Biphasic range expansions with short- and long-distance dispersal

Long-distance dispersal (LDD) has long been recognized as a key factor in determining rates of spread in biological invasions. Two approaches for incorporating LDD in mathematical models of spread are mixed dispersal and heavy-tailed dispersal. In this paper, I analyze integrodifference equation (IDE) models with mixed-dispersal kernels and fat-tailed (a subset of the heavy-tailed class) dispersal kernels to study how short- and long-distance dispersal contribute to the spread of invasive species. I show that both approaches can lead to biphasic range expansions, where an invasion has two distinct phases of spread. In the initial phase of spread, the invasion is controlled by short-distance dispersal. Long-distance dispersal boosts the speed of spread during the ultimate phase, and can have significant effects even when the probability of LDD is vanishingly small. For fat-tailed kernels, I introduce a method of characterizing the "shoulder" of a dispersal kernel, which separates the peak and tail.