  We study a system of PDEs modeling the population dynamics of two competitive
species whose spatial movements are governed by both diffusion and mutually
repulsive chemotaxis effects. We prove that solutions to this system are
globally well-posed, without any smallness assumptions on the chemotactic
coefficients. Moreover, in the weak competition regime, we prove that neither
species can be driven to extinction as the time goes to infinity, regardless of
how strong the chemotaxis coefficients are. Finally, long-time behaviors of the
system are studied both analytically in the weakly nonlinear regime, and
numerically in the fully nonlinear regime.
