  Self-generated gradients have atttracted a lot of attention in the recent
biological literature. It is considered as a robust strategy for a group of
cells to find its way during a long journey. This note is intended to discuss
various scenarios for modeling traveling waves of cells that constantly deplete
a chemical cue, and so create their own signaling gradient all along the way.
We begin with one famous model by Keller and Segel for bacterial chemotaxis. We
present the model and the construction of the traveling wave solutions. We also
discuss the limitation of this approach, and review some subsequent work
addressing stability issues. Next, we review two relevant extensions, which are
supported by biological experiments. They both admit traveling wave solutions
with an explicit value for the wave speed. We conclude by discussing some open
problems and perspectives, and particularly a striking mechanism of speed
determinacy occurring at the back of the wave. All the results presented in
this note are illustrated by numerical simulations.
