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

PI将对具有多个时间尺度和多个长度尺度的科学问题进行研究。第一个项目集中在化学模式的反应扩散模型，包括波，锋，脉冲和斑点。PI帮助开发的半强相互作用理论和重整化群理论的理论框架将用于理解多组分反应扩散系统中散射和钉扎的动力学，以及生物修复反应扩散平流模型中行波的稳定性。 在第二个项目中，PI将研究二维空间中的R-D方程，其中反应项具有截止函数。这些截止值准确地模拟了低浓度区域。主要目标是分析这些截止点对传播锋的速度和稳定性的影响。这将建立在PI开创性地使用几何去奇异化的基础上，以找到1-D中具有截止值的波速的分析公式，该公式与数值数据匹配良好。在第三个项目中，PI将继续他的长期分析和精确模型简化方法的开发。 这些被用于大规模的燃烧，化学和生物化学系统表现出多个时间尺度，找到低维流形，管理有效的动态。新的研究将建立在PI对最常用的模型降阶方法（包括ILDM和CSP方法）的早期分析的基础上，并将通过开发和分析在存在扩散的情况下找到低维流形的方法将该项目提升到一个新的水平。第四个课题研究偏微分方程中的鸭翼新现象，以及数学神经科学模型中的环面鸭翼新现象。PI将以解释最近的实验、分析计算方法、发展新的数学理论为目标，对具有多时间尺度和多长度尺度的科学问题进行研究。第一个项目集中在化学和生物系统中的模式和波，目标是模拟条纹和斑点之间的相互作用。还将研究生物修复模型中的行波。生物修复是诱导微生物降解土壤中对环境有害的有机化合物的过程。PI已经建立了这些行波的基本特性，并将确定操作参数，如注入速率，导致模型内稳定的生物修复。在第二个项目中，PI的目标是在存在截止点的情况下，在两个空间维度中对更具有挑战性和现实性的前向传播问题进行建模。确定截断点如何改变波前的速度和稳定性将在物理学中很有用，特别是在统计物理学中的多粒子系统。在第三个项目中，PI将专注于对燃烧科学，化学和生物化学中的大规模反应网络以及基因调控建模至关重要的模型简化方法。尽管计算方法的速度不断进步，但对这些大规模反应系统的理解关键取决于对易处理的低维简化模型的分析。在第四个项目中，鸭翼是在相对较长的时间内保持在不稳定系统状态附近的解决方案。在数学神经科学中，要研究的新类型鸭对于理解基本状态之间的转换非常重要，例如紧张性尖峰和爆发。所有上述计划中的研究都将产生更广泛的科学影响。 参与这些项目的博士生和博士后中有一半以上是妇女，PI迄今为止监督的20名博士生和博士后研究员也是如此。 此外，计划中的项目是国家实验室和北约国家科学家合作的一部分。

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.