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Apart from Australia and some islands, there are actually land corridors connecting between all the continents on Earth. This is also why we find that the Incas and Mayans reached the Americas despite mankind’s origin in the continent of Asia. It was technically possible to roam from place to place, and people did it without ships. Animals also migrated.
We must also remember that the continental drift is an ongoing process taking place over hundreds and thousands of years, and is taking place even today. We can assume that 4000 years ago the continents were closer together, or at least the areas connecting the various countries were larger, and allowed easy and continuous passage from one continent to another, although such passage is difficult and sometimes impossible today.
The Torah teaches us that the continents separated after the Flood at a period after the Tower of Babel, and it seems that it was a gradual process that occurred in stages during the Flood and over the following centuries.
Genesis 1:9-10 describes the creation of only one continent: “And G-d said, ‘Let the water that is beneath the heavens gather into one place, and let the dry land appear,’ and it was so. And G-d called the dry land earth (sing.)” After the Flood, it says about the son of Eber: “he was named Peleg, because in his days the earth was divided” (Genesis 10:25).
The Zohar, which was revealed 700 years ago, teaches this idea too: “There was one land, and the waters erupted and seven lands came into existence” (Zohar, Bereshit 21:1). This explanation may explain the proliferation of animals all over the world. Animals of all kinds migrated under divine guidance to certain places in the original continent, and were there when the continent split over hundreds or thousands of years into the separate continents we see today.
See more questions and answers about Noah and the Flood.