"And so the lion fell in love with the lamb..." (13.130). When Edward says this, it's pretty obvious that he means that he's the lion and Bella is the lamb in their situation. While most lions wouldn't hesitate to eat a lamb for dinner, what happens if the lion loves the lamb and doesn't want to eat her? A few other predator-prey pairs appear as well, including a shark and an angelfish, and a killer whale and baby seal, and "a bird locked in the eyes of a snake" (13.38). These animal references emphasize how, in the natural scheme of things, Bella is actually Edward's prey.
There's something else we noticed about animals in Twilight: the Cullens are often compared to animals. Emmett is compared to a bear – he's big and powerful. Interestingly, Emmett joined the Cullen clan when he was nearly killed by a bear in the woods, but rescued by Rosalie. Now, Emmett's favorite animal to hunt is a bear.
Edward is frequently compared to big cats. His favorite game to hunt is the mountain lion, and during the Cullen baseball game, Bella says that he runs like a cheetah. From these descriptions, we come to see Edward as rather sleek, elegant, and fast.
Also during the baseball game, Bella compares the graceful Alice to a gazelle, although we're willing to bet that she's a lot more dangerous.
In general, though, humans are compared to benign creatures, and vampires are compared to dangerous predators. Speaking of benign creatures, on more than one occasion, Bella compares Mike to a friendly, tail-wagging dog. This description works pretty well, because Mike is harmless, eager to please Bella, and always hoping for her attention.