Dynamics of an Age-Structured Tumor Cell Population Model with Nutrition

Abstract A nonlinear age-structured tumor population model, in which the tumor population is divided into proliferating and quiescent cells, and the probability of mitosis in proliferating cells are determined by the concentration of nutrient and the proliferating cells, is provided and analyzed. The dynamic behaviors of the model are investigated by means of semigroup operator theory. Firstly, a threshold called $R_{0}$ is proposed, and the model always exists a tumor-free steady state, which is locally asymptotically stable if $R_{0}<1$, and a sufficient condition for its global stability is also obtained. Secondly, a special piecewise function with time delay describing the 'birth' of proliferating cells is introduced, and a particular threshold $R_{0s}$ in accordance with it is obtained. If $R_{0s}>1$ it is shown that there exists a unique positive steady state, which is locally asymptotically stable once it appears. Further, several numerical analyses are carried out to illustrate the validity of the theoretical results, which shows that the nonlinearity of split rate not only halts tumor's exponential growth but also forces the tumor population converge to constant and sometimes even like the Gompertz growth. The results suggest that it may be a good treatment to prolong the split time as long as possible during tumor therapy. Mathematics Subject Classification (2020) 34D20 · 35A25 · 92C37