LAUSR

We study a system of PDEs modelling 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 behaviours of the system are studied both analytically in the weakly nonlinear regime, and numerically in the fully nonlinear regime.