Stationary and non-stationary patterns of the density-suppressed motility model

In this paper, we first explore the stationary problem of the density-suppressed motility (DSM) model proposed in Fu et al. (2012) and Liu et al. (2011) where the diffusion rate of the bacterial cells is a decreasing function (motility function) of the concentration of a chemical secreted by bacteria themselves. We show that the DSM model does not admit non-constant steady states if either the chemical diffusion rate or the intrinsic growth rate of bacteria is large. We also prove that when the decay of the motility function is sub-linear or linear, the DSM model does not admit non-constant steady states if either the chemical diffusion rate or the intrinsic growth rate of bacteria is small. Outside these non-existence parameter regimes, we show that the DSM model will have non-constant steady states under some constraints on the parameters. Furthermore we numerically find the stable stationary patterns only when the parameter values are close to the critical instability regime. Finally by performing a delicate multiple-scale analysis, we derive that the DSM model may generate propagating oscillatory waves whose amplitude is governed by an explicit Ginzburg–Landau equation, which is further verified by numerical simulations.