Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in $\mathbb{R}^3$

Pages: 1111 - 1144, Volume 22, Issue 3, May 2017      doi:10.3934/dcdsb.2017055

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Zhi-Cheng Wang - School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China (email)
Hui-Ling Niu - School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, China (email)
Shigui Ruan - Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States (email)

Abstract: This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space \[ \left\{ \begin{array}{l} \frac \partial {\partial t}u_1 (\mathbf{x}, t)=\Delta u_1(\mathbf{x}, t) + u_1 (\mathbf{x}, t)\left[ 1-\ u_{1 }(\mathbf{x}, t)-k_1u_2(\mathbf{x}, t)\right] , \\ \frac \partial {\partial t}u_2(\mathbf{x}, t)=d\Delta u_2(\mathbf{x}, t)+ru_2(\mathbf{x}, t)\left[ 1-u_2(\mathbf{x}, t)-k_2u_1(\mathbf{x}, t)\right] , \end{array} \right. \] where $\mathbf{x}\in \mathbb{R}^3$ and $t>0$. For the bistable case, namely $k_1,k_2>1$, it is well known that the system admits a one-dimensional monotone traveling front $\mathbf{\Phi}(x+ct)=\left(\Phi_1(x+ct),\Phi_2(x+ct)\right)$ connecting two stable equilibria $\mathbf{E}_u=(1,0)$ and $\mathbf{E}_v=(0,1)$, where $c\in\mathbb{R}$ is the unique wave speed. Recently, two-dimensional V-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption that $c>0$. In this paper it is shown that for any $s>c>0$, the system admits axisymmetric traveling fronts \[ \mathbf{\Psi}(\mathbf{x}^\prime, x_3+st)=\left(\Phi_1(\mathbf{x}^\prime, x_3+st),\Phi_2(\mathbf{x}^\prime, x_3+st)\right) \] in $\mathbb{R}^3$ connecting $\mathbf{E}_u=(1,0)$ and $\mathbf{E}_v=(0,1)$, where $\mathbf{x}^\prime\in\mathbb{R}^2$. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the $x_3$-axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When $s$ tends to $c$, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in $\mathbb{R}^3$. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level sets is discussed.

Keywords:  Lotka-Volterra competition-diffusion system, bistability, axisymmetric traveling front, existence, nonexistence, qualitative properties.
Mathematics Subject Classification:  35K57, 35B35, 35B40.

Received: July 2015;      Revised: April 2016;      Available Online: December 2016.