The settler Thomas Austin released 24 wild rabbits on his Australian farm, called Barwon Park, in 1859, and some other Australian farmers later followed his example. Rabbits are sexually mature at about six months, and they have a 31-day gestation period. Given a favorable environment, rabbits can easily increase their population fourfold in a year. Try to imagine the growth of the rabbit population in Australia: first 24 rabbits, then 100 rabbits after a year, 20,000 in five years, and 25 million after ten years. That's roughly what happened: At the end of ten years, shooting or trapping 2 million a year had no noticeable effect on their population.
At first growth was slow—an increase of 75 in a year doesn't sound that impressive, not in a country the size of the lower 48. But growth speeded up as the rabbit population increased. Another way of putting it is that the percentage of growth per year stayed the same, but that the percentage was multiplied by a larger and larger population as time passed. A process that at first seemed unimpressive left a whole continent literally swarming with rabbits in a single decade. It took only two or three times longer to fill a continent with rabbits than it had to fill a single farm.
A favorable allele, such as the one that confers lactose tolerance, spreads in much the same way, although the process takes thousands of years, largely because human generations are much longer than rabbit generations. But for a sweep to happen rapidly, the population must be "well mixed." This is not always the case, because mixing genes over long distances—over rivers, mountains, deserts, and oceans, or through hostile tribes—is far from automatic. Sometimes it happened, sometimes it didn't. These sweeps were strongly influenced by history—and they influenced history right back.
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