Evolution of passive movement in advective environments: general boundary condition. (English) Zbl 1378.35172
Summary: In a previous work [Discrete Contin. Dyn. Syst. 36, No. 2, 953–969 (2016; Zbl 1322.35072)], P. Zhou et al. studied a Lotka-Volterra competition-diffusion-advection system, where two species are supposed to differ only in their advection rates and the environment is assumed to be spatially homogeneous and closed (no-flux boundary condition), and showed that weaker advective movements are more beneficial for species to win the competition. In this paper, we aim to extend this result to a more general situation, where the environmental heterogeneity is taken into account and the boundary condition at the downstream end becomes very flexible including the standard Dirichlet, Neumann and Robin type conditions as special cases. Our main approaches are to exclude the existence of co-existence (positive) steady state and to provide a clear picture on the stability of semi-trivial steady states, where we introduced new ideas and techniques to overcome the emerging difficulties. Based on these two aspects and the theory of abstract competitive systems, we achieve a complete understanding on the global dynamics.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 37C65 | Monotone flows as dynamical systems |
| 92D25 | Population dynamics (general) |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Keywords:
competition-diffusion-advection; environmental heterogeneity; monotone dynamics; steady states; global stabilityCitations:
Zbl 1322.35072References:
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