### Abstract

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space {∂/∂t u_{1}(x, t) = ▵u_{1}(x, t) + u_{1}(x, t) [1 - u_{1}(x, t) - k_{1}u_{2}(x, t)], ∂/∂t u_{2}(x, t) = d▵u_{2}(x, t) + ru_{2}(x, t) [1 - u_{2}(x, t) - k_{2}u_{1}(x, t)], where x ∈ 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 Φ(x + ct) = (Φ_{1}(x + ct), Φ_{2}(x + ct)) connecting two stable equilibria E_{u} = (1, 0) and E_{v} = (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′, x_{3} + st) = (Φ_{1}(x′, x_{3} + st), Φ_{2}(x′, x_{3} + st)) in ℝ^{3} connecting E_{u} = (1, 0) and E_{v} = (0, 1), where x′ ∈ 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1111-1144 |

Number of pages | 34 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 22 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

^{3}

*Discrete and Continuous Dynamical Systems - Series B*,

*22*(3), 1111-1144. https://doi.org/10.3934/dcdsb.2017055

**On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ ^{3} .** / Wang, Zhi Cheng; Niu, Hui Ling; Ruan, Shigui.

Research output: Contribution to journal › Article

^{3}',

*Discrete and Continuous Dynamical Systems - Series B*, vol. 22, no. 3, pp. 1111-1144. https://doi.org/10.3934/dcdsb.2017055

}

TY - JOUR

T1 - On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3

AU - Wang, Zhi Cheng

AU - Niu, Hui Ling

AU - Ruan, Shigui

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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.

KW - Axisymmetric traveling front

KW - Bistability

KW - Existence

KW - Lotka-Volterra competition-diffusion system

KW - Nonexistence

KW - Qualitative properties

UR - http://www.scopus.com/inward/record.url?scp=85014341731&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014341731&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2017055

DO - 10.3934/dcdsb.2017055

M3 - Article

AN - SCOPUS:85014341731

VL - 22

SP - 1111

EP - 1144

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 3

ER -