As they walk toward Jerusalem, the disciples are afraid of the conflict ahead. Jesus tells the Twelve that the son of man will be mocked, scourged, and killed, and three days later He will rise again.
James and John seem to miss the point, asking to sit on either side of Jesus's throne in the new kingdom.
Maybe we should read between the lines a bit here. Peter, John, and James are frequently mentioned in the gospel of Mark. Mark is apparently Peter's protege', writing all of this down in Rome some 30 to 50 years later.
By that time Peter had established himself as a leader in the church in Rome. In this role he eventually came to be thought of as the first pope.
So according to these verses they were already competing.