He said the ill treatment of the birds is being done by two different people because anyone poisoning birds would not eat them. "There's two contrary things going on here."
Mr Stevens, who is chairman of the council's infrastructure and property committee, said security at Virginia Lake will be stepped up with checks of the area day and night as well as the regular patrols.
He said the council knows how many swans are at the lake because they are counted regularly, and two birds were missing from the wildlife reserve.
It was no coincidence it had happened during the school holidays, he said.
"These acts of violence against animals, this kind of antisocial behaviour steps up during school holidays.
"I don't know how that can be managed, but it's really marked."
Anyone with information about the killing of the swans, or the poisoning, can contact Mr Stevens on 345 2083/027 499 3181 leave a message on his facebook page or call Crimestoppers 0800 555 111.