Spreading Speeds for Reaction–Diffusion Equations with a Shifting Habitat

Abstract

We study the spreading speeds of a species described by a reaction–diffusion equation with a shifting habitat on which the species’ growth rate increases in the positive spatial direction. Persistence and the rightward spreading speed have been previously investigated in the case that the species grows near positive infinity and declines near negative infinity. In the present paper we determine the leftward spreading speed for this case and both the rightward and leftward spreading speeds for the case that the species grows everywhere in space. We show that the spreading speeds depend on c, the speed at which the habitat shifts, and numbers $c^*(\infty )$ and $c^*(-\infty )$ given by the diffusion coefficient and species growth rates at $\infty$ and $-\infty$ respectively. We demonstrate that if the species declines near negative infinity, the leftward spreading speed is $\min \{-c, c^*(\infty )\}$, and if species grows everywhere, under appropriate conditions, the rightward spreading speed is either $c^*(\infty )$ or $c^*(-\infty )$, and the leftward spreading speed is one of $c^*(\infty )$, $c^*(-\infty )$ and $-c$. Our results show that it is possible for a solution to form a two-layer wave, with the propagation speeds analytically determined.