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

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 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′, x₃ + st) = (Φ₁(x′, x₃ + st), Φ₂(x′, x₃ + st)) in ℝ³ connecting E_u = (1, 0) and E_v = (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.