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\section*{Calculus II Project 3: Periodic Population Fluctuations}

\noun{Due Date: Some Date }

\textbf{Guidelines:} This project is a group project which is based
on material found in Section 9.5 (pages 458-468), Fourier Series.
Since this section will not be covered in lecture, you should first
read this material before attempting the problems. Remember that part
of your grade will be based on the quality of your written work. The
paper you turn in should be a mix of equations, formulas and prose.
Graphs may be copied from your calculator, but should be clearly labeled.
Use complete sentences, good grammar, correct spelling and correct
punctuation. You should write your answers in such a way that your
report can be read and understood by anyone who knows the material
for this course. This is your target audience, not the instructor.
Finally, neatness counts, so the project should be neatly typed or
written on good paper (not torn from a notebook). 

\textbf{About Group Projects.} To get everyone involved and the group
functioning smoothly, it is a good idea to meet as early as possible
to arrange meeting times, etc. It might be helpful to bear in mind
that there are at least four roles to be played by various participants
at various times: the chair, reporter, scheduler and scribe. The role
of the chair is to try to get everyone involved and make sure everyone
understands the ideas developed by the group. The reporter jots down
the ideas of the group as they are discussed. The scheduler finds
times and places where everyone in the group can meet, and finally,
the scribe writes up the final report for the group. These jobs can
be rotated on a per meeting basis if the group wishes. However, everyone
must proofread the final draft. 

When the project is turned in, students may be asked to evaluate the
level of participation by other group members by way of a project
participation report to be filled out by each member individually
and turned in to the recitation instructor.

This project comes in the form of a memo from a division manager. 


\section*{North American Ecology Corporation Standard Memo Form}

Date: 10/18/01

\noindent To: Math analysis team

\noindent From: J. Datapoint, Manager, data collection team

\noindent Subject: Melanaplus Bivittatus Life Cycle Project

\vspace{.1in}

As you know, our data collection team was assigned the task of tracking
the life cycle of the \emph{Melanaplus bivittatus} (two-striped grasshopper
in common parlance). We found an uncultivated location in Texas where
the population was fairly stable from year to year, and that is where
we collected population data over a period of three years. The ultimate
goal of the Life Cycles Project is to find ecologically sound ways
to control grasshopper populations and minimize the use of pesticides.
In order to do so, we need to understand how their population changes
and what factors cause these changes. Possible factors include the
weather, predators, parasites, infections and, of course, food supply.
Interestingly, most of these factors tend to be seasonal, and will
spike periodically during the year, possibly more than once. If you're
curious about the biology, you might check out web sites such as \texttt{http://www.sdvc.uwyo.edu/grasshopper,}
but what we want from you is some mathematical analysis. Our biologists
will take it from there.

\textbf{The Model.} We know that the population $P(t)$ as a function
of time is approximately periodic. We are passing on to you the average
monthly population count (in thousands per acre), starting with January
and ending with December. These counts were taken on twelve equally
spaced days over the year, roughly the first day of each month. Our
statisticians assure us that these figures will likely have no more
than $3\%$ error. We want to use Fourier analysis to resolve the
population function into harmonics and determine the energy of each
harmonic. This will give us clues as to what kind of periodic factors
to look for and which are the most important. We think that the actual
population function $P(t)$ should only have a few harmonics. In fact,
there will be none beyond than the first six harmonics since we are
sure that there are no periodic factors with a period of less than
two months.

\textbf{The Problem.} Here is the information we need from you: we
want to know what you think are the significant harmonics. Remember
that the data has as much as $3\%$ noise, so relatively small harmonics
are probably just noise and should be discarded, that is, counted
as zero. We would like some explanation as to how you selected the
significant harmonics. We would also like to see a plot of the resulting
energy spectrum and phase spectrum of $P(t)$. We want to know the
first day in the year that each harmonic peaks (reaches a maximum).
We also want to see a plot of the data points and the Fourier series
approximation for $P(t)$ that results from your analysis using the
significant harmonics. Finally, discuss your results as you present
them.

\textbf{Some Suggestions.} Here are a few suggestions we have for
you. 

(a) We remind you that if the $k$th harmonic of $P(t)$ is $a_{k}\cos kt+b_{k}\sin kt$,
then its \emph{amplitude} is $A_{k}=\sqrt{a_{k}^{2}+b_{k}^{2}}$ and
its \emph{energy} is $A_{k}^{2}.$ The \emph{phase} of the harmonic
is the angle $\phi_{k}$ such that $-\pi\leq\phi_{k}\leq\pi$ and
$a_{k}\cos kt+b_{k}\sin kt=A_{k}\sin(kt+\phi_{k}).$ Thus, $\sin\phi_{k}=a_{k}/A_{k}$
and $\cos\phi_{k}=b_{k}/A_{k}.$ The population function's \emph{energy
and phase spectra} are the plots of $A_{k}^{2}$ and $\phi_{k},$
respectively, against $k.$ Once you know the phase angle, it's easy
to find the peaks of the $k$-th harmonic. The energy of the $0$th
harmonic is defined differently as $A_{0}^{2},$ where $A_{0}=\sqrt{2}a_{0}.$ 

(b) To keep things simple, do your calculations on the interval $[0,2\pi]$
so that $2\pi$ time units represent one year. But answer our peaking
questions in terms of days, day 1 being January 1. 

(b) You will need to approximate the Fourier coefficients using the
integral definition and some numerical integration method. In the
case of a periodic smooth function, the simple left Riemann rule is
highly effective and equivalent to the trapezoidal method. Of course,
answers are only approximate, so there are two sources of error: the
error of our measurements and the error of numerical integration.
Since there are only 12 data points, LEFT(12) is the best choice.

(c) Plotting will help. Certainly, if you omit a significant harmonic,
you should get a poor approximation to the data. Also, the plot of
your approximation to $P(t)$ should give a reasonably good approximation
to the data as should the Fourier series of the data (including insignificant
harmonics). TI-8x users can use 2nd STAT PLOT for displaying discrete
data plots.

\textbf{Data.} Here are the experimentally observed populations for
twelve months, beginning with January 1 and ending with December 1:

0.137, 0.146, 0.220, 0.703, 1.435, 1.508, 1.857, 2.879, 3.154, 3.058,
2.670, 1.192
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