Models for the ecological effects and evolution of dispersal

该项目的目标是利用数学模型来深入了解生物扩散的进化原因和生态影响。分散生物体的个体行为影响其种群的空间分布，从而影响其与资源、捕食者和本物种成员的生态相互作用。这些相互作用反过来影响个体和种群的生存，因此为具有优越分散策略的个体创造了选择优势。选择对个体的分散行为产生了进化压力。数学模型用于理解和描述涉及生态和扩散行为进化的多尺度复杂反馈。主要的建模方法是基于具有变量和/或密度相关系数的反应-平流-扩散方程系统。这样的方程可以从个体使用的运动机制中推导出来，但它们也可以预测物种的相互作用和进化。因此，它们为处理跨越不同时空尺度和不同组织级别的复杂交互提供了一个框架。它们提出了重大的数学挑战，但最近在分岔理论和偏微分方程方面的进展将有助于它们的分析。分析还采用了动力系统理论的思想。除偏微分方程外，还使用了斑块模型和非局部分散的积分微分方程。许多适用于反应-扩散-平流模型的分析方法可以应用于它们，从而导致概念上的统一。传统的扩散模型通常认为扩散本质上是一个随机过程，不依赖于环境条件或个体的内部状态，但已经有大量研究表明，条件扩散是常见的，可能是有利的。因此，研究的重点是个体的扩散和平流率取决于环境条件和/或其物种及其相互作用的物种的种群密度的模型。这导致了涉及非线性抛物型偏微分方程系统的动力学和平衡问题的新的数学研究，这些系统的系数随空间和/或时间变化。生物的扩散影响种群的持久性和相互作用，并推动范围扩大、生物入侵和空旷栖息地的殖民化。因此，了解物种扩散与解决生物保护、入侵物种、害虫控制和其他环境问题有关，包括种群对全球变化的反应。生物的扩散模式影响它们与环境和其他物种相互作用的方式是复杂而微妙的，因此很难评估扩散模式变化对这些相互作用的影响。此外，由于扩散影响生存，从而导致自然选择，进化可能会改变生活在变化环境中的生物的扩散策略。这个过程涉及复杂的反馈循环，很难理解。本研究通过使用数学模型来描述生态效应和扩散的演变来解决这些问题。这些模型是基于偏微分方程的，偏微分方程提供了一个框架来描述量是如何随空间和时间变化和相互影响的。具体来说，这些模型包括描述随机运动、定向运动和生态相互作用的术语，这些术语可能在时间和空间上和/或取决于人口密度。这项研究提供了有关这些过程相互作用方式的新信息。传统的扩散模型通常将其描述为一个不依赖于位置或环境条件的随机过程，因此本研究涉及到新型模型的开发和分析，这些模型可能与研究生态学和相关领域的各种现象有关。

The goal of the project is to use mathematical models to gain insight into the evolutionary causes and ecological effects of the dispersal of organisms. The individual behavior of dispersing organisms influences the spatial distribution of their population, which influences their ecological interactions with resources, predators, and members of their own species. Those interactions in turn influence the survival of individuals and populations, and hence create selective advantages for individuals with superior dispersal strategies. Selection results in evolutionary pressure on the dispersal behavior of individuals. Mathematical models are used to understand and describe the complex feedbacks at multiple scales involved in the ecology and evolution of dispersal behavior. The primary modeling approach is based on systems of reaction-advection-diffusion equations with variable and/or density dependent coefficients. Such equations can be derived from the movement mechanisms used by individuals, but they also make predictions about species interactions and evolution. Thus, they provide a framework for addressing complex interactions across different spatial and temporal scales and different levels of organization. They present significant mathematical challenges but recent advances in bifurcation theory and partial differential equations will facilitate their analysis. The analysis also employs ideas from the theory of dynamical systems. Patch models and integrodifferential equations with nonlocal dispersal are used in addition to partial differential equations. Many of the analytic approaches that work for reaction-diffusion-advection models can be applied to them, leading to conceptual unification. Traditional dispersal models typically view dispersal as an essentially random process that does not depend on environmental conditions or the internal states of individuals, but there have been numerous studies suggesting that conditional dispersal is common and may be advantageous. Thus, the research focuses on models where the diffusion and advection rates of individuals depend on environmental conditions and/or the population densities of their species and species with which they interact. That leads to new mathematical research on questions involving the dynamics and equilibria of nonlinear systems of parabolic partial differential equations with coefficients that vary in space and/or time.The dispersal of organisms influences the persistence and interactions of populations and drives range expansions, biological invasions, and the colonization of empty habitats. Thus, understanding dispersal is relevant to addressing questions about biological conservation, invasive species, pest control, and other environmental issues, including the response of populations to global change. The ways that the dispersal patterns of organisms affect their interactions with the environment and with other species are complex and subtle, so it is difficult to assess the effects of variations in dispersal patterns on those interactions. Also, since dispersal affects survival and thus leads to natural selection, evolution may change the dispersal strategies of organisms inhabiting changing environments. That process involves complex feedback loops which are difficult to understand. The research addresses those problems by using mathematical models to describe the ecological effects and evolution of dispersal. The models are based on partial differential equations, which provide a framework for describing how quantities change and influence each other over space and time. Specifically, the models include terms describing random movements, directed movements, and ecological interactions that may vary in time and space and/or depend on population densities. The research provides new information about the ways that these processes interact. Traditional models for dispersal usually describe it as a random process that does not depend on location or environmental conditions, so this research involves the development and analysis of new types of models that may be relevant in studying various phenomena in ecology and related areas.