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。

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.