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By proving the existence of non-monotone and non-oscillating wavefronts forthe Nicholson's blowflies diffusive equation (the NDE), we answer an openquestion raised in [16]. Surprisingly, these wavefronts can be observed onlyfor sufficiently small delays. Similarly to the pushed fronts, obtained wavesare not linearly determined. In contrast, a broader family of eventuallymonotone wavefronts for the NDE is indeed determined by properties of thespectra of the linearized equations. Our proofs use essentially severalspecific characteristics of the blowflies birth function (its unimodal form andthe negativity of its Schwarz derivative, among others). One of the keyauxiliary results of the paper shows that the Mallet-Paret--Cao--Arino theoryof super-exponential solutions for scalar equations can be extended for someclasses of second order delay differential equations. For the new type ofnon-monotone waves to the NDE, our numerical simulations also confirm theirstability properties established by Mei et al.

Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless