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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Modeling Populations with \\ Death and Immigration Using {\it m}\,{\&}{\it m}s}





\markboth{Modeling Populations with Death and Immigration Using {\it m}\,{\&}{\it m}s}{Modeling Populations with Death and Immigration Using {\it m}\,{\&}{\it m}s}




\author{Brian Winkel, Director SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex



\begin{abstract}
We describe a classroom activity in which students use {\it m}\,{\&}{\it m} candies to simulate changes in a population due to death and immigration. Students collect data, build a mathematical model --- usually a linear, first-order, difference or differential equation,  estimate parameters, and compare their model prediction with their actual data.
\end{abstract}


\teacher{\section*{INTRODUCTION}

The first section of this document, the Scenario Description, is the material provided to students. It describes how to conduct the population model simulations and guides students through model construction. The second section, Notes for Teachers, offers some remarks based on the author's experiences with this material.}


\section*{SCENARIO DESCRIPTION}


For this experiment you will need a regular size bag of {\it m}\,{\&}{\it m}  candies, not peanut, just regular.  There are usually about 55 pieces in each bag. The candies are of different colors, but  (almost) every  piece  has an 
``m" pressed on one side and not the other, and so we can distinguish between the two sides of each piece of candy.  Let us conduct two experiments (life and death experiments) on the {\it m}\,{\&}{\it m}s. 

In each experiment, we will shake the {\it m}\,{\&}{\it m}s from a paper cup onto a paper plate. Those that land {\it m} side up die and are set aside; for the others, there is life! They will go back into the cup for the next iteration.  

\subsection*{A population with death but no birth or immigration}

For  this simulation in which {\it m}\,{\&}{\it m}s either live or die, open the bag of candy  put up to 50 of the 55 or so pieces (using 20-25 {\it m}\,{\&}{\it m}s will make counting faster and get across the  concepts) into a small container; a paper cup will do.   
\begin{itemize}
\item [a)] Describe what you think will happen in each toss.
\item [b)] Offer  some assumptions in support of your description in (a) as well as some which might not have anything to do with what you expect to happen. Often,  assumptions are run into each other, so break the assumptions down to their simplest form. Do not use  words like ``and."  
\end{itemize}



\vspace{.2 true cm} {\bf The simulation (death only)} 
Select a starting number of {\it m}\,{\&}{\it m}s, say 20, and place that number of {\it m}\,{\&}{\it m}s in a container and also write that number in Table~\ref{table:BD} as the number of {\it m}\,{\&}{\it m}s at start of Iteration 0. 

\begin{itemize}
\item [c)] In Table~\ref{table:BD} record what happened and compare this information with what you thought  and described would happen in (a) above.
\end{itemize}

\begin{table}[h]
\large
\begin{center}
\begin{tabular}{|c|c|} \hline\hline
\multicolumn{2}{|c|}{Modeling Death of {\it m}\,{\&}{\it m}s} \\ \hline\hline
Iteration & \# {\it m}\,{\&}{\it m}s at start of iteration\\ \hline\hline
0 & \\ \hline
1 & \\ \hline
2 & \\ \hline
3 & \\ \hline
4 & \\ \hline
5 & \\ \hline
6 & \\ \hline
7 & \\ \hline
8 & \\ \hline
9 & \\ \hline
10 & \\ \hline
\end{tabular}
\caption{\label{table:BD} Modeling death of {\it m}\,{\&}{\it m}s.}
\end{center}
\end{table}

Now, gently shake the {\it m}\,{\&}{\it m}s out onto the desk (you might want to use a paper plate to catch the {\it m}\,{\&}{\it m}s and keep them clean as well). Any m\&m with an {\it m} showing on top has died. Remove them and save them to the side for later. The remaining {\it m}\,{\&}{\it m}s have survived. Count the survivors and write down that number of {\it m}\,{\&}{\it m}s who survive to the next iteration in Table~\ref{table:BD} to start the next generation and place the surviving {\it m}\,{\&}{\it m}s back into the container to begin the next iteration. Repeat ten times and record your results in Table~\ref{table:BD}.




\begin{itemize}
\item [d)] Indicate which of your assumptions from (b) were reasonable and played a relevant role in the experiment.
\item [e)] Make a predication as to what will happen in the future; long term.
\end{itemize}


\subsection*{Building a mathematical model of the population (death only)}

Let us build a mathematical model of this situation. Suppose  the number of {\it m}\,{\&}{\it m}s alive at the start of iteration $n$ is $a(n)$, where $n$ is a whole number between 0 and 10.  What values would $n$ take?  What is $a(0)$? What was  $a(1)$ in your simulation?  

Based on your observations and your assumptions, produce a reasonable formula for $a(n)$, with  $n = 0, 1, 2, \ldots $. 
\begin{itemize}
\item [f)] In what ways is your function model for $a(n)$   unreasonable and in what ways is it  reasonable?
\item [g)] How will you measure  whetcher your model of this situation is successful? (What does it mean for a model to be successful?) 
\item [h)] Compare your model with others in the class.
\item [i)] Just how good is your model at predicting the results of your experiment. Defend your rationale.
\end{itemize}



\section*{A population with death and  immigration}

Each of us will conduct this simulation in which {\it m}\,{\&}{\it m}s either live or die, but this time we will permit immigration to recover  the damage to the m\&m population from death.    
 

\vspace*{.2 true cm} {\bf The Experiment:} Gently shake the {\it m}\,{\&}{\it m}s  out onto the desk. You might want to use a paper plate to catch the {\it m}\,{\&}{\it m}s and keep them clean as well. Then determine for each {\it m}\,{\&}{\it m} if it lives or dies.
Remember, if the {\it m} shows on top this {\it m}\,{\&}{\it m} dies, otherwise there is life for this {\it m}\,{\&}{\it m}. Upon death you should remove the m\&m from the population (set these aside as we will need these for another experiment).

And now for this very important change: 

\vspace*{.4 true cm}

\begin{center}
\parbox{11 true cm}{{\bf VERY IMPORTANT}:  After returning the surviving {\it m}\,{\&}{\it m}s to the cup, add 10 immigrants from your ``death pile''. Repeat this death and immigration step for ten iterations and record your results in Table~\ref{table:BDI}.}
\end{center}
 \vspace*{.1 true cm}


\begin{itemize}
\item [j)] State your assumptions about the physical activity.
\item [k)] Offer up a description of what should  happen.

\item [l)] In Table~\ref{table:BDI} record what happened and compare this with what you thought would happen.

\item [m)] Compare your description/prediction with what actually happened.

\item [n)] Indicate which of your assumptions were reasonable and played a role in the experiment.
\end{itemize}

\begin{table}[!h]
\large
\begin{center}
\begin{tabular}{|c|c|} \hline\hline
\multicolumn{2}{|c|}{Modeling Death of {\it m}\,{\&}{\it m}s} \\ \hline\hline
Iteration & \# {\it m}\,{\&}{\it m}s at start of iteration\\ \hline\hline
0 & \\ \hline
1 & \\ \hline
2 & \\ \hline
3 & \\ \hline
4 & \\ \hline
5 & \\ \hline
6 & \\ \hline
7 & \\ \hline
8 & \\ \hline
9 & \\ \hline
10 & \\ \hline
\end{tabular}
\caption{\label{table:BDI} Modeling death and immigration of {\it m}\,{\&}{\it m}s.}
\end{center}

\end{table}


\subsection*{Building a mathematical model of the population (death and immigration)}

Let us build a mathematical model of this situation. Suppose the number of {\it m}\,{\&}{\it m}s alive at the start of iteration $n$ is $b(n)$, where $n$ is a whole number between 0 and 10.  What values would $n$ take?  What is $b(0)$?  What was $b(1)$ in your simulation?  

Based on your  observations and your assumptions, produce a reasonable formula for $b(n)$, with  $n = 0, 1, 2, \ldots $. 

\begin{itemize}
\item [o)] In what ways is your function model for $b(n)$ reasonable? In what ways is it unreasonable? 
\item [p)] How will you measure  whether your model of this situation is successful? (What does it mean for a model to be successful?) 
\item [q)] Compare your model with others in the class.
\item [r)] Just how good is your model at predicting the results of your experiment. Defend your rationale.
\item [s)]  Compare the two processes and the results of the two simulations: (1) death and (2) death with immigration. What was different about these models? Explain.
\end{itemize}




\vspace*{-.5 true cm}
\section*{Modeling change}

Finding a closed-form solution  for the death and immigration model, $b(n)$, is difficult. (In a closed-form solution we can substitute the independent variable, $n$,  into a forumla that depends only on $n$ and determine $b(n)$.)  Instead, let us think of how we get from generation to generation. At each iteration we lose about one-half of the {\it m}\,{\&}{\it m}s (why?), but   then  10 {\it m}\,{\&}{\it m}s immigrate into the population  as immigrants. What does  the $(n+1)^{\rm th}$ generation, $b(n+1)$, look like in terms of the $n^{\rm th}$ generation, $b(n)$? Try to predict $b(n+1)$ in terms of $b(n)$ and immigration with a formula:


\vspace*{-.3 true cm}
$$b(n+1) = \rule{7 true cm}{.1 mm} \,.$$

\vspace*{-.3 true cm}
\noindent What was easy and what was hard in deciding what to put on the right hand side of the expression above? It is worth pausing here to savor the experiences of discovery and frustration in this modeling activity and to learn from them. Reflect for a moment on what you have accomplished and what it took to build a mathematical model.

Now that you have spent time reflecting on your model let us ask ourselves: is it the only reasonable possibility?

Here are several mathemaical models that {\it do not\/} work. Explain what does not work in each one; is not good about each one. Put each of these ill-conceived models into words and test them against your  understanding of the physical process of m\&m tossing and migration.

\vspace*{-.7 true cm}

\begin{eqnarray*}
b(n+1) &=& 10 \cdot b(n) + .5\\
b(n+1) &=& 0.5 \cdot 10 \cdot b(n) \\
b(n+1) &=& 0.5 \cdot b(n+1) + 10\\
b(n+1) &=& -0.5 \cdot b(n) + 10\\
b(n+1) &=& b(n+1)^n + 10\\
b(n+1) &=& (b(n+1)+10)^n + 10\\
\end{eqnarray*}



Let us say out loud to ourselves what we want the function to do:  ``The value of $(n+1)^{\rm st}$ iteration, $b(n+1)$,  is  one-half that of $b(n)$ (due to death)   plus 10 (due to immigration)."   Does that sentence translate directly into mathematics for you? Does  (\ref{eq:update}) look like a mathematical translation of that sentence? 

\vspace*{-.2 true cm}
\begin{equation}\label{eq:update}
b(n+1) = 0.5 \cdot b(n) + 10 \, . 
\end{equation}

\noindent Another way to describe the population is to model the {\it change\/} at each generation rather than the new population itself. We can do this by subtracting $b(n)$ from each side of (\ref{eq:update}) to obtain the equation:

\begin{equation}\label{eq:dif}
b(n+1) - b(n) = 0.5 \cdot b(n) - b(n) + 10  = -0.5 b(n) + 10 \, . 
\end{equation}

\noindent Equation (\ref{eq:dif}) is called a {\it difference equation\/}.  Now translate (\ref{eq:dif}) into a sentence.



Often we  use the symbols $\Delta b(n)$, read ``change in $b(n)$'', for  $b(n+1) - b(n)$.  Thus (\ref{eq:dif}) could be written as:

\vspace*{-.3 true cm}
\begin{equation}\label{eq:dif2}
\Delta b(n) = -0.5 b(n) + 10 \, . 
\end{equation}

In all three equations, (\ref{eq:update}), (\ref{eq:dif}), and (\ref{eq:dif2}), we need to tell our model where to start. In these cases we might have  $b(0) = 50$ or, say,  $b(0) = 25$, the number of {\it m}\,{\&}{\it m}s we had in the cup at the beginning of the experiment.  We can consider $b(0) = 50$ to be part of the model, and view (\ref{eq:update}) and $b(0) = 50$ as a pair of equations called,  a {\it difference equation with initial condition}. We can do the same for (\ref{eq:dif}) and (\ref{eq:dif2}). The expression in $b(n)$ on the right hand side of (\ref{eq:dif2}) is linear, suggesting that we call such a difference equation {\it linear\/}. So, our death and immigration activity is  a {\it first-order, linear difference equation with initial condition\/}. 

\subsection*{Solving our first-order, linear difference equation with initial condition}

Before we try to solve the difference equation, let us think about the long-term behavior of our model.  One way to this is to use a spreadsheet program, such as {\it Excel\/}, to iterate (\ref{eq:update}). In Figure~\ref{fig:1-1-ModelingOneFirstIteration}  what can you see  about the long term behavior of the model? Does it  match what we saw in our experiment?


\begin{figure}[h]
\centering

\includegraphics[width=7cm]{1-1-ModelingOneFirstIteration.eps}


\vspace* {-3.6 true cm}
\caption{\label{fig:1-1-ModelingOneFirstIteration} Using {\it Excel\/} spreadsheet to iterate (\ref{eq:update}).}
\end{figure}
The powerful {\tt RSolve} command in {\it Mathematica\/}

\centerline{\tt RSolve[\{b[n + 1] == 1/2 b[n] + 10, b[0] == 50\}, b[n], n]}\, ,

\noindent provides a solution for $b(n)$, namely,
\vspace*{-.5 true cm}

$$b(n) = 20 + 15 \cdot 2^{(1-n)} \, .$$
We note that the long term behavior seen in our Table~\ref{table:RSolve} agrees with the iterations of  (\ref{eq:update}) using  {\it Excel\/} displayed in Figure~\ref{fig:1-1-ModelingOneFirstIteration}.

Another approach is to  conjecture a solution, say of the form $b(n) = k_1 + k_2 2^{(1-n)}$, and then use the initial condition and the first iteration  to determine $k_1$ and $k_2$.
 

\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|} \hline\hline
\multicolumn{2}{|c|}{{\tt RSolve} solution of (\ref{eq:update})} \\ \hline\hline
$n$ & $b(n)$ \\ \hline\hline
 0. & 50.0000 \\ \hline
 1. & 35.0000 \\ \hline
 2. & 27.5000 \\ \hline
 3. & 23.7500 \\ \hline
 4. & 21.8750 \\ \hline
 5. & 20.9375 \\ \hline
 6. & 20.4688 \\ \hline
 7. & 20.2344 \\ \hline
 8. & 20.1172 \\ \hline
 9. & 20.0586 \\ \hline
 10. & 20.0293 \\ \hline
\end{tabular}
\end{center}
\caption{\label{table:RSolve} Using {\it Mathematica\/}'s {\tt RSolve} command to solve  (\ref{eq:update}).}
\end{table}




\section*{Moving to a differential equation model}


Equation  (\ref{eq:dif}) describes the change for discrete steps. From calculus, we know quite a bit about how to model change in continuous functions using derivatives which is repeated below.  



\begin{equation} \label{eq:secondtime}
b(n+1) - b(n)   = -0.5 \cdot b(n) + 10 \, . 
\end{equation}

In the simulation we used step sizes of 1 by going from $n$ to $n+1$. Here 1 could be 1 generation or 1 year; it would certainly make sense to measure the population more frequently. Suppose we measured in step sizes of 0.01. Then the change in $n$,  $\Delta n$,  is 0.01. In this case we would not lose half of our population. Instead, we would lose a fraction of that half of our population, namely $\Delta n \cdot 0.5 = 0.01 \cdot 0.5 = 0.005$, every step. Similarly, our population would increase due to immigration by $\Delta n \cdot 10 = 0.01\cdot 10 = 0.1$, every step.

What assumptions  are we making about the process when we say these things? These adjustments yield a new equation, where again $b(n)$ is the size of the population at time $n$:




 \begin{equation}
b(n+0.01) - b(n)   = -0.005 \cdot b(n) + 0.1 \, . 
\end{equation}
More genereally, for a time step size of $\Delta n$ we get: 
%\setcounter{equation}{3}
 \begin{equation}\label{eq:diff}
b(n+\Delta n) - b(n)   = -(\Delta n) \cdot 0.5 \cdot b(n) + \Delta n \cdot 10 \, . 
\end{equation}

\noindent As in  calculus,  we could measure  our  population changes over smaller and smaller time intervals,  $\Delta n$. Indeed, by dividing both sides of in (\ref{eq:diff})  by $\Delta n$, we obtain

 \begin{equation}\label{eq:quo}
\frac{b(n+\Delta n) - b(n)}{\Delta n}   = -0.5 \cdot b(n) + 10 \, . 
\end{equation}

\noindent Look familiar?


\noindent Letting $\Delta n$ approach 0 on both sides of (\ref{eq:quo}), we get the definition of the derivative of $b(n)$ with respect to $n$, namely, 

 \begin{equation}\label{eq:DiffEq}
\frac{db}{dn} = \lim_{\Delta n \to 0}\frac{b(n+\Delta n) - b(n)}{\Delta n}   = -0.5 \cdot b(n) + 10 \, . 
\end{equation}

\vspace*{.2 true cm}
Equation (\ref{eq:DiffEq})  is called a {\it differential equation\/}. If we also include   an initial condition, in this case, $b(0) = 50$, we have in (\ref{eq:IVP}) what is called an {\it initial value problem (IVP)\/}:


\begin{equation}\label{eq:IVP}
\frac{db}{dn} = \lim_{\Delta n \to 0}\frac{b(n+\Delta n) - b(n)}{\Delta n}   = -0.5 \cdot b(n) + 10 \, , \quad b(0) = 50\,  . 
\end{equation}

\vspace*{.2 true cm}
It is conventional  to use $t$ for time-dependent differential equations. Rewriting our differential equation as an IVP  in variable $t$ we get (\ref{eq:IVPt}):


\begin{equation}\label{eq:IVPt}
b'(t) = \frac{db}{dt} = \lim_{\Delta t \to 0}\frac{b(t+\Delta t) - b(t)}{\Delta t}   = -0.5 \cdot b(t) + 10 \, , \quad b(0) = 50\,  . 
\end{equation}


\noindent Using standard notation the  {\it first-order, linear, differential equation with initial condition\/} is written as (\ref{eq:IVPFt}).

\begin{equation}\label{eq:IVPFt}
b'(t) =   -0.5 \cdot b(t) + 10 \, , \quad b(0) = 50\,  . 
\end{equation}

Now what are we supposed to do with this?  Well, this, in part, is what our effort is about, namely solving differential equations, or coming up with a $b(t)$. We saw how hard it was to ``guess" a solution in the discrete case for $b(n)$ from (\ref{eq:dif2}) and it will be as difficult to guess what $b(t)$ is  from (\ref{eq:IVPFt}).



\subsection*{Closed-form solutions to   IVPs}

Well \ldots  at least for IVP (\ref{eq:IVPFt}).

Even before learning to solve IVP's by hand, we can explore their solutions usign a computer algebra system such as {\it Mathematica\/} in which the command to solve the differential equation is {\tt DSolve}. In this case we would input: 


\vspace*{.15 true cm}

 \centerline{\tt DSolve[\{b'[t] == -1/2 b[t] + 10, b[0] == 50\}, b[t], t]}

\vspace*{.15 true cm}

\noindent which gives


\vspace*{-.15 true cm}
\begin{equation}\label{eq:ODE1}
b(t) = 20 + 30 e^{-t/2} \, .
\end{equation}

\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|} \hline\hline
$n$ or $t$ & $b(n)$ & $b(t)$ \\ \hline\hline
 0 & 50.0000 & 50.0000 \\ \hline
 1 & 35.0000 & 38.1959 \\ \hline
 2 & 27.5000 & 31.0364 \\ \hline
 3 & 23.7500 & 26.6939 \\ \hline
 4 & 21.8750 & 24.0601 \\ \hline
 5 & 20.9375 & 22.4625 \\ \hline
 6 & 20.4688 & 21.4936 \\ \hline
 7 & 20.2344 & 20.9059 \\ \hline
 8 & 20.1172 & 20.5495 \\ \hline
 9 & 20.0586 & 20.3333 \\ \hline
 10 & 20.0293 & 20.2021 \\ \hline
\end{tabular}

\parbox{12.4 true cm}{\caption{\label{table:Compare}Comparing the results of our discrete difference equation model $b(n)$ for $n = 0, 1, 2, \ldots, 10$ given in (\ref{eq:dif}) with the results of our continuous differential equation model $b(t)$, $t \ge 0$) (\ref{eq:IVPFt}) .}}
\end{center}
\end{table}


In Table~\ref{table:Compare} we compare the iterated solution to the  discrete difference equation model we found in  {\it Excel} with  the {\it Mathematica\/} solution to the continuous differential equation model.              






There are  slight differences in the early values for of $b(n)$ and $b(t)$  $n = t = 1, 2, 3, 4, {\rm and} \, 5$, but then both models appear to ``settle down'' and head toward  20. What does this tell us?


\subsection*{One model, many applications}



Suppose we are studying  a population of microorganisms in a Petri dish. Half  the population dies each hour, but  10 microorganisms  enter our Petri dish every hour.   Model this situation with (a) a discrete difference equation model and (b) a continuous differential equation model. Be sure to define the  variables and identify their units. 
What are the pros and cons of each model in this  situation?  



\teacher{
\section*{NOTES FOR TEACHERS}


The Project Description  preceding this section is the information and instruction set for the students. It can be found in a separate document 1-001-S-m\&m-DeathImmigration-StudentVersion.pdf. These NOTES  are for the teacher.

This modeling activity described here typically takes fewer  than 30 minutes to conduct.  This simulation makes a great introduction to   modeling. We  give students several  days to do their write-up and schedule follow-up in-class discussions on issues such as communication and mathematics itself to do only one of the models, either death only or death and immigration, to get the essential modeling points across.

The second model has only one slight change from the first: the addition of 10 {\it m}\,{\&}{\it m}s to the living population at every round. 



Ask the students  to describe what happens in their simulation. Soon they see that the  population model with death {\it and}  immigration never  gets to zero. Indeed, just as it appears  it might our rule says, ``add 10 {\it m}\,{\&}{\it m}s   at each iteration." Thus, the population at the next iteration cannot be zero. From the data, neither would we say the population goes to 20. Indeed, it seems to be settling around 20 in most iterations.  It can be helpful to interrupt the class and ask students to summarize their progress after collecting the data and before they start building their models.


This activity introduces of the notion of an {\it Equilibrium Value\/} very naturally for their data seem to get to 20 and  stay roughly ``at'' 20. We can ask them whether  there is some value, say EV, such that  when we start at EV we find ourselves at EV again in the next iteration of the model, i.e. $b(n+1) = b(n) $. Expressed symbolically, can we find EV such that ${\rm EV} = \frac{1}{2}{\rm EV} + 10$?  Using simple algebra to solve for EV we obtain ${\rm EV} = 20$ --- the very value their simulations seem to be dancing about.


This activity can also be used to introduce the use of technology for simulations.  Students can develop the immigration model $a_{n+1} =  \frac{1}{2}a_n + 10$ with $a_0 = 50$ for $n = 0, 1, 2, 3, \ldots$ and iterate it by hand (or in Excel) to compare the model's prediction to the data. A similar approach is to develop the difference equation model $\Delta a_n=a_{n+1} - a_n = -\frac{1}{2}a_n + 10$ or, if the students are ready for it, the differential equation model, $a'(t) = -\frac{1}{2} a(t) + 10$, both with attendant solution strategies and initial conditions of $a_0 = 50$ or  $a(0) = 50$, respectively.

 The closed form solution of the difference equation is $a_n = 20(1 +   2^{2-n})$ while the comparable solution for the differential equation is $a(t) = 20 (1- e^{-t/2})$. Those teaching the closed form solutions can ask students to compare the analytis results with the data.  


If a physical simulation with real (yummy) {\it m}\,{\&}{\it m}s is not possible or feasible, we have provided images of a simulation the students can use to collect data. These images can be found in the Support Docs associates with this Modeling Scenario in files named: 

\qquad \qquad 1-1-m\&m-PureDeathSimulationImageData.pdf (and .ppt)    and

\qquad \qquad  1-1-m\&m-Death-ImmigrationSimulationImageData.pdf (and .ppt). 

Indeed, we offer a video simlulation of this data collection process with each iteration performed and data tallied at \cite{SIMIODEYouTube2017}.



Through physical simulation of a population, students collect data for two mathematical models of a changing population   and then compare the results of the  models to the data. The opportunity is rich with  parallels to modeling in the real world and the students can see at each step the assumptions, data collection, modeling, and comparisons. We recommend the reader try this activity and perhaps add something of their own to it.



\subsection*{Acknowlegements}

The essence and detail of this approach appears in complete detail in \cite{WinkelMM2010}. We acknowledge and appreciate the feedback from many, many colleagues who have tried this activity in their teaching and have thus made the discussion a richer one.  The project {\it never fails} to engage, teach, and interest students, and it often serves as a full-scale first introduction to modeling with  technology in the study of  differential equations.

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\bibitem{SIMIODEYouTube2017} SIMIODE Youtube Channel. 2021.  mANDmDeathImmigrationSimulation.  \url{https://www.youtube.com/watch?v=IOOkVL6o0I8\&list=PLHUyzRr_S0TtP0gAUffhAS8FDccmLh0m8\&index=4}.

\bibitem{WinkelMM2010} Winkel, B. J. 2010. Population Modeling with {\it m}\,{\&}{\it m}s.  {\it International Journal of Mathematical Education in Science and Technology\/}. 40(4): 554-558.


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