Mathematics 162                                  Laboratory 3                            Week of Feb. 8, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

Logistic Growth

Last week we found that the exponential model  does a pretty good job of describing the growth of the population of the U.S. over the early years of its existence, but not over the full two hundred years for which we have data.  In this lab we will develop a more complex model called the logistic or Pearl–Verhulst model.

Before we begin, there are two packages we need to introduce into our Mathematica session, so let's do that right away.  First is a package that contains the rules of algebra for logarithms, called LogAlgebra.m.  To teach Mathematica these, enter

<<i:/course/root/math162/logalgeb.m

Since we will once again be investigating the population growth of the United States, put the population data for the last two hundred years into a variable called popdata with this command

popdata = <<i:/course/root/math162/dataset2.dat

For reference, the data is shown here in a table.

Part 1:  The Logistic Model

The exponential model assumes that the rate of growth of a population is proportional to the population itself:

But this does not take into account crowding or competition for scarce resources. Perhaps the simplest way to take this behavior into account is to assume that deaths due to crowding and competition are roughly proportional the square of the population.  This leads to the logistic equation:

whose solutions we will investigate in this laboratory.  Notice that if we call m = kr (That is, m is a proportion of the basic growth rate k, and r is that proportion.) then we can rewrite the equation as follows:

All of us can easily see that the integral on the right is kt + C, where C is the constant of integration.  Mathematica can perform the integration on the left with the following command:

Integrate[1/(p (1 - r p)),p]

The next command applies one of the basic rules of the algebra of logarithms to simplify this result.  Enter the command, compare its output with the output above, and write down the basic rule Mathematica has applied:

% /. logquot

The basic rule for logarithms applied above was:

So far we have established that solutions of the logistic equation satisfy the relation:

but we haven't actually found the solution p(t).  The next command will do that for us.  Why is it true that the arbitrary constant C can be written as ln c in this command?

p /. Solve[%==k t - Log[c],p][[1]]

Before we use this expression to define the function p, we can simplify it further.  Indicate the algebra rule applied by the next command below.

%/.explog

The rule applied was:

Finally we can ask Mathematica to simplify and define the function:

p[t_] = Simplify[%]

Write the basic form for solutions of the logistic equation below:

p(t) =

Part 2: Logistic and Exponential Models

The basic form of the logistic model is more complicated than that of the exponential model; for instance, there are three constants to be determined instead of two.  We hope that the increased complication results in an even better model of the U.S. population than we obtained last week, using the exponential model.  To solve for these constants we will use a different method that is more efficient, and avoids all the complex solutions.  Your instructor has chosen the three data pairs for 1790 {0, 3.929}, 1850 {60, 23.19}, and 1910 {120, 91.97} to determine the first model. First we will use two of the data pairs to solve for c and r in terms of k.

Solve[{p[0]==3.929,p[60]==23.19},{c,r}]

These are ugly, and we are going to substitute them into the "template" function p(t) to define an ugly function:

ugly[t_] = p[t] /.%[[1]]

Then we use Newton's method (the FindRoot command) and the third data pair to get a good estimate for k.  Before you enter this command, be sure you put in a good first guess for the value of k; look in last week's lab to see what values are reasonable.

FindRoot[ugly[120]==91.97,{k,FIRST GUESS}]

Finally, substitute this value for k to define our first logistic model of the U.S. population.

logist1[t_] = ugly[t] /.%[[1]]

Compare this model with the data on a graph just as we did last week.  Use the commands below.

plot1 = ListPlot[popdata, PlotStyle->Red,
                 PlotRange->{0,250},AxesOrigin->{0,0}]

plot2 = Plot[logist1[t], {t,0,210}, PlotStyle->Blue]

Show[plot1, plot2]

Does your model fit the data well?  Where is the fit the best, and where is it the worst?  Offer reasons why these are reasonable places for the best and worst fits.

Now compare this model with the exponential model we found last week:

model1[t_] = 3.929 E^(0.0293788 t)

plot3 = Plot[model1[t], {t,0,210}, PlotStyle->Green]

Show[plot1, plot2, plot3]

Answer the following questions in a short essay (one paragraph), and turn it in with your report. Which model fits the data best overall?  Over what periods of time is the other model significantly better?  Over what periods is neither model accurate?  Do you think the added complication of the logistic model was worth the trouble? (An essay makes sense when read out of context.)

Part 3:  Improving on the logistic model

Using the steps of the previous part, create your own logistic model for the country since 1840, logist2[t].  That is choose three data points for years 1840 and later, and use the procedure of Part 2 to construct a logistic model.  Plot your results and compare them with the first logistic model.  Write a paragraph answering the same questions as above, except, instead of talking about the "added complication" of the logistic model, explain why you chose the data points you did, and whether using different data points might have given you a better model.

Part 4:  You are an expert now!

Country X has requested a proposal from you for a lucrative contract to estimate their population growth.  Write an essay (no more than one side of a page) discussing the merits of using either an exponential or a logistic model.  Your essay should include the effect on your decision of:  the population density of the country (is it densely or sparsely populated?), the length of time over which population data is available, the length of time over which you will be required to extrapolate the population, the relative difficulty of working with the two models, and any other factors you feel are significant.