Nonlinear Diffusion for Bacterial Traveling Wave Phenomenon

The bacterial traveling waves observed in experiments are of pulse type which is different from the monotone traveling waves of the Fisher–KPP equation. For this reason, the Keller–Segel equations are widely used for bacterial waves. Note that the Keller–Segel equations do not contain the population dynamics of bacteria, but the population of bacteria multiplies and plays a crucial role in wave propagation. In this paper, we consider the singular limits of a linear system with active and inactive cells together with bacterial population dynamics. Eventually, we see that if there are no chemotactic dynamics in the system, we only obtain a monotone traveling wave. This is evidence that chemotaxis dynamics are needed even if population growth is included in the system.