  Hyperbolic transport-reaction equations are abundant in the description of
movement of motile organisms. Here, we focus on system of four coupled
transport-reaction equations that arises from an age-structuring of a species
of turning individuals. The highlight consists of the explicit construction and
characterization of counter-propagating traveling waves, patterns which have
been observed in bacterial colonies. Stability analysis reveals conditions for
the wave formation as well as pulsating-in-time spatially constant solutions.
