Traveling wave solutions in a two-species competition model with density-suppressed motility

In this paper, we investigate the invasion and exclusion phenomena of a foreign bacterium species with density-suppressed motility on a local bacterium species that only has random diffusion. Through introducing an auxiliary problem, constructing relaxed super- and sub-solutions with spatially inhomogeneous decay rates and using energy estimation method, under appropriate parameter conditions, the existence of traveling wave solutions connecting two constant equilibrium points [Formula: see text] and [Formula: see text] with wave speed [Formula: see text] is established, while such traveling wave solutions do not exist when [Formula: see text]. Moreover, the existence of a traveling wave solution with speed [Formula: see text] connecting [Formula: see text] and [Formula: see text] is obtained by a limiting process and energy estimations. Finally, we discuss the possible selection of wave profiles/speeds using linearization method and the possible impact of density-suppressed motility term on the asymptotic spreading speed of traveling waves by numerical simulations.