SignificanceTraveling waves describe diverse natural phenomena from crystal growth in physics to range expansions in biology. Two classes of waves exist with very different properties: pulled and pushed. Pulled waves are driven by high growth rates at the expansion edge, where the number of organisms is small and fluctuations are large. In contrast, fluctuations are suppressed in pushed waves because the region of maximal growth is shifted toward the population bulk. Although it is commonly believed that expansions are either pulled or pushed, we found an intermediate class of waves with bulk-driven growth, but exceedingly large fluctuations. These waves are unusual because their properties are controlled by both the leading edge and the bulk of the front.

Theory predicts rapid genetic drift during invasions, yet many expanding populations maintain high genetic diversity. We find that genetic drift is dramatically suppressed when dispersal rates increase with the population density because many more migrants from the diverse, high‐density regions arrive at the expansion edge. When density dependence is weak or negative, the effective population size of the front scales only logarithmically with the carrying capacity. The dependence, however, switches to a sublinear power law and then to a linear increase as the density dependence becomes strongly positive. We develop a unified framework revealing that the transitions between different regimes of diversity loss are controlled by a single, universal quantity: the ratio of the expansion velocity to the geometric mean of dispersal and growth rates at expansion edge. Our results suggest that positive density dependence could dramatically alter evolution in expanding populations even when its contribution to the expansion velocity is small.

Theory predicts rapid genetic drift in expanding populations due to the serial founder e↵ect at the expansion front. Yet, many natural populations maintain high genetic diversity in the newly colonized regions. Here, we investigate whether density-dependent dispersal could provide a resolution of this paradox. We find that genetic drift is dramatically suppressed when dispersal rates increase with the population density because many more migrants from the diverse, highdensity regions arrive at the expansion edge. When density-dependence is weak or negative, the e↵ective population size of the front scales only logarithmically with the carrying capacity. The dependence, however, switches to a sublinear power law and then to a linear increase as the density-dependence becomes strongly positive. To understand these results, we introduce a unified framework that predicts how the strength of genetic drift depends on the densitydependence in both dispersal and growth. This theory reveals that the transitions between di↵erent regimes of diversity loss are controlled by a single, universal parameter: the ratio of the expansion velocity to the geometric mean of dispersal and growth rates at expansion edge. Importantly, our results suggest that positive density-dependence could dramatically alter evolution in expanding populations even when its contributions to the expansion velocity is small.

Significance Spatial dynamics are important for understanding genetic diversity in many contexts, such as cancer and infectious diseases. Coalescent theory offers a powerful framework for interpreting and predicting patterns of genetic diversity in populations, but incorporating spatial structure into the theory has proven difficult. Here, we address this long-standing problem by studying the coalescent in a spatially expanding population. We find the topology of the coalescent depends on the growth dynamics at the front, but not on the functional form of the growth function. Instead, the transition between coalescent topologies is determined by a single dynamical parameter. Our theory makes precise predictions about the effects of population dynamics on genetic diversity at the expansion front, which we confirm in simulations.