Deterministic and Stochastic Growth Models

0204367 Gravner教授Gravner将继续进一步研究细胞自动机。这项研究将继续在几个新的方向。该项目中的增长模型包括简单的阈值规则和竞争动力学，以及低维和高维空间中的随机游走和渗流模型的相互作用系统。其中的原则议题是：随机扰动的影响，在环境和规则，波动的随机接口，生成的分形图案，行为的大-邻里系统，大规模限制系统的随机游走，和结构之间的关系的基因型空间和遗传多样性。 这个项目的目的是了解各种物理系统传播远离平衡的扰动的微观和宏观原理。这种增长动力学的实例包括晶体生长、流行病传播、波传播、物种之间的竞争和遗传多样性。这种研究有助于理解这样的自然现象：阈值密度的生存和增长与范围的相互作用的物种在一个不友好的环境中，尖锐的几何晶体形状和粗糙的随机之间的过渡，噪声环境对传染病传播的影响，有效性的近似的基础上大范围的相互作用或快速扩散，和中性突变在遗传漂变中的作用。

0204367Gravner Professor Gravner will pursue further research in cellular automata. The research will continue in several new directions. The growth models featured in this project range from simple threshold rules and competition dynamics, to interacting systems of random walks and percolation models in low- and high-dimensional spaces. Among the principle topics are: effects of random perturbations in the environment and in the rule, fluctuations of random interfaces, generation of fractal patterns, behavior of large--neighborhood systems, large scale limits of systems of random walks, and relationship between structure of genotype space and genetic diversity. The aim of this project is to understand microscopic and macroscopic principles by which various physical systems propagate disturbances far from the equilibrium. Among instances of such growth dynamics are crystal growth, spread of epidemics, propagation of waves, competition between species, and genetic diversification. Such research can contribute to understanding of such natural phenomena as: threshold density for survival and growth vs. the range of interaction for species in an unfriendly environment, transition between sharp geometric crystal shapes and rough random ones, effects of noisy environments on spread of infections, validity of approximation based on large range of interaction or fast diffusion, and role of neutral mutations in genetic drift.