Stochastic nonlinear dynamics in the environment and biology

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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.