That's another question I have. I know if an animal is bothering livestock, you can take care of it, but you have to be the farmer or an employee.
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The hunting regs list approved methods for taking game. If airguns aren't on the list they aren't approved. Be carefull with the silencers too as there are illegal in the taking of game. Always er on the side of caution. Having a few hundred dollar fire arm taken away for breaking the rules would suck. Imagine it being your few thousand dollar rig instead. All I'm saying is be diligent and get permission, or denial in writing.
There are no species listed for air guns in the regs, so how does the ASA list all those species that can be taken with an air gun?