Modeling Project 3 MTH 142 Fall 2001

Department of Mathematics, University of Rhode Island

Population Study

Consider the following population data (in millions) that was collected 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

Write a report that has 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 approximation
   
   $$\frac{dP}{dt} \approx \frac{ f(t+h)-f(t-h) }{ 2 h }$$
   
   
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 either trial and error or the Maple fit command to fit a polynomial curve
   
   $$y = a P^2 + b P + c$$
   
   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
   
   $$\frac{dP}{dt} = a P^2 + b P + c \quad \quad {\rm (DE)}$$
   
   is satisfied approximately by the population $P$ as a function of time $t$. 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 $1 \leq t \leq 30$ and $0 \leq P \leq 25$. Comment on whether the plot confirms or not your conclusion on equilibrium values.
7. Generate a slope field plot with $1 \leq t \leq 100$ and $0 \leq P \leq 25$. 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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To plot points load first the package plots.

     with(plots):

In this example, points and a curve are plotted together:

     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.

     with(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 $x(t)$:

     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));

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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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Use the electronic submission system to turn in your work.

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Maple should be used in all calculations and plots.

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For basic information on Plotting, solving equations and calculating integrals in Maple: see the Maple worksheet `` Introduction to Maple in Calculus II''(intro142.mws) , located in www.math.uri.edu/Center/

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MAPLE HELP is be available. The schedule and location is announced in www.math.uri.edu.