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

We consider equation uₜ(t,x) = Δ u(t,x)- u(t,x) + g(u(t-h,x))(*) , when g:₊→ ₊ has exactly two fixed points: x₁= 0 and x₂=κ>0. Assuming that g is unimodal and has negative Schwarzian, we indicate explicitly a closed interval C = C(h,g'(0),g'(κ)) = [ c*, c * ] such that (*) has at least one (possibly, non-monotone) travelling front propagating at velocity c for every c ∈ C. Here c*$ >0$ is finite and c * ∈ ₊ ∪ \+∞\. Every time when 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 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.