Qualitative analysis of a Lotka-Volterra competition system with   advection
  We study a diffusive Lotka-Volterra competition system with advection under Neumann boundary conditions. Our system models a competition relationship that one species escape from the region of high population density of their competitors in order to avoid competitions. We establish the global existence of bounded classical solutions for the system in one-dimensional domain. For multi-dimensional domains, globally bounded classical solutions are obtained for a parabolic-elliptic system under proper assumptions on the system parameters. These global existence results make it possible to study bounded steady states in order to model species segregation phenomenon. We then investigate the stationary problem in one-dimensional domains. Through bifurcation theory, we obtain the existence of nonconstant positive steady states, which are small perturbations from the positive equilibrium; we also study the stability of these bifurcating solutions when the diffusion coefficient of the escaper is large and the diffusion coefficient of its competitor is small. In the limit of large advection rate, we show that the reaction-advection-diffusion system converges to a shadow system involving the competitor population density and an unknown positive constant. The existence and stability of positive solutions to the shadow system have also been obtained through bifurcation theories. Finally, we construct positive solutions with an interior transition layer to the shadow system when the crowding rate of the escaper and the diffusion rate of its interspecific competitors are sufficiently small. The transition-layer solutions can be used to model the species segregation phenomenon. 
Chunyi Gai, Qi Wang, Jingda Yan
math.AP