It's going to be quite awhile before the PSP is gone. Sony's strategy is to keep previous gen consoles around as a lower price option, especially in developing countries. While some people get upset that it's stealing sales away from the Vita, to Sony it's still profit, and probably with a greater margin--and, truth be told, people buying a PSP now probably wouldn't be buying a Vita, anyway. The price difference means that they aren't really direct competitors, for the most part. It's like complaining that the Wii U basic is stealing sales away from the Wii U Deluxe; profit is profit, and the PSP is pretty much the Vita "basic".
The negative has been that some developers have been staying with the PSP, which I don't think Sony anticipated to this degree. That's starting to change, however, with many of the new anime visual novels, one of the main hold-out genres on the PSP, appearing as Vita exclusives. A 'My Teen Romantic Comedy Snafu' Vita game is due in September, for example.
As long as the PSP is still selling well--and it is--I think Sony will keep producing it.
