Persistence and Spread of a Species with a Shifting Habitat Edge

We study a reaction-diffusion model that describes the growth and spread of a species along a shifting habitat gradient on which the species' growth increases. It is assumed that the linearized species growth rate is positive near positive infinity and is negative near negative infinity. We show that the persistence and spreading dynamics depend on the speed of the shifting habitat edge $c$ and a number $c^*(\infty)$ that is determined by the maximum linearized growth rate and the diffusion coefficient. We demonstrate that if $c>c^*(\infty)$, then the species will become extinct in the habitat, and that if $c<c^*(\infty)$, then the species will persist and spread along the shifting habitat gradient at an asymptotic spreading speed $c^*(\infty)$.