Research showed that when teachers were proactive and introduced anti-bullying policies which involved everyone in the school, they would still fail to eliminate all forms of bullying. It was always present.
Mr Mellor, who has become the Scottish specialist over the past 15 years, said he could offer no guarantees to parents that their children would not be bullied after schools took action.
"But when we engage in conversations with parents about bullying we find that we go way beyond the narrow subject and open the door to other things," he said.
There were two distinct types of bullying. The first involved one person persistently bullying another in a power relationship. The second was group bullying or mobbing which could involve name calling.
"You may find that the intent to harm bears no relation to the harm being done," Mr Mellor said. "If someone is significantly hurt, fears it is going to happen again, or has no defence against it, they are probably being bullied and probably need help. We have to make sure the help we give them is appropriate."