In this talk, our objective is tamed and compatible almost complex structures. There are two interesting questions. The classical Nakai-Moishezon theorem (for surfaces) states the duality between ample divisor cone and curve cone for projective surfaces. Demailly-Paun, Buchdahl and Lamari generalized this duality to Kahler surfaces. It is natural to ask for such a duality for almost Kahler surfaces. Another interesting questin is raised by Donaldson. He asked that if there is a J-tamed symplectic form, do we have a J-compatible symplectic form as well? We answered these two questions affirmatively for all tamed almost complex structures on spheres bundles over sphere. We also answer them in many interesting cases for other rational surfaces.