Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system

This paper performs an in-depth qualitative analysis of the dynamic behavior a diffusive Lotka-Volterra type competition system with advection terms under homogeneous Dirichlet boundary condition. First, we obtain existence, multiplicity and explicit structure spatially nonhomogeneous steady-state solutions by using implicit function theorem Lyapunov-Schmidt reduction method. Secondly, analyzing distribution eigenvalues infinitesimal generators, stability positive non-existence Hopf bifurcations at are given. Finally, two concrete examples provided to support our previous theoretical results. It should be noticed that elliptic operator term is not self-adjoint, which causes some trouble in spatial decomposition, expressions deductive processes related generators. Moreover, unlike other work, rate here depends on position, increases difficulties investigation principal eigenvalue.