Cellular Automata and Percolation Models of Growth and Competition

元胞自动机以及增长和竞争的渗透模型

• 批准号: 9703923
• 负责人: Janko Gravner
• 金额: $ 7.5万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703923&HistoricalAwards=false
• 关键词: Cellular; Automata; Percolation; Models; Growth

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.

小行星9703923 这个项目解决了确定性和随机元胞自动机（CA）的行为。 这样的过程描述了通过局部规则的重复更新而演变的格上的配置。 研究目标包括三个主要部分。 第一部分着重于确定性和随机增长模型，强调出现的渐近形状，第一次通过时间的极限定律，以及流体动力学和其他连续极限。 第二部分是专门竞争理论，特别是案件导致近似与运动的平均曲率。 在有限宇宙中，这种动力学的长期演化取决于底层空间的拓扑性质。 第三部分包括研究高维超立方体上的渗流;目标是了解物种的数量是如何由可行的基因型结构和各种交配策略决定的。 这项研究伴随着计算机软件的发展，用于模拟，统计分析，数值计算和组合搜索。 本计画研究几类空间组态之决定性与随机性演化。 这些演化的基本性质是它们受局部规则的支配;它们被称为确定性和随机元胞自动机（CA）。 CA的研究提供了许多科学背景下的基本组织原则的见解。CA补充微分方程，描述物理过程的流行方法，通过展示微分方程如何自然地从局部规则中产生。 在生物学背景下，正在研究的CA模型揭示了环境的恶劣程度如何影响物种的多样性，同样适合的物种如何竞争可用空间以及生物制剂的传播如何受到空间异常的影响。 增长模型中形状的出现通常被视为局部动力学产生的全局结构的基本例子，研究这些形状的正确联系导致了概率论，几何学和复杂性理论的挑战。 最后，CA的计算机分析不仅是数学发展不可或缺的，但它们也提供了一个有价值的测试场并行计算方案和可视化硬件。