When we talk of competition, we usually are referring to athletic competition or business competition, not religious competition. Acts of the Apostles is about different religious groups competing against each other for new members among the God-fearers. We see the first hint of this religious competition in Acts 5:17. Jealousy is a normal response to the success of your opponents.
If we recognize the possibility that Acts of the Apostles is about competing groups of true believers, then we can state that Luke may be writing to Theophilus to win him over to a particular way of life. This means that this writing may be a protreptic letter. The first century witnessed the beginning of a struggle for scripture and covenant that continued for several centuries. The writings of Luke were one of the tools used by the followers of Jesus in that struggle.