eMB: Mathematical Classification of Complexity in Population Dynamics

正如达尔文的著名观察，生命是一场与各种限制其生长的有限资源的斗争。有了无限的营养和空间，一个大肠杆菌细胞可以在两天内繁殖成地球那么大。在现实中，人口增长是一个复杂的非线性过程，受环境因素的影响和众多因素的制约。所有细胞的生长都依赖于各种必需营养物质和空间的可用性。因此，从基因表达到全球生态系统水平，无处不在的多重资源限制（MRL）塑造了生命的复杂动态。这种动态可以用非线性数学模型的形式来描述，该模型基于控制营养限制的守恒定律。这些模型可能体现了生命的规则，表现出适用于多个时间和空间尺度的新兴系统特性。发现这些规律是这组研究人员的目标。该项目将为科学界提供多种生物和数学建模资源。该项目将培养具有生态学、进化生物学和应用数学领域综合研究经验的受过独特训练的研究生和本科生。这项研究将进一步提高社会在实验室和自然环境中预测、设计和工程控制种群动态的能力。此外，该项目将建模与实验工作紧密联系在一起，为科学界提供了一个同步开发建模和实验方法的机会，以更好地理解实验中观察到的复杂性。基于现有和正在进行的面粉甲虫（Tribolium）种群实验的复杂时间序列数据集，预计这项拟议的工作将解决一个具体而引人注目的问题，即MRL如何塑造生命的时空组织。更具体地说，研究者试图根据三个主要阶段对复杂的种群动态模式进行分类：1)可能受到随机因素影响的初始指数增长阶段的瞬态和看似混沌的动力学特征；2)稳定的中间增长阶段；3)最终或渐近增长阶段。由于对共享的有限资源的竞争，预计生物和这些阶段之间将出现新的隐藏的相互作用，导致复杂和高度非线性的特性，这些特性在单一资源限制概念下是罕见的，但可能导致现实世界生态系统中的灾难性问题。了解这些由生命个体、生命系统、它们的环境和相互作用组成的紧急属性的行为规则，将有助于社会识别早期预警信号并制定控制策略，以解决在不断变化的环境中恢复力和可持续性的问题。本文的主要目的是建立一系列的MRL种群增长模型，通过实验数据对其进行验证，并借助新兴的数学理论来理解其复杂的动态。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。

As Darwin famously observed, life is a struggle with various limiting resources constantly inhibiting its growth. With unlimited nutrients and space, a single E. coli cell can multiply into the size of planet Earth in two days. In reality, population growth is a complex nonlinear process influenced by environmental cues and constrained by numerous factors. All cell growth is dependent upon the availability of various essential nutrients and space. The complex dynamics of life is thus shaped by ubiquitous multiple resource limitation (MRL) from gene expression up to the global ecosystem level. Such dynamics may be described in the forms of nonlinear mathematical models based on laws of conservation that govern nutrient limitations. These models may embody rules of life that exhibit emerging systematic properties applicable to multiple temporal and spatial scales. Discovering such rules is the goal of this team of researchers. This project will generate a variety of biological and mathematical modeling resources for the scientific community. The project will produce uniquely trained graduate students and undergraduates with experiences in integrating studies across ecology, evolutionary biology, and applied mathematics fields. The research will further advance society's ability to predict, design and engineer controllable population dynamics in laboratory and natural settings. In addition, the project's intimate association of modeling with experimental work affords the scientific community an opportunity to develop both modeling and experimental approaches in synchrony to better understand the complexity observed in experiments.Motivated by and based on complex time series data sets from existing and ongoing experiments of flour beetle (Tribolium) populations, it is anticipated that this proposed work will address one specific and compelling question about how the MRL shapes the spatiotemporal organization of life. More specifically, the investigators seek to classify complex population dynamical patterns according to three main stages: 1) the transient and seemingly chaotic dynamics characteristic of the initial exponential growth stage that may be subject to influence by random factors to 2) the stable intermediate growth stage, and 3) final or asymptotical growth stage. It is expected that new hidden interactions will emerge between organisms and these stages due to competition for shared limiting resources, leading to complex and highly nonlinear properties that are rare under a single resource limitation concept but could lead to catastrophic problems in real-world ecosystems. Understanding the rules of behavior of these emergent properties consisting of the nutrient state of living individual, living systems, their environments and interactions will help the society to identify early-warning signals and formulate control strategies to address the issues of resilience and sustainability in evolving environments. The main objective of this proposal is to formulate a family of MRL population growth models, validate them via experimental data and understand their complex dynamics with the help of emergent mathematical theories.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.