I was going to attempt to read and comment on a recent paper in PLOS Biology myself, but fortunately smarter people than me, who understand the subject better, got there first. So all I need to do is point you to Razib Khan’s explanation of how recent DNA analyses confirm “tentatively” the idea that farming didn’t spread into Europe as a result of people imitating their neighbours. Instead, the DNA suggests that spread was:
[A] classic demic diffusion process. This is basically a very simple model whereby farmers with larger population growth rates expand into the “space” of hunter-gatherers.
Now to do the same with their crops and livestock?