Invasions with Fat-Tailed and Long-Distance Dispersal

Abstract

Ecologists have recognized fat-tailed and long-distance dispersal (LDD) as critical to our understanding of population spread and invasions; heavy- and fat-tailed kernels fit empirical dispersal data better than classical thin-tailed kernels, and long-distance dispersal has driven some of the most rapid invasions. Despite their importance, researchers have struggled to incorporate fat-tailed and long-distance dispersal into mathematical models of spread. Analytical techniques for fat-tailed dispersal and for LDD in spatially explicit models of spread have seen little development. In this dissertation, I develop new analyses and techniques to study invasions with fat-tailed and long-distance dispersal. I study invasions with two types of dispersal kernels associated with long-distance dispersal: fat-tailed (power-law decay) kernels, which have a propensity for generating extreme events, and thin-tailed mixed-dispersal kernels, which combine multiple dispersal kernels that disperse over different scales. I first characterize the asymptotic rate of spread for invasions with fat-tailed dispersal. I use the tail additivity properties of regularly varying probability densities to analyze invasions with point- and front-release initial conditions, as well as for populations with weak Allee effects. I show the rate of spread to be geometric, with base determined by the net reproductive rate and the degree of fatness of the dispersal kernel tails. I show that the dynamics of fat-tailed invasions have several key qualitative differences from those of thin-tailed invasions. I next turn to the transient, or short-to-intermediate timescale dynamics of invasions with long-distance dispersal. I show that LDD, whether implemented by fat-tailed or mixed-dispersal kernels, can lead to biphasic range expansion, where the invasion has two distinct phases and rates of spread; the initial phase of spread is governed by short-distance dispersal, while long-distance dispersal accelerates the invasion in the ultimate phase of spread. Biphasic linear-linear spread is possible under mixed thin-tailed kernels, while biphasic linear-accelerating spread can occur under fat-tailed kernels. I show that for some families of mixed dispersal, the effects of LDD are persistent even when the probability of long-distance dispersal approaches zero; while reducing the probability delays the onset of the second phase of spread, the ultimate speed of spread remains elevated. For fat-tailed kernels, I show how speed rarefaction curves can be used to delineate between the peak and tail of the dispersal kernel, and to define a "shoulder" of the dispersal kernel separating short- and long-distance dispersal.