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

Dan Wei; Shangjiang Guo

doi:10.3934/dcdsb.2020197

Article Contents

2021, Volume 26, Issue 5: 2599-2623. Doi: 10.3934/dcdsb.2020197

This issue Previous Article A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems Next Article Global classical solutions to two-dimensional chemotaxis-shallow water system

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

Dan Wei^1, and
Shangjiang Guo^2, ,

1.
College of Mathematics, Hunan University, Changsha, Hunan 410082, China
2.
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

^* Corresponding author: Shangjiang Guo
^* Corresponding author: Shangjiang Guo

Received: November 2019

Revised: April 2020

Early access: June 2020

Published: May 2021

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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Abstract

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.

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

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

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

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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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