Did Rainsford decide to stay on the island as a ruler, or did he leave the island?
Ultimately, this answer can go either way. It is up to you.
I think a case could be made for saying that Rainsford chooses to stay on the island and continue hunting people like Zaroff was doing. Rainsford is a very good hunter. It's a passion of his. Beating Zaroff in a game of wits had to be a huge adrenaline rush. Rainsford might decide that he likes the rush of hunting people and being hunted by a fellow human being. The story does end with telling readers that Rainsford sleeps quite well after killing Zaroff. I sleep well when I'm content and happy. Rainsford might be content and happy with his newfound purpose in life.
Despite the previous paragraph, I don't believe that Rainsford would take over the island and Zaroff's position. Rainsford is horrified to learn about what Zaroff is doing. He argues against it on moral grounds. I don't think that being on the run for three days miraculously changes Rainsford's moral position. I think the reason that Rainsford sleeps so well that night is because he is content and happy that his ordeal is over. He can now safely go home.
I am sure that Rainsford would stay on the island you know why?
Coz there are so many political symboles in the story,Zaroff represents for Russia and Cossac and Rainsford represents for Ameriac.the story says that there is no any unity between these two, and a war betwwn these two country is inavoidable. so at last Zarrof is killed by Rainsford(Russia is defeated by America)and Raisford (America) would be definatly the king of his new order world.