Suppose two periodic orbits $\Gamma_1$ and $\Gamma_2$ exist in $D$, both encircling the hole. Then one of the regions between the two orbits — call it $R$ — lies entirely within $D$ and is simply connected (bounded by arcs of $\Gamma_1$ and $\Gamma_2$). Applying Dulac's criterion to $R$ with the same $\phi$ yields \begin{align*} 0 = \oint_{\partial R} \phi\, f \cdot n\, d\ell = \int_R \nabla \cdot (\phi\, f)\, dA, \end{align*} which contradicts $\nabla \cdot (\phi\, f)$ being of one sign in $D \supset R$. No periodic orbit in $D$ can avoid the hole, since any such orbit would lie in a simply connected subset, contradicting the basic Dulac criterion.