On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ<sup>3</sup>

Abstract

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space {partial derivative/partial derivative tu(1) (x, t) = Delta u(1) (x, t) + u(1) (x, t) [1 - u(1) (x, t) - k(1)u(2) (x, t)] , {partial derivative/partial derivative tu(2) (x, t) = d Delta u(2) (x, t) + ru(2) (x, t) [1 - u(2) (x, t) - k(2)u(1) (x, t)] , where x is an element of 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 Phi(x + ct) = (Phi(1) (x + ct), Phi(2) (x + ct)) connecting two stable equilibria E-u = (1, 0) and E-v = (0, 1), where c is an element of R is the unique wave speed. Recently, two-dimensional V-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption c > 0. In this paper it is shown that for any s > c > 0, the system admits axisymmetric traveling fronts psi(x', x(3) + st) = Phi(1)(x ', x(3) + st), Phi(2)(x ', x(3) + st) in R-3 connecting E-u = (1, 0) and E-v = (0, 1), where x' is an element of 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 set is discussed.