Traveling waves for nonlocal Lotka-Volterra competition systems

Han, Byungjin; Wang, Zhicheng; Du, Zhijiang

doi:10.3934/dcdsb.2020011

Cited by 11 publications

(20 citation statements)

References 48 publications

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“…In this paper, we focus on travelling solitary waves of the form

ψ false(t, x false) = e^{i t μ} φ_{v} false(x - v t false)

with some

μ \in ℝ

and travelling velocity

v \in ℝ^{3}

. Different from the travelling waves in reaction‐diffusion equations or nonlinear Schrödinger equation (see previous works 20–24 and the references therein), the travelling waves here possess some special properties. Fröhlich et al 1 proved that for m > 0, the travelling solitary waves (also called boosted ground states) in () above exist if and only if

false| v false| < 1 and 0 < scriptN false(φ_{v} false) = \int_{ℝ^{3}} false| φ_{v} false(x false) {false|}^{2} d x < N_{c} false(v false),

where

scriptN false(φ_{v} false)

denotes the mass (or charge) of the system, and the constant N c ( v ) obeys the bounds

false(1 - false| v false| false) N_{c} \leq N_{c} false(v false) \leq N_{c} false(0 false) = N_{c}

with N c defined in ().…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

Wang

2021

Math Methods in App Sciences

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In this paper, we consider the pseudo‐relativistic Hartree equation i∂tψ=−△+m2ψ−1|x|∗|ψ|2ψonℝ3 and study travelling solitary waves of the form ψ(t, x) = eitμφ(x − v t) , where v∈ℝ3 denotes travelling velocity. Fröhlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1‐30] proved that for |v|<1 there exists a critical constant Nc(v), such that the travelling waves exist if and only if 0 < N < Nc(v), where N denotes particle number. In this paper, we consider v=false(β,0,0false) with 0 < β < 1, and let Ncfalse(βfalse)=Ncfalse(vfalse)false|v=false(β,0,0false). We find that Nc(β) is Lipschitz continuity with respect to β. Based on this fact, we then prove that the boosted ground states φβ with false‖φβfalse‖L22=false(1−βfalse)Ncfalse(βfalse) satisfy limβ→0+false‖φβfalse‖H1false/2→+∞. The explicit blow‐up profile and rate will be computed.

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“…In this paper, we focus on travelling solitary waves of the form

ψ false(t, x false) = e^{i t μ} φ_{v} false(x - v t false)

with some

μ \in ℝ

and travelling velocity

v \in ℝ^{3}

false| v false| < 1 and 0 < scriptN false(φ_{v} false) = \int_{ℝ^{3}} false| φ_{v} false(x false) {false|}^{2} d x < N_{c} false(v false),

where

scriptN false(φ_{v} false)

denotes the mass (or charge) of the system, and the constant N c ( v ) obeys the bounds

false(1 - false| v false| false) N_{c} \leq N_{c} false(v false) \leq N_{c} false(0 false) = N_{c}

with N c defined in ().…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

Wang

2021

Math Methods in App Sciences

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“…And the traveling wave solutions of the non-delayed problem have been studied by previous investigators [8,14]. In this degenerate case the system is four-dimensional, but for > 0, the existence of a traveling front solution of system ( 4) between E 1 and E 2 is equivalent to the existence of a heteroclinic orbit connection between the equilibrium points of the twelve-dimensional system (10). We shall still denote these equilibria by E 1 and E 2 .…”

Section: Definition 3 ([18]mentioning

confidence: 99%

“…Britton [2] considered comprehensively these two factors and introduced the so-called spatio-temporal delay or nonlocal delay. Nowadays, models with spatio-temporal delay or nonlocal delay attracted much attention due to its significant sense in mathematical theory and practical fields [9,10,20,22]. There are some methods which have been used to prove the existence of traveling wave solutions [1,24].…”

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confidence: 99%

“…Systems (4) and (6) are different models, since (6) is a cooperative system and (4) is a competitive system. Recently, Han et al [10] studied the traveling wave solutions of system (4) by using different methods, respectively. They proved that there exists traveling wave solutions of the system connecting equilibrium (0, 0) to some unknown positive steady state for wave speed c > c * = max{2, 2 √ dr} and there is no such traveling wave solutions for c < c * , where with d = d2 d1 and r = r2 r1 .…”

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confidence: 99%

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Traveling wave fronts in a diffusive and competitive Lotka-Volterra system

Du¹,

Yan²,

Zhuang³

2021

DCDS-S

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<p style='text-indent:20px;'>In this paper, we consider a two-species competitive and diffusive system with nonlocal delays. We investigate the existence of traveling wave fronts of the system by employing linear chain techniques and geometric singular perturbation theory. The existence of the traveling wave fronts analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a six-dimensional system without delay.</p>

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“…Due to the influence of the environmental heterogeneity, the mathematical study on the space and/or time periodic traveling wave solutions for various variants of reaction-diffusion equations/systems in periodic habitats have been studied by many people. See, for example, Alikakos et al [1], Bao and Wang [3], Bao et al [4], Berestycki and Hamel [6], Fang et al [12], Han et al [16], Kong et al [20], Liang and Zhao [23], Liang et al [21], Nadin [26], Pan [33], Weinberger [48], Yang et. al [50], Yu and Zhao [51], Zhao and Ruan [52], and so on.…”

mentioning

confidence: 99%