We study an one{dimensional quasilinear system proposed by J. Tello and M.Winkler [19] which models the population dynamics of two competing speciesattracted by the same chemical. The kinetics terms of the interacting speciesare chosen to be the Lotka{Volterra type. We prove the existence of globalbounded and classical solutions for all chemoattraction rates. Underhomogeneous Neumann boundary conditions, we establish the existence ofnonconstant steady states by local bifurcation theory. The stability of thebifurcating solutions is also obtained when the diffusivity of both species islarge. Finally, we perform extensive numerical studies to demonstrate theformation of stable positive steady states with various interesting spatialstructures.
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