Long time dynamics and genealogies of stochastic reaction-diffusion systems

反应扩散型的随机模型对于模拟空间相互作用和动力学系统的随机性至关重要。尽管它们很实用，但由于高维和非线性相互作用等问题，这些模型在数学上具有挑战性。该项目将通过关注空间在影响种群动态方面的关键作用来解决这些挑战，这是生态学、进化生物学和病毒学问题的关键。该项目的结果可能为改善生态系统管理和病毒感染治疗提供见解。这项研究也将有助于发展新的数学方法，并促进不同群体的学生研究人员的参与。我们特别关注的是一类随机偏微分方程（SPDEs），其中空间被建模为一般度量图，允许详细检查空间对人口动态的影响。这种方法不仅解决了理论上的挑战，而且弥合了与微观粒子模型的差距。PI将探讨几个关键现象，包括行波前，随机波的渐近速度，以及扩大种群的谱系。通过整合各种数学分支的创新技术，包括概率和谱图理论，该项目旨在显著推进对度量空间上spde的理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Stochastic models of reaction-diffusion type are crucial for modeling spatial interactions and randomness in dynamical systems across numerous scientific disciplines. Despite their utility, these models are mathematically challenging, due to issues including high dimensionality and nonlinear interactions. This project will address these challenges by focusing on the critical role of space in influencing population dynamics, which is pivotal for questions in ecology, evolutionary biology, and virology. The outcomes of this project may provide insights that improve management of ecosystems and treatments for viral infections. The research will also contribute to the development of novel mathematical methods and promote the participation of a diverse group of student researchers.Our specific focus is on a class of stochastic partial differential equations (SPDEs) where space is modeled as a general metric graph, allowing for a detailed examination of spatial effects on population dynamics. This approach not only addresses the theoretical challenges but also bridges the gap with microscopic particle models. PI will explore several key phenomena, including traveling wavefronts, the asymptotic speed of stochastic waves, and genealogies in expanding populations. By integrating innovative techniques from various branches of mathematics including probability and spectral graph theory, this project aims to significantly advance the understanding of SPDEs on metric spaces.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.