Noise driven evolutionary waves

Adaptation in spatially extended populations entails the propagation of evolutionary novelties across habitat ranges. Driven by natural selection, beneficial mutations sweep through the population in a "wave of advance". The standard model for these traveling waves, due to R. Fisher and A. Kolmogorov, plays an important role in many scientific areas besides evolution, such as ecology, epidemiology, chemical kinetics, and recently even in particle physics. Here, we extend the Fisher-Kolmogorov model to account for mutations that confer an increase in the density of the population, for instance as a result of an improved metabolic efficiency. We show that these mutations invade by the action of random genetic drift, even if the mutations are slightly deleterious. The ensuing class of noise-driven waves are characterized by a wave speed that decreases with increasing population sizes, contrary to conventional Fisher-Kolmogorov waves. When a trade-off exists between density and growth rate, an evolutionary optimal population density can be predicted. Our simulations and analytical results show that genetic drift in conjunction with spatial structure promotes the economical use of limited resources. The simplicity of our model, which lacks any complex interactions between individuals, suggests that noise-induced pattern formation may arise in many complex biological systems including evolution.