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Admissible wavefront speeds for a single species reaction-diffusion equation with delay
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, non-monotone)
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 non-monotone nonlinearity.