Neither pulled nor pushed: Genetic drift and front wandering uncover a new class of reaction-diffusion waves

Epidemics, flame propagation, and cardiac rhythms are classic examples of reaction-diffusion waves that describe a switch from one alternative state to another. Only two types of waves are known: pulled, driven by the leading edge, and pushed, driven by the bulk of the wave. Here, we report a distinct class of semi-pushed waves for which both the bulk and the leading edge contribute to the dynamics. These hybrid waves have the kinetics of pushed waves, but exhibit giant fluctuations similar to pulled waves. The transitions between pulled, semi-pushed, and fully-pushed waves occur at universal ratios of the wave velocity to the Fisher velocity linear spreading velocity. We derive these results in the context of a species invading a new habitat by examining front diffusion, rate of genetic drift, and fluctuation-induced corrections to the expansion velocity. All three quantities decrease as a power law of the population density with the same exponent. We calculate this exponent exactly without neglecting the fluctuations in the shape of the wave front. For fully-pushed waves, the exponent is -1 consistent with the central limit theorem. In semi-pushed waves, however, the fluctuations average out much more slowly, and the exponent approaches 0 towards the transition to pulled waves. As a result, a rapid loss of genetic diversity and large fluctuations in front position occur even for populations with cooperative growth and other forms of an Allee effect. The outcome of spatial spreading in such populations could therefore be less predictable than previously thought.