Collaborative Research: Scaling Properties of Ecological Variation in Complex Dynamical Systems

生态学家预测未来的动物种群，以管理收获的种群，并评估保护种群的灭绝风险。这些预测的一个关键组成部分是了解食物，水或住所等限制性环境因素的变化如何驱动人口波动。在缺乏关于这些因素的具体数据的情况下，生态学家必须使用数据中观察到的变化的特性来预测未来的风险。在生态学中，这些预测通常会对潜在的生物学做出简化的假设，这可能会影响其准确性。该项目将把新的数学思想和模型与经验性的实地观察相结合，以研究经历大波动的种群的特性，使PI能够测试关于种群结构的基本假设。这项工作将侧重于经历有规律周期的种群，这是一种常见的现象，产生于个体之间争夺资源或捕食者-猎物相互作用等因素。这项工作将改善野生动物管理和保护中使用的工具，并了解外部因素如何驱动非线性动力系统的变化。在这个项目中开发的数学思想和模型将使用类似于经济学家用来预测股票市场波动或气象学家用来预测飓风路径的技术;因此，结果可能对许多社会普遍感兴趣的领域产生重要影响。新的研究表明，普遍规律描述了生物系统的波动，无论这些过程在生物尺度上运行。这些定律的一个关键预测是，许多生物系统的波动将随着外在变化的增加而单调增加，这种变化通常被称为“环境噪声”。然而，这种缩放仅在波动很小时有效的线性化模型中进行了研究。当种群出现较大的波动时，这一理论的假设就失效了。这个项目将开发研究非线性生物系统中的方差标度特性所需的数学方法，以增加环境变化。特别是，PI将研究非线性生物系统中外部噪声的缩放特性，重点关注外部扰动幅度的增加如何驱动总系统方差的下降。目的是了解生态系统中的非线性如何导致对环境噪声的鲁棒或次线性响应。研究范围包括发展数学理论来模拟环境变化如何与高度非线性的种群动态相互作用，主要是针对单一物种。 PI还将开发统计方法，以更好地检测来自经验性人口调查的预测方差比例关系的信号。这项工作将是第一个确定外部可变性的增加如何推动生物系统稳定性的增加，这是构建非线性生物系统如何与噪声环境相互作用和响应的更细致描述的重要一步。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Ecologists forecast future animal populations to manage harvested populations and assess extinction risks of populations of conservation concern. A critical component of these forecasts is understanding how variation in limiting environmental factors such as food, water, or shelter drive population fluctuations. In the absence of specific data on these factors, ecologists must use the properties of the observed variation in the data to make projections about future risk. In ecology, these forecasts often make simplifying assumptions about the underlying biology that may impact their accuracy. The project will integrate new mathematical ideas and models with empirical field observations to study the properties of populations that experience large fluctuations, allowing the PIs to test fundamental assumptions about how populations are structured. This work will focus on populations that experience regular cycles, a common phenomenon arising through factors such as competition for resources among individuals or predator-prey interactions. This work will improve both the tools used in the management and conservation of wildlife and the understanding of how external factors drive variation in nonlinear dynamical systems. The mathematical ideas and models developed in this project will use techniques similar to those that economists use to project stock market fluctuations or meteorologists use to predict paths of hurricanes; thus, the results may have important implications for many areas of general societal interest. Emerging work has shown that universal laws describe the fluctuations of biological systems, regardless of the biological scale at which these processes operate. A key prediction of these laws is that the fluctuations of many biological systems will increase monotonically in response to increases in extrinsic variability, often termed “environmental noise”. However, this scaling has only been studied in linearized models that are valid when fluctuations are small. When populations exhibit larger fluctuations, the assumptions underlying this theory break down. This project will develop the mathematics needed to study the variance scaling properties in nonlinear biological systems subject to increased environmental variation. In particular, the PIs will study the scaling properties of extrinsic noise in nonlinear biological systems focusing on how increases in the magnitude of external perturbations may drive declines in total system variance. The goal is to understand how nonlinearities in ecological systems may lead to robust or sublinear responses to environmental noise. The research scope includes developing mathematical theory to model how environmental variation interacts with highly nonlinear population dynamics, mainly for single species. The PIs will also develop the statistical approaches to better detect the signal of the predicted variance scaling relationships from empirical population surveys. This work will be the first to determine how increases in external variability can drive increases in the stability of biological systems, an essential step in constructing a more nuanced description of how nonlinear biological systems interact with and respond to noisy environments.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.