Existence of Traveling Waves for a Diffusive Nicholson Equation with Harvesting

Abstract

We prove the existence of traveling wave solutions for a diffusive Nicholson model with a delayed linear harvesting term and two delayed-response channels. Our approach combines monotone heteroclinic connections of the associated scalar delay equation with the large-speed traveling-wave theorem of Faria, Huang and Wu. For \(0<σ<r\), we use a previous upper- and lower-solution construction under an explicit parameter condition. For \(0<r\leqσ\), we establish a new monotone heteroclinic connection. To fit the abstract framework, we embed the scalar equation into an auxiliary two-dimensional reaction--diffusion system and prove that every relevant traveling wave lies on the invariant diagonal. We then determine an explicit parameter region in which the required spectral hypotheses hold. Consequently, the original diffusive Nicholson equation admits traveling waves connecting the trivial and positive equilibria for all sufficiently large wave speeds.