Fingering in Stochastic Growth Models

Abstract

Motivated by the widespread use of hybrid-discrete cellular automata in modeling cancer, we study two simple growth models 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 θ_c = 0.5, while in the second case, fingering always occurs, i.e., θ_c = 0.

2000 AMS Subject Classification::

• 60K35

Keywords:

• stochastic growth model
• hybrid cellular automaton
• shape theorem
• DLA
• motion by mean curvature