Fingering in Stochastic Growth Models.

Motivated by the widespread use of hybrid-discrete cellular automata in modeling cancer, two simple growth models are studied on the two dimensional lattice that incorporate a nutrient, assumed to be oxygen. In the first model the oxygen concentration (, ) is computed based on the geometry of the growing blob, while in the second one (, ) satisfies a reaction-diffusion equation. A threshold value exists such that cells give birth at rate ((, ) - ) and die at rate ( - (, ). In the first model, a phase transition was found between growth as a solid blob and "fingering" at a threshold = 0.5, while in the second case fingering always occurs, i.e., = 0.