Reaction Networks in Random Environments

Environmental fluctuations are common in biological, chemical, and engineered systems, such as cellular and biochemical systems that are continually affected by changing factors such as temperature, acidity, nutrient levels, and humidity. Understanding how these environmental changes influence system behavior is crucial for predicting and explaining biological processes. Many systems, including epidemic transmission, ecological dynamics, cellular metabolism, and biochemical regulation, can be modeled as chemical reaction networks. This project will address the question of whether qualitative properties of chemical reaction networks, including stability and persistence, are preserved under environmental fluctuations or instead undergo qualitative changes. Such insights may help explain discrepancies between deterministic mathematical models and experimental observations. The project will also study whether random environmental switching can itself create biological patterns, beyond the classic mechanisms like diffusion-driven pattern formation. Overall, the project aims to deepen mathematical understanding of complex networks in biology, with a focus on chemical reaction networks in changing environments. The project will also provide research opportunities for graduate and undergraduate students through mentoring and a new graduate course, as well as foster collaborations in applied mathematics and mathematical biology. The first aim of the project is to establish sufficient conditions for stability and instability of stochastic reaction networks in random environments. The second aim is to characterize bifurcations of stochastic reaction networks in random environments, including switching-induced biological pattern formation. The third aim is to pursue an interdisciplinary project at the interface of applied mathematics, systems biology, and machine learning by integrating chemical reaction network theory with graph neural networks for modeling large biological reaction systems. The project integrates ideas from stochastic processes, dynamical systems, chemical reaction network theory, graph theory, mean-field theory, and scientific computation to develop new analytical and computational tools for studying stochastic dynamical networks. These advances will establish new mathematical foundations for stochastic reaction networks in random environments and support future interdisciplinary research in systems biology and related fields. 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.