Prey drive doesn't have anything to do with being a stray, or sometimes even the breed of dog.

Gypsy never spent time as a stray -- she was loved from the moment we got them out of the dumpster. And her prey drive was INSANE. Nothing was safe.

River has a prey drive as well, though not nearly as obsessively like GKs. She will ignore cats and such while walking but squills ... oh, squills are her nemesis.

Simon chases cats, and he's killed a rat before.

Dogs are dogs. They do dog things. ;)