Math 6410 • ODE

Consider a modified predator-prey system for populations \(x,y\) given by
\[
\begin{aligned}
  \dot{x} &= x\cdot\left(1-\frac{x}{4}-\frac{y}{1+x}\right),\\
  \dot{y} &= y\cdot\left(\frac{x}{1+x}-\frac12\right)
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Show that the triangle with vertices \((0,0)\), \((5,0)\) and \((0,5)\) is a
forward invariant subset.
\item[(b)] Find the fixed points and determine their stability.
\item[(c)] Show that the system has at least one nonconstant periodic solution.
\end{enumerate}