our male did not like our Maisey at first.. she would try get in his bed and well he would snap her face off
we found a rule of if they don't yelp in pain its dogie behavior.. if your dog is yelping in pain lots then separate them as its receiving pain..or clap your hands near them to stop it
its normal for your original dog to put the new dog in its place as the first dog was boss first.. give them time they will sort out the boss line