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