In the beginning, when God created the Earth and humans in His image, until the Fall of mankind, there was no death and humans and animals lived in harmony. Remember, God had Adam name all the animals, and they were subject to him.
This will certainly be the same when God creates the new heaven and new earth, but will it be the case when Jesus rules and reigns on this earth during His Millennial Kingdom? Sadly, I do not think so as the curse is not lifted, and people and animals still die. However, biblical prophecy reveals that Israel, as God’s nation, will be blessed above all others. In the following Scripture, it states that these animals’ shall not hurt or destroy in all my holy mountain, so this may apply in Israel.
“The wolf and the lamb shall graze together; the lion shall eat straw like the ox, and dust shall be the serpent’s food. They shall not hurt or destroy in all my holy mountain,” says the Lord.“ Isaiah 65:25
“You shall be called the priests of the Lord; they shall speak of you as the ministers of our God; you shall eat the wealth of the nations, and in their glory, you shall boast. Instead of your shame, there shall be a double portion; instead of dishonour, they shall rejoice in their lot; therefore, in their land, they shall possess a double portion; they shall have everlasting joy.” Isaiah 61:6-7
This video shows how a man developed a friendly long-term relationship with a shark after he had removed a hook from its mouth. It gives us a snapshot of what it was like before the Fall and what it will be like on the new earth.