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\centerline{{\sc Problem set 3}: Math 207A, Fall 2018}
\centerline{Due: Fri., Oct. 19}


\bigskip\noindent
{\bf 1.} A model for a population $x(t)\ge 0$ with logistic growth and a constant rate of harvesting is
\[
x_t = \mu x\left(1-\frac{x}{K}\right) - H
\]
where the parameters $\mu$, $K$, $H$ are positive constants.

\smallskip\noindent
(a) Show that a nondimensionalized form of the equation is
\begin{equation}
x_t = x(1-x) - h,
\label{ode}
\end{equation}
and express the dimensionless parameter $h$ as a ratio of two times.

\small\noindent
(b) Sketch a graph of the equilibria as functions of $h$, and sketch the phase line of \eqref{ode} for various values of $h > 0$. Determine the stability of the equilibria, both from the phase line and from their linearized stability. For what values of the initial (nondimensionalized) population $x_0 > 0$ and harvesting rate $h>0$ does the population become extinct?


\bigskip\noindent
{\bf 2.} The graph $y=f(x)$ of a Lipschitz function $f : \Rl \to \Rl$ is shown below. The function $f$ has zeros only at certain integer values of $x$ and is never zero outside the $x$-interval shown.

\smallskip\noindent
(a) Sketch the phase line for the ODE $x_t = f(x)$ and state the stability of the equilibria. Which of the equilibria are hyperbolic?


\smallskip\noindent
(b) Sketch the graph of the solution of the initial value problem with $x(0) = 0$.


\smallskip\noindent
(c) Sketch the graph of a potential $E(x)$ such that $f(x) = -E'(x)$.


\centerline{
\includegraphics[width=0.7\textwidth]{scalarf}
}

\bigskip\noindent
{\bf 3.} A spherical raindrop with volume $V(t)$ and surface area $A(t)$ evaporates at a rate proportional to
its surface area, meaning that $V_t = -k A$ for some constant $k > 0$. Write down an ODE for $V$ and show that
the raindrop evaporates completely in finite time. Find an expression for the evaporation time $T$ in terms of
$k$ and the initial volume $V_0$ of the drop, and verify that your result is dimensionally consistent. Why doesn't this result violate the
uniqueness part of the Picard theorem?

\bigskip\noindent
{\bf 4.} (a) Consider a scalar ODE $x_t = f(x)$ where $f:\Rl \to \Rl$ is continuous. Prove that the ODE cannot have
a non-constant periodic solution with minimal period $T > 0$ such that $x(t+T) = x(t)$ for all $t \in \Rl$.
\textsc{Hint.} Consider the integral
\[
\int_0^T f(x) x_t \, dt,
\]
(b) Why doesn't your argument in (a) apply to an ODE $\theta_t = f(\theta)$ on the circle $\Ts$?

\bigskip\noindent
{\bf 5.} (a) A Bernoulli equation is an ODE of the form
\[
x_t = a(t)x + b(t) x^n
\]
where $a$, $b$ are continuous functions and $n\ne1$.
Show that the transformation
\[
u = \frac{1}{x^{n-1}}
\]
reduces a Bernoulli equation to a linear equation for $u$.
Use this transformation to solve the logistic equation $x_t = x(1-x)$.


\smallskip\noindent
(b) A Riccati equation is an ODE of the form
\[
x_t = a(t) + b(t) x + c(t) x^2.
\]
where $a$, $b$, $c$ are continuous functions, with $c\ne 0$. Show that the transformation
\[
x = -\frac{u_t}{cu}
\]
reduces the Riccati equation to a
second order, linear equation for $u$. Use this transformation to solve the logistic equation $x_t = x(1-x)$.
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