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

Zhi Cheng Wang, Hui Ling Niu, Shigui Ruan

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space {∂/∂t u1(x, t) = ▵u1(x, t) + u1(x, t) [1 - u1(x, t) - k1u2(x, t)], ∂/∂t u2(x, t) = d▵u2(x, t) + ru2(x, t) [1 - u2(x, t) - k2u1(x, t)], where x ∈ ℝ3 and t > 0. For the bistable case, namely k1, k2 > 1, it is well known that the system admits a one-dimensional monotone traveling front Φ(x + ct) = (Φ1(x + ct), Φ2(x + ct)) connecting two stable equilibria Eu = (1, 0) and Ev = (0, 1), where c ∈ 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 Ψ(x′, x3 + st) = (Φ1(x′, x3 + st), Φ2(x′, x3 + st)) in ℝ3 connecting Eu = (1, 0) and Ev = (0, 1), where x′ ∈ ℝ2. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the x3-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 ℝ3. 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.

Original languageEnglish (US)
Pages (from-to)1111-1144
Number of pages34
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume22
Issue number3
DOIs
StatePublished - May 1 2017

Keywords

  • Axisymmetric traveling front
  • Bistability
  • Existence
  • Lotka-Volterra competition-diffusion system
  • Nonexistence
  • Qualitative properties

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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