Abstract
This paper is concerned with the traveling waves of a nonlocal dispersal Lotka-Volterra strong competition model with bistable nonlinearity. We first establish the asymptotic behavior of traveling waves at infinity. Then by applying the stronger comparison principle and the sliding method, we prove that the traveling waves with nonzero speed are strictly monotone. Moreover, the uniqueness of wave speeds is also obtained.
Highlights
In this paper, we consider the following Lotka-Volterra competition system with nonlocal dispersal ∂u(x,t) ∂t = d1[(J1 ∗ u)(x, t)− u(x, t)] + r1u(x, t)[1 − u(x, t) − a1v(x, t)],∂v(x,t) ∂t d2[(J2 ∗ v)(x, t)
Li et al [20] studied the asymptotic behavior of traveling waves of (2) connecting E1 and E2 at infinity by using method developed by Zhang, Li and Wang [32]
We study the monotonicity of traveling waves, and the uniqueness of wave speeds in the sense that the wave cannot propagate with two different nonzero wave speeds
Summary
We consider the following Lotka-Volterra competition system with nonlocal dispersal. Li et al [20] studied the asymptotic behavior of traveling waves of (2) connecting E1 and E2 at infinity by using method developed by Zhang, Li and Wang [32]. To the best of our knowledge, for nonlocal dispersal equations and systems, there are two effective methods which have been used to obtain the asymptotic behavior of traveling waves. Motivated by [10], in this paper, we shall continue to adopt the method of Zhang-Li-Wang [32] to establish the asymptotic behavior of traveling waves of (2) with bistable nonlinearity. Similar to Lemma 3.6, we have the following result on the asymptotic behavior of traveling waves at ξ = +∞. We further study the asymptotic behavior of traveling wave profiles of (2) at infinity, which is accurate enough to distinguish the translation differences.
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