Models for the ecological effects and evolution of dispersal

该项目的目标是使用数学模型来洞察生物扩散的进化原因和生态影响。分散生物的个体行为影响其种群的空间分布，从而影响它们与资源、捕食者和自身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，从而为拥有优越分散策略的个体创造选择优势。选择会对个体的扩散行为产生进化压力。数学模型被用来理解和描述涉及扩散行为生态和演化的多尺度的复杂反馈。主要的模拟方法是基于具有可变系数和/或密度相关系数的反应-平流-扩散方程组。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，它们为处理不同空间和时间尺度以及不同组织级别的复杂互动提供了一个框架。它们在数学上提出了巨大的挑战，但最近在分叉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。除了偏微分方程组外，还使用了具有非局部离散的面片模型和积分微分方程组。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，导致概念上的统一。传统的扩散模型通常将扩散视为一个本质上随机的过程，不依赖于环境条件或个体的内部状态，但已有大量研究表明，有条件的扩散是常见的，可能是有利的。因此，研究的重点是模型，其中个体的扩散和平流率取决于环境条件和/或其物种和与其相互作用的物种的种群密度。这导致了对非线性抛物型偏微分方程组的动力学和平衡点问题的新的数学研究，这些问题的系数在空间和/或时间上变化。生物的分散影响种群的持久性和相互作用，并驱动范围扩大、生物入侵和空置栖息地的殖民。因此，了解扩散对于解决有关生物保护、入侵物种、虫害控制和其他环境问题，包括人口对全球变化的反应等问题是相关的。生物体的扩散模式影响它们与环境和其他物种的相互作用的方式是复杂和微妙的，因此很难评估扩散模式的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不断变化的环境中的有机体的扩散策略。这一过程涉及复杂的反馈回路，令人难以理解。这项研究通过使用数学模型来描述扩散的生态效应和演化来解决这些问题。这些模型基于偏微分方程式，它提供了一个框架，描述了量如何在空间和时间上变化和相互影响。具体地说，这些模型包括描述随机迁移、定向迁移和生态相互作用的术语，这些相互作用可能在时间和空间上变化，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，因此本研究涉及开发和分析可能与研究生态学及相关领域中的各种现象相关的新型模型。

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.