Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application

Abstract

In this paper, we provide a general approach to study the asymptotic behavior of traveling wave solutions for a three-component system with nonlocal dispersal. Then as an important application, we establish a new type of entire solutions which behave as two traveling wave solutions coming from both sides of x-axis for a three-species Lotka-Volterra competition system.2010 Mathematics Subject Classification. Primary: 35K57, 37K65; Secondary: 92D30.

Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal

“…From Proposition 1, we can see that c min is the minimal wave speed. By Remark 2 in [8], c min ≥ c * . Note that c min = c * means that the minimal speed is linearly determined.…”

“…As we all know, the traveling waves for nonlocal dispersal equations and systems has been extensively studied, see [3,9,[13][14][15][20][21][22]30,31,36,39,41,43]. In a recent paper, Dong, Li and Wang [8] have established the existence and asymptotic behavior of traveling waves of (1), i.e., solutions of (2) with (3). Based on the asymptotic behavior of traveling waves, they further studied a new type of entire solutions which behave as two traveling waves coming from both sides of x-axis.…”

“…We should point out that a key step in proving uniqueness of traveling waves by the sliding method is to establish rather precise exponential decay rate of wave profiles when x → −∞. Although Dong, Li and Wang [8] have obtained some results on the asymptotic behavior of traveling waves of (1) at ±∞, it is not precise enough for the uniqueness of wave profiles. Inspired by Carr and Chmaj [3] for integro-differential equation without delay, we use a modified version of Ikehara's Theorem to derive more precise asymptotically exponential tails of wave profiles, see also [11,19].…”

“…Then it can be seen from [8] that for c ≥ c * , ∆ 1 (λ, c) = 0 admits two real roots 0 < λ − (c) ≤ λ + (c), and ∆ j (λ, c) = 0 has a real root ν j (c) > 0 with j = 2, 3. Clearly, if…”

“…Proposition 1. (see [8,Theorem 1.4]) Assume that (H1) holds, and J i satisfies (J1) and is compactly supported, i = 1, 2, 3. Then there exists a positive constant c min such that (7) admits a solution (ϕ 1 , ϕ 2 , ϕ 3 ) satisfying ϕ 1 > 0, ϕ 2 > 0 and ϕ 3 > 0 in R if and only if c ≥ c min .…”