Criterion for existence of non-monotone wavefronts in the delayed KPP-Fisher equation

Let $\mathfrak{D}_{n}$ denote the set of parameter pairs $(\tau,c)\in\mathbb{R}_+^2$ for which equation (17) admits a non-monotone wavefront, and let $c^*(\tau)$ and $c^\star(\tau)$ be the lower and upper critical speeds defined in the paper. Existence criterion.

$$\mathfrak{D}_{n}=\{(\tau,c)\in\mathbb{R}_+^2:c^*(\tau)<c\leq c^\star(\tau)\}.$$

The preceding results establish inclusions and existence in substantial parameter regions, but do not prove this full characterization; the equality is proposed as the natural criterion for existence of non-monotone wavefronts.