Uniform boundedness and pattern formation for Keller–Segel systems with two competing species

We consider a fully parabolic reaction–advection–diffusion system over two-dimensional bounded domain endowed with the homogeneous Neumann boundary conditions. This system models the chemotactic movements and population dynamics of two Lotka–Volterra competing microbial species attracted by the same chemical stimulus. We obtain the global existence of classical solutions to this two-dimensional system and prove that the global solutions are uniformly bounded in their L∞-norms. Our result does not require chemotaxis rates to be small or decay rate to be large. Moreover numerical simulations are performed to illustrate the formation and qualitative properties of stable and time-periodic spatially-inhomogeneous patterns of the system. Our theoretical and numerical findings illustrate that this two-dimensional chemotaxis model is able to demonstrate very interesting and complicated spatial-temporal dynamics.