Traveling wave front for a two-component lattice dynamical system arising in competition models

Guo, Jong‐Shenq; Wu, Chang-Hong

doi:10.1016/j.jde.2012.01.009

Cited by 84 publications

(46 citation statements)

References 19 publications

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“…Here c ∈ R is called the wave speed and U , V are wave profiles. For the existence and uniqueness of traveling wave solution of Lotka-Volterra lattice dynamical system with monostable nonlinearity, we refer to [10]. There are many works in corresponding PDE models, for example, see [25,9,7,11,15,12,13] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4, we study the asymptotic behavior of wave tails of traveling wave solutions with nonzero speed. Besides a key lemma (Lemma 4.2 below) which is similar to [10, Lemma 3.4], we shall use a different method from the one used in [10] (for monostable case) to derive the asymptotic behavior of wave tails of traveling waves solutions in bistable case (Propositions 4.1 and 4.6). The main idea of this method is to construct some auxiliary functions to compare with the wave profiles.…”

Section: Introductionmentioning

confidence: 99%

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Wave propagation for a two-component lattice dynamical system arising in strong competition models

Guo

Wu²

2011

Journal of Differential Equations

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We study a Lotka-Volterra type competition system with bistable nonlinearity in which the habitat is divided into discrete niches. We show that there exist non-monotone stationary solutions when the migration coefficients are sufficiently small. Also, we prove that the propagation failure phenomenon occurs. Finally, we focus on the traveling wave with nonzero wave speed. By investigating the asymptotic behavior of tails of wave profiles, we show that nonzero speed wave profiles are monotone. Moreover, the nonzero wave speed is unique in the sense that the wave cannot propagate with two different nonzero wave speeds.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wave propagation for a two-component lattice dynamical system arising in strong competition models

Guo

Wu²

2011

Journal of Differential Equations

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“…Also, we can use a similar argument as [11] to prove the existence and monotone of traveling wave solutions of (1) for case (i). From the viewpoint of ecology, a traveling wave solution satisfying (3) and (4) can model the population invasion process [29]: at any fixed x ∈ R, only v (the resident) can be found long time ago (t → −∞ such that x + ct → −∞), but after a long time (t → ∞ such that x + ct → ∞), only u (the invader) can be seen.…”

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confidence: 99%

“…In Section 2, based on a fundamental theory (Proposition 1) which developed in [41] (see also [11]), we give the asymptotic behavior of traveling wave solutions at infinity. Section 3 is devoted to establishing the existence of invasion entire solutions by using comparison principle and constructing appropriate sub-super solution pairs.…”

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confidence: 99%

Invasion entire solutions in a competition system with nonlocal dispersal

Li¹,

Zhang²,

Zhang³

2015

Discrete &Amp; Continuous Dynamical Systems - A

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This paper is concerned with invasion entire solutions of a Lotka-Volterra competition system with nonlocal dispersal, which formulate a new invasion way of the stronger species to the weaker one. We first give the asymptotic behavior of traveling wave solutions at infinity. Then by the comparison principle and sub-super solutions method, we establish the existence of invasion entire solutions which behave as two monotone waves with different speeds and coming from both sides of x-axis.2010 Mathematics Subject Classification. Primary: 35K57, 37C65; Secondary: 92D30.

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“…For system (1.1) with autonomous nonlinearities, the dynamical behaviors especially for traveling wave solutions have been understood very well, see, e.g., [13,14,20,21,35,36]. Recently, there have been quite a few works focusing on the nonautonomous Lotka-Volterra competition-diffusion system.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

Wang

2018

Journal of Differential Equations

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This paper is concerned with a time periodic competition-diffusion systemwhere u(t, x) and v(t, x) denote the densities of two competing species, d > 0 is some constant, r i (t), a i (t) and b i (t) are T −periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang [J. Differential Equations 255 (2013), 2402-2435] that this system admits a periodic traveling front connecting two stable semi-trivial T −periodic solutions (p(t), 0) and (0, q(t)) associated to the corresponding kinetic system. Assume further that the wave speed is non-zero, we investigate the asymptotic behavior of the periodic bistable traveling front at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then give some key estimates. Finally, by applying super-and subsolutions technique as well as the comparison principle, we establish the existence and various qualitative properties of entire solutions defined for all time and whole space.

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Traveling wave front for a two-component lattice dynamical system arising in competition models

Cited by 84 publications

References 19 publications

Wave propagation for a two-component lattice dynamical system arising in strong competition models

Wave propagation for a two-component lattice dynamical system arising in strong competition models

Invasion entire solutions in a competition system with nonlocal dispersal

Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

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