Math 6410 • ODE

Consider Griffith's model for a genetic control system, where \(x\) and \(y\) are nonnegative
concentrations of a protein and messenger RNA, and \(a\) and \(\mu\) are positive constants.

\[
\dot{x}=y-\mu x,\quad x(0)=x_0
\]
\[
\dot{y}=a+\frac{x^2}{1+x^2}-y,\quad y(0)=y_0. \tag{2}
\]

\begin{enumerate}

\item
[12] Show that the positive quadrant is positively invariant. Show that there is a
rectangle in the positive quadrant that is positively invariant.


\item
[11] Show graphically that there is a region of the \(a-\mu\) parameter space for which there
is a unique stable steady state solution, and a region of parameter space in which
there are three steady state solutions. Identify the stability of the solutions. Give
some identifying features of these regions.


\item
[2] What is the bifurcation structure of the solutions as a function of \(\mu\)?

\end{enumerate}