Hi Steve:
I would have to respectfully disagree depending on the circumstances.
If there were rats running around within a condominium with shared crawl space, walls, etc. then potentially the HOA could be responsible for maintaining the nuisance. However, if a single family home has rats running around, then it would be the homeowner’s responsibility to maintain their property and said nuisance.
If a wild bird decides to sit on my car and take a crap is the HOA then responsible for washing my vehicle? The HOA is not the forest service or other wildlife management service. Peacocks are prevalent in Florida as wildlife and individuals who choose to live around peacocks deal with the circumstances, just as those who live around bears, mountain lions, etc. also deal with those issues.
The HOA does not own, did not bring onto the property, and therefore I believe should not be responsible for said wildlife. If they did own and maintain the Peacocks, then yes they could potentially be responsible.
If the homeowner does not like them, he can feel free to hunt them down, have a big BBQ, and invite all the neighbors.