Probably appointed King over the Strathclyde Welsh about 1018. His Cumbrian subjects harried by Eadulf, Earl of the Northumbrians, 1038. Made yearly progresses through Scotland to restrain oppression. Defeated and slain, some say assassinated, by Macbeth, mormaer of Moray (The concise dictionary of national biography. Part 1 : from the beginnings till 1900. London : Oxford University Press, 1969, pg. 370).
Duncan (Donnchad mac Crínáin) was and son of Crinan, the Abbot of Dunkeld, and Bethroc daughter of Malcolm II. He was the first of the House of Dunkeld.
His incursions south into England were unsuccessful including the siege of Durham in 1139, but he strengthened his hold over Strathclyde. The hereditary rights of his two sons Malcolm and Donald were threatened by Macbeth of Moray who claimed the kingdom. The feud was temporarily settled in 1040 when Duncan was either murdered by Macbeth, or more likely fatally wounded in battle at Pitgaveny near Elgin by Macbeth who then became king. Duncan?s sons escaped and both would later become kings (britroyals.com)