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
    \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 either trial and error or the Maple 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 $t$. 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

To plot points load first the package 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):
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 $x(t)$:

     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:


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.
Use the electronic submission system to turn in your work.
Maple should be used in all calculations and plots.
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
MAPLE HELP is be available. The schedule and location is announced in