The acceleration of evolutionary spread by long-range dispersal

TL;DR

This paper develops a predictive model for the spread of evolutionary traits considering long-range dispersal, revealing that growth follows power-law or stretched exponential patterns depending on dispersal tail behavior, with implications for epidemics.

Contribution

It introduces a simple iterative scaling approximation that accurately predicts spreading dynamics with long-range dispersal, supported by simulations and rigorous bounds.

Findings

Asymptotic growth follows power-law or stretched exponential laws.

Convergence to asymptotic behavior can be slow and complex.

Results apply to epidemic spread on networks with long-range links.

Abstract

The spreading of evolutionary novelties across populations is the central element of adaptation. Unless population are well-mixed (like bacteria in a shaken test tube), the spreading dynamics not only depends on fitness differences but also on the dispersal behavior of the species. Spreading at a constant speed is generally predicted when dispersal is sufficiently short-ranged. However, the case of long-range dispersal is unresolved: While it is clear that even rare long-range jumps can lead to a drastic speedup, it has been difficult to quantify the ensuing stochastic growth process. Yet such knowledge is indispensable to reveal general laws for the spread of modern human epidemics, which is greatly accelerated by aviation. We present a simple iterative scaling approximation supported by simulations and rigorous bounds that accurately predicts evolutionary spread for broad distributions of long distance dispersal. In contrast to the exponential laws predicted by deterministic "mean-field" approximations, we show that the asymptotic growth is either according to a power-law or a stretched exponential, depending on the tails of the dispersal kernel. More importantly, we provide a full time-dependent description of the convergence to the asymptotic behavior which can be anomalously slow and is needed even for long times. Our results also apply to spreading dynamics on networks with a spectrum of long-range links under certain conditions on the probabilities of long distance travel and are thus relevant for the spread of epidemics.