Modeling spatial population dynamics in branching river networks using quantum graphs

缺乏适当的河网模型框架被认为是理解空间结构和环境变化对生态动态的影响的主要障碍。特别需要的是以其自然的、连续的形式来表示河网的空间域。该项目结合了量子图论作为河网建模的数学框架。这个理论是为解决量子力学中的问题而开发的，它将度量图与微分算符配对。度量图分支用连续的区间而不是离散的节点来标识；函数和运算符是沿着这些区间定义的。因此，环境变异性可以解释为系统参数沿网络中的分支的变化，并且可以全局地分析由空间变异性引起的描述种群动态的抛物型和椭圆型微分方程解。虽然量子图在模拟分支网络结构和环境变化如何影响河流生态动态方面具有巨大的潜力，但生态问题的关键数学要求还没有适应这个框架。该项目通过实现两个目标克服了这些障碍：1)扩展量子图以处理河流种群动态模型中通常出现的反应-扩散-平流方程组及其相关的积分核；以及2)开发分支网络解的变换方法，以描述河流种群对环境变化的响应。这种量子图对生态动力学的扩展导致了理论上的进步，也可能影响到科学中涉及连续空间网络的广泛问题，例如捕食者-猎物动力学、神经信号传播以及人体血管中的营养或药物输送。河流如何像树？如果你看地图上的一条河和一棵树，它们看起来非常相似，都有一个大的主干分支成较小的分支。研究河流的生态学家早就知道河流树状结构的重要性，但目前缺乏工具，使研究人员能够将这种结构纳入重要生态过程的模型。例如，这阻碍了预测污染物可能在河流中传播到哪里以及走多远的能力，调查大坝放置或拆除对濒危物种种群的潜在影响，以及了解河流泛滥平原土地利用和气候变化的复杂影响。这个项目导致了一套数学和计算工具的开发，用于研究河流的分支形式。这是通过将被称为量子图理论的数学框架应用于生态问题来实现的。量子图是在量子物理的背景下发展起来的，并得到了大量的研究。然而，这种关注意味着这些技术并不能应用于其他科学领域。该项目消除了这些应用障碍，并扩大了表现河流分支结构所产生的特殊生态模式的能力。

The lack of an appropriate modeling framework for branching river networks has been identified as a major hindrance to understanding the influences of spatial structure and environmental variability on ecological dynamics. Of particular need is a representation of the spatial domain of a river network in its natural, continuous form. This project incorporates quantum graph theory as a mathematical framework for modeling river networks. Developed for problems in quantum mechanics, this theory pairs a metric graph with a differential operator. Metric graph branches are identified with continuous intervals instead of discrete nodes; functions and operators are defined along these intervals. Thus, environmental variability can be interpreted as variation in system parameters along branches in the network, and solutions to parabolic and elliptic differential equations describing population dynamics that result from spatial variability can be analyzed globally. While quantum graphs hold great potential for modeling how branching network structure and environmental variability influence ecological dynamics in rivers, key mathematical demands of the ecological problem have not been adapted to this framework. This project overcomes these obstacles by accomplishing two objectives: 1) Extending quantum graphs to handle systems of reaction-diffusion-advection equations and their associated integral kernels that typically arise in river population dynamic models; and 2) developing transform methods for solutions on branching networks to describe the response of river populations to environmental variability. This extension of quantum graphs to ecological dynamics leads to theoretical advances that may also impact a wide range of problems in science involving continuous spatial networks, for example predator-prey dynamics, nerve-signal propagation, and nutrient or drug delivery in body vessels.How is a river like a tree? If you look at a river on a map and a tree, they look very similar, with a large main stem branching into smaller branches. Ecologists studying rivers have long understood the importance of a rivers tree-like structure, yet tools are currently lacking that allow researchers to incorporate this structure into models of important ecological processes. This hinders, for example, the ability to predict where and how far a pollutant may travel in a river, to investigate the potential impacts of dam placement or removal on populations of endangered species, and to understand the complex effects of land use and climate change in river floodplains. This project leads to the development of a set of mathematical and computational tools for studying rivers in their branching form. This is done by adapting a mathematical framework, known as the theory of quantum graphs, to ecological problems. Quantum graphs were developed and have been largely studied in the context of quantum physics. However, this focus means that these techniques are not available "off the shelf" for application to other fields of science. This project removes these barriers to application, and expands the ability to represent the special ecological patterns that arise from a rivers branching structure.