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Math 107 Project : Periodic Population Fluctuations
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Due Date: Thursday, November 29, 2001 
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\series bold 
Guidelines:
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 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).
 
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About Group Projects.

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 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.
 
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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.
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This project comes in the form of a memo from a division manager.
 
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North American Ecology Corporation Standard Memo Form
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Date: 10/18/01
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\noindent 
To: Math analysis team
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\noindent 
From: J.
 Datapoint, Manager, data collection team
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\noindent 
Subject: Melanaplus Bivittatus Life Cycle Project
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\backslash 
vspace{.1in}
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As you know, our data collection team was assigned the task of tracking
 the life cycle of the 
\emph on 
Melanaplus bivittatus
\emph default 
 (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 
\family typewriter 
http://www.sdvc.uwyo.edu/grasshopper,
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 but what we want from you is some mathematical analysis.
 Our biologists will take it from there.
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The Model.

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 We know that the population 
\begin_inset Formula \( P(t) \)
\end_inset 

 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 
\begin_inset Formula \( 3\% \)
\end_inset 

 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 
\begin_inset Formula \( P(t) \)
\end_inset 

 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.
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The Problem.

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 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 
\begin_inset Formula \( 3\% \)
\end_inset 

 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 
\begin_inset Formula \( P(t) \)
\end_inset 

.
 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 approximat
ion for 
\begin_inset Formula \( P(t) \)
\end_inset 

 that results from your analysis using the significant harmonics.
 Finally, discuss your results as you present them.
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Some Suggestions.

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 Here are a few suggestions we have for you.
 
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(a) We remind you that if the 
\begin_inset Formula \( k \)
\end_inset 

th harmonic of 
\begin_inset Formula \( P(t) \)
\end_inset 

 is 
\begin_inset Formula \( a_{k}\cos kt+b_{k}\sin kt \)
\end_inset 

, then its 
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amplitude
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 is 
\begin_inset Formula \( A_{k}=\sqrt{a_{k}^{2}+b_{k}^{2}} \)
\end_inset 

 and its 
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energy
\emph default 
 is 
\begin_inset Formula \( A_{k}^{2}. \)
\end_inset 

 The
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 phase
\emph default 
 of the harmonic is the angle 
\begin_inset Formula \( \phi _{k} \)
\end_inset 

 such that 
\begin_inset Formula \( -\pi \leq \phi _{k}\leq \pi  \)
\end_inset 

 and 
\begin_inset Formula \( a_{k}\cos kt+b_{k}\sin kt=A_{k}\sin (kt+\phi _{k}). \)
\end_inset 

 Thus, 
\begin_inset Formula \( \sin \phi _{k}=a_{k}/A_{k} \)
\end_inset 

 and 
\begin_inset Formula \( \cos \phi _{k}=b_{k}/A_{k}. \)
\end_inset 

 The population function's 
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energy and phase spectra
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 are the plots of 
\begin_inset Formula \( A_{k}^{2} \)
\end_inset 

 and 
\begin_inset Formula \( \phi _{k}, \)
\end_inset 

 respectively, against 
\begin_inset Formula \( k. \)
\end_inset 

 Once you know the phase angle, it's easy to find the peaks of the 
\begin_inset Formula \( k \)
\end_inset 

-th harmonic.
 The energy of the 
\begin_inset Formula \( 0 \)
\end_inset 

th harmonic is defined differently as 
\begin_inset Formula \( A_{0}^{2}, \)
\end_inset 

 where 
\begin_inset Formula \( A_{0}=\sqrt{2}a_{0}. \)
\end_inset 

 
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(b) To keep things simple, do your calculations on the interval 
\begin_inset Formula \( [0,2\pi ] \)
\end_inset 

 so that 
\begin_inset Formula \( 2\pi  \)
\end_inset 

 time units represent one year.
 But answer our peaking questions in terms of days, day 1 being January
 1.
 
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(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.
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(c) Plotting will help.
 Certainly, if you omit a significant harmonic, you should get a poor approximat
ion to the data.
 Also, the plot of your approximation to 
\begin_inset Formula \( P(t) \)
\end_inset 

 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.
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Data.

\series default 
 Here are the experimentally observed populations for twelve months, beginning
 with January 1 and ending with December 1:
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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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