Analysis of mathematical models for the growth of tumors with time delays in cell proliferation

We study two mathematical models for the growth of tumors with time delays in cell proliferation, one for nonnecrotic tumors in the presence of inhibitors, and the other for necrotic tumors. Mathematical formulations of these models are retarded differential equations. By using a comparison method, we make rigorous analysis of these models. The results show that dynamical behavior of solutions of these models are similar to that of solutions for corresponding nonretarded problems, and the tumor will tend to a dormant state as time goes to infinity in favorable conditions (nutrient sufficient, inhibitor less), while it will finally disappear in unfavorable conditions (nutrient insufficient, inhibitor much).