Cellular Automata and Percolation Models of Growth and Competition

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9703923 Gravner This project addresses the behavior of deterministic and random cellular automata (CA). Such processes describe configurations on lattices which evolve by a repeated update of a local rule. The research goals consist of three main parts. The first part focuses on deterministic and random growth models, emphasizing emergence of asymptotic shapes, limit laws for first passage times, as well as hydrodynamic and other continuum limits. The second part is devoted to competition theory, especially to cases which lead to approximation with the motion by mean curvature. On finite universes, long-time evolution of such dynamics depends on topological properties of the underlying space. The third part consists of studying percolation on hypercubes in high dimensions; the goal is to understand how the number of species is determined by the structure of viable genotypes and various mating strategies. This research is accompanied by development of computer software, which is used for simulation, statistical analyses, numerical computation and combinatorial searches. This project studies several classes of deterministic and random evolutions of spatial configurations. The basic property of these evolutions is that they are governed by local rules; they are referred to as deterministic and random cellular automata (CA). The study of CA offers insights into fundamental organizational principles in many scientific contexts. CA complement differential equations, the prevailing method of describing physical processes, by demonstrating how differential equations arise naturally from local rules. In biological contexts, the CA models under study shed light on how harshness of the environment affects diversity of species, how equally fit species compete for the available space and how spread of biological agents is affected by spatial anomalies. Emergence of shapes in growth models is often seen as the basic example of global structure arising from local dynamics, and studying proper ties of such shapes leads to challenges in probability theory, geometry and complexity theory. Finally, computer analyses of CA are not only indispensable for mathematical development, but they also provide a worthy testing ground for parallel computation schemes and visualization hardware.