Localized Patterns in PDEs: Theory, Computation, and Applications

局部化的时空模式通常会出现在各类线性和非线性扩散过程中。特别是，它们发生在反应扩散（RD）系统建模群体感应（QS）的行为在生物系统中，在植物细胞中的根毛尖端形成的启动，和城市犯罪的空间分布。本地化行为也发生在生物物理的背景下计算第一次通过统计的布朗步行者在一个地区与本地化的陷阱，并在生态学计算的持久性阈值的物种在一个斑块景观。** 我的研究计划的长期目标是在一个统一的框架内开发混合渐近数值方法，以研究生物，化学和社会相互作用的各种新PDE模型中的局部化行为。数学工具将包括渐近，谱分析，偏微分方程和分叉理论，和非线性动力学，我们的小组将与一些数值分析师合作。** 拟议的新研究由四个重叠的主题组成：**I（耦合细胞体模型）：开发和分析新的耦合细胞体ODE-PDE模型，其中空间隔离的动态活性信号“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性单元的稳定和同步振荡，否则不会发生。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约束模式的演变，线性稳定性和平衡。我们还将研究新的类的RD模型，导致从3D体和2D表面扩散过程的耦合，由于体和表面之间的化学交换。**III（混合方法）：已知某些RD系统的点动力学取决于某些绿色函数的梯度，而其线性稳定性特性取决于相关的本征值依赖的绿色函数。这些绿色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斑点动力学的数值实现，支持平衡的分叉和稳定性如何取决于域形状的调查。第四部分：（第一通道问题）我们将分析生物物理学中两个具体的第一通道问题。第一个问题是考虑布朗步行者，其在到达特定目标部位之前经历3D本体与其限制表面之间的间歇性结合。第二个问题是推导出有效的Robin边界条件，以计算涉及大量表面陷阱的平均首次通过时间（MFPT）。需要如此仔细的均质化分析一直是生物物理学中的一个长期问题。

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.***********