Models for the ecological effects and evolution of dispersal

这个项目的目标是使用数学模型来洞察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，以及影响扩散策略演变的因素。研究工作将侧重于了解有条件的扩散策略的影响和演变，即取决于人口密度和环境条件的扩散策略。分散策略将从进化稳定性的角度进行研究。数学模型将主要由反应-扩散-平流方程或此类方程组组成。关于简单扩散的生态模型已经有了大量的研究，但涉及更复杂的条件扩散策略的生态模型的研究则要少得多。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。通过引入沿环境梯度的平流、朝向捕食者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速率等因素，本项目的研究将扩展此类模型，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组理论、动力系统理论和非线性分析。主要的数学思想包括利用分支理论、持久性理论和单调半动力系统理论将对微分算子的特征值的估计转化为关于涉及这些算子的系统的动力学的结论。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)，预计该项目将涉及开发新的数学结果。该项目中的一些研究将旨在了解具体的生态问题。例如，该项目的一个方面将是研究诸如行会内捕食或明显竞争的情况，其中顶级捕食者捕食另外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。这项研究将涉及相互竞争的种群的模型，这些种群在生态上是相同的，除了它们的分散策略，这将从进化稳定性的角度进行研究。以前的研究表明，在这种情况下，扩散策略的一个重要方面是它们允许种群与环境中可用的资源匹配得多好，这样有条件的扩散可能会有利于生物的理想自由分布。这一想法将得到进一步探讨。将被用来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的特例。生物的扩散显然是许多生态过程的一个重要方面。它推动了生物入侵，允许种群殖民空旷的栖息地，并允许个人跟踪资源，避开捕食者或竞争对手。在这个项目中，物种相互作用和扩散的数学模型将被用来获得关于扩散模式如何影响物种的持续、分布和/或灭绝的理论见解，以及当生物进化以适应新的或变化的环境时可能出现的扩散模式。扩散可能反映纯粹的随机运动，也可能取决于环境的性质或其他生物体的存在。这个项目将主要集中在有条件的扩散上。尽管有经验证据表明存在条件扩散，也有一些理论证据表明，在某些情况下，自然选择可能有利于条件扩散，但对条件扩散的研究没有随机扩散那么多。开发包含条件扩散的模型将扩大当前扩散的生态和进化理论的范围。该项目的一个更广泛的动机是，开发扩散的影响和演变的模型应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.