It's an interesting question: animals, we now accept, have the right to be treated humanely, but does that extend to their treatment at the hands (paws) of other animals? And then on the other side, do they have the right to follow their instincts, even when those instincts lead them to kill? I don't have the right to follow all my instincts, but then I know I should control them, unlike a cat.

One way to balance the rights of the cat with the rights of the pigeons, hamster and rodents would be to say that the pigeons and hamster acquired the right to protection when you and your daughter undertook to look after them. So the cat should be allowed to kill as many wild rodents as it likes, but if it kills your animals, it's your fault and you should be put down.