How do we keep competitors from simply copying our app?

Unfortunately, there is no legal way to prevent others from copying your code or replicating the functions of your app. However, if you have a patent covering your app, then you will have the right to seek an injunction, licensing, or damages from those who infringe on your patent. While software patents have come under much more scrutiny with recent case law, there are still software patents being issued. It may be best to start with a patent feasibility study to see if the novel aspects of your app are patentable given the prior art and current law.