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扩散对物种与其环境或其他物种之间的生态相互作用的影响，以及影响扩散策略进化的因素。研究工作将集中在理解条件扩散策略的影响和演变，即依赖于种群密度和环境条件的扩散策略。我们将从进化稳定性的角度来研究扩散策略。数学模型将主要由反应-扩散-平流方程或此类方程的系统组成。对简单扩散的生态模型研究较多，但对复杂条件扩散策略的生态模型研究较少。基于简单扩散的模型描述了不依赖于环境条件的随机扩散。通过结合诸如沿环境梯度、朝向猎物或远离捕食者的平流以及取决于种群密度或环境条件的可变扩散率等因素，本项目的研究将扩展这些模型，以描述各种类型的条件扩散并研究其影响。将使用的一般数学方法包括经典的偏微分方程理论、动力系统理论和非线性分析。关键的数学思想包括使用分岔理论、持续理论和单调半动力系统理论，将微分算子的特征值估计转化为涉及这些算子的系统动力学的结论。由于将要研究的模型包括强耦合的拟线性抛物系统（与通常是弱耦合和半线性的标准反应扩散系统相反），预计该项目将涉及新的数学结果的发展。该项目的一些研究将旨在了解具体的生态问题。举个例子，该项目的一个方面将是研究野生动物内部捕食或明显竞争等情况，其中顶级捕食者捕食其他两个相互作用的物种并影响它们的扩散模式。一些研究将旨在了解影响扩散进化的选择压力。这项研究将涉及竞争种群的模型，这些种群在生态上是相同的，只是它们的分散策略不同，这将从进化稳定性的角度进行研究。先前的研究表明，在这种情况下，扩散策略的一个重要方面是它们如何让种群与环境中可用的资源相匹配，因此，有条件的扩散可能会导致生物体的理想自由分布。这个想法将被进一步探讨。用于研究进化问题的模型在某种意义上是用于研究生态问题的模型类型的特殊案例。生物的扩散显然是许多生态过程的一个重要方面。它驱动生物入侵，允许种群在空旷的栖息地定居，并允许个体追踪资源并避开捕食者或竞争对手。在这个项目中，物种相互作用和分散的数学模型将被用来获得理论见解，以了解扩散模式如何影响物种的持久性、分布和/或灭绝，以及当生物进化以适应新的或不断变化的环境时，可能出现什么样的扩散模式。扩散可能反映了纯粹的随机运动，也可能取决于环境的特性或其他生物体的存在。这个项目将主要集中在条件分散上。尽管有经验证据表明，条件扩散会发生，而且有一些理论证据表明，在某些情况下，它可能受到自然选择的青睐，但对条件扩散的研究不如随机扩散那么多。发展包含条件扩散的模型将扩大当前生态和进化扩散理论的范围。该项目的一个更广泛的动机是，为扩散的影响和进化发展模型应该提供可能有助于制定政策以解决生态问题的见解，例如在环境变化下保护生物多样性或评估引入外来物种或改变环境的可能结果。

The goal of this project is to use mathematical models to gain insight into the effects that dispersal can have on ecological interactions between species and their environment or other species, and into the factors that influence the evolution of dispersal strategies. The research effort will be focused on understanding the effects and evolution of conditional dispersal strategies, that is, dispersal strategies that depend on population densities and environmental conditions. Dispersal strategies will be studied from the viewpoint of evolutionary stability. The mathematical models will primarily consist of reaction-diffusion-advection equations or systems of such equations. There has been a great deal of research on ecological models with simple diffusion, but much less on models involving more complex conditional dispersal strategies. Models based on simple diffusion describe random dispersal that does not depend on environmental conditions. By incorporating factors such as advection along environmental gradients, toward prey, or away from predators, and variable diffusion rates that depend on population densities or environmental conditions, the research in this project will extend such models to describe various types of conditional dispersal and to study their effects. The general mathematical methods that will be used include the classical theory of partial differential equations, dynamical systems theory, and nonlinear analysis. Key mathematical ideas include using bifurcation theory, persistence theory, and the theory of monotone semidynamical systems to translate estimates on eigenvalues of differential operators into conclusions about the dynamics of systems involving those operators. Since the models that will be studied include strongly coupled quasilinear parabolic systems (as opposed to standard reaction-diffusion systems which are usually weakly coupled and semilinear) it is anticipated that the project will involve the development of new mathematical results. Some of the research in the project will be aimed at understanding specific ecological questions. As an example, one aspect of the project will be the study of situations such as intraguild predation or apparent competition where a top predator preys on two other interacting species and influences their dispersal patterns. Some of the research will be aimed at understanding the selective pressures that influence the evolution of dispersal. That research will involve models for competing populations that are ecologically identical except for their dispersal strategies, which will be studied from the viewpoint of evolutionary stability. Previous research suggests that in this context an important aspect of dispersal strategies is how well they allow a population to match the resources available in the environment, so that conditional dispersal leading to something like an ideal free distribution of organisms may be favored. That idea will be explored further. The models that will be used to study evolutionary questions are in some sense special cases of the types of models used to study ecological questions.The dispersal of organisms is clearly an important aspect of many ecological processes. It drives biological invasions, allows populations to colonize empty habitats, and allows individuals to track resources and avoid predators or competitors. In this project mathematical models for interacting and dispersing species will be used to gain theoretical insights into how dispersal patterns can influence the persistence, distribution, and/or extinction of species, and what dispersal patterns are likely to appear as organisms evolve to adapt to new or changing environments. Dispersal may reflect purely random movement or may be conditioned on properties of the environment or the presence of other organisms. This project will be focused largely on conditional dispersal. Conditional dispersal has not been studied as much as random dispersal, even though there is empirical evidence that it occurs and there is some theoretical evidence that it may be favored by natural selection in some situations. Developing models that incorporate conditional dispersal will expand the scope of the current ecological and evolutionary theory of dispersal. A broader motivation for the project is that developing models for the effects and evolution of dispersal should provide insights that may be useful in formulating policy to address ecological issues such as the conservation of biodiversity under environmental change or the assessment of the possible results of introducing exotic species or altering environments.