Under what conditions do auto-turrets stay in play?

Turrets placed by you remain there until the end of the match, or they are destroyed. I've placed several manned and auto turrets on a particular map, and have died several times, noting that they are still there afterward.

You can have as many turrets (of the same type) as you want. You only need to pick up the powerup that grants them, and place them in valid locations. Note that they are favored targets by enemies. (I've had them destroyed seconds after placing them, most players know how powerful they can be.)