Proving Existence of Heteroclinic Orbit between Saddle and Focus

Question

I am working on proving the existence of a travelling wave solution to a PDE. This problem has been reduced to finding a heteroclinic orbit in the phase plane of the following (non-dimensionalized) system: \begin{align*} x'=y, y'=cy+\frac{x(1-x)}{k+x} \end{align*} where c is the constant wave speed, and 0<k<1 is a constant. The case that I'm struggling with is when \begin{align*} -\sqrt{\frac{1}{k+1}}<c<0. \end{align*} In the phase plane, this results in a saddle point at (0,0) and a stable focus at (1,0). It seems obvious that an orbit would exist that asymptotically connects the saddle point to the focus (which would result in a non-monotone travelling wave solution), but this has been challenging to prove analytically. I was able to use the Poincare-Bendixson Theorem to prove the existence in the case where (1,0) is a node, but there is no clear invariant set to work with in this case, due to the nature of the focus. The image [Phase Portrait][1] shows a possible trajectory (in yellow), in the case where c=-0.5 and k=0.5. The other curves are nullclines. I am a third year undergrad student and very new to this field of study, so any ideas/advice on how to prove the existence of such a solution would be greatly appreciated. [1]: https://i.stack.imgur.com/shBnv.png