Why didn't the Dragon slayers disappear when the eclipse gate was destroyed? During the Grand magic games, they were not able to actually defeat all the dragons that came out of Eclipse gate. The dragons as well as future Rouge and Lucy just disappeared and went back to their own timeline when the Eclipse gate was destroyed.

But as Zeref stated, Natsu and the other dragon slayers are from the past and traveled to current timeline using the same Eclipse gate.

So why didn't they disappear as well when the Eclipse gate was destroyed?

Possible plothole??
First (and more importantly): According to the Wikia page of the Eclipse Gate (sorry, I couldn't find the actual chapter where the princess actually explains this):


  Many years later, after a different Eclipse Gate is constructed, following the conclusion of the fourth day of the X791 Grand Magic Games, Arcadios and Yukino bring Lucy, Natsu and their friends before the gate while attempting to persuade Lucy to help with the Eclipse Project.


So this is not the same Eclipse Gate that was constructed by Zeref and used by Anna and Layla to bring the dragon slayers into the present, therefore destroying it can't have an effect on whether the dragon slayers are present in the current timeline or not.

Second: Evil Rogue and his dragons disappeared, because the gate that he used in the future to come back to the present was destroyed in the present, therefore he won't be able to use it to come back to the present, therefore he wouldn't be able to bring the dragons from the past. This same logic doesn't apply for time travel which brings people to the future: no matter what you do with the time travelling device after they arrived, they will be still there as there is no need for the time travelling device to keep them there.

(Other than this, yes, time travel in most of the stories sucks and is usually full of plot holes. Fairy Tail is no different, there are lots of open questions. With Eclipse destroyed why didn't we go back to the original timeline with 10000 dragons destroying everything? Or why did the city remained destroyed after the dragons disappeared? Well, at least we didn't get a "Back To The Future"-like "timelines slowly change, there's still time to save your parents' marriage" plot hole.)