Modeling spatial population dynamics in branching river networks using quantum graphs

缺乏一个适当的模拟框架，分支河流网络已被确定为一个主要的障碍，了解生态动态的空间结构和环境变化的影响。 特别需要的是在其自然的，连续的形式的河流网络的空间域的表示。 该项目采用量子图论作为河流网络建模的数学框架。 这个理论是为解决量子力学中的问题而发展起来的，它将一个度量图与一个微分算子配对。 度量图的分支被标识为连续的区间而不是离散的节点;函数和运算符被沿着这些区间定义。 因此，环境的变化可以被解释为系统参数的变化沿着在网络中的分支，和解决方案的抛物型和椭圆型微分方程描述人口动态，导致空间变异可以在全球范围内进行分析。 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力学方面具有很大的潜力，但生态问题的关键数学要求尚未适应这一框架。 该项目通过实现两个目标克服了这些障碍：1）扩展量子图来处理通常出现在河流种群动态模型中的反应-扩散-平流方程及其相关积分核系统; 2）开发分支网络上的解的转换方法来描述河流种群对环境变化的响应。 量子图在生态动力学中的这种扩展导致了理论上的进步，也可能影响到涉及连续空间网络的广泛科学问题，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。 如果你看地图上的一条河和一棵树，它们看起来非常相似，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了解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的生态过程模型。 例如，这阻碍了预测污染物在河流中可能移动的位置和距离的能力，调查水坝的建造或拆除对濒危物种种群的潜在影响，以及了解河流洪泛区土地使用和气候变化的复杂影响。 该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。这是通过调整一个数学框架来实现的，称为量子图理论，以解决生态问题。量子图是在量子物理学的背景下开发和研究的。然而，这种关注意味着这些技术不能“现成”应用于其他科学领域。 该项目消除了这些应用障碍，并扩展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.