Running tails as codimension two quasi-solitons in excitation taxis waves with negative refractoriness

We describe a new type of wave phenomena observed in reaction-taxis systems of equations. This is “running tail”, a localized stable perturbation steadily moving laterally along the back of a plane wave. This phenomenon is related to “negative refractoriness”, a property observed in some excitable systems with cross-diffusion instead of usual diffusion. We suggest a simple mechanism of such running tails for the Keller–Segel model describing chemotaxis of bacteria on the nutrient substrate. We also demonstrate that collision of running tails may happen by “quasi-soliton” and “half-soliton” scenarios.