On the minimal speed of traveling waves for a nonlocal delayed reaction–diffusion equation

Aguerrea, Maitere; Valenzuela, Gabriel

doi:10.1007/s11072-010-0096-y

Cited by 3 publications

(4 citation statements)

References 10 publications

(25 reference statements)

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“…uniqueness within the class of all wavefronts) of monotone wavefronts to (4) and the conditional uniqueness (i.e. uniqueness within the subclass of monotone wavefronts) of monotone wavefronts to (3) was also proved in these works. As we have mentioned, in general, monotone and non-monotone wavefronts can coexist in (3) [21,36].…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…Proof. For example, see [17,Lemma 22] and [3,Theorem 1.1]. By [3] if, in addition, K(s) = K(−s), s ∈ R, then 0 < c # (h) = O(1/h) at +∞.…”

Section: Main Assumptionsmentioning

confidence: 99%

“…For example, see [17,Lemma 22] and [3,Theorem 1.1]. By [3] if, in addition, K(s) = K(−s), s ∈ R, then 0 < c # (h) = O(1/h) at +∞. In general, however, c # (h) can take negative values, cf.…”

Section: Main Assumptionsmentioning

confidence: 99%

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Monotone waves for non-monotone and non-local monostable reaction–diffusion equations

Trofimchuk¹,

Pinto

Trofimchuk

2016

Journal of Differential Equations

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We propose a new approach for proving existence of monotone wavefronts in non-monotone and non local monostable diffusive equations. This allows to extend recent results established for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (modulo translations) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then each monotone wavefront is unique (modulo translations) within the class of all non-constant traveling waves. Our approach is based on the construction of suitable fundamental solutions for linear integral-differential equations. We consider two alternative scenarios: in the first one, the fundamental solution is negative (typically holds for the Mackey-Glass diffusive equations) while in the second one, the fundamental solution is non-negative (typically holds for the KPP-Fisher diffusive equations).FONDECYT (Chile) 1110309 1120709 115048

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Section: Introduction and Main Resultsmentioning

confidence: 87%

“…Proof. For example, see [17,Lemma 22] and [3,Theorem 1.1]. By [3] if, in addition, K(s) = K(−s), s ∈ R, then 0 < c # (h) = O(1/h) at +∞.…”

Section: Main Assumptionsmentioning

confidence: 99%

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Monotone waves for non-monotone and non-local monostable reaction–diffusion equations

Trofimchuk¹,

Pinto

Trofimchuk

2016

Journal of Differential Equations

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“…We observe that c ⋆ ≥ c * and the function χ 0 (z, c) is associated with the linearization of (1.1) along the trivial equilibrium. Some estimates for c * , c ⋆ can be found in [4,25,34]. For example, if…”

Section: Introductionmentioning

confidence: 99%

On the uniqueness of semi-wavefronts for non-local delayed reaction–diffusion equations

Aguerrea

2015

Journal of Mathematical Analysis and Applications

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Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model

Li,

Guo,

Yuan

2024

MBE

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<abstract><p>The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation.</p></abstract>

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On the minimal speed of traveling waves for a nonlocal delayed reaction–diffusion equation

Abstract: Abstract. In this note, we give constructive upper and lower bounds for the minimal speed of propagation of traveling waves for non-local delayed reaction-diffusion equation.

Cited by 3 publications

References 10 publications

Monotone waves for non-monotone and non-local monostable reaction–diffusion equations

Monotone waves for non-monotone and non-local monostable reaction–diffusion equations

On the uniqueness of semi-wavefronts for non-local delayed reaction–diffusion equations

Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model

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