RUI: Applied dynamics and topology of aggregation systems

在许多自然系统中，粒子、媒介或有机体聚集并表现出集体行为。聚集系统包括纳米粒子自组装、细胞中的肌动蛋白丝网络等。该项目研究聚集系统，着眼于昆虫群，鸟群，鱼群和其他生物群体，其中社会互动起着关键作用。具体目标是：（1）发现计算上具有挑战性的模型是否可以用更简单、更易处理的模型准确地近似;（2）利用这种理解来模拟对环境和经济具有破坏性的蝗虫群;（3）对与聚集有关的大型数据集中的复杂行为进行分类。蝗虫研究影响农业。特别是，它将产生对群体抑制策略的见解。例如，它可能会建议作物种植布局，以避免群居蝗虫的爆发，使国家的农民。该项目以本科院校为基础，其其他关键要素包括：广泛的本科生参与和研究培训;国内外学院和大学网络;从研究到课堂的管道;加强学生研究实验室基础设施;将一名女博士纳入研究和咨询活动。寻求终身职位;关注代表性不足的群体的参与;以及与一位本科女艺术家和生物学专业合作制作的关于聚合的教育性公共艺术展。调查员和他的同事从连续和离散的角度研究聚合系统。一个常见的聚合建模框架是保守型非局部PDE，这是分析和计算上的挑战。退化的Cahn-Hilliard近似的一类典型的模型将研究与线性分析，数值模拟，相平面分析的平衡，和变分分析的最小值，以评估成功的本地模型近似的非本地的。基于这一认识，研究人员将开发一个多型蝗虫与环境相互作用的模型，并利用它来制定抑制破坏性蝗虫群的战略。稳定性分析和数值模拟将揭示环境条件可能抑制滞后分岔的危险蝗虫群。最后，研究人员将使用拓扑数据分析来表征聚集动力学。从数值模拟和生物实验产生的数据集将与拓扑条形码描述其持久的同源性进行分析。这项工作将确定聚合过程中的动态转换。

In many natural systems, particles, agents, or organisms aggregate and display collective behavior. Aggregation systems include nanoparticle self-assembly, actin-filament networks in cells, and more. This project investigates aggregation systems with an eye towards insect swarms, bird flocks, fish schools, and other biological groups in which social interactions play a key role. The specific objectives are (1) to discover whether computationally challenging models can be accurately approximated with simpler, more tractable ones; (2) to use this understanding to model environmentally and economically destructive locust swarms; and (3) to classify complicated behavior in large data sets related to aggregations. The locust research impacts agriculture. In particular, it will yield insight on swarm suppression strategies. For instance, it may suggest crop-planting layouts that would avert the gregarious locust outbreaks that devastate farmers. Other key elements of this project, which is based at an undergraduate institution, include: extensive undergraduate student involvement and research training; a network of domestic and foreign colleges and universities; a pipeline from research to the classroom; enhancement of student research lab infrastructure; inclusion in research and advising activities of a female recent Ph.D. seeking a tenure-track position; a focus on the participation of underrepresented groups; and an educational public art exhibition on aggregations produced collaboratively with an undergraduate female artist and biology major.The investigator and his colleagues study aggregation systems from continuum and discrete perspectives. A common aggregation modeling framework is conservation-type nonlocal PDE, which are analytically and computationally challenging. Degenerate Cahn-Hilliard approximations of a class of canonical models will be investigated with linear analysis, numerical simulation, phase plane analysis of equilibria, and a variational analysis of minimizers in order to evaluate the success of the local model in approximating the nonlocal one. Based on this understanding, the investigator will develop a model of phase polyphenic locusts interacting with the environment and use it to develop strategies that suppress destructive locust swarms. Stability analysis and numerical simulation will reveal environmental conditions likely to suppress a hysteretic bifurcation to a dangerous locust swarm. Finally, the investigator will characterize aggregation dynamics using topological data analysis. Data sets arising from numerical simulations and biological experiments will be analyzed with topological barcodes describing their persistent homology. This work will identify dynamical transitions in aggregation processes.