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

Abstract

This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weightedenergy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → −∞, but can be arbitrarily large in other locations.

Existence and stability of traveling wavefronts for a three‐species Lotka–Volterra competitive‐cooperative system with nonlocal dispersal

“…In past years, there have been extensive investigations on the stability of traveling wave solutions for various evolution systems; see previous studies [23][24][25][26][27][28] for random diffusion systems and previous studies [9,11,22,29,30] for nonlocal dispersal systems. For example, Zhang et al [22] used the weighted energy method together with the comparison principle, which was developed by Mei et al [25], to study the exponential stability of monostable traveling wavefronts for three-species nonlocal dispersal competitive system with special structure. Note that the comparison principle also works for the transformed system of (1.1).…”

“…In this paper, we focus on the existence of monostable traveling wavefronts connecting $$$ {E}_2 $$$ and $$$ {E}_5 $$$, as well as their stability for (). We should point out that the monostable traveling wavefronts of three‐species nonlocal dispersal competitive systems with special structure have been studied by many researchers; see previous studies [20–22]. It is well known that when the component of the Lotka–Volterra competitive system is greater than or equal to three, it is hard to transform the competitive system into a cooperative system any more.…”

“…Furthermore, by a contradiction argument, we obtain the nonexistence of traveling wavefronts with speed $$$ c<{c}_1^{\ast } $$$, which implies that $$$ {c}_1^{\ast } $$$ is the minimal wave speed of traveling wavefronts of (). In past years, there have been extensive investigations on the stability of traveling wave solutions for various evolution systems; see previous studies [23–28] for random diffusion systems and previous studies [9, 11, 22, 29, 30] for nonlocal dispersal systems. For example, Zhang et al [22] used the weighted energy method together with the comparison principle, which was developed by Mei et al [25], to study the exponential stability of monostable traveling wavefronts for three‐species nonlocal dispersal competitive system with special structure.…”

“…Thus, in this paper, we shall employ the weighted energy method together with the comparison principle to establish the stability of traveling wavefronts of (). We should point out that since the system () is more complex than that in [22], the energy estimates are not a simple task. Our result shows that the monostable traveling wavefronts of () with large speeds are exponentially stable, when the initial perturbations around the traveling wavefronts decay exponentially as $$$ x\to -\infty $$$, but can be arbitrarily large in other locations.…”

Propagation dynamics of a mutualistic model of mistletoes and birds with nonlocal dispersal

“…In the past 20 years, nonlocal dispersal equations have been extensively studied. We refer readers to [4,5,7,24,32,34] for travelling wave solutions, [6,14] for asymptotic behaviours of solutions for initial boundary value problems, [8,12,18,33] for spreading speeds and [17,30] for entire solutions. The following hypotheses are imposed in [19]:…”