Fig. 2.

Cooperator populations move as a traveling wave. (A) The experimentally observed one-dimensional expansion of a pure cooperator population over 9 d (m = 0.5 and dilution factor = 600). (B) Density profiles over time for expanding populations of pure cooperators (darker colors indicate later time points). Circles are measurements of population density at a particular time and spatial position. Lines are fits of individual profiles to the hyperbolic tangent function derived in SI Text. (C) An overlay of the density profiles from the last 6 d of the experiment (darker circles indicate later time points). Each profile is normalized to the maximum density found in the bulk population (_max) and shifted by its midpoint position (X_m). The first 3 d are not included, because the expanding population had not yet reached a steady-state profile. The red line shows a theoretical fit to the hyperbolic tangent function derived from a standard reaction–diffusion model of expanding populations (discussed in SI Text). (D) We measure the velocity of the traveling wave by plotting the position of the density profile midpoint (X_m) vs. time and then finding the slope of the line. As in C, the first 3 d are not included.