This paper focuses on a general class of reaction-diffusion-advection system with linear cross-diffusion and cross-advection. 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. The theoretical results are applied to predator-prey and water-vegetation models with cross-diffusion and cross-advection.

This paper focuses on a general class of reaction-diffusion-advection system with linear cross-diffusion and cross-advection. 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. The theoretical results are applied to predator-prey and water-vegetation models with cross-diffusion and cross-advection.