Math 6410 • ODE

\begin{enumerate}
\item Prove Bendixson's Criterion: Let \(\mathbf{f}\) be of class \(C^1\) in a simply
connected region \(E\) in \(\R^2\). Assume the divergence of \(\mathbf{f}\) is not identically
zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then
\(\dot{x}=\mathbf{f}(x)\) has no closed orbit which is entirely in \(E\).
\item Extend the above proof to cover Dulac's Criterion: Let \(B(x_1,x_2)\) and
\(\mathbf{f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\R^2\). Assume the
divergence of \(B\mathbf{f}\) is not identically zero and is either nonnegative in \(E\) or
nonpositive in \(E\). Then \(\dot{x}=\mathbf{f}(x)\) has no closed orbit which is entirely
in \(E\).
\item Consider the nondimensionalized Lotka-Volterra competition equations
\[
\begin{aligned}
  \dot{u}&=ru(1-u-\alpha_1v),\\
  \dot{v}&=v(1-v-\alpha_2u),
\end{aligned}
\]
where \(r\), \(\alpha_1\) and \(\alpha_2\) are positive parameters. Apply Dulac's criterion to
this system for \(u,v>0\). (HINT: try \(B(u,v)=1/uv\).) Does Bendixson's criteria give a
similar result?
\item Draw a biological conclusion.
\end{enumerate}