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.