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There is a thin theme of animal welfare in this story. The story however is primarily escape fiction and we should try not to read too much into the animal welfare theme. Having said that, the author does try to comment how animals might feel being hunted by Rainsford, a famous hunter, experiencing just that.
The question is more complicated than that. He focuses on the feelings of the animals and the fear that they feel which Rainsford initially dismisses. He shows Rainsford that fear is involved; Rainsford may continue to hunt but he will have a different feeling about it when he is stalking his prey.