Stochastic nonlinear dynamics in the environment and biology

现实的生物学和环境模型对于理解对我们未来世界的观察、预测现象和做出改进至关重要。我们专注于随机非线性动力学的领域，这些领域在过去的动力学分析中只得到了很少的关注，但对于现实的环境和生物模型来说是必要的-即，由复杂动力学和噪声的相互作用驱动的非光滑动力学、延迟和过渡。 我们的目标是确定噪声驱动生物和环境动力学发生质变的条件，更好地理解潜在的机制，开发更好的模型和分析，并预测意外行为。我们将基于应用分析、计算以及综合概率、统计、动力学和几何分析的现代方法相结合，以开发可以跨应用领域转换的新技术。我们专注于几个几乎未被探索或未被探索的关键领域，在这些领域，我们需要创造性的新方法来理解、测量和预测随机性和非线性复杂动力学的新相互作用。在整个过程中都有研究培训的机会。 转变：目前针对突然或戏剧性转变的措施，如神经科学中的随机促进、环境背景中的提示、疾病和流行病学中的暴发，并不都能捕捉到非线性动力学和噪声的相互作用。我们提出了捕捉这些影响的新方法。 非光滑动力学：噪声和非光滑系统的新的非线性现象之间的相互作用几乎没有被探索过，即使在具有开关开关、相变或斑驳环境的系统中通常可以看到非光滑行为。 延迟反馈：在具有延迟反馈的模型中，例如光学、人体平衡、疾病和细胞动力学，噪声可以相互作用来稳定或破坏期望的行为。初步结果为研究空间现象指明了新的方向，例如植被模式的弹性或生化和电生理电波。 多尺度：多时间尺度的概念已经成熟，可以转换到新的领域，如流行病学、遗传学和环境，以及具有长时间尺度上罕见事件的疾病动力学，如艾滋病毒或结核病。 优化和控制：控制系统的稳健性通常依赖于我们的新方法捕获的延迟反馈、开关控制和噪声等特征。在随机优化中，我们通过将迭代本身视为一个动态系统，将非线性与采样相互作用，从而开发出新的、有效的方法。

Realistic models of biology and the environment are essential to understanding observations about our world, predicting phenomena, and making improvements. We focus on areas of stochastic nonlinear dynamics that have received minimal attention from past dynamical analyses, but are necessary for realistical environmental and biological models - namely, nonsmooth dynamics, delays, and transitions driven by interaction of complex dynamics and noise. Our objectives are to identify conditions where noise drives qualitative changes in biological and environmental dynamics, to create a better understanding of the underlying mechanisms, to develop better models and analyses, and to predict unexpected behaviours. We combine modern approaches based on applied analysis, computations, and integrated probabilistic, statistical, dynamical and geometrical analyses to develop novel techniques that can translate across application areas. We focus on several key areas that are virtually unexplored or under-explored, where we need creative new approaches for understanding, measuring, and predicting novel interactions of randomness and nonlinear complex dynamics. There are opportunities for research training throughout. Transitions: Present measures for abrupt or dramatic transitions, such as stochastic facilitation in neuroscience, tipping in environmental contexts, and outbreaks in epidemiology, do not all capture the interaction of nonlinear dynamics and noise. We propose new methodologies for capturing these effects. Nonsmooth dynamics: Interactions of noise and novel nonlinear phenomenon for nonsmooth systems are virtually unexplored even though nonsmooth behavior is commonly seen in systems with on-off switches, phase transitions, or patchy environments. Delayed Feedback: In models with delayed feedback, e.g. optics, human balance, disease, and cellular dynamics, noise can interact to either stabilize or destabilize desired behavior. Preliminary results point to new directions for studying spatial phenomenon, such as resilience of vegetation patterns or biochemical and electrophysiological waves. Multiple Scales: Multiple time scale concepts are ripe for translation to new areas such as epidemiology, genetics, and the environment, as well as disease dynamics with rare events on long time scales, such as in HIV or tuberculosis. Optimization and control: Robustness in control systems is often dependent on features such as delayed feedback, on-off control, and noise, that are captured by our novel approaches. In stochastic optimization we develop new, efficient methods by viewing the iterations themselves as a dynamical system, with nonlinearities interacting with sampling.