Spatially explicit models for the population dynamics of a species colonizing an island

We construct reaction-diffusion models for the population dynamics of a species colonizing an island from a source population on a continent. We view the source population as inducing a density or flux of immigrants onto the island and interpret colonization as succeeding if the population on the island is predicted to persist even when immigration from the continent is stopped. To capture the observation that a sufficiently large population or density must be attained for colonization to succeed, we assume Allee (i.e., bistable) dynamics rather than logistic dynamics for the colonizing population. We consider the cases of colonization in both the absence and presence of a competitor. We use reaction-diffusion theory, especially comparison methods and sub- and supersolutions, to determine how parameters such as the distance from the continent to the island and the dispersal, birth and mortality rates, carrying capacity, and minimum viable population density of the colonizing species affect the outcome of the attempted colonization. In the case of colonization in the presence of a competitor we consider a number of scenarios involving different types and strengths of competition. Our analysis permits us to draw conclusions about the characteristics of a species that make it a good colonizer.