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 $g(u)$. 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 making 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.