Do a survey and see how many Villagers have pets. Tell those people to not bring their pets to the town squares. Watch all of the Villages pet owners boycot the town squares. I don't think the squares would survive. Prospective home buyers with pets will start looking elsewhere for retirement options. You're not dealing with wild animals here like buffalo, pets are a part of the familly. If there are isolated incidents involving careless pet owners, then they need to be dealt with individually.