Existence and nonexistence of monotone traveling waves for the delayed Fisher equation

Kwong, Man Kam; Ou, Chunhua

doi:10.1016/j.jde.2010.04.017

Cited by 34 publications

(40 citation statements)

References 13 publications

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“…(3) has been recently considered by Kwong and Ou in [23], where a different approach based on a shooting technique was developed. By presenting a constructive approximation algorithm, indicating asymptotic formulas and proving the uniqueness of monotone fronts, our work complements the interesting investigation in [23]. Next, the existence of fast positive traveling fronts to Eq.…”

Section: Remarkmentioning

confidence: 76%

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Monotone traveling wavefronts of the KPP-Fisher delayed equation

Gomez

Trofimchuk

2011

Journal of Differential Equations

112

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Trofimchuk, S (reprint author), Univ Talca, Inst Matemat & Fis, Casilla 747, Talca, Chile.In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov-Petrovskii-Piskunov-Fisher equation u(t)(t, x) = Delta u(t, x) + u (t, x)(1 - u(t - h, x)), u >= 0, x is an element of R(m). (*) Since then, this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction-diffusion equations. In this paper, we give a complete solution to the problem of existence and uniqueness of monotone waves in Eq. (*). We show that each monotone traveling wave can be found via an iteration procedure. The proposed approach is based on the use of special monotone integral operators (which are different from the usual Wu-Zou operator) and appropriate upper and lower solutions associated to them. The analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach. (c) 2010 Elsevier Inc. All rights reserved

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Section: Remarkmentioning

confidence: 76%

“…Research was supported in part by FONDECYT (Chile), project 1071053. The authors are indebted to the referee for pointing out the reference [23].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Monotone traveling wavefronts of the KPP-Fisher delayed equation

Gomez

Trofimchuk

2011

Journal of Differential Equations

112

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“…Taking the parameter a = 0, (1.9) reduces to the delayed logistic equation 10) which was widely studied in [6,19,28,38]. To our knowledge, the exponential asymptotic behavior at negative infinity of traveling wave fronts for (1.7) cannot be obtained by construction of the upper and lower solutions; see [12].…”

Section: Du(t) Dt = U(t) B -Au(t) -Du(t -1) (13)mentioning

confidence: 99%

Existence and asymptotic of traveling wave fronts for the delayed Volterra-type cooperative system with spatial diffusion

Meng

Zhang

2018

Adv Differ Equ

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In the paper, we are concerned with the existence and the exponential asymptotic behavior of traveling waves for the delayed Volterra-type cooperative system with nonquasimonotone conditionmodeling the variation of the populations. Here, the major contribution to population model is the introduction of population self-regulation depending on not only the populations at time t, but also on the earlier population time t -τ i (i = 1, 3). By constructing a pair of suitable upper and lower solutions we obtain the existence of traveling wave fronts connecting the trivial equilibrium and the positive equilibrium, which indicates that there is a transition zone moving the steady state with no species to the steady state with the coexistence of two species. Furthermore, with the help of Ikehara's theorem, the exponential asymptotic behavior of traveling wave front is exactly derived for this system without quasi-monotone conditions. The results are not only an extension of existing results for the known logistic or cooperative system, but also can extend another type of delayed logistic equation with spatial diffusion ∂u (x, t) ∂t x, t) ∂x 2 + ru(x, t) 1 -au(x, t) -bu(x, t -τ ) . MSC: 35C07; 92D25; 35B35

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“…However, the local asymptotic stability of the positive constant steady-state solution of (1.1) is found in this work. In addition, the traveling wave solutions for an equation in form of (1.2) have been considered in many papers [2,3,[24][25][26]23]. In this paper, by using an iterative technique recently developed by Wang, Li and Ruan [8], sufficient conditions are established for the existence of traveling wave front solution connecting the zero and the positive equilibria in reaction-diffusion equation with spatio-temporal delay (1.1).…”

Section: ð1:2þmentioning

confidence: 99%