On pushed wavefronts of monostable equation with unimodal delayed reaction

We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function \begin{document}$ g(u) $\end{document}. This model, designed to describe evolution of single species populations, is considered here in presence weak Allee effect (\begin{document}$ g(u_0)>g'(0)u_0 $\end{document} for some id="M3">\begin{document}$ u_0>0 $\end{document}). We focus our attention on existence slow monotonic traveling fronts equation: under given assumptions, this problem seems be rather difficult since usual positivity and monotonicity arguments are not effective. First, we solve front small delays, id="M4">\begin{document}$ h \in [0,h_p] $\end{document}, where id="M5">\begin{document}$ h_p by an explicit formula, optimal certain sense. Then take representative piece-wise linear which makes possible computation fronts. In case, find out that a) increase delay can destroy asymptotically stable pushed fronts; b) set all admissible wavefront speeds has structure semi-infinite interval id="M6">\begin{document}$ [c_*, +\infty) $\end{document}; c) each id="M7">\begin{document}$ h\geq 0 unique (if it exists); d) wave oscillate slowly around positive equilibrium sufficiently large delays.