Qualitative analysis of a Lotka-Volterra competition system with advection.

*(English)*Zbl 1327.92050Summary: 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 competition. We establish the global existence of bounded classical solutions to the system over one-dimensional finite domains. 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 one-dimensional stationary problem. Through bifurcation theory, we obtain the existence of nonconstant positive steady states, which are small perturbations from the positive equilibrium; we also rigourously study the stability of these bifurcating solutions when diffusion coefficients of the escaper and its competitor are large and small respectively. 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. Existence and stability of positive nonconstant solutions to the shadow system have also been obtained through bifurcation theories. Finally, we construct infinitely many single interior transition layers to the shadow system when crowding rate of the escapers and diffusion rate of their interspecific competitors are sufficiently small. The transition-layer solutions can be used to model the interspecific segregation phenomenon.

##### MSC:

92D25 | Population dynamics (general) |

35B32 | Bifurcations in context of PDEs |

35B35 | Stability in context of PDEs |

35B36 | Pattern formations in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

##### Keywords:

global existence; Lotka-Volterra competition system; steady state; bifurcation; stability; transition layer##### References:

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