Math 6410 • ODE

Consider Griffith's model for a genetic control system, where \(x\) and \(y\) are proportional
to concentration of protein and the messenger RNA from which it is translated, respectively,
and \(\mu>0\) is a rate constant
\[
\begin{aligned}
  \dot{x} &= y-\mu x,\\
  \dot{y} &= \frac{x^2}{1+x^2}-y.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Show that the system has three fixed points when \(\mu<\mu_c\) and one when
\(\mu>\mu_c\) where \(\mu_c\) is to be determined.
\item[(b)] What is the nature of the bifurcation at \(\mu=\mu_c\)?
\end{enumerate}