The noisy edge of traveling waves

Abstract

Traveling waves are ubiquitous in nature and control the speed of many important dynamical processes, including chemical reactions, epidemic outbreaks, and biological evolution. Despite their fundamental role in complex systems, traveling waves remain elusive because they are often dominated by rare fluctuations in the wave tip, which have defied any rigorous analysis so far. Here, we show that by adjusting nonlinear model details, noisy traveling waves can be solved exactly. The moment equations of these tuned models are closed and have a simple analytical structure resembling the deterministic approximation supplemented by a nonlocal cutoff term. The peculiar form of the cutoff shapes the noisy edge of traveling waves and is critical for the correct prediction of the wave speed and its fluctuations. Our approach is illustrated and benchmarked using the example of fitness waves arising in simple models of microbial evolution, which are highly sensitive to number fluctuations. We demonstrate explicitly how these models can be tuned to account for finite population sizes and determine how quickly populations adapt as a function of population size and mutation rates. More generally, our method is shown to apply to a broad class of models, in which number fluctuations are generated by branching processes. Because of this versatility, the method of model tuning may serve as a promising route toward unraveling universal properties of complex discrete particle systems.