Global existence and steady states of a two competing species Keller-Segel chemotaxis model

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.