Dynamical aspects in nonautonomous and random differential equations and applications

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。这个项目是研究由各种物理和生物问题引起的非自治和随机微分方程的动力学方面。特别是，研究者将研究(1)有界域上非自治和随机线性扩散发展方程的谱理论及其在生态学中扩散演化的应用，增长域上的图灵不稳定性，以及年龄结构/大小结构种群动力学的非线性Leslie型模型；(2)由于相变、神经传播和种群遗传学而产生的无限非均匀和随机介质中的空间传播和前沿传播动力学。这位研究人员和她的合作者推广了几个经典的概念和概念，并建立了一些通用的理论和技术来研究这个项目中的问题和许多其他非自治和随机的微分方程。除了已建立的理论和技术外，研究者还将继续扩展相关的经典概念和概念，并发展通用的理论和技术来研究本项目中的问题和其他相关的非自治和随机微分方程。该项目的研究结果将加深对所研究系统动力学的理解，特别是将加深对时空相关性和随机性对底层问题动力学影响的理解，将为这些系统及其相关问题的进一步研究提供理论和方法论基础，并将拉近几个相互独立但又相互关联的数学分支，包括微分方程、拓扑系统和度规动力系统，并丰富它们各自的内容。现实的物理和生物系统受外部环境变化的影响，经常处于各向异性或非均匀介质中。因此，通过涉及非自治或随机或随机微分方程的模型来研究这类系统得到了越来越多的关注。由于缺乏一般的方法论以及推广经典概念和概念的困难，人们对这些方程中的许多重要的动力学问题仍然知之甚少。研究人员将继续扩展相关的经典概念和概念，并开发新的工具和一般理论来研究这些方程的动力学。该项目的结果将深入了解介质的非均质性和随机性对各种应用问题的动力学的影响，包括相变、神经传播、种群动力学和图案形成。它将为研究生和初级教员提供非自治和随机微分方程领域的研究培训。它还将创造与来自其他学科的学生和科学家互动的机会，并将导致技术进步。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project is to study the dynamical aspects in nonautonomous and random differential equations arising from a variety of physical and biological problems. In particular, the investigator will study (1) spectral theory for nonautonomous and random linear dispersal evolution equations on bounded domains and its applications to evolution of dispersals in ecology, Turing instability on growing domains, and nonlinear Leslie type models for age-structured/size-structured population dynamics; (2) spatial spread and front propagation dynamics in unbounded inhomogeneous and random media arising from phase transition, nerve propagation, and population genetics. The investigator and her collaborators have extended several classical concepts and notions and established a number of general theories and techniques for the study of the problems in this project and many other nonautonomous and random differential equations. In addition to the established theories and techniques, the investigator will continue to extend relevant classical concepts and notions and to develop general theories and techniques for the study of the problems in this project and other related nonautonomous and random differential equations. The results of this project will enhance the understanding of the dynamics of the systems under investigation, specially, will enhance the understanding of the effects of the time/space dependence and randomness on the dynamics of the underling problems, will provide theoretical and methodological foundations for the further study of these systems and related ones, and will bring closer several separate but related branches of mathematics, including differential equations, topological dynamical systems, and metric dynamical systems, and enrich each of them.Realistic physical and biological systems are influenced by variations in the external environment, and are often situated in anisotropic or inhomogeneous media. For this reason, the study of such systems via models involving nonautonomous or random or stochastic differential equations has been gaining more and more attention. Due to a lack of general methodology and difficulties in generalizing classical concepts and notions, there is still little understanding of many important dynamical issues in these equations. The investigator will continue to extend relevant classical concepts and notions and to develop new tools and general theories to study the dynamics of these equations. The results of the project will provide deep insight into the effect of the inhomogeneity and randomness of the media on the dynamics of various applied problems including phase transition, nerve propagation, population dynamics, and pattern formation. It will provide graduate students and junior faculty members research training in the area of nonautonomous and random differential equations. It will also create opportunities to interact with students and scientists from other disciplines and will lead to advance in technology.