The animal kingdom remains where they are, but during the Millennial Reign, they undergo a major change in nature, as Jesus establishes Himself as King over all, even nature will conform to become what it was created to be in the beginning.
I found this excerpt to be informative:
Animals Become Peaceful
Now regarding the character of the kingdom, the book of Isaiah is rich with kingdom prophecies. In Isaiah 11:6-9 it is revealed that "The wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and the young lion and the fatling together; and a little child shall lead them. And the cow and the bear shall feed; their young ones shall lie down together: and the lion shall eat straw like the ox. And the sucking child shall play on the hole of the asp, and the weaned child shall put his hand on the cockatrice' den. They shall not hurt nor destroy in all my holy mountain: for the earth shall be full of the knowledge of the LORD, as the waters cover the sea."
First of all, we see that animals that are presently carnivorous, will begin to eat vegetation only, so that the animals will not fear one another or harm people. Many have taken this passage and tried to spiritualize (allegorically interpret) the truth away from it, saying it is not literal. However, not only will all animals literally be herbivores in the future, but all animals once were herbivores, since neither man nor beast ate animal flesh prior to the great flood.
Genesis 1:29-30 "And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in the which is the fruit of a tree yielding seed; to you it shall be for meat. And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so."
The Hebrew word "oklah" translated "meat" by the King James Version of the Bible (KJV) simply means food or that which is devoured. Notice that the beasts are neither given for Adam's food, nor for one another's food. Both Adam and the beasts are given the herbs and fruits to eat. So man and beast were all vegetarian. However, God changed things after the great flood
In Genesis 9:2-3 God says to Noah, "And the fear of you and the dread of you shall be upon every beast of the earth, and upon every fowl of the air, upon all that moveth upon the earth, and upon all the fishes of the sea; into your hand are they delivered. Every moving thing that liveth shall be meat for you; even as the green herb have I given you all things."
So the animals began to fear man, because man was given permission to eat the animals. This is also when many animals became carnivorous.
Source:
http://www.matthewmcgee.org/millen.html