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@article{54421,
author = {Hasík, Karel and Kopfová, Jana and Nábělková, Petra and Trofimchuk, Sergei},
article_location = {Springfield},
article_number = {12},
doi = {http://dx.doi.org/10.3934/dcds.2021103},
keywords = {Traveling front; pushed wave; minimal speed},
language = {eng},
issn = {1078-0947},
journal = {Discrete and Continuous Dynamical Systems - Series A},
title = {On pushed wavefronts of monostable equation with unimodal delayed reaction},
url = {https://www.aimsciences.org/article/doi/10.3934/dcds.2021103},
volume = {41},
year = {2021}
}
TY - JOUR
ID - 54421
AU - Hasík, Karel - Kopfová, Jana - Nábělková, Petra - Trofimchuk, Sergei
PY - 2021
TI - On pushed wavefronts of monostable equation with unimodal delayed reaction
JF - Discrete and Continuous Dynamical Systems - Series A
VL - 41
IS - 12
SP - 5979-6000
EP - 5979-6000
PB - American Institute of Mathematical Sciences
SN - 10780947
KW - Traveling front
KW - pushed wave
KW - minimal speed
UR - https://www.aimsciences.org/article/doi/10.3934/dcds.2021103
N2 - We study the Mackey-Glass type monostable delayed reaction diffusion equation with a unimodal birth function g(u). This model, designed to describe evolution of single species populations, is considered here in the presence of the weak Allee effect (g(u0) > g'(0)u0 for some u0 > 0). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. First, we solve the front existence problem for small delays, h is an element of [0, hp], where hp, given by an explicit formula, is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function which makes possible explicit computation of traveling fronts. In this case, we find out that a) increase of delay can destroy asymptotically stable pushed fronts; b) the set of all admissible wavefront speeds has usual structure of a semi-infinite interval [c*, +infinity); c) for each h >= 0, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.
ER -
HASÍK, Karel, Jana KOPFOVÁ, Petra NÁBĚLKOVÁ and Sergei TROFIMCHUK. On pushed wavefronts of monostable equation with unimodal delayed reaction. \textit{Discrete and Continuous Dynamical Systems - Series A}. Springfield: American Institute of Mathematical Sciences, 2021, vol.~41, No~12, p.~5979-6000. ISSN~1078-0947. Available from: https://dx.doi.org/10.3934/dcds.2021103.
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