Long-range dispersal, stochasticity and the broken accelerating wave of advance

TL;DR

This study investigates how demographic stochasticity affects the acceleration of invasive species spread driven by long-range dispersal, revealing that stochasticity often disrupts acceleration except in specific power-law kernels.

Contribution

It introduces a stochastic lattice model to systematically analyze the effects of stochasticity on dispersal-driven invasion waves, highlighting conditions where acceleration persists despite stochastic effects.

Findings

Stochasticity generally breaks wave acceleration for fat-tailed kernels.

Acceleration persists in some power-law kernels with $eta < 3$, which lack a second moment.

Long-range dispersal models converge slowly to continuum behavior as stochasticity decreases.

Abstract

Rare long distance dispersal events are thought to have a disproportionate impact on the spread of invasive species. Modelling using integrodifference equations suggests that, when long distance contacts are represented by a fat-tailed dispersal kernel, an accelerating wave of advance can ensue. Invasions spreading in this manner could have particularly dramatic effects. Recently, various authors have suggested that demographic stochasticity disrupts wave acceleration. Integrodifference models have been widely used in movement ecology, and as such a clearer understanding of stochastic effects is needed. Here, we present a stochastic non-linear one-dimensional lattice model in which demographic stochasticity and the dispersal regime can be systematically varied. Extensive simulations show that stochasticity has a profound effect on model behaviour, and usually breaks acceleration for fat-tailed kernels. Exceptions are seen for some power law kernels, $K(l) \propto |l|^{-\beta}$ with $\beta < 3$, for which acceleration persists despite stochasticity. Such kernels lack a second moment and are important in `accelerating' phenomena such as L\'{e}vy flights. Furthermore, for long-range kernels the approach to the continuum limit behaviour as stochasticity is reduced is generally slow. Given that real-world populations are finite, stochastic models may give better predictive power when long-range dispersal is important. Insights from mean-field models such as integrodifference equations should be applied with caution in such circumstances.