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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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