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This paper investigates the formation of time--periodic and stable patternsof a two--competing--species Keller--Segel chemotaxis model with a focus on theeffect of cellular growth. We carry out rigorous Hopf bifurcation analysis toobtain the bifurcation values, spatial profiles and time period associated withthese oscillating patterns. Moreover, the stability of the periodic solutionsis investigated and it provides a selection mechanism of stable time--periodicmode which suggests that only large domains support the formation of theseperiodic patterns. Another main result of this paper reveals that cellulargrowth is responsible for the emergence and stabilization of the oscillatingpatterns observed in the $3\times3$ system, while the system admits a Lyapunovfunctional in the absence of cellular growth. Global existence and boundednessof the system in 2D are proved thanks to this Lyapunov functional. Finally, weprovide some numerical simulations to illustrate and support our theoreticalfindings.

Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth