Abstract: | This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractantunder homogeneous Neumann boundary conditions in a bounded domain () with smooth boundary, where the initial data and are non-negative and the parameters , , and . The chemotactic function () is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for ,(i) and(ii) .Moreover, we prove asymptotic stabilization of solutions in the sense that:? If and , then any global bounded solution exponentially converge to as ;? If and , then any global bounded solution exponentially converge to as ;? If and , then any global bounded solution algebraically converge to as . |