Ecological Modeling: From Individual Utilization of Space to Community Structure

[cn]研究者研究动态空间分布的生物种群的数学模型。该项目的目标是为个体的空间分布和运动如何影响种群的持续或灭绝提供更好的理论理解。其动机是深入了解空间效应对生物多样性维持的贡献方式，以及环境空间方面的变化可能威胁生物多样性的方式。该项目主要关注边缘中介效应模型的构建和分析，以及将关于小空间尺度上个体行为的假设转化为关于大空间尺度上种群动态的结论的方法的发展。当栖息地类型之间的界面对种群动态产生影响时，就会发生边缘中介效应。一个边缘介导效应的例子可能发生，如果一个栖息地的片段被一条道路从一个更大的区域切断，一些物种可以穿过，而其他物种则不能。如果边缘（道路）的存在导致了碎片中生态群落物种组成的变化，这将是边缘介导效应。从局部行为到大规模结论的问题出现在生态学中，因为通过观察少数个体来确定小规模行为通常是可能的，而直接观察大种群的整体动态要困难得多。该项目的方法是为空间分布的种群建立数学模型，然后分析这些模型对它们所描述的物种的持久性或灭绝的预测。数学模型通常基于常微分方程或偏微分方程。分析通常涉及动力系统，非线性泛函分析（特别是分岔理论）和偏微分方程（特别是比较方法和椭圆谱理论）的思想。第二个目标是获得在应用环境中有用的新的数学结果。研究者们研究用数学来理解生态问题的方法。这些问题主要涉及环境的空间方面；例如，如何设计最有效的自然保护区，或者如何预测自然栖息地破碎化对居住在其中的种群可能产生的影响。这些问题还涉及不同生物物种之间的相互作用，例如捕食者和猎物。该方法是用方程来表达生物系统的基本特征，然后研究这些方程来确定它们所预测的东西。这种分析通常需要发展新的数学结果，这些结果可能对其他研究人员有用。分析的一个典型目标是确定各种环境和生物因素如何影响种群的持续或灭绝。使用数学模型的一个原因是，在涉及濒危物种、基因工程生物或大规模环境变化的情况下，通常不可能进行实验，但对预期结果有一定的了解至关重要。有些模型是由避难所设计或虫害控制问题直接引起的。另一些人则是出于一些基本的科学问题，比如环境的空间方面对人口的影响。但即使是这些也可能被证明与环境保护和全球变化的生物效应相关。该项目的潜在好处包括改善保护、避难所设计、土地管理和城市规划方面的决策。

Cosner9973017 The investigators study mathematical models for the dynamicsof spatially distributed biological populations. The goal of theproject is to provide a better theoretical understanding of howthe spatial distribution and movement of individuals affect thepersistence or extinction of populations. The motivation is togain insight into the ways that spatial effects contribute to themaintenance of biodiversity and the ways that changes in thespatial aspects of the environment can threaten biodiversity.The project is primarily concerned with the construction andanalysis of models for edge-mediated effects and the developmentof methods for translating assumptions about the behavior ofindividuals on a small spatial scale into conclusions about thedynamics of populations on a large spatial scale. Edge-mediatedeffects occur when the presence of an interface between habitattypes has an effect on population dynamics. An example of anedge-mediated effect might occur if a fragment of habitat werecut off from a larger region by a road that some species wouldcross but others would not. If the presence of the edge (theroad) thus resulted in a change in the species composition of theecological community in the fragment, that would be anedge-mediated effect. Issues of scaling from local behavior tolarge-scale conclusions arise in ecology because it is oftenpossible to determine small-scale behavior by observing a fewindividuals, while directly observing the overall dynamics of alarge population is much more difficult. The methodology of theproject is to construct mathematical models for spatiallydistributed populations and then to analyze those models'predictions of persistence or extinction for the species theydescribe. The mathematical models are typically based onordinary or partial differential equations. The analysistypically involves ideas from dynamical systems, nonlinearfunctional analysis (especially bifurcation theory), and partialdifferential equations (especially comparison methods andelliptic spectral theory). A secondary goal is to obtain newmathematical results that are useful in the context ofapplications. The investigators study ways of using mathematics tounderstand ecological problems. The problems mostly involvespatial aspects of the environment; for example, how to designthe most effective nature reserves, or how to predict the likelyeffects of the fragmentation of natural habitats on thepopulations that inhabit them. The problems also involveinteractions between different biological species, for examplepredators and prey. The methodology is to express the essentialfeatures of a biological system in terms of equations and then tostudy the equations to determine what they predict. The analysisoften requires the development of new mathematical results, whichmay be of use to other investigators. A typical goal of theanalysis is to determine how various environmental and biologicalfactors affect the persistence or extinction of populations. Areason for using mathematical models is that in situationsinvolving endangered species, genetically engineered organisms,or large scale environmental change, it is often impossible to doexperiments but crucial to have some idea of what to expect.Some of the models are directly motivated by problems in refugedesign or pest control. Others are motivated by basic scientificquestions about the effects of the spatial aspects ofenvironments on populations. But even those may prove to berelevant to issues raised in the contexts of environmentalprotection and the biological effects of global change. Thepotential benefits of the project include improved decisionmaking in conservation, refuge design, land management, and urbanplanning.