultaneously. For the most
part, however, you should be able to reuse your code from MAPLE assignment 2. Pick
an increment size that seems to give you reasonable results. Make sure that
approximate and on a large enough interval that you can see the oscillations when
you answer part (f).
f) Plot the approximations for
and
that you found in part (e).
g) Esimate the period of oscillation for the zebras and lions using part (f).
h) Estimate the maximum and minimum numbers of zebras and the calendar dates
they first appear. (You may need to zoom into the graph to get a better
approximation.)
i) Estimate the maximum and minimum numbers of lions and the calendar dates
they first appear. (You may need to zoom into the graph to get a better
approximation.)