Collaborative Research: Persistence, Stability and Control of Populations in Heterogeneous Networks

环境的空间非均匀性对许多大型生态和流行病学系统的生存或灭绝具有重要意义。本项目研究环境中生物的运动如何影响生态系统的生存或流行病的爆发/灭绝。这种流动的例子包括：海洋幼虫在水流的作用下在不同的珊瑚礁之间传播，假日旅行或环境病原体的流动导致传染病的传播，以及携带疾病的昆虫的流动导致感染。将向生物学家和其他科学家介绍一些用户友好的定量指数及其计算方法，作为检测和防止生态灭绝或流行病爆发的工具。该研究将模拟和优化流行病事件中疫苗接种，医疗和检疫的效果，以及农业/渔业/林业系统的收获和收获限制，以提供有关维持高质量生活环境和预防流行病爆发的重要问题的答案。拟议研究的样本应用包括(a)防止霍乱流行病通过共同水源传播和(b)恢复切萨皮克湾牡蛎种群。应用项目的成果将为政府机构提供可行的战略，并将提高渔业的经济生产力。定量定义和计算空间异质性和分散对关键生态指标和特性的影响，并建立统一的数学框架来研究异质性环境或网络中单一物种和相互作用物种的持久性、稳定性和控制力。重点是不对称运动对生物系统动力学行为的影响，并将图论、矩阵论及其连续理论注入到相应动力系统的研究中是新理论的关键。本文将在更广泛的背景下定义动力学系统和数学生物学中的几个经典概念，如Lyapunov稳定性理论、基本繁殖数、最大可持续产量和目标繁殖数的新概念，并结合额外的网络结构和空间异质性进行分析。该项目将培养研究生和本科生，学生从该项目中获得的建模技能和新的数学技术将使他们成为具有强大定量和分析能力的未来劳动力的一部分。

Spatial nonuniformity of the environment is important in the survival or extinction of many large-scale ecological and epidemiological systems. This project studies how movement of organisms in the environment affects ecosystem survival or epidemic outbreak/extinction. Examples of such movement include dispersal of marine larvae between different reefs under water flow, spreading of infectious diseases via holiday travel or environmental pathogen movement, and infection through movement of insects that carry disease. Several user-friendly quantitative indices and their computation will be introduced to biologists and other scientists as tools for detecting and preventing ecological extinction or epidemic outbreaks. The research will model and optimize the effect of vaccination, medical treatment, and quarantine in epidemic events, and harvesting and harvesting restrictions for agricultural/fishery/forestry systems, to provide answers to important questions about sustaining high quality living environments and preventing epidemic outbreaks. Sample applications of the proposed research include (a) preventing cholera epidemics spreading through a common water source and (b) restoring Chesapeake Bay oyster populations. The outcomes from application projects will provide actionable strategies for government agencies, and the results will improve economic productivity of fisheries. The impact of spatial heterogeneity and dispersal on key ecological indices and properties will be quantitatively defined and calculated, and a unified mathematical framework will be developed to study the persistence, stability and control of both single and interacting species in heterogeneous environment or networks. The focus is the impact of asymmetric movement on the dynamical behavior of the biological systems, and infusing graph theory, matrix theory and their continuous counterparts into the study of corresponding dynamical systems is a key to the new theory. Several classical concepts in dynamical systems and mathematical biology such as Lyapunov stability theory, basic reproduction number, maximum sustainable yield and a new concept of target reproduction number will be defined in a broader context and analyzed with the additional network structure and spatial heterogeneity. The project will train graduate students and undergraduate students, and the modeling skills and new mathematical techniques that students acquire from the project will make them part of a future workforce with strong quantitative and analytical ability.