斗气 is a separable verb (离合词) meaning 'to hold a grudge' or 'to feud over petty matters' — it describes a situation where two people are in a silent conflict, often refusing to speak or cooperate out of stubbornness or pride. The verb 斗 can separate from 气 when inserting a measure word, duration, or object: 斗什么气 ('what are you sulking about?'), 斗了一天气 ('held a grudge for a whole day'). You typically use it with 跟 (跟某人斗气) to specify who you're feuding with.