Stability and Pattern Formation in a General Class of Reaction-Diffusion-Advection System

In this paper, we systematically study two-species reaction-diffusion-advection system with linear cross-diffusion and cross-advection. Firstly, we provide sufficient conditions for cross-diffusion, self-advection and cross-advection driven instability, which implies that cross-diffusion, self-advection and cross-advection can give rise to pattern formation for the same diffusion coefficients. Secondly, we focuses on a class of general reaction-diffusion-advection system. By investigating the linearized stability of the constant equilibrium solution, we prove that the self-diffusion and self-advection terms have no effect on the stabilization of the constant steady state, the linear cross terms favor the destabilization of the constant steady state and mechanism of pattern formation. Furthermore, the theoretical results are applied to predator-prey and water-vegetation systems with cross-diffusion and cross-advection.