I would just get a dog that likes to hunt. My parents have a Dachshund that kills anything that dares to come into their backyard. They are pretty useless other than that though.
It seems that there may be outlying provisions such as competition, recreation, or hunting, however here is what I was able to find in the OCGA.
GEORGIA CODE
*** Current Through the 2010 Regular Session ***
*** Annotations Current Through October 29, 2010 ***
TITLE 16. CRIMES AND...