| Contents |
| We consider a diffusive Lotka$-$Volterra competition system with advection under homogeneous Neumann boundary conditions in smoothly bounded domains. We establish the global existence of bounded classical solutions for the system over one$-$dimensional domains. For multi$-$dimensional domains, globally bounded classical solutions are obtained for a parabolic$-$elliptic system under proper assumptions on the system parameters. Then we investigate the stationary problem over one$-$dimensional domains. Through bifurcation theories, the existence and stability of non$-$constant positive steady states are obtained.
In the limit of large advection rate, we show that the reaction$-$advection$-$diffusion system converges to a shadow system of the competitor population density. The existence and stability of positive solutions to the shadow system have also been studied. Moreover, we construct positive interior$-$layer solutions to the shadow system when the crowding rate of the escaper and the diffusion rate of its inter-specific competitors are sufficiently small. These transition-layer solutions can be used to model the species segregation phenomenon.
This talk is based on the recent work joint with my students Chunyi Gai and Jingda Yan at Southwestern University of Finance and Economics. |
|