Traveling wave solutions to the density-suppressed motility model

Jing Li, Zhi An Wang

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To understand the “self-trapping” mechanism inducing spatio-temporal pattern formations observed in the experiment of [21] for bacterial motion, the following density-suppressed motility model {ut=Δ(γ(v)u)+u(a−bu),vt=Δv+u−v, was proposed in [6,21], where u(x,t) and v(x,t) represent the densities of bacteria and the chemical emitted by the bacteria, respectively; γ(v) is called the motility function satisfying γ(v)<0 and a,b>0 are positive constants accounting for the growth and death rates of bacterial cells. The analysis of the above system is highly non-trivial due to the cross-diffusion and possible degeneracy resulting from the nonlinear motility function γ(v) and mathematical progresses on the global well-posedness and asymptotics of solutions were just made recently. Among other things, the purpose of this paper is to consider the above system with motility function [Formula presented] and investigate the traveling wave solutions which are genuine patterns observed in the experiment of [21]. By introducing an auxiliary parabolic problem to which the comparison principle applies and constructing relaxed super- and sub-solutions with spatially inhomogeneous decay rates, we show that there exist two constants b(m,a) and K(m,a), for b>b(m,a) and K(m,a)<1, the above density-suppressed motility model admits traveling wave solutions (u,v)(x,t)=:(U,V)(x⋅ξ−ct) in RN along the direction ξ∈SN−1 for all wave speed c≥2a connecting the equilibrium (a/b,a/b) to (0,0), while positive traveling wave solutions will not exist if c<2a. As m→0, we have b(m,a)→0 and K(m,a)→0, our results are well consistent with the relevant results for the well-known Fisher-KPP equation (i.e. the first equation of the above system with γ(v)=1). The main novel idea in the analysis of this paper is the construction of super- and sub-solutions with spatially inhomogeneous (i.e. non-constant) decay rates, in contrast to the constant decay rates used in the literature for reaction-diffusion equations, which was developed to cope with the difficulty caused by the density-dependent nonlinear diffusion in the system. We further discuss the selection of wave patterns and wave speeds for given initial value and use numerical simulations to illustrate that both monotone and non-monotone traveling wavefronts exist depending on whether the motility function γ(v) changes its convexity at v=a/b. Two-dimensional simulations demonstrate that the system can generate outward expanding ring (strip) pattern as observed in the experiment.

Original languageEnglish
Pages (from-to)1-36
Number of pages36
JournalJournal of Differential Equations
Publication statusPublished - 15 Nov 2021


  • Auxiliary problem
  • Density-suppressed motility
  • Minimal wave speed
  • Spatially inhomogeneous decay rate
  • Super- and sub-solutions
  • Traveling waves

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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