Analysis of a mathematical model for the growth of tumors under the action of external inhibitors.

In this paper we make rigorous analysis to a mathematical model for the growth of nonnecrotic tumors under the action of external inhibitors. By external inhibitor we mean an inhibitor that is either developed from the immune system of the body or administered by medical treatment to distinguish with that secreted by tumor itself. The model modifies a similar model proposed by H. M. Byrne and M. A. J. Chaplain. After simply establishing the well-posedness of the model, we discuss the asymptotic behavior of its solutions by rigorous analysis. The result shows that an evolutionary tumor will finally disappear, or converge to a stationary state (dormant state), or expand unboundedly, depending on which of the four disjoint regions Delta1,..., Delta4 the parameter vector (A1,A2) belongs to, how large the scaled apoptosis number;sigma is, and how large the initial radius R(0) of the tumor is. Finally, we discuss some biological implications of the result, which reveals how a tumor varies when inhibitor supply is increased and nutrient supply is reduced.