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.

Global well-posedness and pattern formations of the immune system induced by chemotaxis.

This paper studies a reaction-diffusion-advection system describing a directed movement of immune cells toward chemokines during the immune process. We investigate the global solvability of the model based on the bootstrap argument for minimal chemotaxis models. We also examine the stability of nonconstant steady states and the existence of periodic orbits from theoretical aspects of bifurcation analysis. Through numerical simulations, we observe the occurrence of steady or time-periodic pattern formations.

Bacterial chemotaxis without gradient-sensing.

Chemotaxis models are based on spatial or temporal gradient measurements by individual organisms. The key contribution of Keller and Segel (J Theor Biol 30:225-234, 1971a; J Theor Biol 30:235-248, 1971b) is showing that erratic measurements of individuals may result in an accurate chemotaxis phenomenon as a group. In this paper we provide another option to understand chemotactic behavior when individuals do not sense the gradient of chemical concentration by any means. We show that, if individuals increase their dispersal rate to find food when there is not enough food, an accurate chemotactic behavior may be obtained without sensing the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol 75:845-870, 2013) and was called starvation driven diffusion. This model is surprisingly similar to the original Keller-Segel model. A comprehensive picture of traveling bands and fronts is provided.