We consider a Keller-Segel model of two competing species with Lotka–Volterra dynamics and a single chemical which attracts both species. We show that the homogeneous solution loses its stability to, depending on system parameters, either non-constant positive steady states or time-periodic solutions of the system as chemotaxis rate increases. Stability or instability of the bifurcating solutions is investigated rigorously. Our stability results provide a selection mechanism of stable wave mode for these nontrivial patterns. Moreover, it is found that cellular growth is responsible for the oscillating patterns of this model, of which there exists a Lyapunov in the absences of cellular kinetics. Global existence and boundedness of the system in 2D are obtained thanks to this Lyapunov functional. We shall also present some numerical results to demonstrate and verify our theoretical findings.