Influence of cell-cell interactions on the population growth rate in a tumor

The understanding of the macroscopic phenomenological models of the population growth at a microscopic level is important to predict the population behaviors emerged from the interactions between the individuals. In this work we consider the influence of the cell-cell interaction on the population growth rate $R$ in a tumor system, and show that, in most cases especially small proliferative probabilities, the regulative role of the interaction will be strengthened with the decline of the intrinsic proliferative probabilities. For the high replication rates of an individual and the cooperative interactions, the proliferative probability almost has no effect. We compute the dependences of $R$ on the interactions between the cells under the approximation of the nearest neighbor in the rim of an avascular tumor. Our results are helpful to qualitatively understand the influence of the interactions between the individuals on the growth rate in population systems.