Perhaps the best examples of predatory pricing come from the airline industry, when the Department of Justice brought several cases in the 1990's.
Probably our best known airline predation investigation involved Northwest's response to Reno Air's entry into the Reno-Minneapolis city-pair in 1993. Not only did Northwest institute service of its own on this route that it had previously abandoned, it also opened a new mini-hub in Reno that overlaid much of Reno Air's own operation. Our investigation was well under way when the matter was resolved because, with the intervention of the Department of Transportation, Northwest decided to abandon its overlay of Reno Air's hub operation.
See here why these cases are so hard to win.
In other words, the government needs to prove that the low fares and extra flights would prove financially ruinous if continued indefinitely. To make the argument stick, the government will have to prove that American could reasonably expect to recover its losses after Vanguard or Sun Jet exits the market by raising fares -- confident that its high fares would not attract another round of upstarts.