Stochastic reaction-diffusion fronts: applications to ecology and evolution

Abstract

Spatial expansions have shaped the evolutionary history of many organisms, from microbes to humans. These expansions are usually described by two types of reaction-diffusion waves: pulled waves, which are driven by growth at the edge of the expansion, and pushed waves, which are driven by the bulk. In my dissertation, I investigate how demographic fluctuations affect fluctuations in genetic composition and population density when waves transition from pulled to pushed.

First, I show that the variance of the fluctuations decreases with the population size, following a logarithmic dependence for pulled waves or a power law dependence for pushed waves. However, for weakly pushed waves the exponent is small and the fluctuations large, while for strongly pushed waves, the variance of the fluctuations decreases inversely proportional to the population size. I also show that these scaling regimes are present in populations with arbitrary density-dependent growth and dispersal.

Second, I show that the different rates of genetic diversity loss in the different classes of waves are a result of the genealogical structure of the population transitioning from a Bolthausen–Sznitman to a Kingman coalescent as the wave changes from pulled to pushed. Importantly, all of these results are independent of the dispersal and growth models and are controlled by a universal parameter: the ratio of the expansion velocity to the geometric mean of the dispersal and growth rates at low density. Thus, cooperative dispersal and growth could have a large impact on evolutionary dynamics, even when their contributions to the expansion velocity is small.