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.
|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.|
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.