Residence and First Passage Time Functionals in Heterogeneous Ecological Dispersion

异质性生态弥散多尺度问题中的停留和首次通过泛函一直是重要的数学建模和研究的动力。本提案旨在开发和分析与生物学、生态学、海洋学和流行病学的几个例子相关的模型，这些模型涉及由模型中系数值的不连续定义的界面效应。这些现象发生在高度不均匀的区域，在这些区域中，景观的某些物理、化学或生物特性的急剧或突然的不连续性出现在基本方程的系数中。pi将分析相关过程的功能，包括碎片化或斑块化的域以及离散的图形结构，以量化较小尺度的界面不连续对宏观尺度变量（如居住和职业时间函数）的影响。在提案的第一部分，pi将开发具有不连续系数的平流-分散-反应方程的随机方法，以模拟不同的生物过程。与更经典的物理模型不同，微观尺度的界面条件可以由宏观尺度的守恒定律决定，生物对界面边界的反应数据可能是完全不同的。确定适当的模型需要发展新的微观尺度分析方法，涉及当地时间和伊藤-田中随机演算，以揭示适当的宏观尺度方程，控制人口密度和分散的特征函数。在提案的第二部分，Pis将开发数值方法，蒙特卡罗随机粒子格式，以及涉及与关键生物场数据相关的特殊界面几何形状的不连续系数的平流-色散方程的统计参数估计的新方法。自然的物理过程，以及某些人为活动，导致栖息地碎片化，物种（动物、植物和细菌）适应或改变它们的行为。生境结构和/或其条件的变化，对与自然资源有关的科学家、决策者和资源管理者提出了新的挑战和重要的广泛的新问题。据报道，在生物和环境科学与工程领域中发生这种影响的几个当代问题包括：异质景观中受污染沉积物的生物修复；传染病在支离破碎的生境中传播，造成群落结构变化，可能导致外来物种入侵；影响濒危物种持久性的异质环境中的物种扩散和可持续性影响渔业的海洋叶绿素华的空间定位。这些例子中常见的具体数学问题涉及界面过程的适当建模，即。模型方程系数的数学不连续，影响物种运动的大尺度行为。在本研究中发展的数学框架特别旨在评估和量化由这些小尺度突变引起的大尺度界面效应。这项研究将提供一个数学框架和工具，以支持实地和实验室的工作，通过数值和统计算法的结合，以及涉及确定性和随机微积分工具的理论数学分析，量化和解决有关物种扩散的基本问题。

Residence and First Passage Time Functionals in Heterogeneous Ecological Dispersion Multiscale problems continue to motivate important mathematical modeling and research. This proposal aims to develop and analyze models relevant to several examples from biology, ecology, oceanography and epidemiology, that involve interfacial effects defined by discontinuities in values of coefficients in the models. These phenomena occur on highly heterogeneous domains in which sharp or abrupt discontinuities in certain physical, chemical, or biological properties of the landscape occur in the coefficients of the basic equations. The Pis will analyze functionals of the associated processes, both for fragmented or patchy domains and for discrete graphical structures, to quantify the effects that smaller scale interfacial discontinuities have on macro scale variables, such as resident and occupation time functionals. In the first part of the proposal, the PIs will develop stochastic approaches to the advection-dispersion-reaction equations with discontinuous coefficients that model different biological processes. Unlike more classical physical models where the micro-scale interface conditions can be determined by macro-scale conservation laws, data on biological responses to interfacial boundaries can be quite different. The determination of the appropriate models requires the development of new micro-scale methods of analysis involving local time and the Ito-Tanaka stochastic calculus to uncover the appropriate macro-scale equations governing population densities and characteristic functionals of dispersion. In the second part of the proposal the Pis will develop numerical methods, Monte-Carlo stochastic particle schemes, and new methods of statistical parameter estimation for advection-dispersion equations involving discontinuous coefficients with special interface geometries relevant to key biological field data.Natural physical processes, as well as certain anthropogenic activities, result in fragmented habitats to which species (animal, plants and bacteria) adapt or modify their behavior. Changes in the habitat configuration and/or its conditions, present new challenges and pose important broad new questions to scientists, policy makers and resource managers concerned with natural resources. Several contemporary problems in the biological and environmental sciences and engineering where such effects are reported to occur include: Bio-remediation of contaminated sediments in heterogeneous landscapes; Spread of infectious disease over fragmented habitats causing shifts in community structures possibly leading to invasion by exotic species; Species dispersal and sustainability in a heterogeneous environment affecting persistence of endangered species; Spatial localization of oceanic chlorophyll blooms impacting the fisheries industry. The specific mathematical issues common to these examples involve appropriate modeling of interfacial processes,i.e., mathematical discontinuities in the coefficients of the model equations, that affect the large scale behavior of species movement. The mathematical framework to be developed in this research is particularly aimed at assessing and quantifying interfacial effects on the large scale caused by these abrupt small -scale changes. This research will provide a mathematical framework and tools to support field and laboratory efforts to quantify and resolve fundamental questions about species dispersal through a combination of numerical and statistical algorithms, together with a theoretical mathematical analysis involving tools from deterministic and stochastic calculus.