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\begin{document}
\begin{center}
\huge\textbf{Homework 6} \normalsize \\ Due Friday, August 11 2017
\end{center}
\problem{Problem 1 (10 points)}
Suppose that we have a linear system of differential equations given by
\begin{equation} \label{eq:system}
\left\{ \begin{array}{l} \dot{x}_1 = ax_1 + bx_2, \\ \dot{x_2} = cx_1 + dx_2 . \end{array} \right.
\end{equation}
Writing this as a matrix equation, we have $\dot{\mathbf{x}} = A\mathbf{x}$, where
\begin{equation}
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.
\end{equation}
Recall that the trace of a matrix is the sum of its diagonal entries, so $\textrm{Tr}(A) = a + d$ and that the determinant of a $2\times 2$ matrix is $\textrm{det}(A) = ad - bc$.
\begin{enumerate}[(a)]
\item Find the eigenvalues of $A$ in terms of $\textrm{Tr}(A)$ and $\textrm{det}(A)$. (You should not have any $a$'s, $b$'s, $c$'s or $d$'s in your answer.)
\item Find conditions on $\textrm{Tr}(A)$ and $\textrm{det}(A)$ such that the equilibrium of \eqref{eq:system} is a stable node, an unstable node, a stable spiral, an unstable spiral or a saddle. For instance, you should find that the equilibrium is a saddle if and only if $\textrm{det}(A) < 0$.
\item Make a graph with $\textrm{Tr}(A)$ on the $x$-axis and $\textrm{det}(A)$ on the $y$-axis and mark the regions where \eqref{eq:system} has each type of equilibrium.
\end{enumerate}
\problem{Problem 2 (15 points)}
Suppose that we have two species of herbivores in the same region competing with each other for resources. We will let $N(t)$ represent the population density of one of the species at time $t$ and let $M(t)$ represent the population of the other species at time $t$. We will assume that each species would grow logistically in the absence of competition. That is, if $M(t) = 0$, then we would have
\begin{equation}
\deriv{N}{t} = r_N N\left(1 - \frac{N}{K_N}\right),
\end{equation}
where $r_N, K_N > 0$ are constants. Similarly, if $N(t) = 0$, then we would have
\begin{equation}
\deriv{M}{t} = r_M M\left(1 - \frac{M}{K_M}\right),
\end{equation}
where $r_M, K_M > 0$ are constants. However, if the two species are both present, then both of their growth rates are reduced. We will assume that the reduction in per-capita growth rate is proportional to the population density of the competing species. This means that
\begin{align*} \label{eq:lv-comp}
\deriv{N}{t} &= r_N N\left(1 - \frac{N}{K_N} - \frac{\alpha M}{K_N}\right), \numberthis \\
\deriv{M}{t} &= r_M M\left(1 - \frac{M}{K_M} - \frac{\beta N}{K_M}\right),
\end{align*}
where $\alpha, \beta > 0$. For ease of calculation, we will assume that $\alpha\beta \neq 1$.
This is called the Lotka-Volterra competition model.
\begin{enumerate}[(a)]
\item Show that if we let $x = N/K_N$, $y = M/K_M$, $\tau = r_Nt$, $r = r_M/r_N$, $a = \alpha K_M/K_N$ and $b = \beta K_N/K_M$, we can rewrite \eqref{eq:lv-comp} as
\begin{align*} \label{eq:nondim}
\deriv{x}{\tau} &= x\left(1 - x - ay\right), \numberthis \\
\deriv{y}{\tau} &= ry\left(1- y - bx\right).
\end{align*}
\item Find all four of the equilibria of \eqref{eq:nondim}. (They may depend on the parameters $a$, $b$ and $r$.) Three of these should be fairly easy, and the fourth is somewhat more complicated. Under what conditions is there a positive equilibrium? (That is, when is there an equilibrium $(x^*, y^*)$ with both $x^* > 0$ and $y^* > 0$?)
\item Find the Jacobian of this system. That is, if $f_1(x,y) = x(1 - x - ay)$ and $f_2(x,y) = ry(1 - y - bx)$, find $Df(x)$.
\item Use the Jacobian from part (c) to find the linearization of \eqref{eq:nondim} about each equilibrium you found in part (b). Classify each equilibrium as a stable/unstable node, a stable/unstable spiral or a saddle. Since $x$ and $y$ represent populations, you do not need to classify any equilibrium with $x^* < 0$ or $y^* < 0$. Is there any circumstance where the positive equilibrium is stable? (\textbf{Update}: If the positive equilibrium is stable, you do not need to determine whether it is a node or spiral, but doing so will be worth some extra credit.)
\end{enumerate}
\problem{Problem 3 (15 points)}
Consider the model of a pendulum that we discussed in class:
\begin{equation}
\ddot{\theta} = -a\dot{\theta} - b\sin\theta.
\end{equation}
As in class, we will let $x_1 = \theta$ and $x_2 = \dot{\theta}$, so we have the system
\begin{equation} \label{eq:pendulum}
\begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} x_2 \\ -ax_2 - b\sin x_1 \end{pmatrix} = \begin{pmatrix} f_1(x_1, x_2) \\ f_2(x_1, x_2) \end{pmatrix}.
\end{equation}
We already found that this system has equilibria at $(x_1^*, x_2^*) = (n\pi, 0)$, where $n$ is any integer. Moreover, we found that the linearized system is
\begin{equation}
\begin{pmatrix} \dot{y}_1 \\ \dot{y}_2 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -b\cos(n\pi) & -a \end{pmatrix}\begin{pmatrix} y_1 \\ y_2 \end{pmatrix},
\end{equation}
where $y_1 = x_1 - n\pi$ and $y_2 = x_2$.
\begin{enumerate}[(a)]
\item If our pendulum is frictionless, then $a = 0$. What can you conclude about the stability and type of each equilibrium in this case? (You should look at the cases where $n$ is even and where $n$ is odd separately.)
\item Since this is a physical system, we can calculate its energy (kinetic energy plus potential energy). After taking into account all the changes of variables we made, we find that the energy is
\begin{equation}
E(x_1, x_2) = \frac{1}{2}(x_2)^2 - b\cos(x_1).
\end{equation}
Draw the level sets of $E$. That is, on a graph with $x_1$ on the $x$-axis and $x_2$ on the $y$-axis, plot $E(x_1,x_2) = C$ for various values of $C$.
\item Show that if $a = 0$ the energy is conserved. That is,
\begin{equation}
\deriv{}{t}E(x_1(t), x_2(t)) = 0.
\end{equation}
Using this result and your graph from part (b), sketch the phase portrait of \eqref{eq:pendulum} when $a = 0$.
\item Show that if $a > 0$ then the energy is strictly decreasing (unless the pendulum is at equilibrium). That is, show that
\begin{equation}
\deriv{}{t}E(x_1(t), x_2(t)) \leq 0,
\end{equation}
with equality only when $x_1(t) = n\pi$ and $x_2(t) = 0$. What does this tell you about the phase portrait of \eqref{eq:pendulum} when $a > 0$?
\end{enumerate}
\end{document}