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\centerline{{\sc Problem set 2}}
\centerline{Math 207A, Fall 2011}
\centerline{Due: Wed., Oct. 12}
\bigskip\noindent
{\bf 1.} Solve the IVP for the logistic equation
\[
x_t = x(1-x),\qquad x(0) = x_0.
\]
\bigskip\noindent
{\bf 2.} Consider bacterial growth in a closed flask with a fixed initial amount of nutrient, and suppose that the growth rate
of the bacteria is proportional to the amount of available nutrient. If $N(t)$ denotes the population of bacteria
and $C(t)$ denotes the available nutrient at time $t$, explain why the ODEs
\[
N_t = \mu C N,\qquad C_t = - \alpha \mu CN
\]
provide a reasonable model for suitable constants $\alpha, \mu > 0$. Solve the system subject to
the initial conditions
\[
N(0) = N_0,\qquad C(0) = C_0
\]
where $N_0, C_0 > 0$.
Express the limiting population of bacteria
\[
N_\infty = \lim_{t\to\infty} N(t)
\]
in terms of
$\alpha$, $\mu$, $N_0$, $C_0$. Does your answer make sense?
\bigskip\noindent
{\bf 3.} Let
\[
f(x) = \begin{cases}
x^2 \sin(1/x) & x\ne 0,
\\
0 & x= 0.
\end{cases}
\]
Find the equilibria of the ODE
$x_t = f(x)$ and
determine their stability, and sketch the phase line.
\bigskip\noindent
{\bf 4.} Graph the bifurcation diagram for equilibrium solutions of the scalar ODE
\[
x_t = \mu + x - x^3
\]
versus $\mu$ and determine their stability. (You don't have to give an explicit expression for the equilibria.)
Find the values of $(x,\mu)$ at which equilibrium bifurcations occur. What kind of bifurcations
are they? Sketch the phase line of the system for different values of $\mu$, including the values
at which bifurcations occur. Describe what would happen if the system is in equilibrium and
$\mu$ is increased very slowly from $\mu = -1$ to $\mu =1$ and then decreased back to $\mu=-1$.
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