Analysis of Spreading Speeds and Traveling Waves in Multi-species Models of Biological Invasions

本研究致力于发展有关生物入侵重要方面的数学理论。研究者将开发和分析描述多物种生长、相互作用和迁移的数学模型。这些模型将纳入物种之间的竞争和捕食者-猎物相互作用，并将以反应-扩散方程或带有扩散核的积分-差分方程的形式表示。这些模型将用一些可测量的和有生物学意义的参数来代表物种传播的核心方面，并将在非线性积分方程、非线性微分方程和种群建模的定性研究领域提出具有挑战性的问题。研究者将使用常微分方程、偏微分方程、差分方程、积分方程和动力系统的方法来分析模型，并研究物种在与其他物种局部相互作用的同时，以什么速度和什么形式扩大它们的范围或从它们以前占据的区域撤退。研究人员将特别检查一个物种的传播速度，该物种入侵当地的一个物种已经以平衡分布或行波分布的形式建立了自己的区域。研究者还将研究模型中行波的存在性，重点研究行波速度与传播速度的关系。事实上，每一个生态系统都受到外来物种的入侵，往往对本地动植物造成严重后果。生物入侵问题一直是集约化管理和研究活动的重点。这项研究将提供广义的生物入侵解释，并将为预测提供基础。从本项目开发的新型分析技术中获得的知识将使我们有能力制定机制来处理诸如自然资源管理和病虫害控制等问题。这项工作将在生物学以外的领域应用于化学和物理中的波传播现象的研究。此外，通过直接参与研究和在生物入侵研究中开发的以研究为基础的教育材料，这项工作将对学生产生教育效益。

This research is dedicated to the development of mathematical theory regarding important aspects of biological invasions. The investigator will develop and analyze mathematical models that describe the growth, interactions, and migrations of multiple species. The models will incorporate competition and predator-prey interactions between species, and will be formulated in the form of reaction-diffusion equations or integro-difference equations with dispersal kernels. The models will represent the central aspects of spread of species in terms of a few measurable and biologically meaningful parameters, and will pose challenging problems in areas of qualitative studies of nonlinear integral equations, nonlinear differential equations, and population modeling. The investigator will use methods from ordinary and partial differential equations, difference equations, integral equations and dynamical systems to analyze the models and investigate at what speeds and in what forms species expand their ranges or retreat from the areas they previously occupied while interacting with other species locally. The investigator will in particular examine the spreading speed of a species that invades locally an area where a resident species has established itself in the form of an equilibrium distribution or a traveling wave distribution. The investigator will also study the existence of traveling waves in the models with an emphasis on relating traveling wave speeds to spreading speeds. Virtually every ecosystem has been invaded by foreign species with often drastic consequences for the native fauna or flora. The problem of biological invasions has been the focus of intensive management and research activities. This research will offer explanations of biological invasions in broad terms, and will provide a basis for prediction. The knowledge gained from the novel analytical techniques developed in this project will lead to the ability to develop mechanisms to handle problems such as natural resource management, and pest and disease control. This work will have applications outside of biology in areas of studies of wave propagation phenomena in chemistry and physics. In addition, this work will have educational benefits for students through direct involvement in the research and through research-based educational materials developed in the study of biological invasions.