  In this paper, we propose a tumor growth model to incorporate and investigate
the spatial effects of autophagy. The cells are classified into two phases:
normal cells and autophagic cells, whose dynamics are also coupled with the
nutrients. First, we construct a reaction-(cross-)diffusion system describing
the evolution of cell densities, where the drift is determined by the negative
gradient of the joint pressure, and the reaction terms manifest the unique
mechanism of autophagy. Next, in the incompressible limit, such a cell density
model naturally connects to a free boundary system, describing the geometric
motion of the tumor region. Analyzing the free boundary model in a special
case, we show that the ratio of the two phases of cells exponentially converges
to a ``well-mixed" limit. Within this ``well-mixed" limit, we obtain an
analytical solution of the free boundary system which indicates the exponential
growth of the tumor size in the presence of autophagy in contrast to the linear
growth without it. Numerical simulations are also provided to illustrate the
analytical properties and to explore more scenarios.
