So God blessed Noah and his sons, and said to them: “Be fruitful and multiply, and fill the earth. And the fear of you and the dread of you shall be on every beast of the earth, on every bird of the air, on all that move on the earth, and on all the fish of the sea. They are given into your hand. Every moving thing that lives shall be food for you. I have given you all things, even as the green herbs.
(Genesis 9:1-3) NKJV
After Noah’s flood a carnivorous diet was on the menu, so predator-prey relationships developed which, no doubt, spurred the animals to spread out very quickly.
There weren’t that many people around after Noah’s flood. When they spread out to the different parts of the Earth there were a handful of families in each geographical region from which the population developed.
Human culture is one thing, biodiversity is another. The animals which left Noah’s ark were under no compulsion to attempt to stay in one place. This was in large part because God had given the green light to Noah and his descendants that they could eat meat.
Matty's Paradigm 