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\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Feral Cat Control }



\markboth{Feral Cat Control}{Feral Cat Control}

%\author{Rachel L. Bayless\\
%Department of Mathematics\\ 
%Agnes Scott College\\
%Decatur GA USA\\
%rbayless@agnessscott.edu}

%\author{Nathan Pennington\\
%Department of Mathematics\\ 
%Creighton University\\
% Omaha NE USA\\
%nathanpennington@creighton.edu}
%}


\author{
\begin{minipage}[t]{6 true cm}
{Rachel L. Bayless\\
Agnes Scott College\\
Decatur GA 30030 USA\\
rbayless@agnessscott.edu}
\end{minipage}
\ \
\begin{minipage}[t]{6 true cm}
{Nathan Pennington\\
Creighton University\\
Omaha NE 68102 USA\\
nathanpennington@creighton.edu}
\end{minipage}
}

\makeStitlePDFLaTex  











\begin{abstract}
This activity is structured as a letter from a company seeking assistance with a mathematical problem. The students will act as professional mathematical consultants and write a report analyzing the client's problem. The client company is a fictional organization which advocates for the use of trap-neuter-return (TNR) as a control method for feral cat colonies. The students will utilize modified exponential growth models to analyze the efficacy of TNR compared to a euthanasia program. 
\end{abstract}



\section*{SCENARIO DESCRIPTION}

\hspace{4.6in} Alley Cat Advocates \\
\mbox{} \hspace{4.6in} 1 Siamese Drive \\
\mbox{} \hspace{4.6in} Maine Coone, KI 55501 \\ 


Dear Differential Equations Students:

As you may know, rapidly growing feral cat colonies are a serious problem in many towns. Alley Cat Advocates has been working to control the growth of these populations for many years. We have a long track record of implementing humane practices that help slow population growth. 

We were recently approached by Mr.\ Tom Kat, a representative from a town that has a very large feral cat colony. He offered us a wonderful opportunity. Mr. Kat is willing to fund a project where we implement humane methods for population control in his town. We will also have the opportunity to give public presentations nationwide on the proper protocol for controlling feral cat populations. 

Before officially promising us the funds we need, Mr. Kat has asked that we produce a formal report utilizing simulated mathematical data to illustrate the efficacy of our suggestions. If he likes the results of these simulations, then we can begin our new partnership. Thus, we are reaching out to you for help. 

There are two commonly used methods for controlling feral cat populations. In the first method, traps are set for feral cats and when a cat is trapped it is immediately euthanized.  This is called the ``trap-kill" method. We do not believe this method is ethical, and therefore we advocate for a different method, which is commonly called ``trap-neuter-return." In this method traps are set, but when a cat is trapped it is neutered/spayed and returned to the population. We are particularly interested in the relative effectiveness of these two methods at controlling the overall cat population.

As our expert mathematical consultants, we need you to build and analyze mathematical models for the following three scenarios: 
\begin{enumerate}
\item Mr. Kat does not implement any control methods in his town. 
\item Mr. Kat implements the trap-kill method in his town. 
\item Mr. Kat implements the trap-neuter-return method in his town. 
\end{enumerate}

We have a few more pieces of information that might help you build your models. First, it is our experience that the growth rate of a feral cat colony is proportional to the size of the total population. Our research (see \cite{evaluation} and \cite{data}) shows that the per-year per-cat growth-rate coefficient for most feral cat populations is $k=1.08$. Furthermore, Mr. Kat has told us that he estimates the current feral cat population totals around 100 cats.  

In the second and third models we assume a constant number of traps and that each trap catches $12$ cats per year. We also assume that the fertile cats breed first and then are captured. Finally, we have noticed that cats are rarely caught twice. They are crafty little creatures, and once they are caught and neutered/spayed they avoid the traps and are never caught again. 

Using this information, we ask that you first build individual models for each of the three aforementioned scenarios. We need you to provide a detailed  and readable explanation of the development of these models, so we can defend the choices to Mr.\ Kat. In particular, please describe (both mathematically and in English prose) all your notation and each individual step of your analysis. Also, your professor tells us that you have discussed two general approaches to studying differential equations: qualitative techniques and analytic techniques, and you should employ both techniques whenever possible to investigate the models you develop. Please keep in mind that we have not taken mathematics since our kappa kappa kitty kindergarten class, and we have no formal training in cat calculus. Thus, we would appreciate a discussion that is readable for non-mathematicians and includes definitions and citations for all mathematical terminology.

We are particularly curious about the long-term behavior of the cat population in each of the three scenarios. Does the fate of the population depend on the number of traps set? If so, is there a critical number of traps around which the population grows if there are fewer traps and declines if there are more traps? Also, we reiterate that we would really like you to compare how well catch-kill and catch-neuter-return control the cat population. 

If you should find that you have questions during the course of your investigation, then you are to contact your professor. She has already agreed to serve as an extra consultant for this project. Please have your report completed by the beginning of class on DUE DATE to ensure full payment for your services (which will luckily be in the form of a grade).




Sincerely, \\
\\
%\includegraphics[scale=.25]{pawprint.jpg}

Felis Catus \\
President of Alley Cat Advocates 


\begin{thebibliography}{98}


 \bibitem[1]{evaluation} Schmidt, P., T.~Swannack, R.~Lopez, and M.~Slater. 2009.  Evaluation of euthanasia and trap-neutering (TNR) programs in managing free-roaming cat populations. {\it Wildlife Research\/}. 36: 117-125.


\bibitem[2]{data} Schmidt, P., R.~Lopez, R. and B.~Collier. 2007. Survival, Fecundity, and Movements of Free-Roaming Cats. {\it Journal of Wildlife Management\/}. 271(3): 915-919.

\end{thebibliography}


\teacher{
\section*{NOTES FOR TEACHERS}

This project can be a first introduction to using differential equations to model real-world phenomena.  As written, it is appropriate for the first week or two of a differential equations course (or an advanced assignment in Calculus II).  However, a major strength of the project is that it can be easily scaled to a more advanced audience by, for example, using the logistic growth model instead of exponential growth.  Other possible modifications will be discussed below.  Also, as written, the project requires a basic understanding of any software that can plot slope fields for simple differential equations, but this can be easily removed if technology is not available.

The project is completed outside of class and the solution is a written paper (roughly 10-20 pages) describing how the problem was solved. The project should take approximately two weeks. The most challenging part for the students tends to be the written portion. Many of the students have never been asked to write a paper in a mathematics course before, and it can be helpful to have students submit multiple drafts of their written report before the due date. Students may also struggle with the open-ended nature of this project, and it can be useful to encourage them to ask questions of the instructor and their peers. Some instructors may want to require students to meet with them one week after the project is assigned. Although this approach is not necessary for success, it can aid students in development of the models, as well as time management for the project. If, however, encouraging outside meetings is not feasible for the instructor, then they may want to introduce the idea of deriving continuous differential equations from discrete systems in class. This is perhaps the trickiest part of building the models, and teaching this concept should set all the students on the same path. 

The letter format for this project was adapted from \cite{book} which details four different ways to implement these types of writing projects in mathematics courses. Furthermore, \cite{book} also includes assessment tools such as evaluation templates and checklists that the instructor may find helpful.   


\subsection*{Learning Outcomes}
The learning objectives for the project are to:

\begin{enumerate}

\item[\tiny$\bullet$] Describe and justify (in words) the development of a simple mathematical model from a real-world scenario.

\item[\tiny$\bullet$]  Demonstrate a working knowledge of the analytic technique separation of variables.

\item[\tiny$\bullet$] Use mathematical software to generate a slope field or numerical solution and employ qualitative analysis to interpret the information.

\item[\tiny$\bullet$]  Communicate mathematics in formal written prose.

\end{enumerate}

\section*{SOLUTIONS}

We  now give an outline of possible solutions for each of the three tasks in the assignment. 

\subsection*{Method One: No control methods}

In this part of the project, the student should explain that since the feral cat population grows at a rate proportional to its size it is governed by the differential equation
\begin{equation*}
\frac{dp}{dt}=kp(t),
\end{equation*}
where $p(t)$ is the population of the colony $t$ years after the start time, $k$ is the growth rate (given to be $1.08$), and $p(0)$ is the initial population (given to be $100$).  It is important that the students state the units of time, as this is one of the main distinctions between a textbook problem and a real world problem.

Separating variables gives $p(t)=p(0)e^{1.08t}$ as the general solution. Given that the initial population is $100$, the specific solution is $p(t)=100e^{1.08t}$, so the solution grows without bound as time progresses. 

The last step of this part is to graph the solution.   The students should then explain that this solution grows without bound as time progresses, demonstrating that something does need to be done about the cat population. 


\begin{figure}[t]
\begin{center}
\includegraphics[scale=.4]{1-21-FeralCatControl-Model1.eps}
\end{center}
\vspace*{-.3 true cm}
\caption{Solution curve for $\displaystyle \frac{dp}{dt} = 1.08 p$, where $p(0) = 100$.}

\vspace*{-.3 true cm}

\end{figure}



\subsection*{Method 2: Trap and Kill}
To derive the differential equation for this method, we let $p_n$ be the cat population after $n$ time steps in years (with $p_0$ as the initial population), $k$ be the yearly growth rate, and $a$ be the number of traps (recall each trap catches 12 cats per year).  Since trapping occurs after breeding, we have
\begin{equation*}p_n=kp_{n-1}+p_{n-1}-12a,
\end{equation*}
and so the change in population is 
\begin{equation*}p_n-p_{n-1}=kp_{n-1}-12a.
\end{equation*}
To make this a continuous process, we let $t$ be the current time, let $\Delta t$ be the change in time and we then have 
\begin{equation*}p(t+\Delta t)-p(t)=(kp(t)-12a)\Delta t.
\end{equation*}
Dividing by $\Delta t$ and taking the limit as $\Delta t$ goes to zero gives the differential equation
\begin{equation}\label{model2}
\frac{dp}{dt}=kp(t)-12a.
\end{equation}
Equation (\ref{model2}) can be solved by separating variables, and the solution is 
\begin{equation}
p(t)=\frac{1}{k}\left (Ce^{kt}+12a\right ),
\end{equation}
where $C=100k-12a$. Substituting the per-year growth rate of $k=1.08$ yields 
\begin{equation}
p(t)=\frac{1}{1.08}\left (Ce^{1.08t}+12a\right ), \quad \text{where } C = 108 -12a.
\end{equation}

Next, the students should look for the critical number of traps by setting $dp/dt=0$, which in this case holds provided $a=9$.  They should then observe that this means the population will remain constant at 100 cats if exactly $9$ traps are set. Furthermore, if $a<9$, then $dp/dt >0$, so the population is increasing.  Similarly, if $a>9$, then $dp/dt<0$, and the population is decreasing. The student should then graph (and explain) what happens to the solution when $a<9$, $a=9$, and $a>9$.


\begin{figure}[t]
\begin{center}
\includegraphics[scale=.6]{1-21-FeralCatControl-Model2.eps}
\parbox{9.5 true cm}{\caption{Plot of the solutions curves for $dp/dt=1.08p(t)-12a$, where $p(0)=100$ when $a<9$, $a=9$, and $a>9$. }}
\end{center}
\vspace*{-.5 true cm}
\end{figure}





\subsection*{Method 3: Trap-Neuter-Return}

To construct the differential equation in this case, we use the same notation as in Method 2 and let $p_0$ be the initial cat population and $a$ be the number of traps.  Since trapping occurs after breeding, the total population after one unit of time is 
\begin{equation*}p_1=kp_0+p_0.
\end{equation*}

After this first time step, there are $p_1$ total cats and $12a$ sterilized cats, so the number of fertile cats is $p_1-12a$.  This means the population of cats after two time steps is 
\begin{equation*}p_2=k(p_1-12a)+p_1,
\end{equation*}
which means the change in cat population is 
\begin{equation*}p_2-p_1=k(p_1-12a).
\end{equation*}

In general, the change in cat population from time step $n$ to $n+1$ is given by 
\begin{equation*}p_{n+1}-p_n=k(p_n-12na),
\end{equation*}
so the differential equation for this method is 
\begin{equation*}
\frac{dp}{dt}=k\left (p(t)-12at \right ).
\end{equation*}
This equation can be solved by using an integrating factor, and the solution is 

\begin{equation*}
p(t)=12at+\frac{12a}{k}+Ce^{kt},
\end{equation*}
where $C=100-\displaystyle \frac{12a}{k}$.  



As with the previous method, the next step is to find the value of $a$ that yields a qualitative change in the solution.  We begin by looking for a constant solution. Setting $dp/dt=0$ gives 
\begin{equation*}
12a+ k \left (100-\frac{12a}{k}\right )e^{kt}=0,
\end{equation*}
which is a function of $t$, and therefore there is no constant solution.  However, we do see that the solution will have no critical points if $100\geq 12a/k$ and will have one critical point when $100<12a/k$.  This makes $a=25k/3=9$ the critical number of traps.  Plugging this value of $a$ into the equation for $p(t)$ gives $p(t)=100kt+100$, which is a function of $t$, and therefore there is no constant solution.    Indeed, the solution to the differential equation is a line of positive slope. For $a<9$, the solution grows exponentially and for $a>9$, the solution grows until it reaches a maximum, and then decreases.  Thus, for $a > 9$, the population will eventually decay and if $a \leq 9$ the population again grows without bound.









If students have not yet learned the integrating factor method, they can  be directed to choose values of $a$ and graph the results.  They should do enough cases to recognize that there is a boundary value for $a$, and from the previous section, they already have a good candidate ($a=9$). 





\subsection*{Note on Data Collection}

The studies \cite{evaluation} and \cite{data} provide data on survival and fecundity rates for feral and semi-feral cat populations.  Since we are not distinguishing between the two in this project, we will average the results from those two categories to obtain the growth rate constant for our cat population.  From the second paragraph in the Results category of \cite{data}  we have that $1.75$ kittens of a feral cat litter survive for at least four months, and $28.6$ percent of feral female cats have a litter each year.  For the semi-feral population, $2.75$ kittens from each litter survive at least four months, and $50$ percent of semi-feral female cats have $1.6$ litters per year.   From \cite{evaluation}, we have that the survival rate for feral and semi-feral male and female cats is $73$ percent.  So with $y_0$ equal to the initial population of cats, the number of cats after one year is 
\begin{equation*}y_1=\left (1.75*(.286)+1.6*2.75*.5\right )y_0/2+.73y_0,
\end{equation*}
and so the change in population is 
\begin{equation*}y_1-y_0=1.08y_0,
\end{equation*}
giving us a growth rate constant of $1.08$.


\begin{figure}[h]
\begin{center}
\includegraphics[scale=.6]{1-21-FeralCatControl-Model3.eps}
\parbox{9.5 true cm}{\caption{Plot of the solutions curves for $dp/dt=1.08(p(t)-12at)$, where $p(0)=100$ when $a<9$, $a=9$, and $a>9$. }}
\end{center}

\vspace*{-.6 true cm}
\end{figure}

\vspace*{.2 true cm}
\textbf{Suggestions for Future Study and Student Projects:}

\begin{enumerate}
\item Logistic growth could be considered  instead of exponential growth.  This makes the calculations more difficult and gives more interesting dynamics.

\item We currently assume that cats are continuously being caught and that the euthanasia/sterilization happens instantaneously. The model could be modified to include a time delay between the trap, sterilization, healing, and release phases.

\item The assumption that a cat will not be caught more than once could be removed from the model, which would introduce a new term to account for the probability that a trapped cat has already been caught and sterilized.

\item Most trap-neuter-return programs specify that if a trapped cat tests positive for FIV or FeLV, then that cat is euthanized rather than sterilized and returned. The model could be modified to take into account the probability that a cat is actually eligible for sterilization and release.

\item This problem could be discretized and used to study the difference between discrete and continuous dynamical systems.

\item The project could be modified to more closely resemble a research project by giving the students access to \cite{evaluation} and \cite{data} and allowing them to determine the growth rate for the feral cat population.


\end{enumerate}

\begin{thebibliography}{98}



 \bibitem[1]{evaluation} Schmidt, P., T.~Swannack, R.~Lopez, and M.~Slater. 2009.  Evaluation of euthanasia and trap-neutering (TNR) programs in managing free-roaming cat populations. {\it Wildlife Research\/}. 36: 117-125.


\bibitem[2]{data} Schmidt, P., R.~Lopez, R. and B.~Collier. 2007. Survival, Fecundity, and Movements of Free-Roaming Cats. {\it Journal of Wildlife Management\/}. 271(3): 915-919.

 \bibitem[3]{book} Crannell, A., G.~LaRose, T.~Ratliff, E.~Rykken. 2004. \emph{Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go.} Washington 
DC: Mathematical Association of America.


\end{thebibliography}

}










\end{document}
