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\centerline{Mathematics 36, section 1 -- AP Analysis}
\centerline{Lab Day 8: Logistic Models, Modeling World Human Population}
\centerline{November 14 and 17, 1997}
\bigskip
\noindent
{\bf Background}
\bigskip
Last time in our discussion, we introduced the Verhulst, or
{\it logistic\/} population growth model:
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\item{} {\it The ratio of the rate of change of the population 
to the population is proportional at each time to the difference between
$M$ (the maximum sustainable population) and the population.}
\bigskip\noindent
As a differential equation:
$$\eqalign{ {1\over y}\cdot{dy\over dt} &= k(M - y)\qquad {\rm so}\cr
            {dy\over dt} &= ky(M-y)\cr}\leqno(1)$$
\bigskip
\noindent
Today we want to study these equations, their solutions,
and see how an actual population problem might be attacked using this 
model.

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\noindent
{\bf Lab Questions}
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\item{A)} Take $k = 0.1$, $M = 10$ (made-up numbers) in (1), and plot
the slope field for this equation together with some solutions,
using the {\tt phaseportrait}
command from the last lab.  Take $t \in [0,10]$, $P \in [0,15]$,
and plot solutions with $y(0) = 1$, $y(0) = 5$, $y(0) = 12$.
Compare these with graphs of the symbolic solutions 
$y(t) = M/(1+ce^{-Mkt})$ from question E on the Discussion.
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\noindent
Enough of the made-up numbers!  In a ``real-world'' population
study in biology or demographics, the idea might be to 
\bigskip
\item{$\bullet$} Consider the hypothesis that the population being
studied was modelled by a logistic growth law as above,
\item{$\bullet$}  From experimental data, try to determine the 
constants $k$ and $M$ that ``best fit'' the data, 
\item{$\bullet$}  See how well the resulting logistic function
$y(t) = M/(1+ce^{-Mkt})$ matches the observed data.  
If the fit is good, you might make predictions about what the 
population would be at times in the future.

\bigskip
\noindent
We are going to do this for the following table of approximate 
world human population data from the U.N. Population Division.  (These
are all estimates, of course; there has never actually been a world census.)
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$$\matrix{
t {\rm (year)}&1960&1965&1970&1975&1980&1985&1990&1995\cr
y {\rm (pop.\ in\ billions)}&3.049&3.358&3.721&4.103&4.473&4.882&5.249&5.679\cr
}$$


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\noindent
If these points were exactly on some logistic equation solution graph, 
then for each of these $t$ values, the points
$(y, y'/y)$ would lie on some line $z = k(M - y)$
in the $(y,z)$-plane.  To begin, 

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\item{B)}  Using the table of values above, estimate the rate of 
change $y'$ (in billion people per year) at each of the given times, and 
divide by $y$ to get an approximate value for the ratio $y'/y$.  (This can 
be done by hand, or using Maple as a calculator.)  

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\item{C)}  In Maple, make a list of your approximate data points
$(y,y'/y)$ in the following format.  Your first
data point (corresponding to $t = 1960$) should have $y' \doteq .0618$ 
and $y = 3.049$ from the table.  Then $.0618/3.049 \doteq .0203$ is your first
$y'/y$ value, and the first point in the list is as here:
\medskip
\centerline{\tt dl:=[[3.049,.0203],[y2,y2'/y2],$\ldots$,[y7,y7'/y7]];}
\medskip
\item{} (Since there's no obvious good way to estimate $y'(1995)$
from the table, only include data points for $t = 1960$ 
through $t = 1990$.)  You can plot your list of $(y,y'/y)$
points to see how close
they come to lying on a line, using a plot command like this
\medskip
\centerline{\tt plot(dl,style=point);}
\medskip
\item{}  Try this, and try to ``eyeball'' a line fitting these points.
For example, what sign will the slope have?  Approximately where
is the $z$-intercept?
We'll derive the equation of the best fitting line by a 
statistical method in the next question.

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\item{D)}  From (1) again, if our population values were all on the 
graph of a logistic function, then the points we have plotted here
would all lie on a straight line $z = k(M - y)$
in the $(y,z)$-plane.   That isn't
true here, but we can still try to determine the line that
{\it fits the data the best\/}.  You may have seen a method
called {\it linear regression\/} or 
{\it least-squares curve fitting\/} for doing this in another course.
We will do this now, using Maple's least-squares routine.  Enter
the following command to load the {\tt stats} package that contains
this routine:
\medskip
\centerline{\tt with(stats):}
\medskip
\item{}  Next, to put the data into the form Maple wants, we will need
to separate out the $y$- and $z=y'/y$-coordinates of the data
from {\tt dl}, as follows:
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\centerline{\tt ylist := [dl[1][1],dl[2][1],dl[3][1],dl[4][1],dl[5][1],dl[6][1],dl[7][1]];}
\centerline{\tt zlist := [dl[1][2],dl[2][2],dl[3][2],dl[4][2],dl[5][2],dl[6][2],dl[7][2]];}
\medskip
\item{}
Now we're ready to compute the best-fit line.  Enter 
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\centerline{\tt fit[leastsquare[[y,z]]]([ylist,zlist]);}
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\item{}
If all your [, ]'s and (, )'s match, this should generate the equation of 
a line in the form
$$z = {\rm (intercept)} + {\rm (slope)}y$$
From the slope and intercept values, determine the best-fit
$k$ and $M$ for the logistic model, for this data.  (Recall, 
$M$ is the maximum sustainable population in the logistic equation.)

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\item{E)}  Now comes the ``moment of truth''!  Using your $k$ and $M$
values, and the population in 1960, determine an analytic solution
of the logistic equation that should model the world population.
(See question E from the Discussion.)
 Plot your solution curve as a function of 
time for the period 1960 - 2025.  What world population do you 
predict for 2025?  (The U.N. has published an estimate of 
$8.177 \times 10^9$.  How close is your prediction to the one
from the U.N.?)

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\item{F)}  Discuss your results (compare the values of 
your function and the population data above).  As in the last
part of question C above, 
it is possible to plot lists of points in Maple.
Once you have such a plot, you can combine a
point plots with an ordinary plot and display them together
using a command
called {\tt display\/} from the {\tt plots\/} package. 
Plot your exact solution of the logistic equation
together with the $(t,y)$ data points from the table given above.
(Note: This is a different list of points from the 
one you were looking at in question D.)
See the online Help for the info
you need to do this, by entering the command
\medskip
\centerline{\tt ?plots[display]}
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\item{}
The help listing has several examples that you can try to follow.
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\item{G)} discuss any remaining limitations in a model of this kind.  
For instance, is it really reasonable that the maximum sustainable
population is constant over time?  Are there factors that would 
tend to change it?  Are there other models that might be
even better at accurately predicting human population growth?  

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\noindent
{\bf Assignment}
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Solutions to the questions from Discussion 7 and these lab questions 
will be due on Wednesday, November 19.
One set of solutions per group of four.  This assignment will count
for two of the parts of the lab/discussion grade for the course.
\bye