Success, failure, and spreading speeds for invasions on spatial gradients

We study a model that describes the spatial spread of a species along a habitat gradient on which the species' growth increases. Mathematical analysis is provided to determine the spreading dynamics of the model. We demonstrate that the species may succeed or fail in local invasion depending on the species' growth function and dispersal kernel. We delineate the conditions under which a spreading species may be stopped by poor quality habitat, and demonstrate how a species can escape a region of poor quality habitat by climbing a resource gradient to good quality habitat where it spreads at a constant spreading speed. We show that dispersal may take the species from a good quality region to a poor quality region where the species becomes extinct. We also provide formulas for spreading speeds for the model that are determined by the dispersal kernel and linearized growth rates in both directions.