# American Institute of Mathematical Sciences

December  2021, 41(12): 5979-6000. doi: 10.3934/dcds.2021103

## On pushed wavefronts of monostable equation with unimodal delayed reaction

 1 Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic 2 Instituto de Matemática, Universidad de Talca, Casilla 747, Talca, Chile

* Corresponding author: Sergei Trofimchuk

Received  October 2020 Published  December 2021 Early access  June 2021

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 ($g(u_0)>g'(0)u_0$ for some $u_0>0$). 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, $h \in [0,h_p]$, where $h_p$, 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 $[c_*, +\infty)$; c) for each $h\geq 0$, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.

Citation: Karel Hasík, Jana Kopfová, Petra Nábělková, Sergei Trofimchuk. On pushed wavefronts of monostable equation with unimodal delayed reaction. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5979-6000. doi: 10.3934/dcds.2021103
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##### References:
Toy model: piece-wise linear birth function $g$
Vertical asymptote $h = h_*$ followed (in the counter-clockwise direction) by the graphs of $c(h), c_\#(h), c_*(h),c_\kappa(h)$. The cases $k = 1.5$ (left) and $k = 1.2$ (right).
Snapshots of solution $u(t,x)$ to the Cauchy problem (25), $k = 1.2$, converging to the pushed wavefront, at the indicated sequence of times $t = t_j$. The cases $h = 0.5$ (left), $h = 6$ (right).
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