To assess critical parameters controlling tumour growth and response to therapy, competition theory models the tumour-host interface as a network of interacting normal and malignant cell populations using coupled, non-linear differential equations. When the equations are analysed under conditions which simulate tumour development, three phases of tumour growth, each with different critical parameters, can be predicted. Transitions between these phases correspond to the initiation, promotion and invasion stages demonstrated in experimental models of carcinogenesis. Critical cellular properties for each transition are predicted including phenomena already demonstrated experimentally such as the linkage of invasive tumour growth with acquisition of angiogenesis. The model also predicts the previously unknown phenomenon of "functional equivalence" in which disparate tumour traits can play identical roles in tumour growth and invasion. This approach allows the diverse but inconsistent properties of transformed cells to be understood according to their specific contribution to tumorigenesis. The models have significant implications for treatment strategies.