Localized Patterns in PDEs: Theory, Computation, and Applications

局部时空模式通常出现在各种类型的线性和非线性扩散过程中。特别是，它们发生在生物系统中模拟群体感应（QS）行为的反应扩散（RD）系统、植物细胞中根尖形成的起始以及城市犯罪的空间分布中。局部化行为也发生在生物物理背景下，计算布朗步行者在局部陷阱区域的首次通过统计，以及生态学中，计算物种在斑块景观中的持续阈值。******我的研究计划的长期目标是在一个统一的框架内，发展混合渐近数值方法来研究这种定位行为在各种新的生物、化学和社会相互作用的PDE模型中。数学工具将包括渐近，谱分析，偏微分方程和分岔理论，以及非线性动力学，我们的小组将与一些数值分析师合作。******提出的新研究包括四个重叠的主题：******I（耦合细胞-体模型）：开发和分析新型的耦合细胞-体ODE-PDE模型，其中空间分离的动态活性信号“细胞”或膜通过体扩散场耦合。对于一个这样的模型，我们已经表明，体扩散的影响导致动态活动单元的稳定和同步振荡，否则就不会发生这种情况。******II（流形上的模式）：分析封闭流形上RD系统的局部斑点模式，以确定流形的几何形状如何影响这种表面绑定模式的演化、线性稳定性和平衡。我们还将研究由于体和表面之间的化学交换而产生的3D体和2D表面扩散过程耦合的新型RD模型。******III（混合方法）：已知一些RD系统的点动力学依赖于某些格林函数的梯度，而它们的线性稳定性依赖于相关的特征值相关格林函数。这些格林函数的快速多极数值方法的实现将允许在任意平面域中的点动力学的数值实现，支持对平衡的分岔和稳定性如何依赖于区域形状的研究。******IV:（第一题）我们将分析生物物理学中两个具体的第一题。第一个问题是考虑布朗步行者，它在到达特定的目标位置之前，经历了3D体与其围表面之间的间歇性结合。第二个问题是导出有效的Robin边界条件来计算涉及大量表面陷阱的平均首次通过时间（MFPT）。对这种仔细的均质分析的需要一直是生物物理学中一个长期存在的问题。***********

Localized spatial-temporal patterns commonly occur for various classes of linear and nonlinear diffusive processes. In particular, they occur in reaction-diffusion (RD) systems modeling quorum-sensing (QS) behavior in biological systems, the initiation of root-hair tip formation in plant cells, and the spatial distribution of urban crime. Localization behavior also occurs in the biophysical context of calculating first passage statistics for a Brownian walker in a region with localized traps, and in ecology for calculating the persistence threshold of a species in a patchy landscape. ******The long-term goal of my research program is to develop, within a unified framework, hybrid asymptotic-numerical methods to study such localization behavior in a wide variety of new PDE models of biological, chemical, and social interactions. The mathematical tools will include asymptotics, spectral analysis, PDE and bifurcation theory, and nonlinear dynamics, and our group will collaborate with a few numerical analysts. ******The proposed new research consists of four overlapping themes:******I (Coupled Cell-Bulk Models): Develop and analyze new classes of coupled cell-bulk ODE-PDE models in which spatially segregated dynamically active signaling "cells'' or membranes are coupled through a bulk diffusion field. For one such model we have shown that the effect of bulk diffusion leads to stable and synchronous oscillations of the dynamically active units, which otherwise would not occur.******II (Patterns on Manifolds): Analyze localized spot patterns for RD systems on closed manifolds to determine how the geometry of the manifold influences the evolution, linear stability, and equilibria of such surface-bound patterns. We will also study new classes of RD models that result from the coupling of 3D bulk and 2D surface diffusion processes, due to chemical exchanges between the bulk and the surface. ******III (Hybrid Methods): Spot dynamics for some RD systems are known to depend on the gradient of certain Green's functions, while their linear stability properties depend on related eigenvalue-dependent Green's functions. The implementation of fast multipole numerical methods for these Green's functions will allow for numerical realizations of spot dynamics in arbitrary planar domains, supporting investigations of how the bifurcation and stability properties of equilibria depend on the domain shape.******IV: (First Passage Problems) We will analyze two specific first-passage problems in biophysics. The first problem is to consider a Brownian walker that undergoes intermittent binding between a 3D bulk and its confining surface, before reaching a specific target site. The second problem is to derive effective Robin boundary conditions to calculate the mean first passage time (MFPT) involving a large number of surface traps. The need for such a careful homogenization analysis has been a long-standing problem in biophysics.***********