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I think the underlying theme here, if you can call it that, is that humans and animals share the emotions of panic and fear. This is alluded to by the conversation between Whiney and Rainsford at the beginning of the story,
"The best sport in the world," agreed Rainsford.
"For the hunter," amended Whitney. "Not for the jaguar."
"Don't talk rot, Whitney," said Rainsford. "You're a big-game hunter,
not a philosopher. Who cares how a jaguar feels?"
"Perhaps the jaguar does," observed Whitney.
We can pretty much figure out that Rainsford is going to end up getting hunted at some point in the story. Sure enough Rainsford gets what Whitney is talking about later on,
"Then he ran for his life. The hounds raised their voices as they hit the fresh scent. Rainsford knew now how an animal at bay feels."