  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.
