Why did the Apostles not go with the rest of the believers outside of Jerusalem? I noticed that the Apostles stayed in Jerusalem for a long time while others sort of acted like missionaries as a result of being scattered... except the Apostles.


  And there arose on that day a great persecution against the church in Jerusalem, and they were all scattered throughout the regions of Judea and Samaria, except the apostles. (ESV Acts 8:1)


This seems extra strange being that Jesus said:


  Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age. (ESV Matthew 28:19-20)


I am not so much interested in traditions that give reason for their staying in Jerusalem. Rather, are there any scriptures that would explain this unexpected behavior?
John Calvin starts by reminding us that Jesus commanded them to begin at Jerusalem (Acts 1:4,8), and says that it makes sense that they would stay there until "being brought into some other place by his hand":


  But here may a question be asked, forasmuch as they were commanded to preach the gospel throughout the whole world, (Mark 16:16) why they stayed at Jerusalem, even when they were expelled thence with force and hand? I answer, that seeing Christ had commanded them to begin at Jerusalem, they employed themselves there until such time as being brought into some other place by his hand, they might know, for a surety, that he was their guide.


Calvin goes on to argue that it was utterly an act of faith to remain there steadfastly in the midst of danger, and that it would be an act of unbelief to flee just because of tribulation:


  Therefore, seeing they see the gospel so mightily resisted at Jerusalem, they dare go to no other place until such time as they have broken that first huge heap of straits. Assuredly, they provide neither for their ease, nor yet for their own commodities either for being void of care by staying at Jerusalem; for they have a painful charge, they are continually amidst divers dangers they encounter with great troubles. Wherefore, undoubtedly, they are purposed to do their duty; and especially, whereas they stand to it when all the rest fly, that is an evident testimony of valiant constancy. If any man object that they might have divided the provinces amongst them, that they might not all have been occupied in one place, I answer, that Jerusalem alone had business enough for them all.


Note that last sentence, "Jerusalem alone had business enough for them all." It's not as though they built summer homes in Jerusalem -- they were ministering to the lost there, just as they would if they were based in Rome or Ephesus. In particular, they were ministering to the poor, and Paul was "eager" to help them do so (Galatians 2:10; Romans 15:25-26).

G. W. H. Lampe argues that in Luke's theology, as ministers of the new covenant the apostles ought to remain in the capital of Israel:


  Luke simply believes that the apostles must stay in Jerusalem, for now that the Jewish
  hierarchy has remained impenitent, they are the true leaders of Israel. Just as the high priest sent out Saul with an anti-missionary commission to Damascus, so the apostles now direct and control the Christian enterprise from their own capital, Jerusalem, either personally or by sending envoys.


Like Calvin, he also argues that the apostles should remain there "until" Jesus led them out, namely by toppling Jerusalem:


  Luke may have shared the idea of Eusebius (historia ecclesiastica, 3.7.8) that the apostles remained in Jerusalem until the impenitence of the Jews was finally punished by the war with Rome.


Earlier in the same work, Lampe notes that the churches throughout the world were subject to "supervision" by the church of Jerusalem:


  In the beginning the Christian Church is the church of Jerusalem. Later the Church spreads through the world outside Jerusalem, but always, because of its nature as the authentic and renewed Israel, this expansion takes the form of an extension of the Jerusalem church. This does not mean that Luke believes that the mission was everywhere directly organized and pioneered by Jerusalem. On the contrary, Luke makes it perfectly clear that he realizes that he has omitted to tell us about the foundation and early history of churches in many parts of the world, including Rome itself; and there is no reason to think that he believed them all to owe their origin to the direct initiative of Jerusalem. He does, however, hold that these churches all came in due course under the supervision and guidance of the church of Jerusalem.


We see this "supervision" at work when the Jerusalem church sends Peter(!) and John(!) to Samaria (Acts 8:14) and Barnabas to Antioch (Acts 11:22), and when the apostles tell Paul that he should go to the Gentiles and they to the Jews (Galatians 2:9). It's also striking that, though it was Antioch where the controversy originally took place, the actual ruling on the Gentile controversy comes from Jerusalem (Acts 15).