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

TL;DR

This paper constructs and analyzes counter-propagating traveling waves in a system of four coupled hyperbolic transport-reaction equations modeling motile organisms, revealing conditions for wave formation and stability.

Contribution

It explicitly constructs and characterizes counter-propagating waves in a novel four-equation transport-reaction system derived from age-structured species modeling.

Findings

Explicit construction of counter-propagating waves

Conditions for wave stability and formation

Analysis of pulsating-in-time solutions

Abstract

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.