  We investigate the tumor boundary instability induced by nutrient consumption
and supply based on a Hele-Shaw model derived from taking the incompressible
limit of a cell density model. We analyze the boundary stability/instability in
two scenarios: 1) the front of the traveling wave; 2) the radially symmetric
boundary. In each scenario, we investigate the boundary behaviors under two
different nutrient supply regimes, in vitro, and in vivo. Our main conclusion
is that for either scenario, the in vitro regime always stabilizes the tumor's
boundary regardless of the nutrient consumption rate. However, boundary
instability may occur when the tumor cells aggressively consume nutrients, and
the nutrient supply is governed by the in vivo regime.
