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% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
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% Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario only with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
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%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
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% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review, but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
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\begin{document}

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\title{\ifnum\TeacherEdition=1  TEACHER VERSION  \else STUDENT VERSION \fi\\Developing an Invasive Species Model }



\teacher{
    \keywords{Invasive species, fundamental solution, traveling waves, dispersion, migration}%Enter all words that could be considered key, to include words as general as differential equations or as specific as  kinetics. This list is to have no more than  five keywords. These will serve as tags for colleagues to use in searching SIMIODE for location of material. The Editor may suggest/change/add/delete some for more refinement as necessary.}

\tagwords{Partial differential equation model, dispersive migration model, logistic growth, traveling wave solutions, fundamental solutions} %Here you can enter a wider set of words which serve as tags (as do keywords) in the SIMIODE web site. Keep this list to ten words. The Editor may suggest/change/add/delete some for more refinement as necessary.}
}

\markboth{Invasive Species Model}{Invasive Species Model}

\author{Eric Stachura\\
Department of Mathematics\\ 
Kennesaw State University\\
Marietta, GA \teacher{30060} USA\\
\teacher{eric.stachura@kennesaw.edu}
}

\makeStitlePDFLaTex

\teacher{
\begin{abstract}
This scenario takes students through the development of an invasive species partial differential equation model. Basic models are discussed first, which lead students to eventually develop their own model which takes into account dispersion. Students will explore various Mathematica modules along the way to aid in their understanding of each model. Fundamental solutions of differential equations are briefly introduced as well. 
\end{abstract}
}

\teacher{
\vspace*{-.1 true cm}
\listkeywords

\vspace*{-.9 true cm}
\listtags
}

\section*{STATEMENT}
Your goal in this project is to develop a partial differential equation (PDE) model that governs a population (of birds, fish, insects, etc.) that disperses as it migrates. In doing so, you will explore the following:
\begin{itemize}
\item the effects of various nonlinear terms on solutions of PDE models; \\
\vspace*{-.1 true cm}
\item the effects of varying initial population profiles;\\
\vspace*{-.1 true cm}
\item the effects of varying diffusion coefficients; and\\
\vspace*{-.1 true cm}
\item Fundamental solutions of differential equations.\\
\end{itemize}

All Mathematica notebooks are available in the Supporting Documents where the Modeling Scenario resides.

\subsection*{First model: the Logistic Equation}
Consider the logistic equation
\begin{align} \label{logisticequation}
\dfrac{dy}{dt}= ky(A-y), \quad k, A >0, \; y(0)=y_0 \geq 0
\end{align}
Here, $y(t)$ denotes the population size at time $t$. The explicit solution of (\ref{logisticequation}) is given by
\begin{align} \label{logisticSolution}
y(t) = \dfrac{y_0 A e^{Akt}}{y_0 e^{Akt}+A-y_0}
\end{align}
For more details on the model, see e.g., \cite[Section 3.2]{ODEbook}. 
\begin{framed}
\begin{question}
Play around with the Mathematica Module \texttt{Logistic\_Module}. Discuss the effects of changing the growth constant $k$, the carrying capacity $A$, and the initial population $y_0$. Compare your analysis with the solution (\ref{logisticequation}). Does this make sense? 
\end{question}
\end{framed}


\subsection*{Time Evolution of Spatial Profile Curve}
Let's now consider a PDE model by thinking of the population being spread out along a narrow strip of land. Suppose we have an initial population distribution $u_0(x)$, which provides the population at position $x$ along the strip of land. The time evolution of the population is a modification of the logistic equation (\ref{logisticequation})
\begin{align} \label{timeEvolution}
\dfrac{\partial u}{\partial t} =k u\left( A(x) - u\right), \quad k, A(x) >0, \; u(x,0)=u_0(x) \geq 0
\end{align}

\begin{framed}
\begin{question}
How exactly does this generalize (\ref{logisticequation})? How can we recover it as a special case of (\ref{timeEvolution})?
\end{question}
\end{framed}

In (\ref{logisticequation}), the term $A$ was the carrying capacity. The same holds here--except now it depends on the location! In the Mathematica Module \texttt{Time\_Evolution\_of\_Spatial\_Profile\_Module}, we have taken
$$A(x) = 300+100 \cos \left( \dfrac{2\pi x}{50}\right)$$
and the growth constant $k=0.04$. 

\begin{framed}
\begin{question}
Why might the initial population $u_0(x)$ tend towards $A(x)$ as time evolves? Then, see Mathematica module \texttt{Time\_Evolution\_of\_Spatial\_Profile\_Module}.
\end{question}
\end{framed}

\begin{framed}
\begin{question}
Further questions:
\begin{enumerate}
\item Suppose you change the carrying capacity. Would the initial population still evolve towards $A(x)$? Why is this? Verify with some Mathematica demos in the Module \texttt{Time\_Evolution\_of\_Spatial\_Profile\_Module}. PS: For initial populations, we suggest various cosine curves (e.g., \texttt[3+Cos[2*Pi*x/50]) \\
\item What is the effect of taking larger $k$ on the time it takes $u_0(x)$ to approach equilibrium? \\
\item Investigate instead the model
\begin{align}\label{newgrowthmodel}
\dfrac{\partial u}{\partial t} = k(x) u(A-u)
\end{align}
with variable growth parameter. Try various cosine curves for $k(x)$. Explain what you see. %Since we've adapted a logistic model, try $u_0(x)$ being a logistic curve of the form
%$$u_0(x) = \dfrac{A}{1+e^{-k(x)(x-x_0)}}$$
\end{enumerate}

\end{question}
\end{framed}

\subsection*{Migrating Populations}
Next we consider a PDE model where the time rate of change of the population is proportional to its slope:
\begin{align} \label{migratingPopulations}
\dfrac{\partial u}{\partial t} = -c\dfrac{\partial u}{\partial x}, \quad u(x,0)=u_0(x)
\end{align}
This model governs migratory population $u(x,t)$ with a parameter $c$ that controls the speed of the migration. 


\begin{framed}
\begin{question}
Explore the Mathematica Module \texttt{Migrating\_Populations\_Module}. What happens as you increase the speed $c$? What happens when you change the initial population distribution? \\ We suggest trying the following initial population distributions: Probability density function with various means and standard deviations (e.g., $25,6$ respectively):  \\ \texttt{PDF[NormalDistribution[25,6],x]}. \\ 
Note: If $p(x)$ is a density function for some characteristic of a population, then
$$\int_a^b p(x)dx = \text{fraction of population for which } a \leq x \leq b$$
If $p(x)$ is a \textbf{probability} density function, then
$$\int_a^b p(x)dx = \text{probability that }a\leq x \leq b$$
The probability density of a normal distribution is
$$p(x,\mu,\sigma) = \dfrac{1}{2\pi \sigma^2} e^{-\tfrac{(x-\mu)^2}{2\sigma^2}}$$
where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation. Why might this choice of initial population be reasonable?
\end{question}
\end{framed}


We can use Mathematica to solve (\ref{migratingPopulations}) explicitly. Enter the following code into Mathematica:

\noindent \texttt{Module[\{u\}, pdeMigration=D[u[x,t],t]==-cD[u[x,t],x]; \\
uMigrationSoln[c\_,u0\_][x\_,t\_]:= u[x,t] /. \\
First[DSolve[\{pdeMigration, u[x,0]==u0[x]\}, u[x,t], \{x,t\}]]; \\
Row[\{"u[x,t] = ", uMigrationSoln[c,u0][x,t]\}]]
}

\begin{framed}
\begin{question}
Show that, for any $u_0(t)$, the function $u_0(x-ct)$ satisfies (\ref{migratingPopulations}). 
\end{question}
\end{framed}

A solution of the form $u(x,t) = u(x-ct)$ is called a \emph{traveling wave}. 

\begin{framed}
\begin{question}
Explore the Mathematica module \texttt{Traveling\_Wave\_Module}. What can you see? How does the sign of $c$ play a role?
\end{question}
\end{framed}

\subsection*{Seasonal Migration}
To model the periodic nature of certain animals, we can translate the initial population distribution to $u(x,t) = u_0(x-A\sin(ct))$. Then this  $u$ satisfies
\begin{align} \label{seasonalMigration}
\dfrac{\partial u}{\partial t} =-Ac\cos(ct)\dfrac{\partial u}{\partial x}, \quad u(x,0)=u_0(x) \geq 0
\end{align}

\begin{framed}
\begin{question}
Explore the Mathematica module \texttt{Seasonal\_Migration\_Module}. What role does $A$ play in the graph of the solution?
\end{question}
\end{framed}

\subsection*{Dispersion}
If there is overcrowding in one area, a population may disperse over a wider area. One model for this phenomenon is to say the time evolution of the population is proportional to its concavity. In this case the model becomes
\begin{align} \label{dispersion}
\dfrac{\partial u}{\partial t} = D \dfrac{\partial^2u}{\partial x^2}, \quad D>0
\end{align}
where $D$ is called the dispersion coefficient. 


\begin{framed}
\begin{question}
Verify that the fundamental solution of (\ref{dispersion})
\begin{align} \label{fundamentalsolution}
u(x,t) = \dfrac{1}{\sqrt{4\pi D t}} e^{-\tfrac{x^2}{4Dt}}
\end{align}
actually solves (\ref{dispersion}). More details on fundamental solutions are provided below. 
\end{question}
\end{framed}

\begin{framed}
\begin{question}
Now, in the Mathematica module \texttt{Dispersion\_Module}, explore the effects of the dispersion coefficient on the solution. What happens when $D$ becomes larger?
\end{question}
\end{framed}
Why is (\ref{fundamentalsolution}) called a fundamental solution of (\ref{dispersion})? Generally, a \textbf{fundamental solution} of a PDE is a solution with right hand side given by the Dirac Delta function $\delta(x)$ (see \cite[Section 7.9]{ODEbook}). So (\ref{fundamentalsolution}) actually solves the equation
$$\dfrac{\partial u}{\partial t} -D \dfrac{\partial^2u}{\partial x^2} = \delta(x)$$
This is beyond the scope of this course, but if you're interested, take a course in PDEs! 

Actually, one can also discuss the fundamental solution of an ODE as well! Indeed, given a differential operator $L$ (say, $L = \dfrac{d^2}{dx^2}$), the fundamental solution for $L$ is the solution $u(x)$ of $L u(x) = \delta (x)$. It is known that
$$\dfrac{d}{dx} H(x) = \delta(x)$$
where $H(x)$ is the so-called Heaviside function (see \cite[Section 6.3]{BoyceBook}) defined by
$$H(x) = \dfrac{d}{dx} \text{max}\left\{ x, 0 \right\}, \quad x\neq 0\, .$$

Thus, to find a fundamental solution of $L = \dfrac{d^2}{dx^2}$, it is enough to solve
$$\dfrac{du}{dx} =H(x) + C, \quad C \; \text{constant}\, .$$
The solution of this equation can be found to be 
$$u(x) =C_0 |x|$$ 
for some constant $C_0$.  See Figure \ref{fig:heavy}.

\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.5]{Heavisidepic}
\caption{\label{fig:heavy}The Heaviside function $H(x)$.}
\end{center}
\vspace*{-1 true cm}
\end{figure}

\subsection*{Your model}
\begin{framed}
\begin{question}
Develop your own model for a population $u(x,t)$ that disperses as it migrates. How many parameters do you have? Can you formulate a fundamental solution of your equation, based on the ideas of the previous section?
\end{question}
\end{framed}



%This is the STATEMENT of the scenario for the student, with pointers to data sets, supplementary files,  videos, etc. for student consumption. It should have clear statements, instructions, hints, necessary materials and possible reference to original source material, but not to material which ``gives away'' a solution. For example, if there is historical material cited then that should be given in a set of references \cite{Hutchinson1978}.

%The convention in naming scenarios - x-y-NoSpacesInName-StudentVersion or x-y-NoSpacesInName-TeacherVersion - is found in Table~\ref{table:conv}. The x-y designators will be added by the Editor upon acceptance.
%
%\begin{table}[h!]
%\begin{center}
%\begin{tabular}{|l|l|p{7 true cm}|}\hline
%\multicolumn{1}{|c|}{Audience}&\multicolumn{1}{|c|}{Name} & \multicolumn{1}{|c|}{Comment}\\ \hline
%Student & x-y-NameScenario-StudentVersion & x-y- stands for topic or unit (x) and number in that topic or unit (y). NameScenario is a short merged with no spaces two to four word descriptor extracted from the larger title.\\ \hline
%Teacher & x-y-NameScenario-TeacherVersion & x-y- stands for topic or unit (x) and Number in that topic or unit (y).\\ \hline
%Teacher & x-y-NameScenario-Mma-TeacherVersion & Here Mma is a three or four letter abbreviation, in this case for Mathematica file, as an example of supplementary material. Others would include Map for Maple, MatL for MatLab, EXC for EXCEL, etc.\\ \hline
%Student & x-y-NameScenario-Data-StudentVersion & Here Data means this is a data file, in this case for Student, otherwise for Teacher\\ \hline
%\end{tabular}
%\parbox{10.7 true cm}{\caption{\label{table:conv}This is the convention for naming scenarios. The topic or unit (x) will be assigned by the Editor as will the number (y) in that topic or unit.}}
%\end{center}
%\end{table}
%
%
%\noindent x refers  to the Chapter (x = 1 for One, x = 1a, for OneA, x = 2 for Two, x = 3 for Three, etc.) and y refers to the entry or sequence number (just a counter, no meaning) in the more or less  Table of Contents of a traditional differential equation text as shown below.  
%
%\begin{itemize}
%\item [] Modeling One  -  First Order Differential Equations 
%\item [] Modeling OneA -  Numerically Solving Differential Equations 
%\item [] Modeling Two - Limited Growth Models 
%\item [] Modeling Three - Second Order Homogeneous Differential Equation Models 
%\item [] Modeling Four - Second Order NonHomogeneous Differential Equation Models 
%\item [] Modeling Five - Linear Systems of Differential Equation Models 
%\item [] Modeling Six - NonLinear Systems of Differential Equation Models 
%\item [] Modeling Seven - Changing the Venue for Solution Success - Laplace Transforms 
%\item [] Modeling Eight - Representing Natural Phenomena with Sines and Cosines 
%\item [] Modeling Nine - Modeling with Differential Equations in Higher Dimensions
%\end{itemize}
%
%
%This STATEMENT section should be the first section in the Teacher Version  and then followed by the COMMENTS section in which the teacher shares with colleagues issues and possible solution strategies surrounding this modeling scenario.


%\section*{Submission Version}
%
%For submission to the manuscript management system, \url{https://simiode.expressacademic.org} authors should submit a Teacher Version with all author identifying information stripped out. This should be a pdf version of the material. 
%
%Authors may also submit supporting documents such as pdf's of code files, e.g., MatLab, Mathematica, Maple, SAGE, etc. and spreadsheets and data files.  
%
%Video files should be uploaded to YouTube and the author should refer to them by URL in material submitted for consideration. Upon acceptance, the YouTube video (perhaps modified per the referee/edit process) will be uploaded to the SIMIODE  channel, \url{https://www.youtube.com/channel/UC14lC-tyBGkDPmUnKMV3f3w}.
% 
%
%Below is a set of References which appears in the Student Version. Note, there is a second set of References one can use for the Teacher Version.

\begin{thebibliography}{98}




\bibitem{ODEbook}
Nagle, R.~Kent ,  Edward B.~Saff and Arthur D.~Snider. 2018.  \textit{Fundamentals of Differential Equations, 7$^{\rm th}$ Edition}. Boston MA: Addison-Wesley.
%%
\bibitem{BoyceBook}
Boyce, W.~E.~and R.~C.~DiPrima. 2009. \textit{Elementary Differential Equations and Boundary Value Problems, 9$^{\rm th}$ Edition}.  New York NY: John Wiley \& Sons. 

\end{thebibliography}

\teacher{
\section*{COMMENTS}

A complete solution is found in the associated file ``TeacherSolution." One Mathematica notebook (\texttt{Population\_dynamics\_teacher\_version}) with all of the student modules is included, with all of the associated code visible and largely editable. This is not the case for the students' version of the modules. The student modules are made in such a way so that minimal Mathematica knowledge is required. Most notebooks can just be evaluated, or the student needs only to click on a button that takes them through the various questions. The point here is for the students to focus on the mathematical aspects of the model, even if they have not had much exposure to Mathematica. Many of the general ideas from this scenario are taken from \cite{2012Kerckhove} and have been adapted for use in a differential equations course. Another interesting and useful resource is \cite{1994Holmes}, which is probably beyond the scope of a differential equations course, but contains information that may be helpful to students in this project. The last resource we suggest \cite{2014Roos} is included as a PDF and contains a lot of nice information about modeling population dynamics. 

It is expected that this scenario would be most effective at the end of a differential equations course, after the basics of partial differential equations are introduced. It would perhaps be beneficial to spend a few minutes in class discussing the basics of Mathematica if students have never seen or used it before. But as mentioned above, very little knowledge is required, and the students can be up to speed in just a matter of minutes. 

Given the initial data in the Migrating Populations section which is prescribed as a probability density function, the instructor may elect to delve further into this subject, as this is also not frequently encountered in a differential equations course. We have provided a number of (very) basic exercises on this subject; certainly there are many more possible problems. It is up to instructors to determine how far into probability they wish to go. Additionally, the material on fundamental solutions can be used at the discretion of the instructor; however, we think it worth mentioning even if in passing due to its importance in the analysis of differential equations. A module that shows the fundamental solutions of the equations in this project is also included. The parameters $D$ and $c$ are allowed to change so one can observe the effects of wavespeed and the size of diffusion coefficients. 

I suggest implementing this problem as a partner writing project, where students submit a written 2-3 page report. A possible rubric can be found at the end of this document. 

%This is a set of COMMENTS about the scenario, for the teacher only. It might include a ``solution" or a  number of solution strategies. However, the main narrative should be about how to use the scenario in teaching, e.g., what are prerequisites, how long might one devote to it, what resources for students could be expected to be used, what  issues which cause students difficult or ease should be expected to come up, discussion of technology use, pointers to other files, etc.  This set of COMMENTS might have its own  pointers to data sets, supplementary files,  videos, etc. for student and teacher  use, the material for students could be something the teacher might use with discretion. There could be an extended bibliography  for additional teacher resources, analyses,  data, etc.



\begin{thebibliography}{99}


\bibitem{2012Kerckhove} Kerckhove, M. 2012. From Population Dynamics to Partial Differential Equations. {\it The Mathematica Journal\/}.  14. 18 pp.

\bibitem{1994Holmes} Holmes, E.~E., M.~A.~Lewis,  J.~.E.~Banks,  and R.~R.~Veit. 1994. Partial Differential Equations in Ecology: Spatial  Interactions and Population Dynamics. {\it Ecology\/}.  75 (1): 17--29. 

\bibitem{2014Roos} Roos, A. 2012. {\it Modeling Population Dynamics}. Course notes. 222 pp. {\tt https://staff.fnwi.uva.nl/a.m.deroos/downloads/pdf\_readers/syllabus.pdf}. Accessed 7 August 2019.

%\bibitem{Gause1934} Gause, G.~F. 1971. {\it The Struggle for Existence}. New York:  Dover Publications, Inc. First published in 1934 by The Williams \& Wilkins Company and available completely on the world wide web at \url{http://www.ggause.com/Contgau.htm}. Accessed  31 March 2008
%
%
%\bibitem{Hutchinson1978a} Hutchinson, G. Evelyn. 1978. {\it Introduction to Population Ecology\/}. New Haven CT:  Yale University Press.

\end{thebibliography}
}

\newpage

\centerline{\textbf{POSSIBLE WRITING PROJECT RUBRIC} }

\bigskip

Each of the following elements are rated along a range of $0-4$, depending on the completeness and quality of the work on that element.

\begin{enumerate}
\item \textbf{Organization:} You must have the following organizational elements in the final paper, each with a clear header.
\begin{itemize}
%\item Cover Sheet (title of your project, your name, your partners' names (if any) and the date)
\item Problem Introduction Section
\item Model Section
\item Mathematical Analysis Section
\item Prediction and Evaluation Section
%\item Reflection Section
\item Credits: Give credit to people that assisted you or references you used (if any)
\end{itemize}

\item \textbf{Layout and Formatting:} This is a report, not an assignment, so please write this paper in narrative (paragraph) format, except putting equations and formulas on separate lines. Here are further guidelines:
\begin{itemize}
\item Narratives in paragraphs. Typed 11 or 12 point font. 
\item Equations and calculations, etc. are on a separate line and easy to read. 
\item Understandability of graphs. (Suggestion: You may take a picture of hand drawn graphs and of graphs on calculators or other software.)
\item Clear identification of graphs in the narratives through labeling of the graphs (e.g., ``see Figure 1''). 
\item Graphs are used effectively in the narratives and explanations. In other words, you refer to the graphs and use the graphs in your discussions.
\end{itemize}

\item \textbf{Introduction:} Must include: 
\begin{itemize}
\item Introduction of the problem. (Summarize the problem in your own words.) 
\item 	Discuss how this problem may be similar and different from other applied problems you have encountered. 
\end{itemize}
 
\item \textbf{Model:}  Must include: 
\begin{itemize}
\item Clear mathematical statement of the model.
\item 	Definitions of variables. 
\item 	Explanation of the different parts of the model and how they relate to each other.
\item 	Justification of the model and its parts.
\end{itemize}

\newpage

\item \textbf{Mathematical Analysis:} Must include:
\begin{itemize}
\item Qualitative Analysis: Discuss what mathematical information the qualitative analysis gives about the solution. 
\item 	Analytical Analysis: Solve and discuss what mathematical information the analytical analysis gives about the solution.
\item 	Numerical analysis: Solve and discuss what mathematical information the numerical analysis gives about the solution.
\item 	Explain: Each analysis should include an explanation of the steps taken analyzing the model. (But note you don't have to state or show every single basic algebraic step.)
\item 	Model verification: Are the results of the model consistent with any data? How? If there is no data, what else can you do to verify if the model results are accurate? 
\item 	Compare/Contrast: Compare and contrast the three techniques (graphical, analytical, numerical). What is similar and different? Did they agree with each other? For this problem, in what ways were they useful or not? 
\end{itemize}

\item \textbf{Prediction, Interpretation, and Evaluation:}  Must include: 
\begin{itemize}
\item Prediction: Make predictions based on the model. See the project directions. 
\item 	Interpretations: Be sure to state what your solution and predictions mean or say in terms of the applied problem. This should be done throughout the write-up as necessary. 
\item 	Evaluation: Critique the interpretations and predictions. Do the interpretations of the solution and predictions make sense for that applied problem? Why or why not? 
\item 	Evaluation: What might your solutions and predictions imply and why is it important for the people, animals, or objects in the applied problem? Does it imply any actions that should be taken in the future, i.e. do you have any recommendations based on your conclusion? 
\end{itemize}

%\item \textbf{Reflection:}  Each person on the project must include separately a reflection with these elements:
%\begin{itemize}
%\item What was the most important thing you learned mathematically or about solving problems and did you learn anything mathematically new?
%\item 	What was the most important thing you learned personally and did you learn anything new about yourself? (This could be about how you go about your work, how you communicate, your relationship with others, what you enjoy doing, your attitude towards mathematics, etc.)
%\item 	Did you find the project difficult? In what ways? 
%\item 	Did you find any connections to other disciplines or to courses you have had? 
%\item 	Did you encounter any technical or non-technical problems? 
%\item 	Did you encounter or think of other ways to attack the problem? 
%\item 	Did other mathematical or physical questions arise that would be interesting to investigate? 
%\end{itemize}
\end{enumerate}




\end{document}
