Honestly, don't you two read! A Patronus doesn't kill them, it just sends them away. In Hogwarts: A History, it's written that Dementors are impossible to destroy- they are in a way immortal, though they were never alive, to begin with. Dementors die off eventually, the reasons and causes are currently unknown. A Patronus merely protects the caster from the dementors, by scaring them (dementors) away.