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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION\else STUDENT VERSION \fi\\Predator-Prey: Blue Whales and Krill}




\markboth{A Predator-Prey Model}{Blue Whales Love Krill}










\author{Lydia Kennedy\\
Virginia Wesleyan College\\
Norfolk  VA 23502 USA\\
lkennedy@vwc.edu}


\makeStitlePDFLaTex



\begin{abstract}
Students will use Excel to observe qualitative behavior in a simulation of a predator-prey model, with blue whales and krill as the predator and prey populations, respectively.  Students are asked to explain terms in the system of differential equations, compute population values using iteration without a spreadsheet. Then they will implement the simulation in Excel and create charts to observe qualitative behavior.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

Throughout this handout, there are activities to be done and questions to be answered.  In a separate document (Microsoft Word, \LaTeX, etc), answer the questions and include charts where indicated.  You must also submit an electronic copy of your Excel spreadsheet at the end of the project.

One of the favorite foods of the blue whale is krill.  Blue whales are baleen whales and feed almost exclusively on krill. These tiny shrimp-like creatures are devoured in massive amounts to provide the principal food source for the huge whales.  In the absence of predators, in uncrowded conditions, the krill population density grows at a rate of 25$\%$ per year.  The presence of 500 tons/acre of krill increases the blue whale population growth rate by 2$\%$ per year, and the presence of 150,000 blue whales decreases krill growth rate by 10$\%$ per year.  The population of blue whales decreases at a rate of 5$\%$ per year in the absence of krill.\\

These assumptions yield a pair of differential equations (a Lotka-Volterra model) that describe the population of the blue whales ($B$) and the krill population density ($K$) over time given by 

\begin{eqnarray*}
  \frac{dB}{dt} & = & -0.05B + \left(\frac{0.02}{500}\right) BK\\
  \frac{dK}{dt} & = & 0.25K - \left( \frac{0.10}{150000}\right) BK
\end{eqnarray*}

\begin{enumerate}
    \item What are the units for $\frac{dB}{dt}$? for $\frac{dK}{dt}$?
    \item There are two terms on the right hand side in each of the differential equations given above.  Explain (with a complete sentence) what each of the four terms relates to in relation to the description given above. Include a reason for why the term is positive or negative.
    \item Give an explanation of the terms $\frac{0.02}{500}K$ and $-\frac{0.10}{150000}B$ as they relate to the model above.
\end{enumerate}

\teacher{
\begin{center}
\parbox{13 true cm}{\noindent [\textit{The questions given above are intended to help students with ``model creation" by identifying the terms in the model that relate to the description of the relationships between the blue whales and the krill.  The question about the coefficients is likewise intended to help students connect the description with the model itself.}]}
\end{center}}

\vspace*{.2 true cm}
We can estimate the population and population density, respectively, of the two species with a simple simulation method that uses the differential equations to find the change in the function values for a fixed time period (1 year in this case).  This means that starting with an initial population for the blue whales ($B_0$) and an initial population density for the krill ($K_0$), we can estimate the population of blue whales a year later using $B_1=B_0+\frac{dB}{dt} \big|_{B=B_0, K=_0}$ and the population density of krill a year later using $K_1=K_0+\frac{dK}{dt} \big|_{B=B_0, K=K_0}$.


For example, if $B_0=10,000$ and $K_0=200$, then 

\begin{eqnarray*}
  B_1 & = & B_0+\frac{dB}{dt} \bigg|_{B=B_0, K=K_0}\\
  & = & 10,000+(-0.05*10,000)+\left(\frac{0.02}{500}\right)*10,000*200\\
  & = & 9580\\
  \mbox{and} & & \\
  K_1 & = & K_0+\frac{dK}{dt} \bigg|_{B=B_0, K=K_0}\\
   & = & 200 + 0.25*200-\left(\frac{0.10}{150,000}\right)*10,000*200\\
   & = & 249
\end{eqnarray*}

\noindent In a similar manner, $B_2=B_1+\frac{dB}{dt} \big|_{B=B_1, K=K_1}$ and $K_2=K_1+\frac{dK}{dt} \big|_{B=B_1, K=K_1}$, etc.\\

\teacher{\begin{center}
\parbox{13 true cm}{\noindent [\textit{This section is intended to (1) explain how the iteration in the simulation works, and (2) force the students to compute a few iterations so they have some feel for what the Excel simulation is doing and how the population values arise.}]}
\end{center}}


\begin{enumerate}
    \setcounter{enumi}{3}
    \item Using the simple simulation method demonstrated above, fill in the following table.

    \begin{center}
    \renewcommand{\arraystretch}{2}
      \begin{tabular}{|c|c|c|} \hline
      Time (in years) & Blue Whale population & Krill population (in tons per acre) \\ \hline
      0 & 75,000 & 150 \\ \hline
      1 &  & \\ \hline
      2 & & \\ \hline
      3 & & \\ \hline
      4 & & \\ \hline
      5 & & \\ \hline
      \end{tabular}
    \end{center}



    \item This sort of simulation is best done with the aid of technology.  Use an Excel spreadsheet to
    \begin{enumerate}
        \item estimate the population of the blue whales and the population density of the krill each year for 100 years, assuming an initial population density of 150 tons per acre of krill and a population of 75,000 blue whales, and
        \item make a chart (in Excel) that plots the krill population density versus the blue whale population over those 100 years.
    \end{enumerate}

    As part of your answer, give the population of whales and population density of krill (complete sentences!) and a copy of the chart from your spreadsheet.

\teacher{
\begin{center}
\parbox{13 true cm}{\noindent [\textit{My experience has been that students need a lot of direct instruction in how to use Excel.  When I do exercises of this sort with my students, I have found that it works best to do this in a computer lab and have the students create the simulation with me while I demonstrate on a projected screen.}]}
\end{center}}

    \item Using the simulation results in your Excel spreadsheet, what do you notice about the population density of the krill and the population of blue whales over the 100 years?  If one population or the other becomes very endangered (very low population levels), give a plausible explanation as to why.  If one or both populations becomes very endangered, give an estimate of how long it takes for this to happen. (Write complete sentences).

    \item What happens when the initial conditions are changed?
    \begin{enumerate}
        \item Set the initial population density of the krill to be 300 tons per acre and the population of blue whales to be 75,000 blue whales.  Before using Excel to do the simulation, make a prediction about the long-term results for the krill population density and the blue whale population over 100 years.  Then, do the simulation in Excel answer the same questions as in \#5 and \#6.
    \item Set the initial population density of the krill to be 300 tons per acre and the population of blue whales to be 150,000 blue whales.  Before using Excel to do the simulation, make a prediction about the long-term results for the krill population density and the blue whale population over 100 years.  Then, do the simulation in Excel answer the same questions as in \#5 and \#6.
        \item Set the initial population density of the krill to be 300 tons per acre and the population of blue whales to be 175,000 blue whales.  Before using Excel to do the simulation, make a prediction about the long-term results for the krill population density and the blue whale population over 100 years.  Then, do the simulation in Excel answer the same questions as in \#5 and \#6.
    \end{enumerate}
    
    \item Now, change the initial population density of the krill and the population of blue whales again.  Pick different values and experiment.  See what happens.  Answer the same questions as in \#5 and \#6 for at least 3 different sets of initial values.  Do not use the same initial values as in \#7.

    \item Describe what this model seems to say about the interacting species.
    \item What factors that influence the population density of krill and the population of blue whales are included in the model?  What factors are ignored?
    \item Make an improvement in the model to overcome a simplification.  Make this improvement in both the written description of the model and add its mathematical counterpart to the model.
\end{enumerate}


\teacher{\begin{center}
\parbox{13 true cm}{\noindent [\textit{I learn different things about what my students understand when I force them to write in complete sentences.  Having the students write in complete sentences to explain the results they observe reinforces connections between visual (a chart), numerical and verbal.  It also reinforces the ideas that mathematics is not just about numbers and that communications of results is as important as obtaining the results.}]}
\end{center}}


\teacher{
\section*{NOTES FOR TEACHERS}

\begin{itemize}
    \item This exercise is an adaptation of an example in a mathematical modeling book \cite{Meerschaert}, section 4.4, exercise 7.  This example, in a more complicated form, was used in an upper level modeling course.
    \item I have adapted this exercise to work for a differential equations class or a calculus class in which the students have had some exposure to differential equations.  There is no analytical solving in this exercise, so the specific differential equations knowledge could be limited, but understanding of the derivative is necessary (of course!)
    \item An exercise like this should take part of a class period and would be best done in a computer lab.  Ideally, students would answer the first 4 questions before the simulation is demonstrated in Excel.  In this format, (as a written exercise), this would be finished outside of class and turned in later.
    \item This exercise could be completed by students individually or should work also in small groups. However, for the writing part, it is better for students to write individually.
    \item This exercise uses Excel to run the simulation.  It would easily work in a CAS such as \textit{Mathematica} as well, but Excel is more readily accessible and students gain valuable skills working with Excel.  Moreover, iteration is natural in spreadsheets.
    \item This exercise is intended for students to observe qualitative behavior of this system, and particularly to observe changes in the behavior in response to changes in the initial conditions.  Students like to see the instant change in the chart when an initial condition is changed.
    \item Possible extensions/modifications:
    \begin{itemize}
        \item Give students a description of the interaction between two species similar to the description given here and have them write the model themselves.
        \item Include charts that plot time versus population, not just predator population versus prey population.
        \item Have students do a sensitivity analysis on the changes observed in the system in response to changes in the initial conditions.
        \item Have students do a sensitivity analysis on the changes observed in the system when coefficients in the system are changed.
        \item Have students change the system to include more complicated interactions between blue whales and krill.  For example, the following question could be added:
            
            Before the whaling era, there were an estimated $350,000$ blue whales.  In 1966, an international ban on hunting blue whales was established.  Today there are about 11,000 blue whales.  Assuming that all whaling is stopped, what does the model predict for the future of the blue whale species?  (Use a realistic estimate for the initial krill population--you may want to do a little Internet research for that).

        \item Have students change the system to include another source of food for the blue whales.
    \end{itemize}
\end{itemize}


\begin{thebibliography}{99}


\bibitem{Meerschaert} Meerschaert, Mark M. 2013. \textit{Mathematical Modeling}. Fourth Edition. Waltham, MA: Elsevier, Inc.


\end{thebibliography}
}





\end{document}
