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 u(x, t) is computed based on the geometry of the growing blob, while in the second one u(x, t) satisfies a reaction-diffusion equation. A threshold θ value exists such that cells give birth at rate β(u(x, t) - θ)+ and die at rate δ(θ - u(x, t)+. In the first model, a phase transition was found between growth as a solid blob and "fingering" at a threshold θc = 0.5, while in the second case fingering always occurs, i.e., θc = 0.