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\title{\ifnum\TeacherEdition=1 TEACHER VERSION 
\else STUDENT VERSION 
\fi\\Do Runners Synchronize?}




\markboth{Do Runners Synchronize?}{Do Runners Synchronize?}

\author{Eli Goldwyn\\
University of Portland\\
Portland OR 97203 USA \\
goldwyn@up.edu
}

\makeStitlePDFLaTex


\begin{abstract}
In this modeling scenario we practice finding and classifying equilibria of a one-variable differential equation. We do this in the context of a phase model which is often a simpler way of studying oscillatory phenomena. The idea is that instead of describing these oscillatory systems with several differential equations (one for each population or physical quantity), we can reduce the system to one differential equation that describes the phase of oscillation. In this scenario we start with a toy model in which an oscillator moves around the unit circle at a constant speed. Then we add a second oscillator and examine a system where the oscillators attract (or repel) each other (we use the ``Kuramoto Model"). Finally, we extend this concept to a more complicated predator-prey oscillator. By only looking at the difference in phase between these two oscillators, we reduce what is usually modeled with four differential equations to one differential equation which can then be studied with traditional equilibrium techniques. The goal is for students to practice finding and classifying equilibria of a one-variable system in this modeling context while also introducing variables defined on a restricted domain (the circle). This scenario also introduces bifurcations as the phase model described undergoes a saddle node bifurcation as the difference in frequencies between the two oscillators is increased. This scenario should be appropriate immediately after students have been introduced to finding and classifying equilibria of one-variable autonomous differential equations. 
\end{abstract}


\teacher{
\section*{INTRODUCTION TO THE TEACHER}

\centerline{\teacher{\textcolor{red}{Comments for the Teacher only are produced in red italics where appropriate.}}}

Oscillatory systems exist throughout the natural world. Prominent examples of oscillators include the position of a mass in a frictionless mass-spring system, the position of a swinging pendulum, the number of daylight hours over a year, predator-prey population sizes, the action potentials in neurons, light flashes in fireflies, and certain chemical reactions, most notably the Belousov-Zhabotinsky reaction. Each of these systems are often modeled with two (or more) differential equations and frequently appear in differential equations courses. We start with a simple oscillator described by a sine curve and build to the Rosenzweig-MacArthur model for predator-prey dynamics. The Rosenzweig-MacArthur model is similar to the Lotka-Volterra predator-prey model, but assumes logistic growth of the prey population as opposed to exponential growth and assumes that predation saturates at a constant per predator rate (Holling Type II functional response) as opposed to unbounded predation. Given certain parameter values for the growth, predation, and death rates, the system will produce a stable limit cycle (see Figure \ref{fig:rm}), while the Lotka-Volterra model never produces a stable limit cycle. A simple Google search of ``snowshoe hare Canadian lynx Hudson's Bay Company'' will find the data set with figures and illustrations that motivated much of the original predator-prey models. This famous data set contains nearly $100$ years of population data on the hare and lynx and shows oscillatory behavior with a (approximately) 10 year cycle!	

This scenario is designed such that the bulk of it (Questions $\#$1-12) can be completed in one full class day with students working in small groups, though it should be flexible enough for instructors to use in a variety of ways. Specifically, the important concepts appear in Questions $\#$5-8 and $\#$10-12, so other questions or parts of questions can be skipped as necessary. This material draws on basic concepts from trigonometry and some students may benefit from a quick review of the unit circle and the notion of phase (angle). This scenario is designed to give students practice finding equilibria, classifying their stability, and interpreting them in a widely used model. This material is appropriate after students have learned to find the equilibrium of a non-linear differential equation, though with some additional support they could learn that here. To assess learning outcomes, we recommend that students turn in their completed work or certain questions, either individually or as a group. Answers and commentary to the individual questions appear in red and along with Figures 3-7 are only available in the Teacher Version.	


Questions \#9 (b) and (c) involves the use of a computer or graphing calculator and can be either skipped or demonstrated by the instructor if those tools are not available.
}

\section*{SCENARIO DESCRIPTION}
Oscillatory systems exist throughout the natural world. Prominent examples of oscillators include the position of a mass in a frictionless mass-spring system, the position of a swinging pendulum, the number of daylight hours over a year, predator-prey population sizes, electrical activity in brain cells, and light flashes in fireflies. Perhaps the most famous example of oscillatory behavior in ecology comes from data collected by The Hudson's Bay Company on the snowshoe hare and the Canadian lynx. Several different systems of differential equations have been developed to describe these populations. One such system is the Ronsenzweig-MacArthur model, which consists of two differential equations and which under certain parameter values yields the results shown in Figure \ref{fig:rm}.  

\begin{figure}[h]
	\begin{center}
		\scalebox{.5}{\includegraphics{pred_prey_osc3}}
            	\caption{Simulation of the Rosenzweig-MacArthur predator-prey model. The left panel displays the time-series of the prey and predator populations while the right panel displays the limit cycle in the phase plane. The populations move counterclockwise around the closed loop on the right panel as time increases. This figure displays three oscillations,  but these oscillations continue indefinitely. }
		\label{fig:rm}
	\end{center}
	\vspace*{-.5 true cm}
\end{figure}		

A striking phenomenon that has been observed in many of the above systems is that in the presence of a second nearby oscillator, the two will often synchronize. This was first observed by Christiaan Huygens in the 17th century, where pendulum clocks resting on the same support beam will eventually synchronize. Since then, synchrony has been observed in the flashing of fireflies, in spatially distinct predator-prey populations, and in coupled brain cells. In this modeling scenario we will find and classify the equilibria of a differential equation that determines whether or not two oscillators will synchronize.

Let us start by describing a single predator-prey oscillator. The left panel of Figure \ref{fig:rm} shows prey and predator populations as a function of time and shows periodic behavior with a period of $T=19.06$ time units. This means that for any given time $t$ the prey populations is the same at time $t$ and at time $t+19.06$, likewise for the predator. The right panel of Figure \ref{fig:rm} is the phase plane and displays the prey and predator populations relative to each other. As time is increased, the system moves counterclockwise and completes a cycle every $T$ time units. This model consists of two differential equations, one describing the prey population and one describing the predator population. However, because the system has periodic behavior, we don't have to keep track of both the prey and predator populations individually. We can instead study the system by introducing a new variable called the \emph{phase}, $\theta$, which measures how far around the cycle the system is. This one variable describing the phase gives the same information as the two variables describing the prey and the predator. We can now describe our system with one differential equation instead of two! Since the system has period $T$, the phase must also have period $T$, i.e. $\theta\in[0,T)$, with $\theta(0)=\theta(T)$. 

Before we study our predator-prey system further, we start with a simpler system. Our new system also has two differential equations, but instead of the oddly shaped orbit seen in the right side Figure \ref{fig:rm}, our new system moves counterclockwise around the unit circle at a constant velocity of $1$ radian per time unit (right panel Figure \ref{fig:circ}). From the left panel of Figure \ref{fig:circ}, we can see that both $x(t)$ and $y(t)$ are periodic functions with period $2\pi$, i.e. $x(t)=x(t)+2\pi$.  Because $T=2\pi$, we say that $x$, $y$, and $\theta$ are defined on the circle. 

Since we stated above that the system has a constant velocity of  $1$ radian per time unit, 
%$x=\cos(\theta)$, $y=\sin(\theta)$ and 
$\theta$ is described by the following differential equation:

\begin{equation}
	\frac{d\theta}{dt}=1
	\label{eq:circ}
\end{equation}

\begin{figure}[h]
	\begin{center}
            	\scalebox{.4}{\includegraphics{circle_figure3}}
            	\caption{Simulation of an oscillator that moves around the circle counterclockwise with a velocity of $1$ radian per time unit. The left panel displays the time-series of the $x$ and $y$ coordinates. The middle panel displays the phase. The right panel displays the limit cycle in the phase plane. This figure displays three oscillations,  but these oscillations continue indefinitely. }
		\label{fig:circ}
	\end{center}
		\vspace*{-.5 true cm}
\end{figure}

To better visualize this system, let's think of $\theta$ as the position of a runner on this unit-circle track, with the differential equation $\frac{d\theta}{dt}$ describing the velocity of the runner. The left panel of Figure \ref{fig:circ} show the $x$- and $y$-coordinates as a function of time, the middle panel shows the phase, $\theta(t)$. Let $\theta=0$ refer to the right side of the track where $x=1$ and $y=0$. Notice that while (\ref{eq:circ}) would seem to indicate that $\theta$ increases monotonically, because it exists on the circle its domain is restricted to $[0,2\pi)$, with $\theta(0)=\theta(2\pi)$ as can be seen in the middle panel of Figure \ref{fig:circ}.

\begin{enumerate}

\item Assume that the runner has constant velocity measured in radians per hour. Write the differential equation $\frac{d\theta}{dt}$ that satisfies each of the following:
\begin{enumerate}
	\item The runner maintains a  constant velocity of $\omega$.               \\
		\teacher{\textcolor{red}{$\frac{d\theta}{dt}=\omega$  }}	\\
	\item The period of the runner (length of time it takes for the runner to complete one lap) is $1$ hour.           \\
		\teacher{\textcolor{red}{$\frac{d\theta}{dt}=2\pi$  }}	\\
	\item The period of the runner is $10$ hours.          \\
		\teacher{\textcolor{red}{$\frac{d\theta}{dt}=\frac{\pi}{5}$  }}	\\
\end{enumerate}			
Now let us add a second runner to this unit circle track and explore what happens if the two runners are attracted to each other, i.e.  if a runner is behind the other runner she will speed up and if she is ahead of the other runner she will slow down. For now let us assume that the two runners are identical in their individual velocity, $\omega$, and in how their velocity changes due to the presence of another runner. In the context of our predator-prey model, the two runners could represent the populations in the phase plane of animals in two separate habitats and the change in runner velocity could take the form of animals migrating between from the more dense habitat to the less dense one. If we assume that the runners attraction to each other is the sine of their difference in phase, then the velocities of the two runners can now be described by the following system of two differential equations (this formulation is known as the Kuramoto Model):
	
	%\begin{equation}
	\begin{align}
		\frac{d\theta_1}{dt}&=\omega+\frac{a}{2}\sin(\theta_2-\theta_1)	\nonumber	\\
		\frac{d\theta_2}{dt}&=\omega+\frac{a}{2}\sin(\theta_1-\theta_2)
		\label{eq:phase}
	\end{align}
	The parameter $a$ describes how much a runner slows down or speeds up due to the presence of the other runner.

	\item For each scenario below sketch the position of the two runners on the unit circle and calculate their velocities. Let $\omega=3$ and $a=1$.
	\begin{enumerate}
		\item Runner 1 is at $\theta_1=\frac{\pi}{6}$ and runner 2 is at $\theta_2=0$            \\
			\teacher{\textcolor{red}{Runner 1 has velocity 2.75 and runner 2 has velocity of 3.25  }}	
		\item Runner 1 is at $\theta_1=\frac{\pi}{3}$ and runner 2 is at $\theta_2=\frac{\pi}{6}$            \\
			\teacher{\textcolor{red}{same as part (a)  }}	
		\item Runner 1 is at $\theta_1=\frac{\pi}{2}$ and runner 2 is at $\theta_2=0$            \\
			\teacher{\textcolor{red}{Runner 1 has velocity 2.5 and runner 2 has velocity of 3.5  }}	
		\item Runner 1 is at $\theta_1=0$ and runner 2 is at $\theta_2=\frac{\pi}{2}$            \\
			\teacher{\textcolor{red}{Runner 1 has velocity 3.5 and runner 2 has velocity of 2.5  }}			
		\item Runner 1 is at $\theta_1=0$ and runner 2 is at $\theta_2=\pi$            \\
			\teacher{\textcolor{red}{Runners 1 and 2 have velocity 3 }}		
	\end{enumerate}
		
	\item What values of $a$ would correspond to the opposite situation, where the runners are repelled by each other? We see this phenomenon in inhibitory neurons, whose action potentials inhibits other nearby neurons from having an action potential.	\\
		\teacher{\textcolor{red}{$a<0$  }}		\\	
It is often the case in these types of systems that we would not care about the phase of each individual oscillator (the position of each runner), only about the difference between the two phases (distance between the two runners). Let's define the \emph{phase difference}, $\phi(t)\in[0,2\pi)$, as:
	\begin{equation}
		\phi=\theta_1 - \theta_2
		\label{eq:phi}
	\end{equation}
		
	\item Phase difference:
	\begin{enumerate}
		\item If $\phi=0$, what does that say about the two runners?		\\	\teacher{\textcolor{red}{The runners are together.  }}	
		\item If $\phi=\pi$, what does that say about the two runners?		\\	\teacher{\textcolor{red}{The runners are exactly half a lap apart from each other. }}	
		\item If $\phi=2\pi$, what does that say about the two runners?		\\	\teacher{\textcolor{red}{The runners are together.  }}	
		\item If $\phi=\pi/2$, what does that say about the two runners?		\\	\teacher{\textcolor{red}{Runner 1 is a quarter of a lap ahead of runner 2. }}		
		\item If $\phi=3\pi/2$, what does that say about the two runners?	\\	\teacher{\textcolor{red}{Runner 1 is a three quarters of a lap ahead of runner 2, or equivalently one quarter of a lap behind runner 2.  }}			
		\item Sometimes $\phi$ is defined as $\phi\in[-\pi,\pi)$. Why is this sometimes considered more intuitive?	\hspace{2em}	
			\teacher{\textcolor{red}{It is easier to see who is ahead, $\phi$ describe the lead that runner 1 has. Positive $\phi$ indicates runner 1 is ahead while negative $\phi$  indicates runner 1 is behind (answers may vary). }}	
	\end{enumerate}
	
	\item Find $\frac{d\phi}{dt}$. First take the derivative of both sides of  (\ref{eq:phi}), then substitute using (\ref{eq:phase}), and then simplify. (Hint: sine is an odd function ) \\				\teacher{\textcolor{red}{$\frac{d\phi}{dt}=-a\sin \phi $ }}	

	\item Find all equilibria for the phase difference.\\	
		\teacher{\textcolor{red}{$\phi^*=0,\pi$ }}
	\item Classifying the equilibria.
	\begin{enumerate}
		\item Sketch a graph with $\frac{d\phi}{dt}$ on the vertical axis and $\phi$ on the horizontal axis. Use this graph to classify the stability of all equilibria.	\\
			\teacher{	\textcolor{red}{See Figure 3.}
	       			\begin{figure*}[h].
					\begin{center}
            					\scalebox{.5}{\includegraphics{g_func}}
            					\caption{		
						\textcolor{red}{The solid curve is the graph of $G(\phi)=\frac{d\phi}{dt}$ from Question 5 and the dotted line is the horizontal axis. When $\frac{d\phi}{dt}$ is negative, $\phi$ decreases and when $\frac{d\phi}{dt}$ is positive, $\phi$ increases. Therefore, $\phi^*=0$ is stable and $\phi^*=\pi$ is unstable	}}
						\label{fig:phi}
					\end{center}
				\end{figure*}
			}
		
		\item Let $G(\phi)=\frac{d\phi}{dt}$. Classify each equilibria by finding $G'(\phi)$ and evaluating it at each equilibria. 	\\	
			\teacher{\textcolor{red}{
			$G'(\phi)=-a\cos \phi $ 	\\
			$G'(0)<0$ stable, $G'(\pi)>0$ unstable}}	
		\item What would change in (a) and (b) if we let $a<0$?	\\
			\teacher{\textcolor{red}{The graph would be reflected over the horizontal axis and the stability would switch. 	\\
			$G'(0)>0$ unstable, $G'(\pi)<0$ stable}}
			
	\end{enumerate}
	
	\item Based on our equilibria calculations above, what do we expect to happen to the runners if we let them run for a long time? Assume they are in excellent physical shape and never get tired.
		\\	\teacher{\textcolor{red}{They will eventually be running together (unless they start exactly 1/2 a lap apart from each other!!)}}		
	
	\item Let's find an expression for $\phi$ based on the differential equation you found in Question 5 to see if it agrees with your above equilibria analysis.
	\begin{enumerate}
		\item	Use the method of separation of variables to integrate the differential equation you found in Question 5. Solving for $\phi$ explicitly requires some tricky trigonometric identities so it may be easier to leave as an implicit expression.
			\\	\teacher{\textcolor{red}{
					Implicit: $\ln |\csc \phi - \cot \phi|=-t+C$	\hspace{2em} $C=\ln(\csc(\phi_0)-\cot(\phi_0))$ \\ 
					Explicit: $\phi(t)=2 \arccot \left(e^{at}\cot(\frac{\phi_0}{2})\right)$}}
		\item Use a graphing utility to graph your solution to (a). If necessary, make sure the graphing utility can handle implicit expression (such as desmos.com).\\ 
			\teacher{	\textcolor{red}{See Figure 4.}
	         \begin{figure*}[h!]
		\begin{center}
            		\scalebox{.23}{\includegraphics{co_trig3}}
			\label{fig:co_trig}
			\caption{ \textcolor{red}{This is a screen shot using desmos.com to graph the implicit (blue) and explicit (red) solutions to $\frac{d\phi}{dt}$ as found above. The horizontal axis of the graph is $t$ and the vertical axis is $\phi$. Because of limitations in Desmos, the explicit solution uses $f(t)$ instead of $\phi$ and $x_0$ as the initial condition $\phi_0$. The implicit solution uses $y$ instead of $\phi$ and $x$ instead of $t$.}}
		\end{center}
		\end{figure*}
	}
		\item Does this graphical solution agree with the equilibria you found?	
			\\	\teacher{\textcolor{red}{Yes, the phase difference decreases to zero which means runners get closer and closer to each other. }}	\\
	\end{enumerate}
	What if the runners no longer have the same independent velocities? We can model this by adding the terms $\omega_1$ and $\omega_2$ to (\ref{eq:phase}) such that  $\omega_1 \neq \omega_2$:
		
	\begin{align}
		\frac{d\theta_1}{dt}&=\omega_1+\frac{a}{2}\sin(\theta_2-\theta_1)	\nonumber	\\
		\frac{d\theta_2}{dt}&=\omega_2+\frac{a}{2}\sin(\theta_1-\theta_2)
		\label{eq:phase_hetero}
	\end{align}
	
	\item Let runner 1 be slightly faster than runner 2: 
	$$\omega_1-\omega_2=1/3$$
	Let $a=1$.
	
	\begin{enumerate}
		\item Find $\frac{d\phi}{dt}$.		
			\\	\teacher{\textcolor{red}{$\frac{d\phi}{dt}=\frac{1}{3}-\sin \phi $ }}
		\item  Find and classify  all equilibria (sketch $\frac{d\phi}{dt}$ on the vertical axis and $\phi$ on the horizontal axis).	
			\\	\teacher{\textcolor{red}{This is a vertical shift of the graph from when the runners had the same speed $\phi^*=.253$ is stable, $\phi^*=2.89$ is unstable.  }}
		\item What do we expect to happen to the runners if we let them run for a long time (assume they are in excellent physical shape and never get tired).
			\\	\teacher{\textcolor{red}{Runner 1 will be slightly ahead of runner 2 (.253 radians).}}
	\end{enumerate}
	
	\item Consider the situation where runner 1 is much faster than runner 2: $$\omega_1-\omega_2=2$$ 
	Let $a=1$.
	\begin{enumerate}
		\item Find $\frac{d\phi}{dt}$.
					\\	\teacher{\textcolor{red}{$\frac{d\phi}{dt}=2-\sin \phi $ }}
		\item Find and classify all equilibria.
					\\	\teacher{\textcolor{red}{There are no equilibria!! }}		
		\item What do we expect to happen to the runners if we let them run for a long time (assume they are in excellent physical shape and never get tired)
					\\	\teacher{\textcolor{red}{Runner 1 is always faster than runner 2 and will keep lapping her. This is known as \emph{phase drift}.}}	
	\end{enumerate}

	\item Bifurcation
	\begin{enumerate}
		\item Considering Questions 10 and 11, what can we say about the relationship between $\omega_1-\omega_2$ and $a$ that is required for equilibria to exist?
					\\	\teacher{\textcolor{red}{Need $|\omega_1-\omega_2| \le a$ for equilibria to exist}}	
		\item Describe how the equilibria change as the difference in speeds is increased from $0$ to $2$.
						\\	\teacher{\textcolor{red}{The two equilibria get closer and closer to each other and eventually collide \\ (when $|\omega_1-\omega_2|= a$) and then disappear. This is a saddle node bifurcation. }}	\\
	\end{enumerate}	
Let us return to our predator-prey system of Figure \ref{fig:rm}. Assume that coupling between the oscillators takes the form of migration and that prey and predators leave the higher population oscillator and move to the lower population oscillator.
	\item If we reduce the predator-prey to a phase model, what can we say about the value of $a$?	\\
	\teacher{\textcolor{red}{$a>0$. The two oscillators will be attracted to each other.}}		\\	\\
In our previous phase model  (\ref{eq:phase_hetero}), the change in a runner's speed due to the other runner was only a function of the difference in phases between the two runners. In other words, if runner 1 was $1/4$ lap behind runner 2, she would speed up the same amount regardless of where on the track they both were. This is not going to be true in our predator-prey model because the movement around the cycle (track) is not constant, see the right panel of Figure \ref{fig:rm}. Therefore, the effect of migration on the oscillator depends on the phase difference between the oscillators and also on the phase of that oscillator. Additionally, we want to allow for a different coupling function as it is unlikely that animal migration should follow a sine curve. Now the differential equations describing the phase of each oscillator (\ref{eq:phase_hetero}) are written:
					$$	\frac{d\theta_1}{dt}=\omega_1+\frac{a}{2}h_1(\theta_2-\theta_1)$$		
					$$	\frac{d\theta_2}{dt}=\omega_2+\frac{a}{2}h_2(\theta_1-\theta_2)$$	
	The effect of migration on the velocity around the cycle of the predator-prey populations in patch 1 and 2 is described by the functions $h_1$ and $h_2$, respectively. To reduce the original predator-prey system to this phase model requires that migration be small compared to the predation, reproduction, and death rates of the populations. Actually calculating or approximating the functions $h_1$ and $h_2$ requires asymptotic analysis or perturbation theory and it is beyond the scope of a first differential equations course. However, we will skip the details of the mathematical derivation and as in Question $\#10$, we can find a differential equation describing the difference in phase between the two oscillators using the above phase model and \ref{eq:phi}. Letting $\omega=\omega_1-\omega_2$ and $H(\phi)=\frac{a}{2}(h_1(-\phi)-h_2(\phi))$, we get:
	$$\frac{d\phi}{dt}=\omega+H(\phi)$$
	The function $H(\phi)$ is often called the interaction function and depending on its properties, the above differential equations describing the change in phase difference can exhibit a variety of behaviors. Recall that because $\phi=0$ is identical to $\phi=2\pi$, we have previously found $\frac{d\phi}{dt}$ that lead to two different equilibria (Questions 6 and 10) and to no equilibria (Question 11). 
	
	\item Assume that the two predator-prey systems are identical, i.e.  $\omega_1=\omega_2$ and $h_1=h_2$. Sketch a graph of $G(\phi)=\frac{d\phi}{dt}$ that leads to three or more equilibrium on the interval $[0,2\pi)$. Describe the behavior of this equilibria behavior of this system and how it depends on initial conditions. Note that  $G(\phi)$ and $H(\phi)$ are periodic functions with period $2\pi$. \\
		\teacher{
		\textcolor{red}{There are many possible answers. The requirements are that $G(\phi)$ has the symmetry that $G(a)=-G(2\pi-a)$ and that $G(\pi)=0=G(2\pi)$. These two properties are consequences of $h_1=h_2$ and $\omega_1=\omega_2$ respectively (these two properties require $G(\pi)=0$). See Figure 5 for one such example.}
	       		\begin{figure*}[h]
				\begin{center}
            				\scalebox{.5}{\includegraphics{g_func4}}
            				\caption{		
					\textcolor{red}{This example $G(\phi)=\frac{d\phi}{dt}$ appeared in a recent peer-reviewed article using the same predator-prey model and similar assumptions to what has been discussed so far with $\omega_1 = \omega_2$. There are four equilibria values for the phase. There are stable equilibria when the oscillators are at the same phase, $\phi^*=0$, and at opposite phases, $\phi^*=\pi$. Unstable equilibria occur at  $\phi^* \approx 1, 2\pi-1$. Depending on the initial conditions, the two oscillators will either synchronize together or always be at opposite ends of the cycle from each other.}}
						%\label{fig:phi}
				\end{center}
			\end{figure*}
		}
		
		
	\item Assume $\omega_1>\omega_2$ and $H(\phi)$ as in Question 14. Sketch several graphs of $G(\phi)$ as the difference between $\omega_1$ and $\omega_2$ increases. Describe what happens to the equilibrium.
			\teacher{
		\textcolor{red}{
		These graph are vertical shifts of the functions from the previous problem. As the difference between $\omega_1$ and $\omega_2$ is increase, the vertical shift increase. This leads to the elimination of two of the zeros of the function $G(\phi)$ at a time. The corresponds to to two equilibria colliding and dissapearing. As in problem 12, these are saddle node bifurcations. See Figure 6 for a smaller vertical shift where the system goes from four equilibria to two equilibria and Figure 7 for a large shift such that the system has zero equilibria.
		}
	       		\begin{figure*}[h]
				\begin{center}
            				\scalebox{.5}{\includegraphics{g_func4_shift_small}}
            				\caption{		
					\textcolor{red}{
					A small vertical shift of the function that is graphed in Figure 5 leading to two equilibria.
					}}
				%\label{fig:phi}
				\end{center}
			\end{figure*}
			
			\begin{figure*}[h]
				\begin{center}
            				\scalebox{.5}{\includegraphics{g_func4_shift_large}}
            				\caption{		
					\textcolor{red}{
					A large vertical shift of the function that is graphed in Figure 5 leading to no equilibria.
					}}
				%\label{fig:phi}
				\end{center}
			\end{figure*}
		}
	
	
	\item Assume $\omega_1 \neq \omega_2$ and $h_1 \neq h_2$. Sketch a graph of $G(\phi)$ and find all equilibrium.	\\
				\teacher{
		\textcolor{red}{
		The function $G(\phi)$ needs to be continuous and periodic with period $2\pi$, but otherwise any shape is possible as long as $G(\phi)\neq 0$ and it does not have the symmetry described above. It should make sense ecologically that $G(\phi)$ is differentiable everywhere, though we did not explicitly assume that in the text.
		}
		}

\end{enumerate}


\end{document}
