He will call when you exit the mahogany gym. If you beat Pryce and got the badge then will invade and take over.
If you want to check if they are invading then you can check the redio and on every channel it will appear,
Team Rocket Annoucement-Testing, testing 123
To battle team rocket(I was stuck too at this point for a while-___-) go to near the top of the underground place and a team rocket guy will be there and and change you into a team rocket suit.Then go back to the R-tower and try to go upstairs and you rival will come and notice you and you will change you back into yourself. After you go to the top(You need to do alot of other stuff)and beat Archer, He will disband Team Rocket and you will get the silver wing(which unlocks Lugia).
Hope this helps!=)