Modeling Project 3 MTH 142 Fall 2004
Department of Mathematics, University
of Rhode Island
Consider the following bird population data (in millions) that was collected in an island in the Pacific Ocean during a period of several years, beginning in 1970 (t=0) and ending in 2000 (t=30)
14.99295, 14.62998, 14.28663, 13.95963, 13.64571, 13.34487, 13.05384, 12.77589, 12.50775, 12.24942, 11.99763, 11.75565, 11.52021, 11.29131, 11.07222, 10.85640, 10.64712, 10.44111, 10.24491, 10.04871, 9.85905, 9.67266, 9.48954, 9.31296, 9.13638, 8.96307, 8.79630, 8.62953, 8.46603, 8.30253, 8.14230
Your job will be to find a suitable model for this data, and
to use it to determine some trends.
Write a Maple worksheet report with the following parts.
- Title, Author, Date, Course and Section, Instructor.
- Calculate numerical estimates of for t =1, 6, 11, 16, 21, 26. For this,
use the "central difference " approximation
- Plot of (vertical axis)
versus (horizontal axis)
for the points you have information on (see previous question)
- Use Maple's fit command to fit a polynomial curve
to the points in the previous question by choosing suitable values
of the parameters , , and .
Plot together the points and the curve that you have chosen,
so that it can be verified (visually) that you have a ``reasonable
- The differential equation
is satisfied approximately by the population as a function of time (in months). Explain why.
- Use algebra and Maple to determine if there exist equilibrium
values of . Verify
your answer by producing a slope field plot with 1 < t <
30 and 0 < P < 25. Comment on whether the plot confirms
or not your conclusion on equilibrium values.
- Generate a slope field plot with
0< t < 100 and 0 < P < 25 .
does the model predict the population will be in the year 2030?
What will be the population in the year 2060?
Tips, comments, and additional information
- In this example, points and a curve are plotted together:
p2 := plot(x^2,x=0..2):
To use differential equations related commands in Maple you must
first load the package DEtools. Here is an example of how one
plots in Maple a direction field and a solution of the logistic
differential equation .
Note the x(t) term:
de1 := diff(x(t),t) = 3*x(t)*(1-x(t)/300);
Here is how to plot the slope field and a particular solution:
Maple can solve some differential equations, for example:
Curve fitting in Maple may be done with the fit command. Here
is one example:
xvals := [-2.,-1.,0.,1.,2.];
yvals := [-3,0,1,0,-3];
fit[leastsquare[[x,y], y=a*x^2+b*x+c]]( [xvals,yvals]);
- The final project should have only one author. You may discuss
the project with your classmates, but what you turn in should
contain your own original designs.
- Before each calculation, describe in English what you
are about to do.
- Maple should be used in all calculations and plots.
- MAPLE HELP is be available. The schedule and location is
announced in www.math.uri.edu.