Collaborative Research: RUI: Topological methods for analyzing shifting patterns and population collapse

由于气候变化、生境破坏和过度使用自然资源，全球正在发生生态系统深刻和不可逆转的变化，如人口崩溃，预计未来只会变得更加频繁。为了防止即将到来的崩溃，我们必须认识到早期的预警信号。这在生态系统中尤其具有挑战性，因为它们在空间和时间上的自然复杂行为，以及嘈杂和/或分辨率较差的数据。在这个项目中，研究人员将使用一种新的方法来早期检测即将到来的种群崩溃，并将该方法应用于空间分布的种群，例如草原。他们利用一种称为计算拓扑学的方法，可以量化人口分布模式的特征，例如模式中的斑块水平。在以前的工作中，研究人员使用空间种群模型来量化种群灭绝时发生的种群分布模式的变化，并观察到“灭绝的拓扑路径”。在这个项目中，研究人员将开发和扩展用于随机人口模型和真实世界数据集的方法，这些数据集预计将包含高水平的噪音和/或丢失/损坏的数据。开发的方法将作为预测即将发生的人口崩溃的另一个工具。保护生物学家和自然资源管理人员可以使用这一工具，以协助保护脆弱物种和生态系统。该项目还支持本科生的研究，并包括招聘工作针对学生从代表性不足的群体。在以前的工作中产生的数据由一个确定性的人口模型，研究人员测量的拓扑特征的变化（通过立方同源）的人口分布模式的路线灭绝，并观察到清晰的拓扑签名即将崩溃。确定性模型的结果作为概念的证明，但在这个项目中，研究人员将研究随机种群模型和真实的生态数据集的动态变化。从确定性系统过渡到随机性系统将需要大量开发方法，并需要使用更复杂的工具，例如，多参数持续同调所开发的方法必须能够在噪声数据、损坏的数据、丢失的数据以及在空间和/或时间上稀疏的数据中检测信号。由于拓扑方法可以区分细尺度随机噪声和大尺度确定性空间模式，因此它是分析噪声生态数据的一种有前途的工具，利用多参数持续性的初步工作表明，它能够恢复“真实”的动力学信号（一种种群分布模式）。本项目由数学科学部数学生物学项目联合资助（DMS）在数学和物理科学理事会（MPS），既定计划，以刺激竞争性研究（EPSCoR），以及生物科学理事会（BIO）环境生物学司（DEB）的人口和社区生态组（PEC）该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的评估被认为值得支持。影响审查标准。

Profound and irreversible changes in ecosystems, such as population collapse, are occurring globally due to climate change, habitat destruction, and overuse of natural resources, and are only expected to become more frequent in the future. To prevent an impending collapse, we must recognize the early warning signs. This is particularly challenging in ecological systems due to their naturally complex behavior in both space and time, as well as noisy and/or poorly resolved data. In this project, the investigators will use a novel approach for early detection of impending population collapse, and apply the methodology to spatially distributed populations, for example, a grassland. They utilize a method called computational topology, which can quantify features of a population distribution pattern, such as the level of patchiness in the pattern. In previous work, the investigators used a spatial population model to quantify the changes in a population distribution pattern that occurred as the population went extinct and observed a "topological route to extinction". In this project, the investigators will develop and extend the methodology for use in stochastic population models and real-world data sets, which are expected to contain high levels of noise and/or missing/corrupted data. The developed methodology will serve as an additional tool for the prediction of impending population collapse. This tool can then be used by conservation biologists and natural resource managers in order to assist in preserving vulnerable species and ecosystems. The project also supports undergraduate research, and includes recruitment efforts directed at students from underrepresented groups.In previous work on data generated by a deterministic population model, the investigators measured changes in topological features (via cubical homology) of population distribution patterns en route to extinction, and observed clear topological signatures of impending collapse. Results with the deterministic model serve as a proof of concept, but in this project, the investigators will study dynamical changes in stochastic population models and real ecological data sets. Transitioning from deterministic to stochastic systems will require substantial development of the methodology, and will require the use of more sophisticated tools, e.g., multiparameter persistent homology. The developed methodology must be able to detect signal in noisy data, corrupted data, missing data, and data that is sparse in space and/or time. Because the topological approach can distinguish fine-scale stochastic noise from large-scale deterministic spatial patterns, it is a promising tool for the analysis of noisy ecological data, and preliminary work using multiparameter persistence shows that it is capable of recovering "true” dynamical signal (a population distribution pattern) from noise.This project is jointly funded by the Mathematical Biology program of the Division of Mathematical Sciences (DMS) in the Directorate for Mathematical and Physical Sciences (MPS), the Established Program to Stimulate Competitive Research (EPSCoR), and the Population and Community Ecology Cluster (PEC) of the Division of Environmental Biology (DEB) in the Directorate for Biological Sciences (BIO).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.