The Saharan Pump
How big? Pleistocene lions stood five feet tall at the shoulder. Really big.
There's also something called the "Saharan Pump," which is alternating wet (savannah) and dry (extremely arid deserts) periods in what is now (temporarily) the Saharan Desert. The idea is, wet periods are inviting and allow humans to expand out of Africa, while dry periods turn huge rivers into wadis, slam the door shut on the way out and force people south and west in Africa, or into Europe and Asia. Vast time scales. About two million years ago, humans appeared in Africa, and got pumped out into the wider world. Actual modern humans appeared about 100,000 years ago, and got pumped out of Africa the same way.
The Saharan Pump is still active.
What happened to the big cats during this same period? Are tigers a kind of lion? What about leopards, which seem pretty much the same everywhere? How in the world did humans "adapt" to big cats? They're not wolves. You can't adopt lion puppies... I'm puzzled. Actually, there's a story (I think told by natural historian Raymond L. Ditmars, or possibly Gerald Durrell) that men stink, so far as lions are concerned; they find us disgusting and unpalatable, like human females. Thank gosh for Old Spice.