RUI: Collaborative Research-Nonlinear Demographic Dynamics: Mathematical Models, Biological Experiments, Data Analyses

9319073 Desharnais研究者和他的同事们进行了一个跨学科的研究项目来测试非线性人口理论：他们构建和分析数学模型，设计和实施生物实验，开发和应用统计技术来分析数据。在生物实验中，使用了一种易于培养、世代时间短（即产生长时间序列数据）并允许对动物数量进行准确普查的生物：Tribolium属的粉甲虫。在研究的第一部分，目标是模型识别和参数估计。在第二部分，关注的是记录人口动态的定性行为的转变。繁殖率和成人死亡率被操纵，以便在参数空间中跨越边界，从稳定的平衡，到周期循环，到混乱。在第三阶段，目标是检验关于这些不寻常的人口动态存在的假设，并开发在实验数据中识别这些现象的方法。这个项目的一个主要贡献是一个明确的例子，通过数学模型预测了生物种群在质量上不同的动态行为之间的实验操纵转换。在过去十年左右的时间里，人们认识到简单的方程可以产生复杂的动力学，这导致了大量解释人口时间序列数据的迷人理论可能性。理解观察到的动物种群数量的波动是种群生物学的一个核心问题；它在从粮食生产和病虫害防治、可再生资源管理到物种多样性保护等领域具有深远的应用。由于难以收集足够的生态数据、难以通过实验操纵生态系统以及难以用生态数据评估复杂的数学模型，波动是非线性动力的结果这一假设已被证明是难以捉摸的。在本研究项目中，研究者检验非线性种群理论的方法是采用新发展的非线性时间序列统计方法，将数学模型与实验生物数据紧密联系起来。该项目在跨学科方法上是独一无二的，因为它涉及理论和实验，并利用了生物学家、统计学家和数学家的才能。从实验设计和实施，到理论建模和分析，再到统计测试和验证，所有这些学科的研究人员都参与到项目的各个方面，这在人口生物学领域是不寻常的。最终目标是证明非线性数学在获得对动物种群动态的严格理解方面的有用性和重要性，特别是种群数量的波动，无论这些波动是规则的还是“混沌的”。

9319073 Desharnais The investigator and his colleagues conduct an interdisciplinary research program to test nonlinear population theory: they construct and analyze mathematical models, design and implement biological experiments, develop and apply statistical techniques for the analysis of data. For the biological experiments an organism is used that is easy to culture, has a short generation time (i.e. yields long time series data) and allows an accurate census of animal numbers: flour beetles of the genus Tribolium. In the first part of the study the objectives are model identification and parameter estimation. In the second part, the concern is to document transitions in the qualitative behavior of the demographic dynamics. Rates of reproduction and adult mortality are manipulated in order to cross boundaries in parameter space from stable equilibria, to periodic cycles, to chaos. In phase three the objectives are to test hypotheses concerning the existence of these unusual demographic dynamics and develop methods for identifying these phenomena in experimental data. A major contribution of this project is an unequivocal example of experimentally manipulated transitions between qualitatively different dynamical behaviors of a biological population as predicted by a mathematical model. In the last ten years or so, the recognition that simple equations can generate complex dynamics has led to an outpouring of fascinating theoretical possibilities for the explanation of population time series data. Understanding the observed fluctuations in animal population numbers is a central question in population biology; it has far-reaching applications in areas ranging from food production and pest control, to the management of renewable resources, to the conservation of species diversity. The hypothesis that fluctuations are the result of nonlinear dynamic forces has proved to be elusive to test due to the difficulties of gathering adequate ecological da ta, of experimentally manipulating ecological systems, and of evaluating complex mathematical models with ecological data. In this research project the investigators' approach to testing nonlinear population theory is to connect mathematical models rigorously with experimental biological data by means of newly developed statistical methods for nonlinear time series. The project is unique in its interdisciplinary approach because it involves both theory and experimentation and utilizes the talents of the biologist, statistician, and mathematician. It is unusual in the field of population biology to have an interdisciplinary effort in which investigators from all of these disciplines are involved in all aspects of the project, from experimental design and implementation, through theoretical modeling and analysis, to statistical testing and verification. The ultimate goal is to demonstrate the usefulness and importance of nonlinear mathematics in gaining a rigorous understanding of the dynamics of animal populations and in particular of fluctuations in population numbers, be these fluctations regular or "chaotic."