title: Fingering in Stochastic Growth Models.
abstract: 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 concentrationu(x,t) is computed based on the geometry of the growing blob, while in the second oneu(x,t) satisfies a reaction-diffusion equation. A thresholdθvalue exists such that cells give birth at rateβ(u(x,t) -θ)+and die at rateδ(θ-u(x,t)+. In the first model, a phase transition was found between growth as a solid blob and "fingering" at a thresholdθc= 0.5, while in the second case fingering always occurs, i.e.,θc= 0.
authors.affiliation: Department of Mathematics, Duke U., Box 90320, Durham, NC 27708-0320.
authors.forename: Andreas C
authors.initials: AC
authors.lastname: Aristotelous
authors.affiliation: Department of Mathematics, Duke U., Box 90320, Durham, NC 27708-0320.
authors.forename: Richard
authors.initials: R
authors.lastname: Durrett