Long time dynamics and genealogies of stochastic reaction-diffusion systems

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.