  We study an one{dimensional quasilinear system proposed by J. Tello and M.
Winkler [19] which models the population dynamics of two competing species
attracted by the same chemical. The kinetics terms of the interacting species
are chosen to be the Lotka{Volterra type. We prove the existence of global
bounded and classical solutions for all chemoattraction rates. Under
homogeneous Neumann boundary conditions, we establish the existence of
nonconstant steady states by local bifurcation theory. The stability of the
bifurcating solutions is also obtained when the diffusivity of both species is
large. Finally, we perform extensive numerical studies to demonstrate the
formation of stable positive steady states with various interesting spatial
structures.
