How do we keep competitors from simply copying our app?

Protectable IP assets of your app may include the functionality accomplished by the app (potentially protectable under patent), the artwork, code, and U/X you use (protectable in varying degrees under copyright), and the use of the app's name and associated logos (protectable under trademark law).

Preventing others from pragmatically "replicating [y]our software" will require obtaining patent protection, but whether you may do so is a very open question. In the absence of patent protection, your market advantage will need to come by brand awareness and simply having a superior product, customer service, etc.

To determine whether some part of your app may be patent protectable and how best to brand the venture, you should contact a patent/trademark attorney in private.

Disclaimer:

This communication: (i) is not a substitute for professional legal advice; (ii) does not create any attorney-client relationship; and (iii) is not a solicitation to offer legal advice. There is no duty to keep any confidential information you disclose in this forum confidential or refrain from providing representation adverse to your interests. You should seek the advice of a licensed attorney in an appropriate jurisdiction before taking any action that may affect your rights.