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Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system

  • * Corresponding author: Shangjiang Guo

    * Corresponding author: Shangjiang Guo
The first and second authors are supported by NSF of China (Grant Nos. 11671123 and 11801089) and by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan)
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  • 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.

    Mathematics Subject Classification: Primary: 92D25; Secondary: 35K57.

    Citation:

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  • Figure 1.  Solutions of (58) with parameters (59) and $ r = 1.1 $ tend to a spatially nonhomogeneous boundary steady-state

    Figure 2.  Solutions of (58) with parameters (60) and $ r = 2.1 $ tend to a spatially nonhomogeneous positive steady-state

    Table 1.  The Corresponding Table of Bifurcation Parameter and Open Set

    Theorem Bifurcation Value Open Set $ \Lambda $
    Theorem 2.3 $ \mu_1 $ $ (\mu_1-\delta_1, \mu_1)\bigcup(\mu_1, \mu_1+\delta_1) $
    Theorem 2.4 $ \mu_2 $ $ (\mu_2-\delta_2, \mu_2)\bigcup(\mu_2, \mu_2+\delta_2) $
    Theorem 2.5 $ \mu_\ast $ $ (\mu_\ast-\delta_3, \mu_\ast)\bigcup(\mu_\ast, \mu_\ast+\delta_3) $
    Theorem 2.6 $ (\mathbf{i}) $ $ \mu_1 $ $ (\mu_1-\delta_5, \mu_1)\bigcup(\mu_1, \mu_1+\delta_5) $
    Theorem 2.6 $ (\mathbf{ii}) $ $ \mu_2 $ $ (\mu_2-\delta_6, \mu_2)\bigcup(\mu_2, \mu_2+\delta_6) $
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