I guess with "desktop system" you actually mean PC desktops. In my definition a desktop system is a system on a desktop. In the past decades this meant PCs but in the last couple of years many users chose tablets as their favorite desktops. They just replaced their PC desktops by mobile desktops for gaming, surfing and emails.
On the PC desktop world Windows is still leading with Win7 and XP, followed by OSX, followed by Linux and others. Linux won't get leadership here but who cares? Linux is a very good PC desktop system anyway, absolutely competitive to Win 7.
OK, since your definition seems remove every aspect of "desktop system" that makes it a meaningful distinction, we'll just have to agree to disagree here. As for linux on PC desktop, competitive in what sense? I prefer a good Linux based OS over Windows any day, but in terms of market share, it's just a ridiculous thing to say.
On the PC desktop world Windows is still leading with Win7 and XP, followed by OSX, followed by Linux and others. Linux won't get leadership here but who cares? Linux is a very good PC desktop system anyway, absolutely competitive to Win 7.