A THREE-DIMENSIONAL ANGIOGENESISMODEL WITH TIME-DELAY

In this paper, we study the effect of time delays on the dynamicsof angiogenesis tumor growth. The delays represent the time taken for themitosis. The model is based on the reaction-diffusion equation described in theform of concentration of nutrients sigma and the distribution of internal pressurepcaused by movement of cells. The vasculature supplies nutrients to the tumor,so that partial derivative sigma/partial derivative n +beta(sigma- sigma) = 0 holds on the boundary, where a positive constant beta is the rate of nutrient supply to the tumor and sigma is the nutrient concentrationoutside the tumor. A parameter mu in the model expresses the "aggressiveness"of the tumor. It is proved that under non-radially symmetric perturbations,there exists a mu & lowast;>0 such that the stationary solution is linearly stable for mu < mu & lowast;, and is linearly unstable for mu > mu & lowast;. Moreover, we also found thatadding time delay to the model would lead to a large stationary tumor. Thebigger the tumor proliferation intensity mu and angiogenesis intensity beta are, thegreater the effect of time delay on tumor size