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There is no biblical support for that position, the only biblical quotation that comes close to what you describe refers to the situation in heaven, not in the pre-fall garden of eden.
There is prophetic scripture of the coming millennial messianic era when animals will again be vegetarian and will not hurt one another nor mankind in Isaiah 11:1-10. Most fundamentalist Biblical scholars believe this is also descriptive of how it was before the flood.
I have a friend who once had a dog which loved raw garden squash. Most cats eat some grass. With a super climate and lush vegitation I'm quite sure animals would soon adapt. This is not to mention the supernatural factor which will has been and will be in play.