Stability and existence of selected monotonic nontrivial traveling waves

The systems under consideration admit traveling waves, including trivial traveling waves and, in some parameter regimes, nontrivial traveling waves with profiles $N$ and $P$. The numerical experiments select monotonic nontrivial traveling waves in certain regions of parameter space. Stability and existence conjecture. In the whole parameter region where monotonic nontrivial traveling waves are numerically selected, trivial traveling waves are unstable, and monotonic nontrivial traveling waves exist and are stable. This is a rough conjecture motivated by numerical experiments; the existence and stability of nontrivial traveling waves remain open, and the appropriate notion of stability would need to be specified before a rigorous proof.