RUI: Applied dynamics and topology of aggregation systems

在许多自然系统中，粒子、试剂或生物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-细丝网络等。这个项目调查聚集系统，着眼于昆虫群、鸟群、鱼群和其他社会互动发挥关键作用的生物群体。具体目标是(1)发现是否可以用更简单、更容易处理的模型来准确地近似计算上具有挑战性的模型；(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；以及(3)对与聚集相关的大型数据集中的复杂行为进行分类。蝗虫研究对农业的影响。特别是，它将对蜂群抑制策略产生洞察力。例如，它可能会建议进行农作物种植布局，以避免摧毁农民的群居蝗虫暴发。这个以本科院校为基础的项目的其他关键要素包括：广泛的本科生参与和研究培训；国内外学院和大学的网络；从研究到课堂的管道；加强学生研究实验室基础设施；将一名寻求终身教职的女性博士纳入研究和咨询活动；关注未被充分代表的群体的参与；以及与一名本科生女艺术家和生物专业合作制作的关于聚合的教育公共艺术展。研究人员和他的同事从连续和离散的角度研究聚合系统。一种常见的聚集建模框架是守恒型非局部PDE，这在分析和计算上都是具有挑战性的。通过线性分析、数值模拟、平衡点的相平面分析和极小点的变分分析，研究了一类正则模型的退化Cahn-Hilliard近似，以评价局部模型逼近非局部模型的成功程度。基于这一认识，调查者将开发一个多相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危险蝗群滞回分叉的环境条件。最后，研究人员将使用拓扑数据分析来表征聚集动力学。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。这项工作将识别聚集过程中的动态转变。

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.