Collaborative Research: Mathematical and Experimental Analysis of Ecological Models: Patches, Landscapes and Conditional Dispersal on the Boundary

该项目整合了无脊椎动物捕食者-被捕食系统的数学模型和实验分析，以探索栖息地破碎化、条件扩散、捕食和种间竞争对从斑块水平到景观水平的草食动物种群动态的影响。它代表了两位数学家、生态学家、本科生和博士生之间的独特合作。该项目预计将提供种群生态学方面急需的信息，了解有条件扩散对破碎景观中物种种群动态的影响。该项目的结果将回答几个关键的生态问题，例如密度依赖性扩散的存在是否有助于减轻潜在的有害因素，如栖息地破碎或更糟，加剧其影响。该项目还将为分析具有非线性边界条件的椭圆边值问题做出重大贡献，因为将开发新的数学工具来更好地理解这些总体模型的动态。最后，该项目将为如何构建实证研究提供明确的指导方针，以根据这些理论模型的预测来评估密度相关扩散的存在和后果。研究人员将通过各种媒体向生态和数学界传播该项目的结果，包括同行评审的数学和生态期刊、国内和国际会议上的演讲以及展示该研究的用户友好网站。该项目的一个重要方面将涉及通过研究人员主办的研讨会和独立研究项目的指导来培训研究生和本科生。此外，还将针对本科生和高级高中生开发涵盖基本种群生态学的种群动态课程，通过数学工具和有趣的例子来探索与密度依赖扩散相关的种群模型，并通过该项目的网站免费向公众开放。这个合作项目的目的是通过反应扩散模型、数学分析和无脊椎动物系统的实验分析来整合种群动态建模，以探索栖息地破碎化、条件扩散、捕食和种间竞争对草食动物种群的影响从斑块级别到景观级别的动态。这项研究将有助于回答重要的生物学问题，例如 1) 密度依赖性扩散可以预期产生什么斑块水平效应，特别是正、负或 U 形密度依赖性扩散，2) 密度依赖性扩散是否会减轻甚至加剧栖息地破碎化、Allee 效应、种间竞争或捕食对种群局部或区域稳定性/持久性的影响，以及 3) 应如何构建实证研究来评估密度依赖性扩散的存在和后果根据这些理论模型的预测。在存在捕食、种间竞争和栖息地破碎化等复杂因素的情况下，更全面地了解密度依赖性扩散对斑块和景观水平的影响本身就很重要，但也可能导致制定更好的种群管理策略，特别是在种群因捕食、栖息地破碎化和全球气候变化而面临各种生态挑战的环境中。该项目预计将具有重要意义，因为它提供了种群生态学中急需的关于条件扩散（即作为同种、种间竞争者和捕食者密度的函数）对破碎景观中物种种群动态的影响的信息。这项研究之所以新颖，是因为迄今为止，碎片系统的理论和实证研究忽略了自然界中常见的其他形式的密度依赖扩散（负或 U 形）。该项目的结果将回答几个关键的生态问题，即负或 U 形密度依赖性扩散的存在是否有助于减轻潜在的有害因素，因为栖息地破碎或更糟的是，会加剧其影响。该项目还将为分析具有非线性边界条件的椭圆边值问题做出重大贡献，因为将开发新的数学工具来更好地理解这些总体模型的动态。此外，基于反应扩散方程建立的真实景观水平建模框架的开发将为理论模型中景观动力学的加强研究奠定基础。研究人员计划通过各种媒体向生态和数学界传播该项目的结果，包括：ArXiv、同行评审的数学、数学生物学和生态学期刊，以及在数学生物学和生态会议上的演讲。

This project is an integration of mathematical modeling and experimental analysis of an invertebrate predator-prey system to explore the effects of habitat fragmentation, conditional dispersal, predation, and interspecific competition on herbivore population dynamics from the patch level to the landscape level. It represents a unique collaboration between two mathematicians, and ecologist, and undergraduate and PhD students. This project is expected to provide much-needed information in population ecology on the consequences of conditional dispersal to population dynamics of species in fragmented landscapes. Results from this project will answer several key ecological questions such as will the presence of density dependent dispersal help to moderate potentially detrimental factors as habitat fragmentation or worse, exacerbate their effects. The project will also provide a significant contribution towards the analysis of elliptic boundary value problems with nonlinear boundary conditions, as new mathematical tools will be developed to better understand the dynamics of these population models. Finally, the project will provide clear guidelines for how empirical studies should be constructed to evaluate the presence and consequences of density dependent dispersal in light of the predictions of these theoretical models. The investigators will disseminate the results of this project to both the ecological and mathematical communities through various media including peer-reviewed mathematical and ecological journals, talks at national and international conferences, and a user-friendly website showcasing the research. An important aspect of this project will involve the training of graduate and undergraduate students through workshops hosted by the investigators and mentorship of independent research projects. Moreover, a population dynamics curriculum covering basic population ecology through mathematical tools and interesting examples for exploring population models related to density dependent dispersal will be developed targeting undergraduate and advanced level high school students and freely available to the public via the project's website.The purpose of this collaborative project between will be an integration of modeling of population dynamics via reaction diffusion models, mathematical analysis, and experimental analysis of an invertebrate system to explore the effects of habitat fragmentation, conditional dispersal, predation, and interspecific competition on herbivore population dynamics from the patch level to the landscape level. This study will help answer important biological questions such as 1) what patch level effects can be expected from density dependent dispersal, specifically of positive, negative or U-shaped density dependent dispersal, 2) does density dependent dispersal moderate or even exacerbate the effects of habitat fragmentation, Allee effects, interspecific competition, or predation on local or regional stability/persistence of a population, and 3) how should empirical studies be constructed to evaluate the presence and consequences of density dependent dispersal in light of the predictions of these theoretical models. A more comprehensive understanding of the patch and landscape level consequences of density dependent dispersal in the presence of such complicating factors as predation, interspecific competition, and habitat fragmentation is important by itself, but may also lead to the development of better population management strategies, especially in an environment where populations face diverse ecological challenges due to predation, habitat fragmentation, and global climate change. This project is expected to be significant by providing much-needed information in population ecology on the consequences of conditional dispersal (i.e., as a function of the density of conspecifics, interspecific competitors, and predators) to population dynamics of species in fragmented landscapes. The research is novel because, to date, theoretical and empirical studies in fragmented systems have ignored other forms of density dependent dispersal (negative or U-shaped) that are commonly found in nature. Results from this project will answer several key ecological questions as to whether the presence of negative or U-shaped density dependent dispersal helps to moderate potentially detrimental factors as habitat fragmentation or worse, exacerbate their effects. The project will also provide a significant contribution towards the analysis of elliptic boundary value problems with nonlinear boundary conditions, as new mathematical tools will be developed to better understand the dynamics of these population models. Further, development of a true landscape level modeling framework built on reaction diffusion equations will serve as a foundation for enhanced study of landscape dynamics in theoretical models. The investigators plan to disseminate the results of this project to both the ecological and mathematical communities through various media including: the ArXiv, peer-reviewed mathematics, mathematical biology, and ecology journals, and in talks at mathematical biology and ecological conferences.