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 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 |
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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
On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3 . / Wang, Zhi Cheng; Niu, Hui Ling; Ruan, Shigui.
In: Discrete and Continuous Dynamical Systems - Series B, Vol. 22, No. 3, 01.05.2017, p. 1111-1144.Research output: Contribution to journal › Article
}
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 -