Fingering in stochastic growth models

AC Aristotelous, R Durrett - Experimental mathematics, 2014 - Taylor & Francis
Experimental mathematics, 2014Taylor & Francis
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 …
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 θc = 0.5, while in the second case, fingering always occurs, i.e., θc = 0.
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