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A stable is a nice thing to have... but if you really want critters to reproduce waaay beyond the limits of your stable (eight for most types), you gotta let them roam free. BUT, and here's the important bit, you have to make it so the hatches have a way back into the stable, and will eventually wander back into the rancher's warm grooming hands.

That way you get the best of both worlds: when a hatch is inside a stable, it gets all the bonuses of grooming by that poor, overworked rancher.

But let the hatch (or other critter) wander outside its stable after being groomed, it simply stops feeling overcrowded. It's no longer restricted by a stable's hard limits anymore, which allows "fertility" numbers to tick up...

... and hey look, it's groomed and tamed, so those numbers will tick up a lot faster than a wild hatch (or other critters).