On the existence of coexistence states for an advection-cooperative system with spatial heterogeneities

This paper is devoted to the analysis of the existence of coexistence states for cooperative systems under homogeneous Dirichlet boundary conditions. We assume rather general spatial heterogeneities for the coefficients of the non-linearities and the cooperative terms. In order to obtain sharp existence results, we add advection terms related to the cooperative terms so that the system can be transformed into a variational one. Applying then several variational methods and bifurcation theory, we arrive at necessary and sufficient condition on the parameter λ for the existence and the uniqueness of coexistence states. We also compare the conditions for the heterogeneous case and the constant case.