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 fit''.

- 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 . What 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:

with(plots):

p1 :=PLOT(POINTS([1,2],[2,4],[1.5,3]));

p2 := plot(x^2,x=0..2):

display([a,b]);

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:

with(DEtools):

de1 := diff(x(t),t) = 3*x(t)*(1-x(t)/300);

DEplot(de1,x(t),t=0..3,x=0..400);

Here is how to plot the slope field and a particular solution:

DEplot(de1,x(t),t=0..3,x=0..400,[[x(0)=50]]);

Maple can solve some differential equations, for example:

dsolve({de1,x(0)=50},x(t));

Curve fitting in Maple may be done with the fit command. Here is one example:

with(stats):

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*.