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

Abstract

Short AbstractTraveling 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 towards 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.\n\nLong AbstractEpidemics, 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. We derive these results in the context of a species invading a new habitat by examining front diffusion, rate of diversity loss, 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 analytically calculate this exponent taking into account 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 the position of the front occur even for populations with cooperative growth and other forms of an Allee effect. The evolutionary outcome of spatial spreading in such populations could therefore be less predictable than previously thought.