Admissible wavefront speeds for a single species reaction-diffusion equation with delay

Abstract

We consider equation $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x)) (*) $, when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=\kappa>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(\kappa)) = [c_*,c^*]$ such that $(*)$ has at least one (possibly, nonmonotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here $c_*>0$ is finite and $c^* \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound $c_*$ is sharp so that there are not wavefronts moving with speed $c < c_*$. In contrast to reported results, the interval $\mathcal{C}$ can be compact, and we conjecture that some of equations $(*)$ can indeed have an upper bound for propagation speeds of travelling fronts. As particular cases, Eq. $(*)$ includes the diffusive Nicholson's blowflies equation and the Mackey-Glass equation with nonmonotone nonlinearity.