We study an integro-difference equation model that describes the spatial dynamics of a species in an expanding or contracting habitat. We give conditions under which the species disperses to a region of poor quality where the species eventually becomes extinct. We show that when the species persists in the habitat, the rightward and leftward spreading speeds are determined by c, the speed at which the habitat quality increases or decreases in time, as well as [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], which are formulated in terms of the dispersal kernel and species growth rates in both directions. We demonstrate that in the case that the species grows everywhere in space, the rightward spreading speed is [Formula: see text] if c is relatively small and is [Formula: see text] if c is large, and the leftward spreading speed is one of [Formula: see text], [Formula: see text], or [Formula: see text]. We also show that it is possible for a solution to form a two-layer wave, with the propagation speeds of the two layers analytically determined.
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