  We investigate a minimal model for cell propagation involving migration along
self-generated signaling gradients and cell division, which has been proposed
in an earlier study. The model consists in a system of two coupled parabolic
diffusion-advection-reaction equations. Because of a discontinuous advection
term, the Cauchy problem should be handled with care. We first establish
existence and uniqueness locally in time through the reduction of the problem
to the well-posedness of an ODE, under a monotonicity condition on the
signaling gradient. Then, we carry out an asymptotic analysis of the system.
All positive and bounded traveling waves of the system are computed and an
explicit formula for the minimal wave speed is deduced. An analysis on the
inside dynamics of the wave establishes a dichotomy between pushed and pulled
waves depending on the strength of the advection. We identified the minimal
wave speed as the biologically relevant speed, in a weak sense, that is, the
solution propagates slower, respectively faster, than the minimal wave speed,
up to time extraction. Finally, we extend the study to a hyperbolic
two-velocity model with persistence.
