Localized Patterns in PDEs: Theory, Computation, and Applications

局域化的时空模式通常发生在各种线性和非线性扩散过程中。特别是，它们发生在反应-扩散(RD)系统中，模拟生物系统中的群体感应(QS)行为，植物细胞中根毛尖端形成的启动，以及城市犯罪的空间分布。局部化行为也发生在生物物理背景下，计算布朗步行者在具有局部陷阱的区域中的首次通过统计，以及在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开发渐近-数值混合方法，以研究生物、化学和社会相互作用的各种新的PDE模型中的这种局部化行为。数学工具将包括渐近、谱分析、偏微分方程和分叉理论，以及非线性动力学，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)：发展和分析一类新的细胞-体模型-PDE模型，在该模型中，空间上分离的动态活性信号“细胞”或膜通过体散射场耦合。对于一个这样的模型，我们已经证明了体扩散的效应导致了动态活动单元的稳定和同步振荡，否则这是不会发生的。*II(流形上的图案)：分析闭合流形上RD系统的局域光斑图案，以确定流形的几何形状如何影响这种表面边界图案的演化、线性稳定性和平衡性。我们还将研究新的RD模型，这些模型是由于体和表面之间的化学交换而导致的3D体和2D表面扩散过程的耦合。*III(混合方法)：已知某些RD系统的光斑动力学依赖于某些格林函数的梯度，而它们的线性稳定性则依赖于相关的特征值相关的格林函数。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区域中实现光斑动力学的数值实现，支持对平衡的分叉和稳定性如何依赖于区域形状的研究。*IV：(首次通过问题)我们将分析生物物理学中的两个具体的首次通过问题。第一个问题是考虑一个布朗漫游者，它在到达特定的目标位置之前，在3D块体和它的限制表面之间经历间歇性的结合。第二个问题是推导有效的Robin边界条件，以计算包含大量表面陷阱的平均首次通过时间(MFPT)。对如此仔细的均质分析的需要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.***********