Pack means family. It's a happy-go-lucky, trusting, relaxed state of mind and being.
There are two types of space:
1. Intimate space.
2. Social space.
In our family, we can be close to one another and comfortably even touching. In social situations with others, we naturally afford 3 to 5 feet of respectful space. Our dogs need to do the same. Any dog who is invading social space is asserting dominance.
"I'd rather be the hunter than the prey." - Imagine Dragons
Hunt. Be hunted. This is the Prey Predator Drive. Dogs are hard wired with certain instincts which are corralled with strategy calls by the pack leader. One is to hunt and kill for food. Another is to eradicate intruders in their territory. A third is to assemble the pack to protect themselves from threats.
An animal who percives grave risk, who feels cornered with peril for life, is in the fight or flight Defense Drive.
This dog will not be distracted by treats, praise, gimmicks. He believes his safety and even life are in jeopardy.
Strategy; get in front of and block before the thoughts even start forming. He has lost rationality, the ability to be dealt with by reason, once he's in this feral primal zone.
The two things that dogs will most often kill over sex and food. Your dog needs to perceive that you, her leader own all food and treats. And that is totally within your discretion to make decisions about sharing.
You should be able to take a bone away from your dog with no resistance. And if there are other dogs in the house, each dog must understand that he is not allowed to look at and contemplate claiming the others' food as his own. And they must each walk away from the food area as they finish their meals.
This is the second drive that dogs will kill over, although we humans manipulate and control by fixing our dogs. But even fixed dogs relate sexually to other dogs as well as people.
It is not unusual for a male dog to claim a female owner as his own and even object to a human male's intimacy with her, particularly if the dog sleeps in the bed.