Counter-propagating waves in a system of transport-reaction equations

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.