I am using Devise gem for web authentication in my Application.
Now i am about to write a mobile app for my Application which includes Sign in / Sign up process.
What is the correct way i should use to sign in a user and authenticate each call made by the user from the mobile app?
Which of the below strategy is correct? (i am not sure which method to follow to be more secure)

Note : You can view the above image in https://i.stack.imgur.com/I13uT.png (will be more clear)
FYI : I am using Titanium to develop mobile app and my backend server runs Rails app
Model #1 isn’t secure, you aren’t passing any sort of authentication on subsequent requests to validate that the user is still who they say they are.
What I’m presuming you really want to know is, what’s the best way to verify the user is who they say they are, after logging in. I’ve answered this previously, Exposing Rails/Devise Authentication to iOS application and the same answer applies here.
Using token authentication in Devise will match model #2, and is also the most secure since you exchange the username/password for a token rather than having to store their username and password and reuse it with every request.