Spatial Spread with Allee Effect

Abstract

Many species exhibit an Allee effect, where population growth rates are highest at intermediate rather than low density, and small populations may even decline. Determining the spread rates of these species turns out to be much more difficult than the theory in the preceding chapter, where there was no Allee effect. Mathematically, this difficulty arises since—just as in the case of steady states—we cannot expect the linearization at zero to give useful information about the behavior of solutions for larger density, and hence we cannot expect the linearization-based spread formulas from the previous chapter to hold. One of the most interesting biological results here is that with the Allee effect, a population may spread or retreat. Hence, eradication of an invading pest species seems possible if management measures could turn an invasion into a retreat. We begin this chapter with a caricature model for which all relevant quantities can be explicitly calculated. Then we present a general condition for whether a population will spread or retreat. Finally, we present a theorem about the existence of traveling waves and the uniqueness of their speed.