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.

Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Toward this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gröbner basis computations.