Collaborative Research: Nonlinear Population Dynamics: Mathematical Models, Biological Experiments, Data Analyses

了解动物数量的波动是种群生物学的核心问题，它对从粮食生产到物种保护等一系列问题具有深远的影响和含义。非线性动力学为种群研究开辟了一个新的阶段，在这个阶段中，理论和实验集中在循环和准周期性、混沌和奇异吸引子、多吸引子和复杂吸引盆地、鞍集及其稳定流形等现象上。这个跨学科项目由Robert Costantino， Jim Cushing, Brian Dennis，Robert descharais和Shandelle Henson合作，涵盖了测试非线性人口理论所必需的活动的各个方面：将生物学转化为数学语言，再转化为数学语言，分析确定性和随机模型，开发和应用用于分析数据的统计技术，以及设计和实施生物学实验。对粉甲虫属的一系列实验为非线性现象提供了严格的实验检验。主题包括随机背景下的非线性，混沌和种群控制，周期性环境对动物丰度的影响，人口动态和自然选择，非平衡物种相互作用，以及有关模型参数化和验证的统计问题。有了对动物种群潜在动态的充分了解，生态学家可以预测环境退化的后果，并学习更好的方法来管理自然种群和控制害虫种群。例如，其中一项实验表明，通过利用混沌系统的特征“对初始条件的敏感性”，管理者可以在不使用化学杀虫剂的情况下，通过在关键时刻制造小扰动，大幅减少害虫种群数量。跨学科研究团队将生态理论与实验相结合，为数学应用于生态问题带来了新技术。该项目由MPS的应用数学、计算数学和统计学专业以及BIO的种群生物学和生态学专业支持。

Costantino9981374 Understanding the fluctuations in animal numbers is acentral issue in population biology that has far-reaching impacton and implications for problems ranging from food production tothe conservation of species. Nonlinear dynamics opens the way toa new phase of population research in which theory andexperimentation focus on phenomena such as cycles andquasi-periodicity, chaos and strange attractors, multipleattractors and complicated basins of attraction, saddle sets andtheir stable manifolds, and so on. This interdisciplinaryproject, in which Robert Costantino, Jim Cushing, Brian Dennis,Robert Desharnais, and Shandelle Henson collaborate, covers aspectrum of activities essential to testing nonlinear populationtheory: the translation of the biology into the language ofmathematics and back again, the analysis of deterministic andstochastic models, the development and application of statisticaltechniques for the analysis of data, and the design andimplementation of biological experiments. A series of experimentswith flour beetles of the genus Tribolium provide rigorousexperimental tests of nonlinear phenomena. Topics includenonlinearity in the context of stochasticity, chaos andpopulation control, the impact of periodic environments on animalabundance, demographic dynamics and natural selection,nonequilibrium species interactions, and statistical questionsconcerning parameterization and validation of models. With a sound understanding of the underlying dynamics ofanimal populations, ecologists can anticipate the consequences ofenvironmental degradation and learn better ways to manage naturalpopulations and control populations of pests. For example, one ofthe experiments suggests that, by taking advantage of the"sensitivity to initial conditions" that is the hallmark of achaotic system, managers can dramatically decrease pestpopulation numbers without the use of chemical pesticides bymaking small perturbations at critical times. The marriage ofecological theory and experiments by the interdisciplinaryresearch team leads to new techniques for the application ofmathematics to ecological problems. The project is supported bythe Applied Mathematics, Computational Mathematics, andStatistics programs in MPS and the Population Biology and Ecologyprograms in BIO.