On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3

Zhi-Cheng Wang; Hui-Ling Niu; Shigui Ruan

doi:10.3934/dcdsb.2017055

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2017, Volume 22, Issue 3: 1111-1144. Doi: 10.3934/dcdsb.2017055

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On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ³

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
2.
School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, China
3.
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

Corresponding author
Corresponding author

Dedicated to Professor Robert Stephen Cantrell on the occasion of his 60th birthday

Received: June 30, 2015

Revised: March 31, 2016

Published: September 2017

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Abstract

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space

$\left\{ {\begin{array}{*{20}{l}}{\frac{\partial }{{\partial t}}{u_1}({\bf{x}},t) = \Delta {u_1}({\bf{x}},t) + {u_1}({\bf{x}},t)\left[ {1 - \;{u_1}({\bf{x}},t) - {k_1}{u_2}({\bf{x}},t)} \right],}\\{\frac{\partial }{{\partial t}}{u_2}({\bf{x}},t) = d\Delta {u_2}({\bf{x}},t) + r{u_2}({\bf{x}},t)\left[ {1 - {u_2}({\bf{x}},t) - {k_2}{u_1}({\bf{x}},t)} \right],}\end{array}} \right.$

where $\mathbf{x}∈ \mathbb{R}^3$ 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 $c∈\mathbb{R}$ 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 $\mathbf{x}^\prime∈\mathbb{R}^2$ . 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.

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