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、细胞内的肌动蛋白丝网络等。本项目着眼于昆虫群、鸟群、鱼群和其他生物群体的聚集系统，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、更容易处理的模型精确地近似；(2)利用这一认识对具有环境和经济破坏性的蝗群进行建模；(3)对与聚合相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，它将产生对蜂群抑制策略的见解。例如，它可能建议作物种植布局，以避免破坏农民的群居蝗灾。该项目以本科院校为基础，其他关键要素包括：广泛的本科生参与和研究培训；拥有国内外高校网络；从研究到课堂的管道；加强学生研究实验室的基础设施；将一名寻求终身职位的女博士纳入研究和指导活动；注重代表性不足群体的参与；与一位生物学专业的女艺术家合作举办了一个关于聚合体的教育公共艺术展。研究者和他的同事从连续和离散的角度研究聚合系统。一种常见的聚合建模框架是守恒型非局部偏微分方程，它在分析和计算上都具有挑战性。一类典型模型的退化Cahn-Hilliard近似将通过线性分析，数值模拟，平衡的相平面分析和最小化的变分分析来研究，以评估局部模型在近似非局部模型方面的成功。基于这一认识，研究者将开发一个阶段多食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群的策略。稳定性分析和数值模拟将揭示可能抑制蝗群迟滞分岔的环境条件。最后，研究者将使用拓扑数据分析来描述聚合动态。来自数值模拟和生物实验的数据集将用描述其持久同源性的拓扑条形码进行分析。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.