  Cell colonies of bacteria, tumour cells and fungi, under nutrient limited
growth conditions, exhibit complex branched growth patterns. In order to
investigate this phenomenon we present a simple hybrid cellular automaton model
of cell colony growth. In the model the growth of the colony is limited by a
nutrient that is consumed by the cells and which inhibits cell division if it
falls below a certain threshold. Using this model we have investigated how the
nutrient consumption rate of the cells affects the growth dynamics of the
colony. We found that for low consumption rates the colony takes on a Eden-like
morphology, while for higher consumption rates the morphology of the colony is
branched with a fractal geometry. These findings are in agreement with previous
results, but the simplicity of the model presented here allows for a linear
stability analysis of the system. By observing that the local growth of the
colony is proportional to the flux of the nutrient we derive an approximate
dispersion relation for the growth of the colony interface. This dispersion
relation shows that the stability of the growth depends on how far the nutrient
penetrates into the colony. For low nutrient consumption rates the penetration
distance is large, which stabilises the growth, while for high consumption
rates the penetration distance is small, which leads to unstable branched
growth. When the penetration distance vanishes the dispersion relation is
reduced to the one describing Laplacian growth without ultra-violet
regularisation. The dispersion relation was verified by measuring how the
average branch width depends on the consumption rate of the cells and shows
good agreement between theory and simulations.
