They are all going to survive, though if one would lose, it would be either the Xbox or PS2. Why:
PS2 - has the massive amount of fanboys, and some big name titles. Other than that its technically inferior to both the Gamecube and Xbox, and is a pain in the ass to develop for.
Gamecube - Awesome tech specs, easy to program for, low cost, and has a huge fanbase, though it is mainly kids. But honestly, how isn't wetting their pants thinking about a Mario, Zelda, Metroid, etc on a next-gen system?
Xbox - most powerful of the three, has MS behind it, but other than that not much. If MS makes some big game announcements and shows us some truley amazing graphics at E3, their set. They need to convince us at E3 that they are a player, otherwise their done.