Traveling waves with dispersive variability and time delay

We first determine an approximate traveling wave profile for the Cook model [J. Murray, Mathematical Biology I: An Introduction (Springer, New York, 2002), pp. 471-478] for the case in which the number of dispersers is small relative to the number of nondispersers. The results are consistent with the previous linearized wavefront analysis that predicts, counterintuitively, that relatively few dispersers can drive the population expansion wave with a wavespeed not too different from that for the case of a single dispersing population as described by the Fisher equation. The method of solution differs from that used in the latter case since here the dimensionless wavespeed is close to unity. We next generalize the Cook model to include time-delay effects. While the Cook model, like the Fisher equation, does not adequately describe the wave of advance during the Neolithic transition in Europe, we show that the generalized Cook model provides a close agreement with the historical record.