On pushed wavefronts of monostable equation with unimodal delayed reaction

Abstract

<p style='text-indent:20px;'>We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function <inline-formula><tex-math id="M1">\begin{document}$ g(u) $\end{document}</tex-math></inline-formula>. This model, designed to describe evolution of single species populations, is considered here in the presence of the weak Allee effect (<inline-formula><tex-math id="M2">\begin{document}$ g(u_0)>g'(0)u_0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M3">\begin{document}$ u_0>0 $\end{document}</tex-math></inline-formula>). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. First, we solve the front existence problem for small delays, <inline-formula><tex-math id="M4">\begin{document}$ h \in [0,h_p] $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ h_p $\end{document}</tex-math></inline-formula>, given by an explicit formula, is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function which makes possible explicit computation of traveling fronts. In this case, we find out that a) increase of delay can destroy asymptotically stable pushed fronts; b) the set of all admissible wavefront speeds has usual structure of a semi-infinite interval <inline-formula><tex-math id="M6">\begin{document}$ [c_*, +\infty) $\end{document}</tex-math></inline-formula>; c) for each <inline-formula><tex-math id="M7">\begin{document}$ h\geq 0 $\end{document}</tex-math></inline-formula>, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.</p>