Walking the trails in some of our larger parks, to see 2-5 dogs running in a pack is not unusual. Its there park and you are on their trail. It is wise to step to the side and let them pass. Of course, occasionally they are coming from behind and if you have your ears turned off, you might end up like I did once. Felt the impact behind my knees, looked up and saw might feet, and had just enough time to think, 'I hope the landing doesn't hurt.' It happens. It is a dog park.From experience I can say that treats and personal toys can sometimes be a problem especially the toy as that is a possession and the dog will react if another dog tries to take the toy. Our park provides a constant supply of tennis balls that are so generic no dog assumes ownership.
Treats are only a problem when other dogs, read Duke, sense that you have them, then you will look like the Pied Piper, only leading a pack of dogs.
As for the running rule, thats plain ridiculous. Thats the main reason for providing an off leash area. My guess is someone probably got knocked over by a careening pack. You do have to be alert at all times and brace yourself when you see a herd approaching.