Reading Ian Morris Geography Is Destiny. His subject is Britain and geography over 10,000 years.
An interesting point he makes is in the domestication of wild grains in the Middle East--his opinion is that it began with women, since modern hunter-gatherers have women doing most of the gathering while the men do the hunting.
That makes sense to me. But he posits that men were involved in domesticating animals--livestock. He doesn't give his logic, but the implication is hunting would lead into domestication. I'm not so sure. We know, or I think we know, that the maternal instinct lives in both sexes and in many mammals. We've seen the cute pictures of animals of different species being "friends", grooming each other, sleeping with each other, playing, etc., which I'd ascribe to the maternal instinct at work.
I'd assume domestication proceeded by human adoption/seduction of young mammals, young girls perhaps saving a young animal from being eaten.