It's not just property rights. The framework for protecting animals is more complex than "animal rights < property rights". You can't for example go and shoot or poison a stray dog out in the city - but you can a rat or a cockroach. For fishing, there are typically limits on how many fish you can catch and of what kinds, even though they aren't anyone's property. You can't harm your own animals in certain ways, at least in certain states - you may be allowed to shoot them, but you're not allowed to torture them almost anywhere.

The general point is that animal protection are almost entirely subsumed to human rights - animals are protected in so far as their protection helps humans in some way (either specifically, such as your chicken being useful to you so that no one else can kill them; or environmentally, such as elk being important for the health of certain forests). Given the human need to consume or displace other living beings, this is the only tenable moral position that can be held anyway.