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This paper studies a fully parabolic chemotaxis system with competitive kinetics and fractional diffusion of orderon two dimensional periodic torus. It is proved that ( 1 ) has a unique global bounded solution for all appropriately regular nonnegative initial dataand. Moreover, ifand(or) are large enough, we can reach the following conclusions by constructing appropriate energy functional: (i)uniformly inas. (ii)uniformly inas. (iii)uniformly inas.

Existence and asymptotic stability in a fractional chemotaxis system with competitive kinetics