QEIB - Nonlinearities in Stochastic Population Dynamics

大多数自然过程，从亚原子粒子之间的相互作用到决定天气的大气过程，都是非线性的，这意味着变量之间的关系不能用简单的线条来描述。非线性会导致诸如混沌和巨灾这样的数学现象，即系统输入的微小变化会导致输出的巨大变化。生态系统充满了非线性，此外，它们还会经历随机的、不可预测的环境所产生的随机噪声。非线性动力学和随机噪声的结合有可能使生态系统无法预测，因为自然系统的噪声加剧了非线性的混乱和灾难。这项工作将解决模型生态系统中非线性动力学和随机噪声之间的相互作用，试图找到使自然系统更可预测的一般特性。尽管非线性动力学和随机噪声之间可能存在极其复杂的相互作用，但来自真实生态系统的许多数据集显示出近似于有噪声但线性系统的行为模式。该研究项目的一部分将通过发展统计技术来检测生态数据中的非线性，以证实或反驳这一观察结果。研究的另一部分将从理论角度提出，非线性随机系统期望表现出何种类型的动力学，以及这些期望实际上是否与真正的线性随机动力学相似。许多关键的环境问题需要随机人口模型。这项提议的工作将为生态学家提供解释数据的新工具。一个博士后有很好的培训，本科生有很多机会

Most natural processes, from interactions among subatomic particles to atmospheric processes that dictate weather, are nonlinear, meaning that the relationships among variables cannot be described by simple lines. Nonlinearities lead to such mathematical phenomena as chaos and catastrophes, in which small changes to the inputs to a system lead to large changes in the outputs. Ecological systems are full of nonlinearities, and in addition they experience stochastic noise created by a random, unpredictable environment. The combination of nonlinear dynamics and stochastic noise has the potential for making ecological systems impossible to predict, because the chaos and catastrophes of nonlinearities are exacerbated by the noisiness of natural systems.The proposed work will address the interactions between nonlinear dynamics and stochastic noise in model ecological systems in an attempt to find general properties that will make natural systems more predictable. Even though there is the potential for extremely complex interactions between nonlinear dynamics and stochastic noise, many data sets from real ecological systems show patterns approximating the behavior of noisy but linear systems. Part of the research project will be directed towards confirming or refuting this observation by developing statistical techniques to detect nonlinearities in ecological data. The other part of the research will ask from a theoretical perspective what types of dynamics are expected to be exhibited by nonlinear stochastic systems, and whether these expectations are in fact similar to truly linear stochastic dynamics. Many key environmental problems require stochastic population models. The proposed work will offer ecologists new tools for interpreting data. There will be excellent training for one post-doc, and many opportunities for undergraduates