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\begin{document}

\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\\STRUGGLE FOR EXISTENCE}





\markboth{Struggle for Existence}{Struggle for Existence}






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

\makeStitlePDFLaTex



\begin{abstract}
 We use historical data from the 1930's in the Soviet Union and model competition between two species  of yeast after modeling each species separately and estimate parameters.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

 


We apply an approach to parameter estimation, using two logistic growth models for competing species or strains of one species and historical  data from  G.~F.~Gause's work \cite{Gause1934} and \cite{GauseNastukovaAlpatov1934}, in the Soviet  Union of the 1930's, for studying paramecia and yeast populations. 


We point out that Gause was interested in  modeling yeast growth (separate species and combined populations) with respect to the production of vodka in state institutions of Mother Russia! In this activity we attack his logistic problem for competition by first  determining the $(r, K)$ parameters in separate logistic population models for  two yeast  species he initially studied. Gause was interested in determining what he called the ``coefficients for the struggle for existence," ($\alpha$ and $\beta$) in (\ref{eq:species1}) and {\ref{eq:species2}), respectively, which model the intensity of the influence of one species upon the other's carrying capacity, all using his data and our several approaches. We shall leave that for another time.

In (\ref{eq:species1}) and (\ref{eq:species2}) we have the classical competition model found in  \cite{Hutchinson1978}, for example, in which two populations (different species or strains of the same species), $N_1$ and $N_2$, with respective growth rates, $r_1$ and $r_2$, and carrying capacities, $K_1$ and $K_2$, are in the same environment. We shall study this model in more detail later in our studies of differential equations as a modeling tool.

The ``coefficients for the struggle for existence" ($\alpha$ and $\beta$) in (\ref{eq:species1}) and (\ref{eq:species2}), as Gause called them, were of great interest to him and what he did was to model each species separately (that is EXACTLY what we shall ask you to do) to discover their $r$ and $K$ values and later in our studies, we shall use  (\ref{eq:species1}) and (\ref{eq:species2})  to estimate the parameters $\alpha$ and $\beta$. 
 
 \hspace*{-.5 true cm}
\begin{eqnarray}
\frac{dN_1}{dt} &=& r_1 N_1\frac{K_1 - \alpha N_2 - N_1}{K_1} \label{eq:species1} \\
\frac{dN_2}{dt} &=& r_2 N_2\frac{K_2 - \beta N_1 - N_2}{K_2}\label{eq:species2}
\end{eqnarray}


\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
 & {\it Saccharomyces\/} & Mixed Population& {\it Schizosaccharomyces\/} & Mixed Population\\ \cline{2-5}
Age in  & Volume & Volume & Volume & Volume \\ 
hours & of yeast & of yeast &of yeast &of yeast  \\ \hline
6 & 0.37     & 0.375 & -     & 0.291 \\
16 & 8.87    & 3.99  & 1.00  & 0.98 \\
24 & 10.66   &  4.69 & -     & 1.47  \\  
29 & 12.50   &  6.15 & 1.70  & 1.46 \\
40 & 13.27   &   -   & -     &- \\
48 & 12.87   &  7.27 &2.73   & 1.71 \\
53 & 12.70   &  8.30 & -     & 1.84 \\
72 & -       & -     &  4.87 & - \\
93 & -       & -     &  5.67 & - \\
117 & -      & -     &  5.80 & - \\
141 & -      & -     &  5.83 & - \\
7.5 & 1.63   &  0.923&-      & 0.371 \\
15.0 & 6.20  &  3.082&1.27   &  0.630 \\
24.0 & 10.97 &  5.780& -     & 1.220 \\
31.5 & 12.60 &  9.910& 2.33  & 1.112 \\
33.0 & 12.90 &  9.470& -     & 1.225 \\
44.0 & 12.77 & 10.570& -     & 1.102 \\
51.5 & 12.90 &  9.883& 4.56  & 0.961 \\ \hline
\end{tabular}

\vspace*{.2 true cm}
\parbox{10 true cm}{\caption{\label{table: GauseData}The growth of  the yeast volume and the number of cells in pure cultures of {\it Saccharomyces cerevisiae\/} (column 1),
{\it Schizosaccaromyces kefir\/} (column 3) and in the mixed population  of these species (column 2 and 4 respectively). \cite[p.~395]{Gause1932}}}
\end{center}
\end{table}

In Table~\ref{table: GauseData} we provide the data that Gause obtained from his observations for each population when alone and when mixed  in their  environment and we will ask you to estimate  the respective parameter sets, $r_1$ and $K_1$   and then  $r_2$ and $K_2$, for his two separate populations in the two separate logistics situations (\ref{eq:N1}) and (\ref{eq:N2}). 


\begin{eqnarray}
\frac{dN_1}{dt} &=& r_1 N_1\frac{K_1  - N_1}{K_1} \label{eq:N1} \\
\frac{dN_2}{dt} &=& r_2 N_2\frac{K_2  - N_2}{K_2}\label{eq:N2}
\end{eqnarray}



We need the data for the species in isolation in order to determine their respective $r$ and $K$ values. Table~\ref{table: GauseData}  presents Gause's data of two combined ``runs" for each species, in isolation and in mixed population.




\begin{itemize}
\item [a)] Estimate  the respective parameter sets, $r_1$ and $K_1$   and then  $r_2$ and $K_2$, for Gause's two separate populations in the two separate logistics situations (\ref{eq:N1}) and (\ref{eq:N2}) using the appropriate data in Table~\ref{table:GauseYourData}. 

We find Gause's estimates for $r$ and $K$ for each of the two population models in his published works (see Table~\ref{table:GauseYourData}) and we include their values below for you to compare your work with his.\cite[p.~398]{Gause1932}

\begin{table}[h]
\begin{center}

\begin{tabular}{|c||p{1.5 true cm}|p{1.5 true cm}||c|c|}\hline
& \multicolumn{2}{|c||}{Your Analysis} & \multicolumn{2}{|c|}{Gause's Analysis} \\ \hline \hline
Species & \multicolumn{1}{c}{$r$} & \multicolumn{1}{c||}{$K$}  & $r$ & $K$ \\ \hline \hline
{\it Saccharomyces\/}&  &  & 0.21827& 13.0\\ \hline
{\it Schizosaccharomyces\/}  &  &  & 0.06069& 5.8\\ \hline
\end{tabular}

\parbox{8.7 true cm}{\caption{\label{table:GauseYourData} Gause's parameter estimates for the two separate populations of yeast and your estimates for comparison.}}

\end{center}
\vspace*{-.5 true cm}
\end{table}

\item [b)] Using your parameters plot the data and the solution to each of (\ref{eq:N1}) and (\ref{eq:N2}) on the same respective axes. Comment on how good your model is for the data.
\item [c)] Using Gause's parameters found in Table~\ref{table:GauseYourData} plot the data and the solution to each of (\ref{eq:N1}) and (\ref{eq:N2}) on the same respective axes. Comment on how good Gause's model is for the data.
\item [d)] Compare your model with Gause's. You should know that he did all of his calculations by hand using a ruler to estimate slopes from the data and then used these slopes to advance his ``solutions" for his differential equations. There were no computers to assist in the analyses.
\item [e)] Now estimate the parameters $\alpha$ and $\beta$ and compare your model using (\ref{eq:N1}) and  (\ref{eq:N2})  with all your estimated parameters and compare your model result with the data in Table~\ref{table: GauseData}.
\end{itemize}



\teacher{
\section*{NOTES FOR TEACHERS}

There are basically two components to this activity. First, students determine the parameters $r$, growth rate,  and $K$, carrying capacity, for two sets of populations of yeast living alone. Second, students determine the coefficients of interaction when these two populations are placed in the same environment using a competition model.

Much of the material for this scenario appeared in \cite{Winkel2011}. More immediately we offer a Mathematica notebook 6-040-T-Mma-StruggleForExistence-ToyData-TeacherVersion.nb and a pdf file for those who do not read Mathematica in which we offer the analyses. This notebook shows  methods appropriate for (a) above  with  toy data set. 
}

\begin{thebibliography}{99}


\bibitem{Gause1932} Gause, G.~F.  1932. Experimental Studies on the Struggle for Existence. {\it Journal of Experimental Biology\/}. 9(4): 389-402. Available on-line at \href{http://jeb.biologists.org/cgi/reprint/9/4/389.pdf}{http://jeb.biologists.org/cgi/reprint/9/4/389.pdf}. Accessed  21 September 2016.


\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.


\bibitem{GauseNastukovaAlpatov1934} Gause, G. F., O. K. Nastukova, and W. W. Alpatov.  1934. The Influence of Biologically Conditioned Media on the Growth of a Mixed Population of {\it Paramecium cadatum\/} and {\it P. aureliax\/}.  {\it Journal of Animal Ecology\/}.  3(2): 222-230. 

\bibitem{Hutchinson1978} Hutchinson, G. Evelyn. {\it Introduction to Population Ecology\/}. New Haven CT:  Yale University Press.

\teacher{
\bibitem{Winkel2011} Winkel, B. 2011. Parameter Estimation in Differential Equation Models for Population Growth. {\it PRIMUS\/}.  21(2): 101-129.
}

\end{thebibliography}






\end{document}
