Short answer: Because God took care of Noah and everything else on the Ark.
Bible reference: Genesis 8:1
“And God remembered Noah, and every living thing, and all the cattle that was with him in the ark: and God made a wind to pass over the earth, and the waters assuaged”
The Bible says that “God remembered Noah” and those aboard the Ark during the Flood. When we think of remembering, we usually think of something like not forgetting, but the word “remembered” has a special meaning here. It means that God was thinking of, watching over and taking care of the Ark the entire time.
What’s this have to do with hungry meat-eating dinosaurs and man-eating tigers?
Well, God sent all of the animals to Noah and commanded him to bring them on board. This verse tells us that He took care of them. We can only guess how God kept everyone safe on the Ark during that voyage – including safe from each other! But the Biblle does give us some clues.
For example, the Book of Daniel records that God’s prophet was thrown into a den of hungry lions, but he wasn’t eaten because because God locked the jaws of the lions. By His power, He kept them from eating Daniel.
The Book of Genesis also tells us that God originally created the animals to eat plants. A lot of animals have sharp teeth, but sharp teeth doesn’t always mean that animal eats meat. Pandas, mountain gorillas, lemurs and fruit bats have big sharp teeth, but they don’t eat meat as a rule. also, you may have a pet, such as a dog or a cat. They do eat meat, but you may have noticed that sometimes they eat grass. creatiuon scientists believe that God may have caused meat eaters to eat plants again like they were originally created to do, before Adam sinned and the world was cursed. After they left the Ark, God would have stopped suppressing their fallen urge to eat meat and let them return to being predators.
We could also add that when animals go through a disaster, such as a forest fire or earthquake, they go into a state of near hibernation. And many animals hibernate [or sleep for a season] and do not need to eat during this time. It’s possible that many animals, including meat eaters like lions and T-rexes could have gone into hibernation for long periods of the Ark’s voyage.
While the Bible doesn’t tell us the details of how God “remembered” those on the Ark, there are reasonable answers if we think about it.