Dynamical systems and singular perturbation theory for multi-scale reaction-diffusion phenomena

Trustees Of Boston University, Boston

Investigators

Abstract

The PI will conduct research on scientific problems exhibiting multiple time scales and multiple length scales. The first project centers on reaction-diffusion models for chemical patterns involving waves, fronts, pulses, and spots. The theoretical framework of semi-strong interaction theory and renormalization group theory, which the PI helped develop, will be used to understand the dynamics of scattering and pinning in multi-component reaction-diffusion systems, as well as the stability of traveling waves in a reaction-diffusion-advection model of bioremediation. In the second project, the PI will study R-D equations in two space dimensions with cut-off functions on the reaction terms. These cut-offs accurately model regions of low concentrations. The principal goal is to analyze the impact of these cut-offs on the speeds and stability of propagating fronts. This will build on the PI's pioneering use of geometric desingularization to find analytical formulas for wave speeds with cut-offs in 1-D, which match well the numerical data. In the third project, the PI will continue his long-term analysis and development of accurate model reduction methods. These are used in large-scale combustion, chemical, and biochemical systems exhibiting multiple time scales, to find low-dimensional manifolds that govern the effective dynamics. The new research will build on the PI's earlier analysis of the most commonly-used model reduction methods including the ILDM and CSP methods and will take this project to the next level by developing and analyzing methods that find low-dimensional manifolds in the presence of diffusion. In the fourth project, the new phenomena of canards in PDEs will be investigated, as will the new phenomenon of torus canards in mathematical neuroscience models. The PI will conduct research on scientific problems exhibiting multiple time scales and multiple length scales, with the goals of explaining recent experiments, of analyzing computational methods, and of developing new mathematical theory. The first project centers on patterns and waves in chemical and biological systems, with the goal of modeling the interactions between stripes and spots. Traveling waves in models of bioremediation will also be studied. Bioremediation is the process by which microorganisms are induced to degrade environmentally-harmful organic compounds in soil. The PI has established fundamental properties of these traveling waves, and will determine the operating parameters, such as injection rates, that lead to stable bioremediation within the models. In the second project, the PI aims to model the more challenging and realistic problem of front propagation in two space dimensions in the presence of cut-offs. Determination of how cut-offs change the speeds and stability of the fronts will be useful in physics, in particular for many-particle systems in statistical physics. In the third project, the PI will focus on model reduction methods that are critical for combustion science, large-scale reaction networks in chemistry and biochemistry, and in gene regulatory modeling. Despite the ever-advancing speed of computational methods, understanding of these large-scale reaction systems depends critically on the analysis of tractable low-dimensional reduced models. In the fourth project, canards are solutions that stay near unstable system states for relatively long periods of time. The new types of canards to be studied are important in mathematical neuroscience for understanding the transitions between fundamental states, such as tonic spiking and bursting. All of the above planned research will have broader scientific impacts. More than half of the PhD students and postdocs who will work on these projects are women, as is the case for the twenty PhD students and postdoctoral fellows whom the PI has supervised to date. Moreover, the planned projects are parts of collaborations involving scientists at national laboratories and in NATO countries.