  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
parameter: 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 contributions to the expansion velocity is small.
