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\centerline{{\sc Problem set 7}}
\centerline{Math 207A, Fall 2011}
\centerline{Due: Wed., Nov. 23}




\bigskip\bigskip\noindent
{\bf 1.} Classify the equilibrium $(x,y) = (0,0)$ of the system
\[
x_t = x, \qquad y_t = -y + x^2.
\]
Is the equilibrium hyperbolic? Find an equation for the trajectories
in $(x,y)$-phase space, and sketch the phase plane. What are the stable and unstable subspaces
$E^s$ and $E^u$ and the stable and unstable manifolds
$W^s(0,0)$ and $W^u(0,0)$ of the origin?



\bigskip\noindent
{\bf 2.} Write the system
\begin{align*}
&x_t = x - y - x\left(x^2 + y^2\right)
\\
&y_t = x + y - y\left(x^2 + y^2\right)
\end{align*}
in polar coordinates. Classify the equilibrium $(x,y) = (0,0)$ and sketch the phase portrait.
How do solutions behave as $t\to \infty$?

\bigskip\noindent
{\bf 2.} Find and classify the equilibria of the system
\[
x_t = \mu x - x^2, \qquad y_t = -y.
\]
Sketch the phase portraits for $\mu < 0$, $\mu = 0$, and $\mu > 0$.
In each case, say if
the equilibria are hyperbolic and describe their stable and unstable subspaces
$E^s$ and $E^u$ and their stable and unstable manifolds
$W^s$ and $W^u$.


\bigskip\noindent
{\bf 4.} Consider the following model for the dynamics of a predator with population $x(t)$ and a prey with population $y(t)$
\textit{e.g.} pikes and eels, or foxes and rabbits:
\begin{align*}
&x_t = x \left(-1 + y\right),
\\
&y_t = y\left(1 - x\right).
\end{align*}
Explain why this is a reasonable qualitative model for a predator-prey system. Find the equilibria and classify them.
Sketch the phase portrait. How do solutions behave?

\smallskip\noindent
\textsc{Hint.} To find the trajectories, solve the first order ODE for $y$ as a function of $x$ that is obtained from
\[
\frac{d y}{dx} = \frac{y_t}{x_t}.
\]
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