Forced Traveling Waves in a Reaction-Diffusion Equation with Strong Allee Effect and Shifting Habitat

Abstract

We study a reaction-diffusion equation that describes the growth of a population with a strong Allee effect in a bounded habitat which shifts at a speed [Formula: see text]. We demonstrate that the existence of forced positive traveling waves depends on habitat size L, and [Formula: see text], the speed of traveling wave for the corresponding reaction-diffusion equation with the same growth function all over the entire unbounded spatial domain. It is shown that for [Formula: see text] there exists a positive number [Formula: see text] such that for [Formula: see text] there are two positive traveling waves and for [Formula: see text] there is no positive traveling wave. It is also shown if [Formula: see text] for any [Formula: see text] there is no positive traveling wave. The dynamics of the equation are further explored through numerical simulations.

Keywords: Allee effect; Reaction-diffusion equation; Shifting habitat; Traveling wave.