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.

  1. Title, Author, Date, Course and Section, Instructor.
  2. Calculate numerical estimates of $\frac{dP}{dt}$ for  t =1, 6, 11, 16, 21, 26. For this, use the "central difference " approximation
    \begin{displaymath}\frac{dP}{dt} \approx \frac{ f(t+h)-f(t-h) }{ 2 h }\end{displaymath}
  3. Plot of $y=\frac{dP}{dt}$ (vertical axis) versus $P$ (horizontal axis) for the points you have information on (see previous question)
  4. Use Maple's fit command to fit a polynomial curve
    \begin{displaymath}y = a P^2 + b P + c\end{displaymath}
    to the points in the previous question by choosing suitable values of the parameters $a$$b$, and $c$. Plot together the points and the curve that you have chosen, so that it can be verified (visually) that you have a ``reasonable fit''.

  5. The differential equation
    \begin{displaymath}\frac{dP}{dt} = a P^2 + b P + c \quad \quad {\rm (DE)}\end{displaymath}
    is satisfied approximately by  the population  $P$ as a function of time $t$(in months). Explain why.

  6. Use algebra and Maple to determine if there exist equilibrium values of $P( t )$. 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.
  7. 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

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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 $\frac{dx}{dt} = 3 x (1- x/300 )$. 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]);
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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.
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Before each calculation, describe in English what you are about to do.
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Maple should be used in all calculations and plots.
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MAPLE HELP is be available. The schedule and location is announced in www.math.uri.edu.