QUALITATIVE ANALYSIS OF A LOTKA-VOLTERRA COMPETITION-DIFFUSION-ADVECTION SYSTEM

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

Automated summary

This paper provides an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms. The study focuses on the existence and stability of spatially nonhomogeneous steady-state solutions, as well as the non-existence of Hopf bifurcations at these solutions. Two concrete examples are provided to support the theoretical results. The presence of advection terms that depend on spatial position complicates the investigation of the principal eigenvalue compared to previous work.