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

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) ∝ |l| −β with β < 3, for which acceleration persists despite stochasticity. Such kernels lack a second moment and are important in 'accelerating' phenomena such as Lévy 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.

The importance of being atomic: Ecological invasions as random walks instead of waves

“…(6.6) diverge for all z. This is analytic confirmation of the observations of Jacobs and Sluckin [2015] that Cauchy kernels lead to accelerating invasions, even in the presence of density-dependence.…”

“…For the case of density-independent proliferation, invasions accelerate faster than geometrically [Kot et al, 1996]. Méndez et al [2010] predicted and Jacobs and Sluckin [2015] observed numerically that invasions with carrying capacities exhibited finite speeds as long as β > 3, but that infinite speeds appeared when β < 3, even in systems with hard carrying capacities. The subsequent analysis provides further explanation.…”

“…Hallatschek and Fisher [2015] provide a scaling-law approach to the analysis of long-range dispersal. Jacobs and Sluckin [2015] provide an extensive simulation analysis of lattice theories, and conclude that while finite-population stochasticity slows invasions and can stop acceleration in many heavy-tailed kernels, the heaviest-tailed power-law kernels can still exhibit accelerations.…”

Population genetic models for the spatial spread of adaptive variants: A review in light of SARS-CoV-2 evolution

“…The underlying process of long-distance dispersal in a finite-size population is inherently stochastic, which may result in the inaccuracy of deterministic models when long-distance dispersal is frequent [ 30 , 41 ]. Ralph and Coop [ 22 ] show that under some conditions, properties of a stochastic traveling wave can be approximated well by considering a single “mean” wave that is representative of the wave’s path (their work in this area arises in the case of multiple adaptive lineages and is further described in the next section).…”

Fractional Diffusion Emulates a Human Mobility Network during a Simulated Disease Outbreak

“…A similar approach, space-fractional diffusion, arises from a generalization of a random walk when the distribution of step sizes in the walk is described by a power law, x −(1+α) with α ≤ 2, rather than a Gaussian function of the step sizes. This type of model has general application across fields from plasma physics [12] to ecology [13,14]. Mathematically, this heavy-tailed distribution of step sizes with an infinite variance produces superdiffusive dispersion, also known as a Lévy flight.…”