Stability and Dynamics of Invasion Fronts in Spatially Extended Systems

这个项目涉及到数学技术的发展，以预测和解释前沿传播到不稳定状态的特征，并在物理、生物和化学中得到应用。随着系统参数的改变或新物种的引入，在应用问题中经常会出现不稳定状态。随后的动力学通常由称为入侵前锋的移动界面的形成所主导，这些界面以固定的速度传播到不稳定状态，并在它们的尾流中选择一些次要状态。这一建议是建立在扩大这些过程的知识基础上的，具体目标是阐明导致入侵前线出现的机制，然后利用这些知识在实际感兴趣的系统中做出预测。这项工作的一部分将发展新的数学方法来研究局部强迫对偏微分方程(PDE)模型中界面传播速度的影响。第二部分将为在复杂网络上传播的不稳定性的动力学发展预测和一般原理。提出了两条互补的研究途径。拟议工作的一个领域涉及入侵前锋的稳定性，重点是了解外部强迫在选择界面速度和随后的动力学中的作用。由于入侵前锋传播到不稳定状态，稳定性分析要求这些前锋的扰动充分局部化。这一建议将研究强迫函数也是局部化的，但不是足够局部化的问题，以适应齐次方程的稳定性框架。上述项目的成功实施将为行波稳定性的研究带来新的数学方法，并将澄清和解释一些新的现象，这将是从事应用问题的研究人员感兴趣的。第二个拟议的工作领域涉及入侵前沿，其中空间被视为一个复杂的网络，其局部动态发生在每个节点上，节点之间通过图拉普拉斯耦合。这里的目标是利用PDE设置中的前沿传播知识来做出与网络上不稳定传播相关的具体预测。应用包括全球流行病的传播和离散环境中的入侵物种的传播。这项工作的成功完成将为这一领域的研究人员提供新的工具来估计到达时间，提供一个严格的理论框架来理解这些传播现象，并深入了解复杂网络的特征如何影响入侵的到达时间。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns the development of mathematical techniques to predict and explain features of front propagation into unstable states with applications in physics, biology and chemistry. Unstable states arise frequently in applied problems as system parameters are changed or new species are introduced into the problem. The subsequent dynamics are often dominated by the formation of moving interfaces called invasion fronts that propagate into the unstable state at fixed speed and select some secondary state in their wake. This proposal is built around expanding knowledge of these processes with a specific goal of elucidating mechanisms leading to the emergence of invasion fronts and then leveraging this knowledge to make predictions in actual systems of interest. One portion of this work will develop new mathematical approaches to study the effect of localized forcing on the spreading speed of interfaces in partial differential equation (PDE) models. The second portion will develop predictions and general principles for the dynamics of instabilities spreading over complex networks. Two complementary avenues of research are proposed. One area of proposed work concerns stability of invasion fronts with a focus on understanding the role of external forcing in the selection of the inter-facial velocity and subsequent dynamics. Since invasion fronts propagate into unstable states, stability analysis requires perturbations of these fronts to be sufficiently localized. This proposal will study problems where the forcing function is also localized, but not sufficiently localized to fit into the stability framework of the homogeneous equation. Success in carrying out the projects described will lead to new mathematical approaches to the study of stability of traveling waves and will clarify and explain some novel phenomena that will be of interest to researchers working in applied problems. A second area of proposed work concerns invasion fronts where space is taken to be a complex network with local dynamics occurring at each node with coupling between nodes via a graph Laplacian. The goal here is to leverage knowledge of front propagation in the PDE setting to make concrete predictions related to the spread of instabilities over networks. Applications include the spread of global epidemics and that of invasive species in discrete environments. Successful completion of this work will provide researchers working in this area with new tools to estimate arrival times, provide a rigorous theoretical framework by which to understand these spreading phenomena and provide insight into the way features of complex networks influence arrival times of invasions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.