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He was the great-grandson of Jadzia Dax. The crew of the Defiant wanted to attempt to get home, which would erase the existence of Gaia. Yedrin said he had a plan to duplicate the Defiant when it reentered the barrier, allowing the original crew to return home while the duplicate crew could fulfill its destiny by crashing on the planet. Jadzia found out that this was a ruse, and that Yedrin was trying to ensure the existence of himself and the rest of Gaia's inhabitants. Yedrin defended his actions by pointing out that he was essentially responsible for the colony's existence, since Jadzia could have warned the Defiant away from the anomaly that originally caused them to land on Gaia, but decided not to due to the possibility of making a scientific discovery.
His existence was erased when the Defiant avoided the crash and that timeline was averted. Captain Benjamin Sisko and Jadzia initially assumed that Yedrin was the one responsible for their changed flight path when the Defiant missed the anomaly that would have sent them back in time, since only a member of the crew would have possessed the necessary command codes, speculating that he had been unable to go through with the accident again given his prior memories of the crew. Odo later revealed to Major Kira Nerys that it was actually the alternate Odo who was responsible. (DS9 episode: "Children of Time")