\documentclass[12pt]{article}
\pagestyle{empty}
\usepackage{amsfonts,amsmath}
\def\Rl{\mathbb{R}}
\begin{document}
\centerline{{\sc Problem set 3}}
\centerline{Math 207A, Fall 2011}
\centerline{Due: Wed., Oct. 19}
\bigskip\noindent
{\bf 1.} For each of the the following systems, find the equilibria and their stability.
Determine what bifurcations occurs, sketch the bifurcation diagram, and sketch the qualitatively different phase
lines:
\[
\mbox{(a)}\ x_t = \mu - x^2 + x^4;
\qquad
\mbox{(b)}\ x_t = \mu x + x^3 - x^5;
\qquad
\mbox{(c)}\ x_t = \mu x - e^{x}.
\]
\bigskip\noindent
{\bf 2.} (a) Consider a pair of rigid rods of length $L$ connected by a torsional spring
with spring constant $k$ that resists bending. If the rods are subject to a compressive force $\lambda$,
and $x$ is the angle of the rods
to the applied force, explain why
\[
V(x) = \frac{1}{2} k x^2 + 2\lambda L (\cos x -1)
\]
is a reasonable expression for the potential energy of the system.
\smallskip\noindent
(b) Show that equilibrium solutions such that $V^\prime(x)=0$ satisfy the equation
\[
x-\mu \sin x = 0
\]
where $\mu>0$ is a suitable dimensionless parameter. Find and classify the bifurcation point
on the branch $x=0$ and give a physical interpretation. Sketch the behavior of the potential $V(x)$ as
$\mu$ passes through the bifurcation value.
\bigskip\noindent
{\bf 3.} (a) A model of a fishery with harvesting is
\[
N_t = \mu N\left(1-\frac{N}{K}\right) - \frac{HN}{A + N}
\]
where $N(t)$ is the population of fish at time $t$ and $\mu$, $K$, $H$, $A$ are positive parameters.
Explain why this is a reasonable model and give a biological interpretation of
each of the parameters.
\smallskip\noindent
(b) Show that the ODE can be put in dimensionless form
\[
x_t = x(1-x) - \frac{hx}{a+x}
\]
where $t$ is a suitably rescaled time and $a$, $h$ are dimensionless parameters. Give expressions for $a$, $h$
in terms of the original dimensional parameters.
\smallskip\noindent
(c) Carry out a bifurcation analysis of the ODE in (b). Discuss the implications of your results for the
original fish-harvesting problem.
\smallskip\noindent
\end{document} % End of document.