Stochastic Systems for Interacting Populations

这项研究项目调查在种群动态和多物种相互作用的谱系研究中出现的概率问题。后一项研究是了解生态系统中的生物多样性、流行病学传播、癌症肿瘤的异质性以及自然界许多复杂系统背后的机制的关键。这个项目是朝着开发更精细的模型迈出的第一步，这些模型不仅解决了病毒学和生态学中的悬而未决的问题，而且还催化了新数学方法的产生。本科生参与随机分析和模拟将增强他们在概率和应用科学之间工作的能力。主要的研究目标是设计各种空间随机模型，包括非标准相互作用的粒子系统、相互作用的超过程和一般度量图上的耦合随机偏微分方程组，以帮助更深入地理解空间结构和随机性对不同观测尺度下的各种种群动力学及其谱系的影响。在这些模型的适定性和时间渐近性质的研究中出现了新的挑战。该项目将集中在两个研究方向。(I)探索这些模型之间的新联系。这项研究的结果有望包括标度极限定理和对偶公式，它们严格地证明了新的联系。(Ii)研究这些空间随机模型的长期行为。将开发新的概率方法来计算感兴趣的数量，如竞争结果的渐近形状和行波在网络上的传播速度。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project investigates probability questions arising in the study of population dynamics and genealogies with multi-species interactions. The latter study is a key to understanding biodiversity in ecosystems, epidemiological spread, heterogeneity of cancer tumors, and the mechanisms underlying many complex systems in nature. This project is the first step towards the development of more refined models that not only address standing questions in virology and ecology, but also catalyze the generation of new mathematical methods. The involvement of undergraduate students in stochastic analysis and simulations will enhance their ability to work at the interface between probability and applied sciences. The principal investigator aims to design various spatial stochastic models, including nonstandard interacting particle systems, interacting superprocesses, and coupled stochastic partial differential equations on general metric graphs, to contribute to a deeper understanding of the effect of spatial structures and randomness on various population dynamics and their genealogies in different scales of observation. New challenges arise in the study of well-posedness and time asymptotic properties of these models. The project will focus on two research directions.(i) Explore new connections among these models. Outcomes of this research are expected to include scaling limit theorems and duality formulas that rigorously justify the new connections.(ii) Study the long-time behavior of these spatial stochastic models. New probabilistic methods will be developed to compute quantities of interest, such as the asymptotic shape of competition outcome and the propagation speeds of travelling waves on networks.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.