Traveling wave solutions for a bacteria system with density-suppressed motility

Roger Lui; Hirokazu Ninomiya

doi:10.3934/dcdsb.2018213

Article Contents

2019, Volume 24, Issue 2: 931-940. Doi: 10.3934/dcdsb.2018213

This issue Previous Article Global weak solutions for a coupled chemotaxis non-Newtonian fluid Next Article Hermite spectral method for Long-Short wave equations

Traveling wave solutions for a bacteria system with density-suppressed motility

Roger Lui^1, and
Hirokazu Ninomiya^2,

1.
Department of Mathematical Sciences, WPI, 100 Institute Road, Worcester, MA 01609, USA
2.
School of Interdisciplinary Mathematical Sciences, Meiji University 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan

Received: October 31, 2017

Revised: December 31, 2017

Early access: June 2018

Published: January 2019

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

In 2011, Liu et. al. proposed a three-component reaction-diffusion system to model the spread of bacteria and its signaling molecules (AHL) in an expanding cell population. At high AHL levels the bacteria are immotile, but diffuse with a positive diffusion constant at low distributions of AHL. In 2012, Fu et. al. studied a reduced system without considering nutrition and made heuristic arguments about the existence of traveling wave solutions. In this paper we provide rigorous proofs of the existence of traveling wave solutions for the reduced system under some simple conditions of the model parameters.

Keywords:

Cell motility,
degenerate reaction-diffusion equations,
traveling wave equations,
pattern formations,
nonhomogeneous linear differential equations.

Mathematics Subject Classification: 35K57, 35K65, 92C17.

Citation:

Full Text(HTML)

Figure 1. Traveling wave solutions with parameter values up to four places after decimal: α = 2.4862, ρ₋₀ = 0.5130, γ = 0.1565, D = 0.3439. Wave speed is approximately c = 0.6430. Note that h(z) lies below 1 and is not monotone for z > 0.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	E. Ben-Jacob , I. Cohen and H. Levine , Cooperative self-organization of microorganisms, Adv. Phys., 49 (2000) , 395-554. doi: 10.1080/000187300405228.
	E. O. Budrene and H. C. Berg , Complex patterns formed by motile cells of Escherichia coli, Nature, (London), 349 (1991) , 630-633. doi: 10.1038/349630a0.
	X. Fu, L. H. Tang, C. Liu, J. D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial systems with density-suppressed motility. Physical Review Letters, 108 (2012), 198102. Supplementary Material. doi: 10.1103/PhysRevLett.108.198102.
	C. Liu et al., Sequential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238–241, Supporting Online Material at http://www.sciencemag.org/cgi/content/full/334/6053/238/DC1. doi: 10.1126/science.1209042.
	J. D. Murray, Mathematical Biology I. An Introduction, Springer-Verlag, New York, 2002.