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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Bifurcations in fish harvesting}


\markboth{Bifurcations in fish harvesting}{Bifurcations in fish harvesting}

\author{Jennifer Crodelle\\ 
Middlebury College\\
Middlebury  VT 05753 USA\\
jcrodelle@middlebury.edu
}

\makeStitlePDFLaTex


\begin{abstract}
This modeling scenario will introduce students to the concept of a bifurcation through a fish harvesting model. This short activity will walk students through a guided list of questions to help them to understand how the stability of equilibrium changes with changes in a model parameter, in this case the rate of harvesting fish.
\end{abstract}




\section*{SCENARIO DESCRIPTION}
This modeling scenario introduces the idea of a {\bf bifurcation}, or a qualitative change in dynamics. Many differential equations have  a {\bf bifurcation point}, or a parameter value for which the differential equation exhibits a qualitative change in dynamics. This can be a change in the stability of an equilibrium solution or the number of equilibrium solutions. We will learn how to study bifurcations using a population model.

Suppose there is a stream with salmon nearby to a family of bears. In the absence of the bears (when they are on vacation, for example), the salmon grow according to logistic growth at a rate $r>0$ to a carrying capacity $K>0$. Suppose that the bears can remove a constant amount of salmon, $R>0$, from the stream at any time $t$.  We can write the following differential equation to model the number of salmon in a stream, $S(t)$, at any given time $t$:
  \begin{equation}
   \frac{dS}{dt} =  rS\left(1 - \frac{S}{K}\right) - R, \label{eq1}
   \end{equation}

We will use qualitative analysis to help the bears understand how the number of salmon in the stream change depending on their rate of removal $R$. 

\section*{Assignment}
   \begin{enumerate}
   \item First, set $K=20$ and $r=4$ in (\ref{eq1}). What do these values say about the growth of the salmon population? 
   \item Plot $\frac{dS}{dt}$ vs $S$ for $R=0$. Note that this is the typical logistic growth equation we've seen before, without any harvesting. From this plot, sketch the plot of $S$ vs $t$, making sure to indicate stability of the equilibrium solutions. Discuss all possible long-term behaviors of the salmon population given any initial condition.
   \item Now let our harvest rate $R= 15$ salmon per unit time. Plot $\frac{dS}{dt}$ vs $S$ for this case. How many equilibrium solutions are there? Sketch the corresponding plot of $S$ vs $t$, including the stability of the equilibrium solutions.  Discuss the long-term behavior.
   \item Find the value of $R$ for which there is only one equilibrium solution. What would this correspond to physically? Plot  $\frac{dS}{dt}$ vs $S$ for this case and sketch the corresponding $S$ vs $t$ plot. Explain why this value of $R$ is a {\bf bifurcation point}.
   \item We can draw a {\bf bifurcation diagram} to illustrate how the dynamics change with continuous changes in $R$. To do this, we sketch the equilibrium solutions, $S^*$, on one axis and $R$ on the other. A solid line indicates that the equilibrium point is stable and a dashed line indicates that it's unstable. See Figure \ref{fig1} below for the bifurcation diagram of (\ref{eq1}).
  
  \begin{figure}[h!]
  \centering
  \includegraphics[scale=0.4]{bifurcationPopSalmon.png}
  \caption{Bifurcation diagram for (\ref{eq1}) with $r=4$ and $K = 20$ as we change the rate of removal, $R$.}
  \label{fig1}
  \end{figure}
  
  Use your answers from questions 2 - 4 to explain this bifurcation diagram. Imagine that the nearby family of bears would like to use this bifurcation diagram to figure out the rate at which they can take salmon from the stream. Explain to the bears how this rate of removal, $R$, is related to the initial number of salmon and what the long-term consequences of their removal will be.
   \end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}
I would suggest allowing the students 50 minutes (roughly one class period) to complete this scenario in groups. Perhaps the groups may need additional time outside of class to complete the last problem and write up. Students will need to be familiar with the logistic growth equation, including the idea of a carrying capacity and growth rate. The students will also need to be well-versed in qualitative analysis and finding the stability of equilibrium solutions. 

Solutions to this scenario are provided as a \textsc{Matlab} live script, but the scenario is written to be used with any graphing software. In fact, students can just draw the parabolas by hand. A possible modification might be to include an axis for the students to draw on to make it a bit easier to do. You may also want to expand the space between questions to allow students to write answers on the worksheet itself. 

If the students are not very familiar with population modeling, one could extend the introduction to include an example of the logistic growth with a detailed explanation of the growth rate and carrying capacity. It might also be useful to include a clear description of the difference between a parameter and the dependent variable. 


}



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