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

由于气候变化、栖息地破坏和自然资源的过度使用，人口崩溃等生态系统正在全球范围内发生深刻而不可逆转的变化，预计未来只会变得更加频繁。为了防止即将到来的崩溃，我们必须识别早期预警信号。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发生的种群崩溃，并将该方法应用于空间分布的种群，例如草原。他们利用一种叫做计算拓扑的方法，这种方法可以量化种群分布模式的特征，比如模式中的斑块程度。在之前的工作中，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，这些数据集可能包含高水平的噪声和/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额外工具。然后，保护生物学家和自然资源管理者可以使用该工具，以协助保护脆弱物种和生态系统。该项目还支持本科生研究，并包括针对代表性不足群体的学生的招生工作。在先前对确定性种群模型产生的数据的研究中，研究人员测量了种群分布模式在灭绝过程中拓扑特征的变化（通过立方体同源性），并观察到即将崩溃的清晰拓扑特征。确定性模型的结果作为概念的证明，但在本项目中，研究者将研究随机种群模型和真实生态数据集的动态变化。从确定性系统过渡到随机系统将需要方法论的大量发展，并且需要使用更复杂的工具，例如，多参数持久同调。所开发的方法必须能够在有噪声的数据、损坏的数据、缺失的数据以及在空间和/或时间上稀疏的数据中检测信号。由于拓扑方法可以将精细尺度的随机噪声与大尺度的确定性空间模式区分开来，因此它是一种有前途的工具，用于分析有噪声的生态数据，并且使用多参数持久性的初步工作表明它能够从噪声中恢复“真正的”动态信号（种群分布模式）。该项目由数学和物理科学理事会（MPS）数学科学部（DMS）的数学生物学计划、刺激竞争研究的既定计划（EPSCoR）以及生物科学理事会（BIO）环境生物学学部（DEB）的人口和社区生态集群（PEC）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.